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+<li class="chapter" data-level="10.5.2" data-path="linear-model-1.html"><a href="linear-model-1.html#misc-5"><i class="fa fa-check"></i><b>10.5.2</b> Misc</a></li>
 </ul></li>
 <li class="chapter" data-level="10.6" data-path="linear-model-1.html"><a href="linear-model-1.html#session-info-9"><i class="fa fa-check"></i><b>10.6</b> Session info</a></li>
 </ul></li>
@@ -478,7 +446,7 @@
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 <ul>
 <li class="chapter" data-level="11.8.1" data-path="linear-model-2.html"><a href="linear-model-2.html#datacamp-4"><i class="fa fa-check"></i><b>11.8.1</b> Datacamp</a></li>
-<li class="chapter" data-level="11.8.2" data-path="linear-model-2.html"><a href="linear-model-2.html#misc-5"><i class="fa fa-check"></i><b>11.8.2</b> Misc</a></li>
+<li class="chapter" data-level="11.8.2" data-path="linear-model-2.html"><a href="linear-model-2.html#misc-6"><i class="fa fa-check"></i><b>11.8.2</b> Misc</a></li>
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 <li class="chapter" data-level="11.9" data-path="linear-model-2.html"><a href="linear-model-2.html#session-info-10"><i class="fa fa-check"></i><b>11.9</b> Session info</a></li>
 <li class="chapter" data-level="11.10" data-path="linear-model-2.html"><a href="linear-model-2.html#references"><i class="fa fa-check"></i><b>11.10</b> References</a></li>
@@ -509,7 +477,7 @@
 <li class="chapter" data-level="12.7" data-path="linear-model-3.html"><a href="linear-model-3.html#additional-resources-10"><i class="fa fa-check"></i><b>12.7</b> Additional resources</a>
 <ul>
 <li class="chapter" data-level="12.7.1" data-path="linear-model-3.html"><a href="linear-model-3.html#datacamp-5"><i class="fa fa-check"></i><b>12.7.1</b> Datacamp</a></li>
-<li class="chapter" data-level="12.7.2" data-path="linear-model-3.html"><a href="linear-model-3.html#misc-6"><i class="fa fa-check"></i><b>12.7.2</b> Misc</a></li>
+<li class="chapter" data-level="12.7.2" data-path="linear-model-3.html"><a href="linear-model-3.html#misc-7"><i class="fa fa-check"></i><b>12.7.2</b> Misc</a></li>
 </ul></li>
 <li class="chapter" data-level="12.8" data-path="linear-model-3.html"><a href="linear-model-3.html#session-info-11"><i class="fa fa-check"></i><b>12.8</b> Session info</a></li>
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@@ -529,7 +497,7 @@
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 <li class="chapter" data-level="13.6" data-path="linear-model-4.html"><a href="linear-model-4.html#additional-resources-11"><i class="fa fa-check"></i><b>13.6</b> Additional resources</a>
 <ul>
-<li class="chapter" data-level="13.6.1" data-path="linear-model-4.html"><a href="linear-model-4.html#misc-7"><i class="fa fa-check"></i><b>13.6.1</b> Misc</a></li>
+<li class="chapter" data-level="13.6.1" data-path="linear-model-4.html"><a href="linear-model-4.html#misc-8"><i class="fa fa-check"></i><b>13.6.1</b> Misc</a></li>
 </ul></li>
 <li class="chapter" data-level="13.7" data-path="linear-model-4.html"><a href="linear-model-4.html#session-info-12"><i class="fa fa-check"></i><b>13.7</b> Session info</a></li>
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@@ -552,7 +520,7 @@
 <ul>
 <li class="chapter" data-level="14.7.1" data-path="power-analysis.html"><a href="power-analysis.html#datacamp-6"><i class="fa fa-check"></i><b>14.7.1</b> Datacamp</a></li>
 <li class="chapter" data-level="14.7.2" data-path="power-analysis.html"><a href="power-analysis.html#cheatsheets-5"><i class="fa fa-check"></i><b>14.7.2</b> Cheatsheets</a></li>
-<li class="chapter" data-level="14.7.3" data-path="power-analysis.html"><a href="power-analysis.html#misc-8"><i class="fa fa-check"></i><b>14.7.3</b> Misc</a></li>
+<li class="chapter" data-level="14.7.3" data-path="power-analysis.html"><a href="power-analysis.html#misc-9"><i class="fa fa-check"></i><b>14.7.3</b> Misc</a></li>
 </ul></li>
 <li class="chapter" data-level="14.8" data-path="power-analysis.html"><a href="power-analysis.html#session-info-13"><i class="fa fa-check"></i><b>14.8</b> Session info</a></li>
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@@ -607,7 +575,7 @@
 <ul>
 <li class="chapter" data-level="16.8.1" data-path="causation.html"><a href="causation.html#books"><i class="fa fa-check"></i><b>16.8.1</b> Books</a></li>
 <li class="chapter" data-level="16.8.2" data-path="causation.html"><a href="causation.html#tutorials-1"><i class="fa fa-check"></i><b>16.8.2</b> Tutorials</a></li>
-<li class="chapter" data-level="16.8.3" data-path="causation.html"><a href="causation.html#misc-9"><i class="fa fa-check"></i><b>16.8.3</b> Misc</a></li>
+<li class="chapter" data-level="16.8.3" data-path="causation.html"><a href="causation.html#misc-10"><i class="fa fa-check"></i><b>16.8.3</b> Misc</a></li>
 </ul></li>
 <li class="chapter" data-level="16.9" data-path="causation.html"><a href="causation.html#session-info-15"><i class="fa fa-check"></i><b>16.9</b> Session info</a></li>
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@@ -653,19 +621,20 @@
 <li class="chapter" data-level="19.2" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#load-packages-and-set-plotting-theme-12"><i class="fa fa-check"></i><b>19.2</b> Load packages and set plotting theme</a></li>
 <li class="chapter" data-level="19.3" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#load-data-sets-4"><i class="fa fa-check"></i><b>19.3</b> Load data sets</a>
 <ul>
-<li class="chapter" data-level="19.3.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#sleep-data"><i class="fa fa-check"></i><b>19.3.1</b> Sleep data</a></li>
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+<li class="chapter" data-level="19.3.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#reasoning-data"><i class="fa fa-check"></i><b>19.3.1</b> Reasoning data</a></li>
 </ul></li>
 <li class="chapter" data-level="19.4" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#understanding-the-lmer-syntax"><i class="fa fa-check"></i><b>19.4</b> Understanding the lmer() syntax</a></li>
-<li class="chapter" data-level="19.5" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#anova-vs.-lmer"><i class="fa fa-check"></i><b>19.5</b> ANOVA vs. Lmer</a>
+<li class="chapter" data-level="19.5" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#anova-vs.-lmer"><i class="fa fa-check"></i><b>19.5</b> ANOVA vs. lmer</a>
 <ul>
 <li class="chapter" data-level="19.5.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#between-subjects-anova"><i class="fa fa-check"></i><b>19.5.1</b> Between subjects ANOVA</a></li>
 <li class="chapter" data-level="19.5.2" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#repeated-measures-anova"><i class="fa fa-check"></i><b>19.5.2</b> Repeated-measures ANOVA</a></li>
 <li class="chapter" data-level="19.5.3" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#mixed-anova"><i class="fa fa-check"></i><b>19.5.3</b> Mixed ANOVA</a></li>
 </ul></li>
-<li class="chapter" data-level="19.6" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#session-info-18"><i class="fa fa-check"></i><b>19.6</b> Session info</a></li>
+<li class="chapter" data-level="19.6" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#additional-resources-17"><i class="fa fa-check"></i><b>19.6</b> Additional resources</a>
+<ul>
+<li class="chapter" data-level="19.6.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#readings-2"><i class="fa fa-check"></i><b>19.6.1</b> Readings</a></li>
+</ul></li>
+<li class="chapter" data-level="19.7" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#session-info-18"><i class="fa fa-check"></i><b>19.7</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="20" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html"><i class="fa fa-check"></i><b>20</b> Linear mixed effects models 4</a>
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@@ -673,10 +642,10 @@
 <li class="chapter" data-level="20.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#load-packages-and-set-plotting-theme-13"><i class="fa fa-check"></i><b>20.2</b> Load packages and set plotting theme</a></li>
 <li class="chapter" data-level="20.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#load-data-sets-5"><i class="fa fa-check"></i><b>20.3</b> Load data sets</a>
 <ul>
-<li class="chapter" data-level="20.3.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#sleep-data-1"><i class="fa fa-check"></i><b>20.3.1</b> Sleep data</a></li>
+<li class="chapter" data-level="20.3.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#sleep-data"><i class="fa fa-check"></i><b>20.3.1</b> Sleep data</a></li>
 <li class="chapter" data-level="20.3.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#reasoning-data-1"><i class="fa fa-check"></i><b>20.3.2</b> Reasoning data</a></li>
-<li class="chapter" data-level="20.3.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#weight-loss-data-1"><i class="fa fa-check"></i><b>20.3.3</b> Weight loss data</a></li>
-<li class="chapter" data-level="20.3.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politness-data-1"><i class="fa fa-check"></i><b>20.3.4</b> Politness data</a></li>
+<li class="chapter" data-level="20.3.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#weight-loss-data"><i class="fa fa-check"></i><b>20.3.3</b> Weight loss data</a></li>
+<li class="chapter" data-level="20.3.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politness-data"><i class="fa fa-check"></i><b>20.3.4</b> Politness data</a></li>
 </ul></li>
 <li class="chapter" data-level="20.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#follow-up-tests-with-emmeans"><i class="fa fa-check"></i><b>20.4</b> Follow-up tests with emmeans</a>
 <ul>
@@ -685,11 +654,17 @@
 <li class="chapter" data-level="20.4.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politeness-study"><i class="fa fa-check"></i><b>20.4.3</b> Politeness study</a></li>
 </ul></li>
 <li class="chapter" data-level="20.5" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#mixtures-of-participants"><i class="fa fa-check"></i><b>20.5</b> Mixtures of participants</a></li>
-<li class="chapter" data-level="20.6" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#bootstrapping"><i class="fa fa-check"></i><b>20.6</b> Bootstrapping</a>
+<li class="chapter" data-level="20.6" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#simulating-different-random-effects-structures"><i class="fa fa-check"></i><b>20.6</b> Simulating different random effects structures</a>
+<ul>
+<li class="chapter" data-level="20.6.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#two-level-model"><i class="fa fa-check"></i><b>20.6.1</b> Two-level model</a></li>
+<li class="chapter" data-level="20.6.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#three-level-model"><i class="fa fa-check"></i><b>20.6.2</b> Three-level model</a></li>
+</ul></li>
+<li class="chapter" data-level="20.7" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#bootstrapping"><i class="fa fa-check"></i><b>20.7</b> Bootstrapping</a>
 <ul>
-<li class="chapter" data-level="20.6.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-model"><i class="fa fa-check"></i><b>20.6.1</b> Linear model</a></li>
+<li class="chapter" data-level="20.7.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-model"><i class="fa fa-check"></i><b>20.7.1</b> Linear model</a></li>
+<li class="chapter" data-level="20.7.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-mixed-effects-model"><i class="fa fa-check"></i><b>20.7.2</b> Linear mixed effects model</a></li>
 </ul></li>
-<li class="chapter" data-level="20.7" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#session-info-19"><i class="fa fa-check"></i><b>20.7</b> Session info</a></li>
+<li class="chapter" data-level="20.8" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#session-info-19"><i class="fa fa-check"></i><b>20.8</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="21" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html"><i class="fa fa-check"></i><b>21</b> Generalized linear model</a>
 <ul>
@@ -712,7 +687,7 @@
 <li class="chapter" data-level="21.7" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#logistic-mixed-effects-model"><i class="fa fa-check"></i><b>21.7</b> Logistic mixed effects model</a></li>
 <li class="chapter" data-level="21.8" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#additional-information"><i class="fa fa-check"></i><b>21.8</b> Additional information</a>
 <ul>
-<li class="chapter" data-level="21.8.1" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#misc-10"><i class="fa fa-check"></i><b>21.8.1</b> Misc</a></li>
+<li class="chapter" data-level="21.8.1" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#misc-11"><i class="fa fa-check"></i><b>21.8.1</b> Misc</a></li>
 <li class="chapter" data-level="21.8.2" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#datacamp-7"><i class="fa fa-check"></i><b>21.8.2</b> Datacamp</a></li>
 </ul></li>
 <li class="chapter" data-level="21.9" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#session-info-20"><i class="fa fa-check"></i><b>21.9</b> Session info</a></li>
@@ -733,7 +708,7 @@
 <li class="chapter" data-level="22.4.2" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#frequentist-analysis"><i class="fa fa-check"></i><b>22.4.2</b> Frequentist analysis</a></li>
 <li class="chapter" data-level="22.4.3" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#bayesian-regression"><i class="fa fa-check"></i><b>22.4.3</b> Bayesian regression</a></li>
 </ul></li>
-<li class="chapter" data-level="22.5" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#additional-resources-17"><i class="fa fa-check"></i><b>22.5</b> Additional resources</a>
+<li class="chapter" data-level="22.5" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#additional-resources-18"><i class="fa fa-check"></i><b>22.5</b> Additional resources</a>
 <ul>
 <li class="chapter" data-level="22.5.1" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#books-and-chapters-5"><i class="fa fa-check"></i><b>22.5.1</b> Books and chapters</a></li>
 </ul></li>
@@ -776,7 +751,7 @@
 <li class="chapter" data-level="23.7.2" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#specify-and-fit-the-model-3"><i class="fa fa-check"></i><b>23.7.2</b> 2. Specify and fit the model</a></li>
 <li class="chapter" data-level="23.7.3" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#test-specific-hypotheses-3"><i class="fa fa-check"></i><b>23.7.3</b> 5. Test specific hypotheses</a></li>
 </ul></li>
-<li class="chapter" data-level="23.8" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#additional-resources-18"><i class="fa fa-check"></i><b>23.8</b> Additional resources</a></li>
+<li class="chapter" data-level="23.8" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#additional-resources-19"><i class="fa fa-check"></i><b>23.8</b> Additional resources</a></li>
 <li class="chapter" data-level="23.9" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#session-info-22"><i class="fa fa-check"></i><b>23.9</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="24" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html"><i class="fa fa-check"></i><b>24</b> Bayesian data analysis 3</a>
@@ -803,7 +778,7 @@
 <li class="chapter" data-level="24.6.4" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#gaussian-regression-unequal-variance"><i class="fa fa-check"></i><b>24.6.4</b> Gaussian regression (unequal variance)</a></li>
 <li class="chapter" data-level="24.6.5" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#model-comparison-2"><i class="fa fa-check"></i><b>24.6.5</b> Model comparison</a></li>
 </ul></li>
-<li class="chapter" data-level="24.7" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#additional-resources-19"><i class="fa fa-check"></i><b>24.7</b> Additional resources</a></li>
+<li class="chapter" data-level="24.7" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#additional-resources-20"><i class="fa fa-check"></i><b>24.7</b> Additional resources</a></li>
 <li class="chapter" data-level="24.8" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#session-info-23"><i class="fa fa-check"></i><b>24.8</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="25" data-path="model-assumptions.html"><a href="model-assumptions.html"><i class="fa fa-check"></i><b>25</b> Model assumptions</a>
@@ -818,7 +793,7 @@
 <li class="chapter" data-level="25.3.4" data-path="model-assumptions.html"><a href="model-assumptions.html#non-parametric-tests"><i class="fa fa-check"></i><b>25.3.4</b> Non-parametric tests</a></li>
 <li class="chapter" data-level="25.3.5" data-path="model-assumptions.html"><a href="model-assumptions.html#bootstrapping-regressions"><i class="fa fa-check"></i><b>25.3.5</b> Bootstrapping regressions</a></li>
 </ul></li>
-<li class="chapter" data-level="25.4" data-path="model-assumptions.html"><a href="model-assumptions.html#additional-resources-20"><i class="fa fa-check"></i><b>25.4</b> Additional resources</a></li>
+<li class="chapter" data-level="25.4" data-path="model-assumptions.html"><a href="model-assumptions.html#additional-resources-21"><i class="fa fa-check"></i><b>25.4</b> Additional resources</a></li>
 <li class="chapter" data-level="25.5" data-path="model-assumptions.html"><a href="model-assumptions.html#session-info-24"><i class="fa fa-check"></i><b>25.5</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="26" data-path="reporting-statistics.html"><a href="reporting-statistics.html"><i class="fa fa-check"></i><b>26</b> Reporting statistics</a>
@@ -835,7 +810,11 @@
 <ul>
 <li class="chapter" data-level="26.2.1" data-path="reporting-statistics.html"><a href="reporting-statistics.html#frequentist-statistics"><i class="fa fa-check"></i><b>26.2.1</b> Frequentist statistics</a></li>
 </ul></li>
-<li class="chapter" data-level="26.3" data-path="reporting-statistics.html"><a href="reporting-statistics.html#session-info-25"><i class="fa fa-check"></i><b>26.3</b> Session info</a></li>
+<li class="chapter" data-level="26.3" data-path="reporting-statistics.html"><a href="reporting-statistics.html#additional-resources-22"><i class="fa fa-check"></i><b>26.3</b> Additional resources</a>
+<ul>
+<li class="chapter" data-level="26.3.1" data-path="reporting-statistics.html"><a href="reporting-statistics.html#misc-12"><i class="fa fa-check"></i><b>26.3.1</b> Misc</a></li>
+</ul></li>
+<li class="chapter" data-level="26.4" data-path="reporting-statistics.html"><a href="reporting-statistics.html#session-info-25"><i class="fa fa-check"></i><b>26.4</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="27" data-path="cheatsheets-6.html"><a href="cheatsheets-6.html"><i class="fa fa-check"></i><b>27</b> Cheatsheets</a>
 <ul>
@@ -878,278 +857,279 @@ <h2><span class="header-section-number">22.1</span> Learning goals<a href="bayes
 </div>
 <div id="load-packages-and-set-plotting-theme-15" class="section level2 hasAnchor" number="22.2">
 <h2><span class="header-section-number">22.2</span> Load packages and set plotting theme<a href="bayesian-data-analysis-1.html#load-packages-and-set-plotting-theme-15" class="anchor-section" aria-label="Anchor link to header"></a></h2>
-<div class="sourceCode" id="cb1228"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1228-1"><a href="bayesian-data-analysis-1.html#cb1228-1" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;knitr&quot;</span>)      <span class="co"># for knitting RMarkdown</span></span>
-<span id="cb1228-2"><a href="bayesian-data-analysis-1.html#cb1228-2" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;janitor&quot;</span>)    <span class="co"># for cleaning column names</span></span>
-<span id="cb1228-3"><a href="bayesian-data-analysis-1.html#cb1228-3" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;patchwork&quot;</span>)  <span class="co"># for figure panels</span></span>
-<span id="cb1228-4"><a href="bayesian-data-analysis-1.html#cb1228-4" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;tidybayes&quot;</span>)  <span class="co"># tidying up results from Bayesian models</span></span>
-<span id="cb1228-5"><a href="bayesian-data-analysis-1.html#cb1228-5" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;greta&quot;</span>)      <span class="co"># for writing Bayesian models</span></span>
-<span id="cb1228-6"><a href="bayesian-data-analysis-1.html#cb1228-6" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;gganimate&quot;</span>)  <span class="co"># for animations</span></span>
-<span id="cb1228-7"><a href="bayesian-data-analysis-1.html#cb1228-7" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;extraDistr&quot;</span>) <span class="co"># additional probability distributions</span></span>
-<span id="cb1228-8"><a href="bayesian-data-analysis-1.html#cb1228-8" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;broom&quot;</span>)      <span class="co"># for tidy regression results</span></span>
-<span id="cb1228-9"><a href="bayesian-data-analysis-1.html#cb1228-9" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;tidyverse&quot;</span>)  <span class="co"># for wrangling, plotting, etc. </span></span></code></pre></div>
-<div class="sourceCode" id="cb1229"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1229-1"><a href="bayesian-data-analysis-1.html#cb1229-1" tabindex="-1"></a><span class="fu">theme_set</span>(<span class="fu">theme_classic</span>() <span class="sc">+</span> <span class="co">#set the theme </span></span>
-<span id="cb1229-2"><a href="bayesian-data-analysis-1.html#cb1229-2" tabindex="-1"></a>    <span class="fu">theme</span>(<span class="at">text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">20</span>))) <span class="co">#set the default text size</span></span>
-<span id="cb1229-3"><a href="bayesian-data-analysis-1.html#cb1229-3" tabindex="-1"></a></span>
-<span id="cb1229-4"><a href="bayesian-data-analysis-1.html#cb1229-4" tabindex="-1"></a>opts_chunk<span class="sc">$</span><span class="fu">set</span>(<span class="at">comment =</span> <span class="st">&quot;&quot;</span>,</span>
-<span id="cb1229-5"><a href="bayesian-data-analysis-1.html#cb1229-5" tabindex="-1"></a>               <span class="at">fig.show =</span> <span class="st">&quot;hold&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="bayesian-data-analysis-1.html#cb1-1" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;knitr&quot;</span>)      <span class="co"># for knitting RMarkdown</span></span>
+<span id="cb1-2"><a href="bayesian-data-analysis-1.html#cb1-2" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;janitor&quot;</span>)    <span class="co"># for cleaning column names</span></span>
+<span id="cb1-3"><a href="bayesian-data-analysis-1.html#cb1-3" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;patchwork&quot;</span>)  <span class="co"># for figure panels</span></span>
+<span id="cb1-4"><a href="bayesian-data-analysis-1.html#cb1-4" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;tidybayes&quot;</span>)  <span class="co"># tidying up results from Bayesian models</span></span>
+<span id="cb1-5"><a href="bayesian-data-analysis-1.html#cb1-5" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;greta&quot;</span>)      <span class="co"># for writing Bayesian models</span></span>
+<span id="cb1-6"><a href="bayesian-data-analysis-1.html#cb1-6" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;gganimate&quot;</span>)  <span class="co"># for animations</span></span>
+<span id="cb1-7"><a href="bayesian-data-analysis-1.html#cb1-7" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;extraDistr&quot;</span>) <span class="co"># additional probability distributions</span></span>
+<span id="cb1-8"><a href="bayesian-data-analysis-1.html#cb1-8" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;broom&quot;</span>)      <span class="co"># for tidy regression results</span></span>
+<span id="cb1-9"><a href="bayesian-data-analysis-1.html#cb1-9" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;tidyverse&quot;</span>)  <span class="co"># for wrangling, plotting, etc. </span></span></code></pre></div>
+<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="bayesian-data-analysis-1.html#cb2-1" tabindex="-1"></a><span class="fu">theme_set</span>(<span class="fu">theme_classic</span>() <span class="sc">+</span> <span class="co">#set the theme </span></span>
+<span id="cb2-2"><a href="bayesian-data-analysis-1.html#cb2-2" tabindex="-1"></a>    <span class="fu">theme</span>(<span class="at">text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">20</span>))) <span class="co">#set the default text size</span></span>
+<span id="cb2-3"><a href="bayesian-data-analysis-1.html#cb2-3" tabindex="-1"></a></span>
+<span id="cb2-4"><a href="bayesian-data-analysis-1.html#cb2-4" tabindex="-1"></a>opts_chunk<span class="sc">$</span><span class="fu">set</span>(<span class="at">comment =</span> <span class="st">&quot;&quot;</span>,</span>
+<span id="cb2-5"><a href="bayesian-data-analysis-1.html#cb2-5" tabindex="-1"></a>               <span class="at">fig.show =</span> <span class="st">&quot;hold&quot;</span>)</span></code></pre></div>
 </div>
 <div id="doing-bayesian-inference-by-hand" class="section level2 hasAnchor" number="22.3">
 <h2><span class="header-section-number">22.3</span> Doing Bayesian inference “by hand”<a href="bayesian-data-analysis-1.html#doing-bayesian-inference-by-hand" class="anchor-section" aria-label="Anchor link to header"></a></h2>
 <div id="sequential-updating-based-on-the-beta-distribution" class="section level3 hasAnchor" number="22.3.1">
 <h3><span class="header-section-number">22.3.1</span> Sequential updating based on the Beta distribution<a href="bayesian-data-analysis-1.html#sequential-updating-based-on-the-beta-distribution" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1230"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1230-1"><a href="bayesian-data-analysis-1.html#cb1230-1" tabindex="-1"></a><span class="co"># data </span></span>
-<span id="cb1230-2"><a href="bayesian-data-analysis-1.html#cb1230-2" tabindex="-1"></a>data <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>)</span>
-<span id="cb1230-3"><a href="bayesian-data-analysis-1.html#cb1230-3" tabindex="-1"></a></span>
-<span id="cb1230-4"><a href="bayesian-data-analysis-1.html#cb1230-4" tabindex="-1"></a><span class="co"># whether observation is a success or failure </span></span>
-<span id="cb1230-5"><a href="bayesian-data-analysis-1.html#cb1230-5" tabindex="-1"></a>success <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="fu">cumsum</span>(data)) </span>
-<span id="cb1230-6"><a href="bayesian-data-analysis-1.html#cb1230-6" tabindex="-1"></a>failure <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="fu">cumsum</span>(<span class="dv">1</span> <span class="sc">-</span> data))</span>
-<span id="cb1230-7"><a href="bayesian-data-analysis-1.html#cb1230-7" tabindex="-1"></a><span class="co"># I&#39;ve added 0 at the beginning to show the prior</span></span>
-<span id="cb1230-8"><a href="bayesian-data-analysis-1.html#cb1230-8" tabindex="-1"></a></span>
-<span id="cb1230-9"><a href="bayesian-data-analysis-1.html#cb1230-9" tabindex="-1"></a><span class="co"># plotting function</span></span>
-<span id="cb1230-10"><a href="bayesian-data-analysis-1.html#cb1230-10" tabindex="-1"></a>fun.plot_beta <span class="ot">=</span> <span class="cf">function</span>(success, failure){</span>
-<span id="cb1230-11"><a href="bayesian-data-analysis-1.html#cb1230-11" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> <span class="fu">tibble</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">1</span>)),</span>
-<span id="cb1230-12"><a href="bayesian-data-analysis-1.html#cb1230-12" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> x)) <span class="sc">+</span></span>
-<span id="cb1230-13"><a href="bayesian-data-analysis-1.html#cb1230-13" tabindex="-1"></a>    <span class="fu">stat_function</span>(<span class="at">fun =</span> dbeta,</span>
-<span id="cb1230-14"><a href="bayesian-data-analysis-1.html#cb1230-14" tabindex="-1"></a>                  <span class="at">args =</span> <span class="fu">list</span>(<span class="at">shape1 =</span> success <span class="sc">+</span> <span class="dv">1</span>, <span class="at">shape2 =</span> failure <span class="sc">+</span> <span class="dv">1</span>),</span>
-<span id="cb1230-15"><a href="bayesian-data-analysis-1.html#cb1230-15" tabindex="-1"></a>                  <span class="at">geom =</span> <span class="st">&quot;area&quot;</span>,</span>
-<span id="cb1230-16"><a href="bayesian-data-analysis-1.html#cb1230-16" tabindex="-1"></a>                  <span class="at">color =</span> <span class="st">&quot;black&quot;</span>,</span>
-<span id="cb1230-17"><a href="bayesian-data-analysis-1.html#cb1230-17" tabindex="-1"></a>                  <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1230-18"><a href="bayesian-data-analysis-1.html#cb1230-18" tabindex="-1"></a>    <span class="fu">coord_cartesian</span>(<span class="at">expand =</span> F) <span class="sc">+</span></span>
-<span id="cb1230-19"><a href="bayesian-data-analysis-1.html#cb1230-19" tabindex="-1"></a>    <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="fl">0.25</span>, <span class="fl">0.75</span>, <span class="fl">0.25</span>)) <span class="sc">+</span> </span>
-<span id="cb1230-20"><a href="bayesian-data-analysis-1.html#cb1230-20" tabindex="-1"></a>    <span class="fu">theme</span>(<span class="at">axis.title =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1230-21"><a href="bayesian-data-analysis-1.html#cb1230-21" tabindex="-1"></a>          <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1230-22"><a href="bayesian-data-analysis-1.html#cb1230-22" tabindex="-1"></a>          <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1230-23"><a href="bayesian-data-analysis-1.html#cb1230-23" tabindex="-1"></a>          <span class="at">plot.margin =</span> <span class="fu">margin</span>(<span class="at">r =</span> <span class="dv">1</span>, <span class="at">t =</span> <span class="fl">0.5</span>, <span class="at">unit =</span> <span class="st">&quot;cm&quot;</span>))</span>
-<span id="cb1230-24"><a href="bayesian-data-analysis-1.html#cb1230-24" tabindex="-1"></a>}</span>
-<span id="cb1230-25"><a href="bayesian-data-analysis-1.html#cb1230-25" tabindex="-1"></a></span>
-<span id="cb1230-26"><a href="bayesian-data-analysis-1.html#cb1230-26" tabindex="-1"></a><span class="co"># generate the plots </span></span>
-<span id="cb1230-27"><a href="bayesian-data-analysis-1.html#cb1230-27" tabindex="-1"></a>plots <span class="ot">=</span> <span class="fu">map2</span>(success, failure, <span class="sc">~</span> <span class="fu">fun.plot_beta</span>(.x, .y))</span>
-<span id="cb1230-28"><a href="bayesian-data-analysis-1.html#cb1230-28" tabindex="-1"></a></span>
-<span id="cb1230-29"><a href="bayesian-data-analysis-1.html#cb1230-29" tabindex="-1"></a><span class="co"># make a grid of plots</span></span>
-<span id="cb1230-30"><a href="bayesian-data-analysis-1.html#cb1230-30" tabindex="-1"></a><span class="fu">wrap_plots</span>(plots, <span class="at">ncol =</span> <span class="dv">3</span>)</span></code></pre></div>
-<p><img src="22-bayesian_data_analysis1_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
+<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="bayesian-data-analysis-1.html#cb3-1" tabindex="-1"></a><span class="co"># data </span></span>
+<span id="cb3-2"><a href="bayesian-data-analysis-1.html#cb3-2" tabindex="-1"></a>data <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>)</span>
+<span id="cb3-3"><a href="bayesian-data-analysis-1.html#cb3-3" tabindex="-1"></a></span>
+<span id="cb3-4"><a href="bayesian-data-analysis-1.html#cb3-4" tabindex="-1"></a><span class="co"># whether observation is a success or failure </span></span>
+<span id="cb3-5"><a href="bayesian-data-analysis-1.html#cb3-5" tabindex="-1"></a>success <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="fu">cumsum</span>(data)) </span>
+<span id="cb3-6"><a href="bayesian-data-analysis-1.html#cb3-6" tabindex="-1"></a>failure <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="fu">cumsum</span>(<span class="dv">1</span> <span class="sc">-</span> data))</span>
+<span id="cb3-7"><a href="bayesian-data-analysis-1.html#cb3-7" tabindex="-1"></a><span class="co"># I&#39;ve added 0 at the beginning to show the prior</span></span>
+<span id="cb3-8"><a href="bayesian-data-analysis-1.html#cb3-8" tabindex="-1"></a></span>
+<span id="cb3-9"><a href="bayesian-data-analysis-1.html#cb3-9" tabindex="-1"></a><span class="co"># plotting function</span></span>
+<span id="cb3-10"><a href="bayesian-data-analysis-1.html#cb3-10" tabindex="-1"></a>fun.plot_beta <span class="ot">=</span> <span class="cf">function</span>(success, failure){</span>
+<span id="cb3-11"><a href="bayesian-data-analysis-1.html#cb3-11" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> <span class="fu">tibble</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">1</span>)),</span>
+<span id="cb3-12"><a href="bayesian-data-analysis-1.html#cb3-12" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> x)) <span class="sc">+</span></span>
+<span id="cb3-13"><a href="bayesian-data-analysis-1.html#cb3-13" tabindex="-1"></a>    <span class="fu">stat_function</span>(<span class="at">fun =</span> dbeta,</span>
+<span id="cb3-14"><a href="bayesian-data-analysis-1.html#cb3-14" tabindex="-1"></a>                  <span class="at">args =</span> <span class="fu">list</span>(<span class="at">shape1 =</span> success <span class="sc">+</span> <span class="dv">1</span>, <span class="at">shape2 =</span> failure <span class="sc">+</span> <span class="dv">1</span>),</span>
+<span id="cb3-15"><a href="bayesian-data-analysis-1.html#cb3-15" tabindex="-1"></a>                  <span class="at">geom =</span> <span class="st">&quot;area&quot;</span>,</span>
+<span id="cb3-16"><a href="bayesian-data-analysis-1.html#cb3-16" tabindex="-1"></a>                  <span class="at">color =</span> <span class="st">&quot;black&quot;</span>,</span>
+<span id="cb3-17"><a href="bayesian-data-analysis-1.html#cb3-17" tabindex="-1"></a>                  <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
+<span id="cb3-18"><a href="bayesian-data-analysis-1.html#cb3-18" tabindex="-1"></a>    <span class="fu">coord_cartesian</span>(<span class="at">expand =</span> F) <span class="sc">+</span></span>
+<span id="cb3-19"><a href="bayesian-data-analysis-1.html#cb3-19" tabindex="-1"></a>    <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="fl">0.25</span>, <span class="fl">0.75</span>, <span class="fl">0.25</span>)) <span class="sc">+</span> </span>
+<span id="cb3-20"><a href="bayesian-data-analysis-1.html#cb3-20" tabindex="-1"></a>    <span class="fu">theme</span>(<span class="at">axis.title =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb3-21"><a href="bayesian-data-analysis-1.html#cb3-21" tabindex="-1"></a>          <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb3-22"><a href="bayesian-data-analysis-1.html#cb3-22" tabindex="-1"></a>          <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb3-23"><a href="bayesian-data-analysis-1.html#cb3-23" tabindex="-1"></a>          <span class="at">plot.margin =</span> <span class="fu">margin</span>(<span class="at">r =</span> <span class="dv">1</span>, <span class="at">t =</span> <span class="fl">0.5</span>, <span class="at">unit =</span> <span class="st">&quot;cm&quot;</span>))</span>
+<span id="cb3-24"><a href="bayesian-data-analysis-1.html#cb3-24" tabindex="-1"></a>}</span>
+<span id="cb3-25"><a href="bayesian-data-analysis-1.html#cb3-25" tabindex="-1"></a></span>
+<span id="cb3-26"><a href="bayesian-data-analysis-1.html#cb3-26" tabindex="-1"></a><span class="co"># generate the plots </span></span>
+<span id="cb3-27"><a href="bayesian-data-analysis-1.html#cb3-27" tabindex="-1"></a>plots <span class="ot">=</span> <span class="fu">map2</span>(success, failure, <span class="sc">~</span> <span class="fu">fun.plot_beta</span>(.x, .y))</span>
+<span id="cb3-28"><a href="bayesian-data-analysis-1.html#cb3-28" tabindex="-1"></a></span>
+<span id="cb3-29"><a href="bayesian-data-analysis-1.html#cb3-29" tabindex="-1"></a><span class="co"># make a grid of plots</span></span>
+<span id="cb3-30"><a href="bayesian-data-analysis-1.html#cb3-30" tabindex="-1"></a><span class="fu">wrap_plots</span>(plots, <span class="at">ncol =</span> <span class="dv">3</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
 </div>
 <div id="coin-flip-example" class="section level3 hasAnchor" number="22.3.2">
 <h3><span class="header-section-number">22.3.2</span> Coin flip example<a href="bayesian-data-analysis-1.html#coin-flip-example" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <p>Is the coin biased?</p>
-<div class="sourceCode" id="cb1231"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1231-1"><a href="bayesian-data-analysis-1.html#cb1231-1" tabindex="-1"></a><span class="co"># data </span></span>
-<span id="cb1231-2"><a href="bayesian-data-analysis-1.html#cb1231-2" tabindex="-1"></a>data <span class="ot">=</span> <span class="fu">rep</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">1</span>, <span class="fu">c</span>(<span class="dv">8</span>, <span class="dv">2</span>))</span>
-<span id="cb1231-3"><a href="bayesian-data-analysis-1.html#cb1231-3" tabindex="-1"></a></span>
-<span id="cb1231-4"><a href="bayesian-data-analysis-1.html#cb1231-4" tabindex="-1"></a><span class="co"># parameters </span></span>
-<span id="cb1231-5"><a href="bayesian-data-analysis-1.html#cb1231-5" tabindex="-1"></a>theta <span class="ot">=</span> <span class="fu">c</span>(<span class="fl">0.1</span>, <span class="fl">0.5</span>, <span class="fl">0.9</span>)</span>
-<span id="cb1231-6"><a href="bayesian-data-analysis-1.html#cb1231-6" tabindex="-1"></a></span>
-<span id="cb1231-7"><a href="bayesian-data-analysis-1.html#cb1231-7" tabindex="-1"></a><span class="co"># prior </span></span>
-<span id="cb1231-8"><a href="bayesian-data-analysis-1.html#cb1231-8" tabindex="-1"></a>prior <span class="ot">=</span> <span class="fu">c</span>(<span class="fl">0.25</span>, <span class="fl">0.5</span>, <span class="fl">0.25</span>)</span>
-<span id="cb1231-9"><a href="bayesian-data-analysis-1.html#cb1231-9" tabindex="-1"></a><span class="co"># prior = c(0.1, 0.1, 0.8) # alternative setting of the prior</span></span>
-<span id="cb1231-10"><a href="bayesian-data-analysis-1.html#cb1231-10" tabindex="-1"></a><span class="co"># prior = c(0.000001, 0.000001, 0.999998) # another prior setting </span></span>
-<span id="cb1231-11"><a href="bayesian-data-analysis-1.html#cb1231-11" tabindex="-1"></a></span>
-<span id="cb1231-12"><a href="bayesian-data-analysis-1.html#cb1231-12" tabindex="-1"></a><span class="co"># likelihood </span></span>
-<span id="cb1231-13"><a href="bayesian-data-analysis-1.html#cb1231-13" tabindex="-1"></a>likelihood <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="fu">sum</span>(data <span class="sc">==</span> <span class="dv">1</span>), <span class="at">size =</span> <span class="fu">length</span>(data), <span class="at">prob =</span> theta)</span>
-<span id="cb1231-14"><a href="bayesian-data-analysis-1.html#cb1231-14" tabindex="-1"></a></span>
-<span id="cb1231-15"><a href="bayesian-data-analysis-1.html#cb1231-15" tabindex="-1"></a><span class="co"># posterior </span></span>
-<span id="cb1231-16"><a href="bayesian-data-analysis-1.html#cb1231-16" tabindex="-1"></a>posterior <span class="ot">=</span> likelihood <span class="sc">*</span> prior <span class="sc">/</span> <span class="fu">sum</span>(likelihood <span class="sc">*</span> prior)</span>
-<span id="cb1231-17"><a href="bayesian-data-analysis-1.html#cb1231-17" tabindex="-1"></a></span>
-<span id="cb1231-18"><a href="bayesian-data-analysis-1.html#cb1231-18" tabindex="-1"></a><span class="co"># store in data frame </span></span>
-<span id="cb1231-19"><a href="bayesian-data-analysis-1.html#cb1231-19" tabindex="-1"></a>df.coins <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">theta =</span> theta,</span>
-<span id="cb1231-20"><a href="bayesian-data-analysis-1.html#cb1231-20" tabindex="-1"></a>                  <span class="at">prior =</span> prior,</span>
-<span id="cb1231-21"><a href="bayesian-data-analysis-1.html#cb1231-21" tabindex="-1"></a>                  <span class="at">likelihood =</span> likelihood,</span>
-<span id="cb1231-22"><a href="bayesian-data-analysis-1.html#cb1231-22" tabindex="-1"></a>                  <span class="at">posterior =</span> posterior) </span></code></pre></div>
+<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="bayesian-data-analysis-1.html#cb4-1" tabindex="-1"></a><span class="co"># data </span></span>
+<span id="cb4-2"><a href="bayesian-data-analysis-1.html#cb4-2" tabindex="-1"></a>data <span class="ot">=</span> <span class="fu">rep</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">1</span>, <span class="fu">c</span>(<span class="dv">8</span>, <span class="dv">2</span>))</span>
+<span id="cb4-3"><a href="bayesian-data-analysis-1.html#cb4-3" tabindex="-1"></a></span>
+<span id="cb4-4"><a href="bayesian-data-analysis-1.html#cb4-4" tabindex="-1"></a><span class="co"># parameters </span></span>
+<span id="cb4-5"><a href="bayesian-data-analysis-1.html#cb4-5" tabindex="-1"></a>theta <span class="ot">=</span> <span class="fu">c</span>(<span class="fl">0.1</span>, <span class="fl">0.5</span>, <span class="fl">0.9</span>)</span>
+<span id="cb4-6"><a href="bayesian-data-analysis-1.html#cb4-6" tabindex="-1"></a></span>
+<span id="cb4-7"><a href="bayesian-data-analysis-1.html#cb4-7" tabindex="-1"></a><span class="co"># prior </span></span>
+<span id="cb4-8"><a href="bayesian-data-analysis-1.html#cb4-8" tabindex="-1"></a>prior <span class="ot">=</span> <span class="fu">c</span>(<span class="fl">0.25</span>, <span class="fl">0.5</span>, <span class="fl">0.25</span>)</span>
+<span id="cb4-9"><a href="bayesian-data-analysis-1.html#cb4-9" tabindex="-1"></a><span class="co"># prior = c(0.1, 0.1, 0.8) # alternative setting of the prior</span></span>
+<span id="cb4-10"><a href="bayesian-data-analysis-1.html#cb4-10" tabindex="-1"></a><span class="co"># prior = c(0.000001, 0.000001, 0.999998) # another prior setting </span></span>
+<span id="cb4-11"><a href="bayesian-data-analysis-1.html#cb4-11" tabindex="-1"></a></span>
+<span id="cb4-12"><a href="bayesian-data-analysis-1.html#cb4-12" tabindex="-1"></a><span class="co"># likelihood </span></span>
+<span id="cb4-13"><a href="bayesian-data-analysis-1.html#cb4-13" tabindex="-1"></a>likelihood <span class="ot">=</span> <span class="fu">dbinom</span>(<span class="fu">sum</span>(data <span class="sc">==</span> <span class="dv">1</span>), <span class="at">size =</span> <span class="fu">length</span>(data), <span class="at">prob =</span> theta)</span>
+<span id="cb4-14"><a href="bayesian-data-analysis-1.html#cb4-14" tabindex="-1"></a></span>
+<span id="cb4-15"><a href="bayesian-data-analysis-1.html#cb4-15" tabindex="-1"></a><span class="co"># posterior </span></span>
+<span id="cb4-16"><a href="bayesian-data-analysis-1.html#cb4-16" tabindex="-1"></a>posterior <span class="ot">=</span> likelihood <span class="sc">*</span> prior <span class="sc">/</span> <span class="fu">sum</span>(likelihood <span class="sc">*</span> prior)</span>
+<span id="cb4-17"><a href="bayesian-data-analysis-1.html#cb4-17" tabindex="-1"></a></span>
+<span id="cb4-18"><a href="bayesian-data-analysis-1.html#cb4-18" tabindex="-1"></a><span class="co"># store in data frame </span></span>
+<span id="cb4-19"><a href="bayesian-data-analysis-1.html#cb4-19" tabindex="-1"></a>df.coins <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">theta =</span> theta,</span>
+<span id="cb4-20"><a href="bayesian-data-analysis-1.html#cb4-20" tabindex="-1"></a>                  <span class="at">prior =</span> prior,</span>
+<span id="cb4-21"><a href="bayesian-data-analysis-1.html#cb4-21" tabindex="-1"></a>                  <span class="at">likelihood =</span> likelihood,</span>
+<span id="cb4-22"><a href="bayesian-data-analysis-1.html#cb4-22" tabindex="-1"></a>                  <span class="at">posterior =</span> posterior) </span></code></pre></div>
 <p>Visualize the results:</p>
-<div class="sourceCode" id="cb1232"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1232-1"><a href="bayesian-data-analysis-1.html#cb1232-1" tabindex="-1"></a>df.coins <span class="sc">%&gt;%</span> </span>
-<span id="cb1232-2"><a href="bayesian-data-analysis-1.html#cb1232-2" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span>theta,</span>
-<span id="cb1232-3"><a href="bayesian-data-analysis-1.html#cb1232-3" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
-<span id="cb1232-4"><a href="bayesian-data-analysis-1.html#cb1232-4" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1232-5"><a href="bayesian-data-analysis-1.html#cb1232-5" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">factor</span>(index, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior&quot;</span>, <span class="st">&quot;likelihood&quot;</span>, <span class="st">&quot;posterior&quot;</span>)),</span>
-<span id="cb1232-6"><a href="bayesian-data-analysis-1.html#cb1232-6" tabindex="-1"></a>         <span class="at">theta =</span> <span class="fu">factor</span>(theta, <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;p = 0.1&quot;</span>, <span class="st">&quot;p = 0.5&quot;</span>, <span class="st">&quot;p = 0.9&quot;</span>))) <span class="sc">%&gt;%</span> </span>
-<span id="cb1232-7"><a href="bayesian-data-analysis-1.html#cb1232-7" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1232-8"><a href="bayesian-data-analysis-1.html#cb1232-8" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> theta,</span>
-<span id="cb1232-9"><a href="bayesian-data-analysis-1.html#cb1232-9" tabindex="-1"></a>                       <span class="at">y =</span> value,</span>
-<span id="cb1232-10"><a href="bayesian-data-analysis-1.html#cb1232-10" tabindex="-1"></a>                       <span class="at">fill =</span> index)) <span class="sc">+</span> </span>
-<span id="cb1232-11"><a href="bayesian-data-analysis-1.html#cb1232-11" tabindex="-1"></a>  <span class="fu">geom_bar</span>(<span class="at">stat =</span> <span class="st">&quot;identity&quot;</span>,</span>
-<span id="cb1232-12"><a href="bayesian-data-analysis-1.html#cb1232-12" tabindex="-1"></a>           <span class="at">color =</span> <span class="st">&quot;black&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1232-13"><a href="bayesian-data-analysis-1.html#cb1232-13" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
-<span id="cb1232-14"><a href="bayesian-data-analysis-1.html#cb1232-14" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>,</span>
-<span id="cb1232-15"><a href="bayesian-data-analysis-1.html#cb1232-15" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1232-16"><a href="bayesian-data-analysis-1.html#cb1232-16" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1232-17"><a href="bayesian-data-analysis-1.html#cb1232-17" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1232-18"><a href="bayesian-data-analysis-1.html#cb1232-18" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
-<span id="cb1232-19"><a href="bayesian-data-analysis-1.html#cb1232-19" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1232-20"><a href="bayesian-data-analysis-1.html#cb1232-20" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1232-21"><a href="bayesian-data-analysis-1.html#cb1232-21" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1232-22"><a href="bayesian-data-analysis-1.html#cb1232-22" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1232-23"><a href="bayesian-data-analysis-1.html#cb1232-23" tabindex="-1"></a>        <span class="at">axis.title.x =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1232-24"><a href="bayesian-data-analysis-1.html#cb1232-24" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
-<p><img src="22-bayesian_data_analysis1_files/figure-html/unnamed-chunk-6-1.png" width="672" /></p>
+<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="bayesian-data-analysis-1.html#cb5-1" tabindex="-1"></a>df.coins <span class="sc">%&gt;%</span> </span>
+<span id="cb5-2"><a href="bayesian-data-analysis-1.html#cb5-2" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span>theta,</span>
+<span id="cb5-3"><a href="bayesian-data-analysis-1.html#cb5-3" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
+<span id="cb5-4"><a href="bayesian-data-analysis-1.html#cb5-4" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb5-5"><a href="bayesian-data-analysis-1.html#cb5-5" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">factor</span>(index, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior&quot;</span>, <span class="st">&quot;likelihood&quot;</span>, <span class="st">&quot;posterior&quot;</span>)),</span>
+<span id="cb5-6"><a href="bayesian-data-analysis-1.html#cb5-6" tabindex="-1"></a>         <span class="at">theta =</span> <span class="fu">factor</span>(theta, <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;p = 0.1&quot;</span>, <span class="st">&quot;p = 0.5&quot;</span>, <span class="st">&quot;p = 0.9&quot;</span>))) <span class="sc">%&gt;%</span> </span>
+<span id="cb5-7"><a href="bayesian-data-analysis-1.html#cb5-7" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb5-8"><a href="bayesian-data-analysis-1.html#cb5-8" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> theta,</span>
+<span id="cb5-9"><a href="bayesian-data-analysis-1.html#cb5-9" tabindex="-1"></a>                       <span class="at">y =</span> value,</span>
+<span id="cb5-10"><a href="bayesian-data-analysis-1.html#cb5-10" tabindex="-1"></a>                       <span class="at">fill =</span> index)) <span class="sc">+</span> </span>
+<span id="cb5-11"><a href="bayesian-data-analysis-1.html#cb5-11" tabindex="-1"></a>  <span class="fu">geom_bar</span>(<span class="at">stat =</span> <span class="st">&quot;identity&quot;</span>,</span>
+<span id="cb5-12"><a href="bayesian-data-analysis-1.html#cb5-12" tabindex="-1"></a>           <span class="at">color =</span> <span class="st">&quot;black&quot;</span>) <span class="sc">+</span></span>
+<span id="cb5-13"><a href="bayesian-data-analysis-1.html#cb5-13" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
+<span id="cb5-14"><a href="bayesian-data-analysis-1.html#cb5-14" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>,</span>
+<span id="cb5-15"><a href="bayesian-data-analysis-1.html#cb5-15" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb5-16"><a href="bayesian-data-analysis-1.html#cb5-16" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb5-17"><a href="bayesian-data-analysis-1.html#cb5-17" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb5-18"><a href="bayesian-data-analysis-1.html#cb5-18" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
+<span id="cb5-19"><a href="bayesian-data-analysis-1.html#cb5-19" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb5-20"><a href="bayesian-data-analysis-1.html#cb5-20" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb5-21"><a href="bayesian-data-analysis-1.html#cb5-21" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb5-22"><a href="bayesian-data-analysis-1.html#cb5-22" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb5-23"><a href="bayesian-data-analysis-1.html#cb5-23" tabindex="-1"></a>        <span class="at">axis.title.x =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb5-24"><a href="bayesian-data-analysis-1.html#cb5-24" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-6-1.png" width="672" /></p>
 </div>
 <div id="bayesian-inference-by-discretization" class="section level3 hasAnchor" number="22.3.3">
 <h3><span class="header-section-number">22.3.3</span> Bayesian inference by discretization<a href="bayesian-data-analysis-1.html#bayesian-inference-by-discretization" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="effect-of-the-prior" class="section level4 hasAnchor" number="22.3.3.1">
 <h4><span class="header-section-number">22.3.3.1</span> Effect of the prior<a href="bayesian-data-analysis-1.html#effect-of-the-prior" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1233"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1233-1"><a href="bayesian-data-analysis-1.html#cb1233-1" tabindex="-1"></a><span class="co"># grid</span></span>
-<span id="cb1233-2"><a href="bayesian-data-analysis-1.html#cb1233-2" tabindex="-1"></a>theta <span class="ot">=</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.01</span>)</span>
-<span id="cb1233-3"><a href="bayesian-data-analysis-1.html#cb1233-3" tabindex="-1"></a></span>
-<span id="cb1233-4"><a href="bayesian-data-analysis-1.html#cb1233-4" tabindex="-1"></a><span class="co"># data</span></span>
-<span id="cb1233-5"><a href="bayesian-data-analysis-1.html#cb1233-5" tabindex="-1"></a>data <span class="ot">=</span> <span class="fu">rep</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">1</span>, <span class="fu">c</span>(<span class="dv">8</span>, <span class="dv">2</span>))</span>
-<span id="cb1233-6"><a href="bayesian-data-analysis-1.html#cb1233-6" tabindex="-1"></a></span>
-<span id="cb1233-7"><a href="bayesian-data-analysis-1.html#cb1233-7" tabindex="-1"></a><span class="co"># calculate posterior</span></span>
-<span id="cb1233-8"><a href="bayesian-data-analysis-1.html#cb1233-8" tabindex="-1"></a>df.prior_effect <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">theta =</span> theta, </span>
-<span id="cb1233-9"><a href="bayesian-data-analysis-1.html#cb1233-9" tabindex="-1"></a>                  <span class="at">prior_uniform =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">1</span>, <span class="at">shape2 =</span> <span class="dv">1</span>),</span>
-<span id="cb1233-10"><a href="bayesian-data-analysis-1.html#cb1233-10" tabindex="-1"></a>                  <span class="at">prior_normal =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">5</span>, <span class="at">shape2 =</span> <span class="dv">5</span>),</span>
-<span id="cb1233-11"><a href="bayesian-data-analysis-1.html#cb1233-11" tabindex="-1"></a>                  <span class="at">prior_biased =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">8</span>, <span class="at">shape2 =</span> <span class="dv">2</span>)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1233-12"><a href="bayesian-data-analysis-1.html#cb1233-12" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span>theta,</span>
-<span id="cb1233-13"><a href="bayesian-data-analysis-1.html#cb1233-13" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;prior_index&quot;</span>,</span>
-<span id="cb1233-14"><a href="bayesian-data-analysis-1.html#cb1233-14" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;prior&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1233-15"><a href="bayesian-data-analysis-1.html#cb1233-15" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">likelihood =</span> <span class="fu">dbinom</span>(<span class="fu">sum</span>(data <span class="sc">==</span> <span class="dv">1</span>),</span>
-<span id="cb1233-16"><a href="bayesian-data-analysis-1.html#cb1233-16" tabindex="-1"></a>                             <span class="at">size =</span> <span class="fu">length</span>(data),</span>
-<span id="cb1233-17"><a href="bayesian-data-analysis-1.html#cb1233-17" tabindex="-1"></a>                             <span class="at">prob =</span> theta)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1233-18"><a href="bayesian-data-analysis-1.html#cb1233-18" tabindex="-1"></a>  <span class="fu">group_by</span>(prior_index) <span class="sc">%&gt;%</span> </span>
-<span id="cb1233-19"><a href="bayesian-data-analysis-1.html#cb1233-19" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">posterior =</span> likelihood <span class="sc">*</span> prior <span class="sc">/</span> <span class="fu">sum</span>(likelihood <span class="sc">*</span> prior)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1233-20"><a href="bayesian-data-analysis-1.html#cb1233-20" tabindex="-1"></a>  <span class="fu">ungroup</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1233-21"><a href="bayesian-data-analysis-1.html#cb1233-21" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, prior_index),</span>
-<span id="cb1233-22"><a href="bayesian-data-analysis-1.html#cb1233-22" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
-<span id="cb1233-23"><a href="bayesian-data-analysis-1.html#cb1233-23" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>)</span>
-<span id="cb1233-24"><a href="bayesian-data-analysis-1.html#cb1233-24" tabindex="-1"></a></span>
-<span id="cb1233-25"><a href="bayesian-data-analysis-1.html#cb1233-25" tabindex="-1"></a><span class="co"># make the plot</span></span>
-<span id="cb1233-26"><a href="bayesian-data-analysis-1.html#cb1233-26" tabindex="-1"></a>df.prior_effect <span class="sc">%&gt;%</span> </span>
-<span id="cb1233-27"><a href="bayesian-data-analysis-1.html#cb1233-27" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">factor</span>(index, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior&quot;</span>, <span class="st">&quot;likelihood&quot;</span>, <span class="st">&quot;posterior&quot;</span>)),</span>
-<span id="cb1233-28"><a href="bayesian-data-analysis-1.html#cb1233-28" tabindex="-1"></a>         <span class="at">prior_index =</span> <span class="fu">factor</span>(prior_index,</span>
-<span id="cb1233-29"><a href="bayesian-data-analysis-1.html#cb1233-29" tabindex="-1"></a>                              <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior_uniform&quot;</span>, <span class="st">&quot;prior_normal&quot;</span>, <span class="st">&quot;prior_biased&quot;</span>),</span>
-<span id="cb1233-30"><a href="bayesian-data-analysis-1.html#cb1233-30" tabindex="-1"></a>                              <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;uniform&quot;</span>, <span class="st">&quot;symmetric&quot;</span>, <span class="st">&quot;asymmetric&quot;</span>))) <span class="sc">%&gt;%</span> </span>
-<span id="cb1233-31"><a href="bayesian-data-analysis-1.html#cb1233-31" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1233-32"><a href="bayesian-data-analysis-1.html#cb1233-32" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> theta,</span>
-<span id="cb1233-33"><a href="bayesian-data-analysis-1.html#cb1233-33" tabindex="-1"></a>                       <span class="at">y =</span> value,</span>
-<span id="cb1233-34"><a href="bayesian-data-analysis-1.html#cb1233-34" tabindex="-1"></a>                       <span class="at">color =</span> index)) <span class="sc">+</span></span>
-<span id="cb1233-35"><a href="bayesian-data-analysis-1.html#cb1233-35" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
-<span id="cb1233-36"><a href="bayesian-data-analysis-1.html#cb1233-36" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">cols =</span> <span class="fu">vars</span>(prior_index),</span>
-<span id="cb1233-37"><a href="bayesian-data-analysis-1.html#cb1233-37" tabindex="-1"></a>             <span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
-<span id="cb1233-38"><a href="bayesian-data-analysis-1.html#cb1233-38" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>,</span>
-<span id="cb1233-39"><a href="bayesian-data-analysis-1.html#cb1233-39" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1233-40"><a href="bayesian-data-analysis-1.html#cb1233-40" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.2</span>)) <span class="sc">+</span></span>
-<span id="cb1233-41"><a href="bayesian-data-analysis-1.html#cb1233-41" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1233-42"><a href="bayesian-data-analysis-1.html#cb1233-42" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1233-43"><a href="bayesian-data-analysis-1.html#cb1233-43" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
-<span id="cb1233-44"><a href="bayesian-data-analysis-1.html#cb1233-44" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1233-45"><a href="bayesian-data-analysis-1.html#cb1233-45" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1233-46"><a href="bayesian-data-analysis-1.html#cb1233-46" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1233-47"><a href="bayesian-data-analysis-1.html#cb1233-47" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1233-48"><a href="bayesian-data-analysis-1.html#cb1233-48" tabindex="-1"></a>        <span class="at">axis.text.x =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">10</span>),</span>
-<span id="cb1233-49"><a href="bayesian-data-analysis-1.html#cb1233-49" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
+<div class="sourceCode" id="cb6"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb6-1"><a href="bayesian-data-analysis-1.html#cb6-1" tabindex="-1"></a><span class="co"># grid</span></span>
+<span id="cb6-2"><a href="bayesian-data-analysis-1.html#cb6-2" tabindex="-1"></a>theta <span class="ot">=</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.01</span>)</span>
+<span id="cb6-3"><a href="bayesian-data-analysis-1.html#cb6-3" tabindex="-1"></a></span>
+<span id="cb6-4"><a href="bayesian-data-analysis-1.html#cb6-4" tabindex="-1"></a><span class="co"># data</span></span>
+<span id="cb6-5"><a href="bayesian-data-analysis-1.html#cb6-5" tabindex="-1"></a>data <span class="ot">=</span> <span class="fu">rep</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">1</span>, <span class="fu">c</span>(<span class="dv">8</span>, <span class="dv">2</span>))</span>
+<span id="cb6-6"><a href="bayesian-data-analysis-1.html#cb6-6" tabindex="-1"></a></span>
+<span id="cb6-7"><a href="bayesian-data-analysis-1.html#cb6-7" tabindex="-1"></a><span class="co"># calculate posterior</span></span>
+<span id="cb6-8"><a href="bayesian-data-analysis-1.html#cb6-8" tabindex="-1"></a>df.prior_effect <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">theta =</span> theta, </span>
+<span id="cb6-9"><a href="bayesian-data-analysis-1.html#cb6-9" tabindex="-1"></a>                  <span class="at">prior_uniform =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">1</span>, <span class="at">shape2 =</span> <span class="dv">1</span>),</span>
+<span id="cb6-10"><a href="bayesian-data-analysis-1.html#cb6-10" tabindex="-1"></a>                  <span class="at">prior_normal =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">5</span>, <span class="at">shape2 =</span> <span class="dv">5</span>),</span>
+<span id="cb6-11"><a href="bayesian-data-analysis-1.html#cb6-11" tabindex="-1"></a>                  <span class="at">prior_biased =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">8</span>, <span class="at">shape2 =</span> <span class="dv">2</span>)) <span class="sc">%&gt;%</span> </span>
+<span id="cb6-12"><a href="bayesian-data-analysis-1.html#cb6-12" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span>theta,</span>
+<span id="cb6-13"><a href="bayesian-data-analysis-1.html#cb6-13" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;prior_index&quot;</span>,</span>
+<span id="cb6-14"><a href="bayesian-data-analysis-1.html#cb6-14" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;prior&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb6-15"><a href="bayesian-data-analysis-1.html#cb6-15" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">likelihood =</span> <span class="fu">dbinom</span>(<span class="fu">sum</span>(data <span class="sc">==</span> <span class="dv">1</span>),</span>
+<span id="cb6-16"><a href="bayesian-data-analysis-1.html#cb6-16" tabindex="-1"></a>                             <span class="at">size =</span> <span class="fu">length</span>(data),</span>
+<span id="cb6-17"><a href="bayesian-data-analysis-1.html#cb6-17" tabindex="-1"></a>                             <span class="at">prob =</span> theta)) <span class="sc">%&gt;%</span> </span>
+<span id="cb6-18"><a href="bayesian-data-analysis-1.html#cb6-18" tabindex="-1"></a>  <span class="fu">group_by</span>(prior_index) <span class="sc">%&gt;%</span> </span>
+<span id="cb6-19"><a href="bayesian-data-analysis-1.html#cb6-19" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">posterior =</span> likelihood <span class="sc">*</span> prior <span class="sc">/</span> <span class="fu">sum</span>(likelihood <span class="sc">*</span> prior)) <span class="sc">%&gt;%</span> </span>
+<span id="cb6-20"><a href="bayesian-data-analysis-1.html#cb6-20" tabindex="-1"></a>  <span class="fu">ungroup</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb6-21"><a href="bayesian-data-analysis-1.html#cb6-21" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, prior_index),</span>
+<span id="cb6-22"><a href="bayesian-data-analysis-1.html#cb6-22" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
+<span id="cb6-23"><a href="bayesian-data-analysis-1.html#cb6-23" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>)</span>
+<span id="cb6-24"><a href="bayesian-data-analysis-1.html#cb6-24" tabindex="-1"></a></span>
+<span id="cb6-25"><a href="bayesian-data-analysis-1.html#cb6-25" tabindex="-1"></a><span class="co"># make the plot</span></span>
+<span id="cb6-26"><a href="bayesian-data-analysis-1.html#cb6-26" tabindex="-1"></a>df.prior_effect <span class="sc">%&gt;%</span> </span>
+<span id="cb6-27"><a href="bayesian-data-analysis-1.html#cb6-27" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">factor</span>(index, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior&quot;</span>, <span class="st">&quot;likelihood&quot;</span>, <span class="st">&quot;posterior&quot;</span>)),</span>
+<span id="cb6-28"><a href="bayesian-data-analysis-1.html#cb6-28" tabindex="-1"></a>         <span class="at">prior_index =</span> <span class="fu">factor</span>(prior_index,</span>
+<span id="cb6-29"><a href="bayesian-data-analysis-1.html#cb6-29" tabindex="-1"></a>                              <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior_uniform&quot;</span>, <span class="st">&quot;prior_normal&quot;</span>, <span class="st">&quot;prior_biased&quot;</span>),</span>
+<span id="cb6-30"><a href="bayesian-data-analysis-1.html#cb6-30" tabindex="-1"></a>                              <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;uniform&quot;</span>, <span class="st">&quot;symmetric&quot;</span>, <span class="st">&quot;asymmetric&quot;</span>))) <span class="sc">%&gt;%</span> </span>
+<span id="cb6-31"><a href="bayesian-data-analysis-1.html#cb6-31" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb6-32"><a href="bayesian-data-analysis-1.html#cb6-32" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> theta,</span>
+<span id="cb6-33"><a href="bayesian-data-analysis-1.html#cb6-33" tabindex="-1"></a>                       <span class="at">y =</span> value,</span>
+<span id="cb6-34"><a href="bayesian-data-analysis-1.html#cb6-34" tabindex="-1"></a>                       <span class="at">color =</span> index)) <span class="sc">+</span></span>
+<span id="cb6-35"><a href="bayesian-data-analysis-1.html#cb6-35" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
+<span id="cb6-36"><a href="bayesian-data-analysis-1.html#cb6-36" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">cols =</span> <span class="fu">vars</span>(prior_index),</span>
+<span id="cb6-37"><a href="bayesian-data-analysis-1.html#cb6-37" tabindex="-1"></a>             <span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
+<span id="cb6-38"><a href="bayesian-data-analysis-1.html#cb6-38" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>,</span>
+<span id="cb6-39"><a href="bayesian-data-analysis-1.html#cb6-39" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>) <span class="sc">+</span></span>
+<span id="cb6-40"><a href="bayesian-data-analysis-1.html#cb6-40" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.2</span>)) <span class="sc">+</span></span>
+<span id="cb6-41"><a href="bayesian-data-analysis-1.html#cb6-41" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb6-42"><a href="bayesian-data-analysis-1.html#cb6-42" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb6-43"><a href="bayesian-data-analysis-1.html#cb6-43" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
+<span id="cb6-44"><a href="bayesian-data-analysis-1.html#cb6-44" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb6-45"><a href="bayesian-data-analysis-1.html#cb6-45" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb6-46"><a href="bayesian-data-analysis-1.html#cb6-46" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb6-47"><a href="bayesian-data-analysis-1.html#cb6-47" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb6-48"><a href="bayesian-data-analysis-1.html#cb6-48" tabindex="-1"></a>        <span class="at">axis.text.x =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">10</span>),</span>
+<span id="cb6-49"><a href="bayesian-data-analysis-1.html#cb6-49" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
 <pre><code>Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
 ℹ Please use `linewidth` instead.
 This warning is displayed once every 8 hours.
-Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.</code></pre>
+Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
+generated.</code></pre>
 <div class="figure"><span style="display:block;" id="fig:unnamed-chunk-7"></span>
-<img src="22-bayesian_data_analysis1_files/figure-html/unnamed-chunk-7-1.png" alt="Illustration of how the prior affects the posterior." width="672" />
+<img src="psych252_files/figure-html/unnamed-chunk-7-1.png" alt="Illustration of how the prior affects the posterior." width="672" />
 <p class="caption">
-Figure 2.6: Illustration of how the prior affects the posterior.
+Figure 22.1: Illustration of how the prior affects the posterior.
 </p>
 </div>
 </div>
 <div id="effect-of-the-likelihood" class="section level4 hasAnchor" number="22.3.3.2">
 <h4><span class="header-section-number">22.3.3.2</span> Effect of the likelihood<a href="bayesian-data-analysis-1.html#effect-of-the-likelihood" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1235"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1235-1"><a href="bayesian-data-analysis-1.html#cb1235-1" tabindex="-1"></a><span class="co"># grid</span></span>
-<span id="cb1235-2"><a href="bayesian-data-analysis-1.html#cb1235-2" tabindex="-1"></a>theta <span class="ot">=</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.01</span>)</span>
-<span id="cb1235-3"><a href="bayesian-data-analysis-1.html#cb1235-3" tabindex="-1"></a></span>
-<span id="cb1235-4"><a href="bayesian-data-analysis-1.html#cb1235-4" tabindex="-1"></a>df.likelihood_effect <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">theta =</span> theta, </span>
-<span id="cb1235-5"><a href="bayesian-data-analysis-1.html#cb1235-5" tabindex="-1"></a>                              <span class="at">prior =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">2</span>, <span class="at">shape2 =</span> <span class="dv">8</span>),</span>
-<span id="cb1235-6"><a href="bayesian-data-analysis-1.html#cb1235-6" tabindex="-1"></a>                              <span class="at">likelihood_left =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">1</span>, <span class="at">shape2 =</span> <span class="dv">9</span>),</span>
-<span id="cb1235-7"><a href="bayesian-data-analysis-1.html#cb1235-7" tabindex="-1"></a>                              <span class="at">likelihood_center =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">5</span>, <span class="at">shape2 =</span> <span class="dv">5</span>),</span>
-<span id="cb1235-8"><a href="bayesian-data-analysis-1.html#cb1235-8" tabindex="-1"></a>                              <span class="at">likelihood_right =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">9</span>, <span class="at">shape2 =</span> <span class="dv">1</span>)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1235-9"><a href="bayesian-data-analysis-1.html#cb1235-9" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, prior),</span>
-<span id="cb1235-10"><a href="bayesian-data-analysis-1.html#cb1235-10" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;likelihood_index&quot;</span>,</span>
-<span id="cb1235-11"><a href="bayesian-data-analysis-1.html#cb1235-11" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;likelihood&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1235-12"><a href="bayesian-data-analysis-1.html#cb1235-12" tabindex="-1"></a>  <span class="fu">group_by</span>(likelihood_index) <span class="sc">%&gt;%</span> </span>
-<span id="cb1235-13"><a href="bayesian-data-analysis-1.html#cb1235-13" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">posterior =</span> likelihood <span class="sc">*</span> prior <span class="sc">/</span> <span class="fu">sum</span>(likelihood <span class="sc">*</span> prior)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1235-14"><a href="bayesian-data-analysis-1.html#cb1235-14" tabindex="-1"></a>  <span class="fu">ungroup</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1235-15"><a href="bayesian-data-analysis-1.html#cb1235-15" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, likelihood_index),</span>
-<span id="cb1235-16"><a href="bayesian-data-analysis-1.html#cb1235-16" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
-<span id="cb1235-17"><a href="bayesian-data-analysis-1.html#cb1235-17" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>)</span>
-<span id="cb1235-18"><a href="bayesian-data-analysis-1.html#cb1235-18" tabindex="-1"></a></span>
-<span id="cb1235-19"><a href="bayesian-data-analysis-1.html#cb1235-19" tabindex="-1"></a>df.likelihood_effect <span class="sc">%&gt;%</span> </span>
-<span id="cb1235-20"><a href="bayesian-data-analysis-1.html#cb1235-20" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">factor</span>(index, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior&quot;</span>, <span class="st">&quot;likelihood&quot;</span>, <span class="st">&quot;posterior&quot;</span>)),</span>
-<span id="cb1235-21"><a href="bayesian-data-analysis-1.html#cb1235-21" tabindex="-1"></a>         <span class="at">likelihood_index =</span> <span class="fu">factor</span>(likelihood_index,</span>
-<span id="cb1235-22"><a href="bayesian-data-analysis-1.html#cb1235-22" tabindex="-1"></a>                                   <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;likelihood_left&quot;</span>,</span>
-<span id="cb1235-23"><a href="bayesian-data-analysis-1.html#cb1235-23" tabindex="-1"></a>                                              <span class="st">&quot;likelihood_center&quot;</span>,</span>
-<span id="cb1235-24"><a href="bayesian-data-analysis-1.html#cb1235-24" tabindex="-1"></a>                                              <span class="st">&quot;likelihood_right&quot;</span>),</span>
-<span id="cb1235-25"><a href="bayesian-data-analysis-1.html#cb1235-25" tabindex="-1"></a>                                   <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;left&quot;</span>, <span class="st">&quot;center&quot;</span>, <span class="st">&quot;right&quot;</span>))) <span class="sc">%&gt;%</span> </span>
-<span id="cb1235-26"><a href="bayesian-data-analysis-1.html#cb1235-26" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1235-27"><a href="bayesian-data-analysis-1.html#cb1235-27" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> theta,</span>
-<span id="cb1235-28"><a href="bayesian-data-analysis-1.html#cb1235-28" tabindex="-1"></a>                       <span class="at">y =</span> value,</span>
-<span id="cb1235-29"><a href="bayesian-data-analysis-1.html#cb1235-29" tabindex="-1"></a>                       <span class="at">color =</span> index)) <span class="sc">+</span></span>
-<span id="cb1235-30"><a href="bayesian-data-analysis-1.html#cb1235-30" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
-<span id="cb1235-31"><a href="bayesian-data-analysis-1.html#cb1235-31" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">cols =</span> <span class="fu">vars</span>(likelihood_index),</span>
-<span id="cb1235-32"><a href="bayesian-data-analysis-1.html#cb1235-32" tabindex="-1"></a>             <span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
-<span id="cb1235-33"><a href="bayesian-data-analysis-1.html#cb1235-33" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>,</span>
-<span id="cb1235-34"><a href="bayesian-data-analysis-1.html#cb1235-34" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1235-35"><a href="bayesian-data-analysis-1.html#cb1235-35" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.2</span>)) <span class="sc">+</span></span>
-<span id="cb1235-36"><a href="bayesian-data-analysis-1.html#cb1235-36" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1235-37"><a href="bayesian-data-analysis-1.html#cb1235-37" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1235-38"><a href="bayesian-data-analysis-1.html#cb1235-38" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
-<span id="cb1235-39"><a href="bayesian-data-analysis-1.html#cb1235-39" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1235-40"><a href="bayesian-data-analysis-1.html#cb1235-40" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1235-41"><a href="bayesian-data-analysis-1.html#cb1235-41" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1235-42"><a href="bayesian-data-analysis-1.html#cb1235-42" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1235-43"><a href="bayesian-data-analysis-1.html#cb1235-43" tabindex="-1"></a>        <span class="at">axis.text.x =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">10</span>),</span>
-<span id="cb1235-44"><a href="bayesian-data-analysis-1.html#cb1235-44" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1235-45"><a href="bayesian-data-analysis-1.html#cb1235-45" tabindex="-1"></a>        <span class="at">strip.text.x =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
+<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="bayesian-data-analysis-1.html#cb8-1" tabindex="-1"></a><span class="co"># grid</span></span>
+<span id="cb8-2"><a href="bayesian-data-analysis-1.html#cb8-2" tabindex="-1"></a>theta <span class="ot">=</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.01</span>)</span>
+<span id="cb8-3"><a href="bayesian-data-analysis-1.html#cb8-3" tabindex="-1"></a></span>
+<span id="cb8-4"><a href="bayesian-data-analysis-1.html#cb8-4" tabindex="-1"></a>df.likelihood_effect <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">theta =</span> theta, </span>
+<span id="cb8-5"><a href="bayesian-data-analysis-1.html#cb8-5" tabindex="-1"></a>                              <span class="at">prior =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">2</span>, <span class="at">shape2 =</span> <span class="dv">8</span>),</span>
+<span id="cb8-6"><a href="bayesian-data-analysis-1.html#cb8-6" tabindex="-1"></a>                              <span class="at">likelihood_left =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">1</span>, <span class="at">shape2 =</span> <span class="dv">9</span>),</span>
+<span id="cb8-7"><a href="bayesian-data-analysis-1.html#cb8-7" tabindex="-1"></a>                              <span class="at">likelihood_center =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">5</span>, <span class="at">shape2 =</span> <span class="dv">5</span>),</span>
+<span id="cb8-8"><a href="bayesian-data-analysis-1.html#cb8-8" tabindex="-1"></a>                              <span class="at">likelihood_right =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">9</span>, <span class="at">shape2 =</span> <span class="dv">1</span>)) <span class="sc">%&gt;%</span> </span>
+<span id="cb8-9"><a href="bayesian-data-analysis-1.html#cb8-9" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, prior),</span>
+<span id="cb8-10"><a href="bayesian-data-analysis-1.html#cb8-10" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;likelihood_index&quot;</span>,</span>
+<span id="cb8-11"><a href="bayesian-data-analysis-1.html#cb8-11" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;likelihood&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb8-12"><a href="bayesian-data-analysis-1.html#cb8-12" tabindex="-1"></a>  <span class="fu">group_by</span>(likelihood_index) <span class="sc">%&gt;%</span> </span>
+<span id="cb8-13"><a href="bayesian-data-analysis-1.html#cb8-13" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">posterior =</span> likelihood <span class="sc">*</span> prior <span class="sc">/</span> <span class="fu">sum</span>(likelihood <span class="sc">*</span> prior)) <span class="sc">%&gt;%</span> </span>
+<span id="cb8-14"><a href="bayesian-data-analysis-1.html#cb8-14" tabindex="-1"></a>  <span class="fu">ungroup</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb8-15"><a href="bayesian-data-analysis-1.html#cb8-15" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, likelihood_index),</span>
+<span id="cb8-16"><a href="bayesian-data-analysis-1.html#cb8-16" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
+<span id="cb8-17"><a href="bayesian-data-analysis-1.html#cb8-17" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>)</span>
+<span id="cb8-18"><a href="bayesian-data-analysis-1.html#cb8-18" tabindex="-1"></a></span>
+<span id="cb8-19"><a href="bayesian-data-analysis-1.html#cb8-19" tabindex="-1"></a>df.likelihood_effect <span class="sc">%&gt;%</span> </span>
+<span id="cb8-20"><a href="bayesian-data-analysis-1.html#cb8-20" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">factor</span>(index, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior&quot;</span>, <span class="st">&quot;likelihood&quot;</span>, <span class="st">&quot;posterior&quot;</span>)),</span>
+<span id="cb8-21"><a href="bayesian-data-analysis-1.html#cb8-21" tabindex="-1"></a>         <span class="at">likelihood_index =</span> <span class="fu">factor</span>(likelihood_index,</span>
+<span id="cb8-22"><a href="bayesian-data-analysis-1.html#cb8-22" tabindex="-1"></a>                                   <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;likelihood_left&quot;</span>,</span>
+<span id="cb8-23"><a href="bayesian-data-analysis-1.html#cb8-23" tabindex="-1"></a>                                              <span class="st">&quot;likelihood_center&quot;</span>,</span>
+<span id="cb8-24"><a href="bayesian-data-analysis-1.html#cb8-24" tabindex="-1"></a>                                              <span class="st">&quot;likelihood_right&quot;</span>),</span>
+<span id="cb8-25"><a href="bayesian-data-analysis-1.html#cb8-25" tabindex="-1"></a>                                   <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;left&quot;</span>, <span class="st">&quot;center&quot;</span>, <span class="st">&quot;right&quot;</span>))) <span class="sc">%&gt;%</span> </span>
+<span id="cb8-26"><a href="bayesian-data-analysis-1.html#cb8-26" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb8-27"><a href="bayesian-data-analysis-1.html#cb8-27" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> theta,</span>
+<span id="cb8-28"><a href="bayesian-data-analysis-1.html#cb8-28" tabindex="-1"></a>                       <span class="at">y =</span> value,</span>
+<span id="cb8-29"><a href="bayesian-data-analysis-1.html#cb8-29" tabindex="-1"></a>                       <span class="at">color =</span> index)) <span class="sc">+</span></span>
+<span id="cb8-30"><a href="bayesian-data-analysis-1.html#cb8-30" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
+<span id="cb8-31"><a href="bayesian-data-analysis-1.html#cb8-31" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">cols =</span> <span class="fu">vars</span>(likelihood_index),</span>
+<span id="cb8-32"><a href="bayesian-data-analysis-1.html#cb8-32" tabindex="-1"></a>             <span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
+<span id="cb8-33"><a href="bayesian-data-analysis-1.html#cb8-33" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>,</span>
+<span id="cb8-34"><a href="bayesian-data-analysis-1.html#cb8-34" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>) <span class="sc">+</span></span>
+<span id="cb8-35"><a href="bayesian-data-analysis-1.html#cb8-35" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.2</span>)) <span class="sc">+</span></span>
+<span id="cb8-36"><a href="bayesian-data-analysis-1.html#cb8-36" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb8-37"><a href="bayesian-data-analysis-1.html#cb8-37" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb8-38"><a href="bayesian-data-analysis-1.html#cb8-38" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
+<span id="cb8-39"><a href="bayesian-data-analysis-1.html#cb8-39" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb8-40"><a href="bayesian-data-analysis-1.html#cb8-40" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb8-41"><a href="bayesian-data-analysis-1.html#cb8-41" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb8-42"><a href="bayesian-data-analysis-1.html#cb8-42" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb8-43"><a href="bayesian-data-analysis-1.html#cb8-43" tabindex="-1"></a>        <span class="at">axis.text.x =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">10</span>),</span>
+<span id="cb8-44"><a href="bayesian-data-analysis-1.html#cb8-44" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb8-45"><a href="bayesian-data-analysis-1.html#cb8-45" tabindex="-1"></a>        <span class="at">strip.text.x =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
 <div class="figure"><span style="display:block;" id="fig:unnamed-chunk-8"></span>
-<img src="22-bayesian_data_analysis1_files/figure-html/unnamed-chunk-8-1.png" alt="Illustration of how the likelihood of the data affects the posterior." width="672" />
+<img src="psych252_files/figure-html/unnamed-chunk-8-1.png" alt="Illustration of how the likelihood of the data affects the posterior." width="672" />
 <p class="caption">
-Figure 2.7: Illustration of how the likelihood of the data affects the posterior.
+Figure 22.2: Illustration of how the likelihood of the data affects the posterior.
 </p>
 </div>
 </div>
 <div id="effect-of-the-sample-size" class="section level4 hasAnchor" number="22.3.3.3">
 <h4><span class="header-section-number">22.3.3.3</span> Effect of the sample size<a href="bayesian-data-analysis-1.html#effect-of-the-sample-size" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1236"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1236-1"><a href="bayesian-data-analysis-1.html#cb1236-1" tabindex="-1"></a><span class="co"># grid</span></span>
-<span id="cb1236-2"><a href="bayesian-data-analysis-1.html#cb1236-2" tabindex="-1"></a>theta <span class="ot">=</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.01</span>)</span>
-<span id="cb1236-3"><a href="bayesian-data-analysis-1.html#cb1236-3" tabindex="-1"></a></span>
-<span id="cb1236-4"><a href="bayesian-data-analysis-1.html#cb1236-4" tabindex="-1"></a>df.sample_size_effect <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">theta =</span> theta, </span>
-<span id="cb1236-5"><a href="bayesian-data-analysis-1.html#cb1236-5" tabindex="-1"></a>                               <span class="at">prior =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">5</span>, <span class="at">shape2 =</span> <span class="dv">5</span>),</span>
-<span id="cb1236-6"><a href="bayesian-data-analysis-1.html#cb1236-6" tabindex="-1"></a>                               <span class="at">likelihood_low =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">2</span>, <span class="at">shape2 =</span> <span class="dv">8</span>),</span>
-<span id="cb1236-7"><a href="bayesian-data-analysis-1.html#cb1236-7" tabindex="-1"></a>                               <span class="at">likelihood_medium =</span> <span class="fu">dbeta</span>(theta,</span>
-<span id="cb1236-8"><a href="bayesian-data-analysis-1.html#cb1236-8" tabindex="-1"></a>                                                         <span class="at">shape1 =</span> <span class="dv">10</span>,</span>
-<span id="cb1236-9"><a href="bayesian-data-analysis-1.html#cb1236-9" tabindex="-1"></a>                                                         <span class="at">shape2 =</span> <span class="dv">40</span>),</span>
-<span id="cb1236-10"><a href="bayesian-data-analysis-1.html#cb1236-10" tabindex="-1"></a>                               <span class="at">likelihood_high =</span> <span class="fu">dbeta</span>(theta,</span>
-<span id="cb1236-11"><a href="bayesian-data-analysis-1.html#cb1236-11" tabindex="-1"></a>                                                       <span class="at">shape1 =</span> <span class="dv">20</span>,</span>
-<span id="cb1236-12"><a href="bayesian-data-analysis-1.html#cb1236-12" tabindex="-1"></a>                                                       <span class="at">shape2 =</span> <span class="dv">80</span>)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1236-13"><a href="bayesian-data-analysis-1.html#cb1236-13" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, prior),</span>
-<span id="cb1236-14"><a href="bayesian-data-analysis-1.html#cb1236-14" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;likelihood_index&quot;</span>,</span>
-<span id="cb1236-15"><a href="bayesian-data-analysis-1.html#cb1236-15" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;likelihood&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1236-16"><a href="bayesian-data-analysis-1.html#cb1236-16" tabindex="-1"></a>  <span class="fu">group_by</span>(likelihood_index) <span class="sc">%&gt;%</span> </span>
-<span id="cb1236-17"><a href="bayesian-data-analysis-1.html#cb1236-17" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">posterior =</span> likelihood <span class="sc">*</span> prior <span class="sc">/</span> <span class="fu">sum</span>(likelihood <span class="sc">*</span> prior)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1236-18"><a href="bayesian-data-analysis-1.html#cb1236-18" tabindex="-1"></a>  <span class="fu">ungroup</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1236-19"><a href="bayesian-data-analysis-1.html#cb1236-19" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, likelihood_index),</span>
-<span id="cb1236-20"><a href="bayesian-data-analysis-1.html#cb1236-20" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
-<span id="cb1236-21"><a href="bayesian-data-analysis-1.html#cb1236-21" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>)</span>
-<span id="cb1236-22"><a href="bayesian-data-analysis-1.html#cb1236-22" tabindex="-1"></a></span>
-<span id="cb1236-23"><a href="bayesian-data-analysis-1.html#cb1236-23" tabindex="-1"></a>df.sample_size_effect <span class="sc">%&gt;%</span> </span>
-<span id="cb1236-24"><a href="bayesian-data-analysis-1.html#cb1236-24" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">factor</span>(index, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior&quot;</span>, <span class="st">&quot;likelihood&quot;</span>, <span class="st">&quot;posterior&quot;</span>)),</span>
-<span id="cb1236-25"><a href="bayesian-data-analysis-1.html#cb1236-25" tabindex="-1"></a>         <span class="at">likelihood_index =</span> <span class="fu">factor</span>(likelihood_index,</span>
-<span id="cb1236-26"><a href="bayesian-data-analysis-1.html#cb1236-26" tabindex="-1"></a>                                   <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;likelihood_low&quot;</span>,</span>
-<span id="cb1236-27"><a href="bayesian-data-analysis-1.html#cb1236-27" tabindex="-1"></a>                                              <span class="st">&quot;likelihood_medium&quot;</span>,</span>
-<span id="cb1236-28"><a href="bayesian-data-analysis-1.html#cb1236-28" tabindex="-1"></a>                                              <span class="st">&quot;likelihood_high&quot;</span>),</span>
-<span id="cb1236-29"><a href="bayesian-data-analysis-1.html#cb1236-29" tabindex="-1"></a>                                   <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;n = low&quot;</span>, <span class="st">&quot;n = medium&quot;</span>, <span class="st">&quot;n = high&quot;</span>))) <span class="sc">%&gt;%</span> </span>
-<span id="cb1236-30"><a href="bayesian-data-analysis-1.html#cb1236-30" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1236-31"><a href="bayesian-data-analysis-1.html#cb1236-31" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> theta,</span>
-<span id="cb1236-32"><a href="bayesian-data-analysis-1.html#cb1236-32" tabindex="-1"></a>                       <span class="at">y =</span> value,</span>
-<span id="cb1236-33"><a href="bayesian-data-analysis-1.html#cb1236-33" tabindex="-1"></a>                       <span class="at">color =</span> index)) <span class="sc">+</span></span>
-<span id="cb1236-34"><a href="bayesian-data-analysis-1.html#cb1236-34" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
-<span id="cb1236-35"><a href="bayesian-data-analysis-1.html#cb1236-35" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">cols =</span> <span class="fu">vars</span>(likelihood_index),</span>
-<span id="cb1236-36"><a href="bayesian-data-analysis-1.html#cb1236-36" tabindex="-1"></a>             <span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
-<span id="cb1236-37"><a href="bayesian-data-analysis-1.html#cb1236-37" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>,</span>
-<span id="cb1236-38"><a href="bayesian-data-analysis-1.html#cb1236-38" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1236-39"><a href="bayesian-data-analysis-1.html#cb1236-39" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.2</span>)) <span class="sc">+</span></span>
-<span id="cb1236-40"><a href="bayesian-data-analysis-1.html#cb1236-40" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1236-41"><a href="bayesian-data-analysis-1.html#cb1236-41" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1236-42"><a href="bayesian-data-analysis-1.html#cb1236-42" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
-<span id="cb1236-43"><a href="bayesian-data-analysis-1.html#cb1236-43" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1236-44"><a href="bayesian-data-analysis-1.html#cb1236-44" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1236-45"><a href="bayesian-data-analysis-1.html#cb1236-45" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1236-46"><a href="bayesian-data-analysis-1.html#cb1236-46" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1236-47"><a href="bayesian-data-analysis-1.html#cb1236-47" tabindex="-1"></a>        <span class="at">axis.text.x =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">10</span>),</span>
-<span id="cb1236-48"><a href="bayesian-data-analysis-1.html#cb1236-48" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
-<p><img src="22-bayesian_data_analysis1_files/figure-html/unnamed-chunk-9-1.png" width="672" /></p>
+<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="bayesian-data-analysis-1.html#cb9-1" tabindex="-1"></a><span class="co"># grid</span></span>
+<span id="cb9-2"><a href="bayesian-data-analysis-1.html#cb9-2" tabindex="-1"></a>theta <span class="ot">=</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.01</span>)</span>
+<span id="cb9-3"><a href="bayesian-data-analysis-1.html#cb9-3" tabindex="-1"></a></span>
+<span id="cb9-4"><a href="bayesian-data-analysis-1.html#cb9-4" tabindex="-1"></a>df.sample_size_effect <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">theta =</span> theta, </span>
+<span id="cb9-5"><a href="bayesian-data-analysis-1.html#cb9-5" tabindex="-1"></a>                               <span class="at">prior =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">5</span>, <span class="at">shape2 =</span> <span class="dv">5</span>),</span>
+<span id="cb9-6"><a href="bayesian-data-analysis-1.html#cb9-6" tabindex="-1"></a>                               <span class="at">likelihood_low =</span> <span class="fu">dbeta</span>(theta, <span class="at">shape1 =</span> <span class="dv">2</span>, <span class="at">shape2 =</span> <span class="dv">8</span>),</span>
+<span id="cb9-7"><a href="bayesian-data-analysis-1.html#cb9-7" tabindex="-1"></a>                               <span class="at">likelihood_medium =</span> <span class="fu">dbeta</span>(theta,</span>
+<span id="cb9-8"><a href="bayesian-data-analysis-1.html#cb9-8" tabindex="-1"></a>                                                         <span class="at">shape1 =</span> <span class="dv">10</span>,</span>
+<span id="cb9-9"><a href="bayesian-data-analysis-1.html#cb9-9" tabindex="-1"></a>                                                         <span class="at">shape2 =</span> <span class="dv">40</span>),</span>
+<span id="cb9-10"><a href="bayesian-data-analysis-1.html#cb9-10" tabindex="-1"></a>                               <span class="at">likelihood_high =</span> <span class="fu">dbeta</span>(theta,</span>
+<span id="cb9-11"><a href="bayesian-data-analysis-1.html#cb9-11" tabindex="-1"></a>                                                       <span class="at">shape1 =</span> <span class="dv">20</span>,</span>
+<span id="cb9-12"><a href="bayesian-data-analysis-1.html#cb9-12" tabindex="-1"></a>                                                       <span class="at">shape2 =</span> <span class="dv">80</span>)) <span class="sc">%&gt;%</span> </span>
+<span id="cb9-13"><a href="bayesian-data-analysis-1.html#cb9-13" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, prior),</span>
+<span id="cb9-14"><a href="bayesian-data-analysis-1.html#cb9-14" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;likelihood_index&quot;</span>,</span>
+<span id="cb9-15"><a href="bayesian-data-analysis-1.html#cb9-15" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;likelihood&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb9-16"><a href="bayesian-data-analysis-1.html#cb9-16" tabindex="-1"></a>  <span class="fu">group_by</span>(likelihood_index) <span class="sc">%&gt;%</span> </span>
+<span id="cb9-17"><a href="bayesian-data-analysis-1.html#cb9-17" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">posterior =</span> likelihood <span class="sc">*</span> prior <span class="sc">/</span> <span class="fu">sum</span>(likelihood <span class="sc">*</span> prior)) <span class="sc">%&gt;%</span> </span>
+<span id="cb9-18"><a href="bayesian-data-analysis-1.html#cb9-18" tabindex="-1"></a>  <span class="fu">ungroup</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb9-19"><a href="bayesian-data-analysis-1.html#cb9-19" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(theta, likelihood_index),</span>
+<span id="cb9-20"><a href="bayesian-data-analysis-1.html#cb9-20" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
+<span id="cb9-21"><a href="bayesian-data-analysis-1.html#cb9-21" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>)</span>
+<span id="cb9-22"><a href="bayesian-data-analysis-1.html#cb9-22" tabindex="-1"></a></span>
+<span id="cb9-23"><a href="bayesian-data-analysis-1.html#cb9-23" tabindex="-1"></a>df.sample_size_effect <span class="sc">%&gt;%</span> </span>
+<span id="cb9-24"><a href="bayesian-data-analysis-1.html#cb9-24" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">factor</span>(index, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;prior&quot;</span>, <span class="st">&quot;likelihood&quot;</span>, <span class="st">&quot;posterior&quot;</span>)),</span>
+<span id="cb9-25"><a href="bayesian-data-analysis-1.html#cb9-25" tabindex="-1"></a>         <span class="at">likelihood_index =</span> <span class="fu">factor</span>(likelihood_index,</span>
+<span id="cb9-26"><a href="bayesian-data-analysis-1.html#cb9-26" tabindex="-1"></a>                                   <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;likelihood_low&quot;</span>,</span>
+<span id="cb9-27"><a href="bayesian-data-analysis-1.html#cb9-27" tabindex="-1"></a>                                              <span class="st">&quot;likelihood_medium&quot;</span>,</span>
+<span id="cb9-28"><a href="bayesian-data-analysis-1.html#cb9-28" tabindex="-1"></a>                                              <span class="st">&quot;likelihood_high&quot;</span>),</span>
+<span id="cb9-29"><a href="bayesian-data-analysis-1.html#cb9-29" tabindex="-1"></a>                                   <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;n = low&quot;</span>, <span class="st">&quot;n = medium&quot;</span>, <span class="st">&quot;n = high&quot;</span>))) <span class="sc">%&gt;%</span> </span>
+<span id="cb9-30"><a href="bayesian-data-analysis-1.html#cb9-30" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb9-31"><a href="bayesian-data-analysis-1.html#cb9-31" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> theta,</span>
+<span id="cb9-32"><a href="bayesian-data-analysis-1.html#cb9-32" tabindex="-1"></a>                       <span class="at">y =</span> value,</span>
+<span id="cb9-33"><a href="bayesian-data-analysis-1.html#cb9-33" tabindex="-1"></a>                       <span class="at">color =</span> index)) <span class="sc">+</span></span>
+<span id="cb9-34"><a href="bayesian-data-analysis-1.html#cb9-34" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
+<span id="cb9-35"><a href="bayesian-data-analysis-1.html#cb9-35" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">cols =</span> <span class="fu">vars</span>(likelihood_index),</span>
+<span id="cb9-36"><a href="bayesian-data-analysis-1.html#cb9-36" tabindex="-1"></a>             <span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
+<span id="cb9-37"><a href="bayesian-data-analysis-1.html#cb9-37" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>,</span>
+<span id="cb9-38"><a href="bayesian-data-analysis-1.html#cb9-38" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>) <span class="sc">+</span></span>
+<span id="cb9-39"><a href="bayesian-data-analysis-1.html#cb9-39" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="fu">seq</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="fl">0.2</span>)) <span class="sc">+</span></span>
+<span id="cb9-40"><a href="bayesian-data-analysis-1.html#cb9-40" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb9-41"><a href="bayesian-data-analysis-1.html#cb9-41" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb9-42"><a href="bayesian-data-analysis-1.html#cb9-42" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
+<span id="cb9-43"><a href="bayesian-data-analysis-1.html#cb9-43" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb9-44"><a href="bayesian-data-analysis-1.html#cb9-44" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb9-45"><a href="bayesian-data-analysis-1.html#cb9-45" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb9-46"><a href="bayesian-data-analysis-1.html#cb9-46" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb9-47"><a href="bayesian-data-analysis-1.html#cb9-47" tabindex="-1"></a>        <span class="at">axis.text.x =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">10</span>),</span>
+<span id="cb9-48"><a href="bayesian-data-analysis-1.html#cb9-48" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-9-1.png" width="672" /></p>
 </div>
 </div>
 </div>
@@ -1158,24 +1138,24 @@ <h2><span class="header-section-number">22.4</span> Doing Bayesian inference wit
 <p>You can find out more about how get started with “greta” here: <a href="https://greta-stats.org/articles/get_started.html">https://greta-stats.org/articles/get_started.html</a>. Make sure to install the development version of “greta” (as shown in the “install-packages” code chunk above: <code>devtools::install_github("greta-dev/greta")</code>).</p>
 <div id="attitude-data-set" class="section level3 hasAnchor" number="22.4.1">
 <h3><span class="header-section-number">22.4.1</span> Attitude data set<a href="bayesian-data-analysis-1.html#attitude-data-set" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1237"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1237-1"><a href="bayesian-data-analysis-1.html#cb1237-1" tabindex="-1"></a><span class="co"># load the attitude data set </span></span>
-<span id="cb1237-2"><a href="bayesian-data-analysis-1.html#cb1237-2" tabindex="-1"></a>df.attitude <span class="ot">=</span> attitude</span></code></pre></div>
+<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="bayesian-data-analysis-1.html#cb10-1" tabindex="-1"></a><span class="co"># load the attitude data set </span></span>
+<span id="cb10-2"><a href="bayesian-data-analysis-1.html#cb10-2" tabindex="-1"></a>df.attitude <span class="ot">=</span> attitude</span></code></pre></div>
 <p>Visualize relationship between how well complaints are handled and the overall rating of an employee</p>
-<div class="sourceCode" id="cb1238"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1238-1"><a href="bayesian-data-analysis-1.html#cb1238-1" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.attitude,</span>
-<span id="cb1238-2"><a href="bayesian-data-analysis-1.html#cb1238-2" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> complaints,</span>
-<span id="cb1238-3"><a href="bayesian-data-analysis-1.html#cb1238-3" tabindex="-1"></a>                     <span class="at">y =</span> rating)) <span class="sc">+</span></span>
-<span id="cb1238-4"><a href="bayesian-data-analysis-1.html#cb1238-4" tabindex="-1"></a>  <span class="fu">geom_point</span>()</span></code></pre></div>
-<p><img src="22-bayesian_data_analysis1_files/figure-html/unnamed-chunk-11-1.png" width="672" /></p>
+<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb11-1"><a href="bayesian-data-analysis-1.html#cb11-1" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.attitude,</span>
+<span id="cb11-2"><a href="bayesian-data-analysis-1.html#cb11-2" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> complaints,</span>
+<span id="cb11-3"><a href="bayesian-data-analysis-1.html#cb11-3" tabindex="-1"></a>                     <span class="at">y =</span> rating)) <span class="sc">+</span></span>
+<span id="cb11-4"><a href="bayesian-data-analysis-1.html#cb11-4" tabindex="-1"></a>  <span class="fu">geom_point</span>()</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-11-1.png" width="672" /></p>
 </div>
 <div id="frequentist-analysis" class="section level3 hasAnchor" number="22.4.2">
 <h3><span class="header-section-number">22.4.2</span> Frequentist analysis<a href="bayesian-data-analysis-1.html#frequentist-analysis" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1239"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1239-1"><a href="bayesian-data-analysis-1.html#cb1239-1" tabindex="-1"></a><span class="co"># fit model </span></span>
-<span id="cb1239-2"><a href="bayesian-data-analysis-1.html#cb1239-2" tabindex="-1"></a>fit.lm <span class="ot">=</span> <span class="fu">lm</span>(<span class="at">formula =</span> rating <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> complaints, </span>
-<span id="cb1239-3"><a href="bayesian-data-analysis-1.html#cb1239-3" tabindex="-1"></a>            <span class="at">data =</span> df.attitude)</span>
-<span id="cb1239-4"><a href="bayesian-data-analysis-1.html#cb1239-4" tabindex="-1"></a></span>
-<span id="cb1239-5"><a href="bayesian-data-analysis-1.html#cb1239-5" tabindex="-1"></a><span class="co"># print summary</span></span>
-<span id="cb1239-6"><a href="bayesian-data-analysis-1.html#cb1239-6" tabindex="-1"></a>fit.lm <span class="sc">%&gt;%</span> </span>
-<span id="cb1239-7"><a href="bayesian-data-analysis-1.html#cb1239-7" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb12-1"><a href="bayesian-data-analysis-1.html#cb12-1" tabindex="-1"></a><span class="co"># fit model </span></span>
+<span id="cb12-2"><a href="bayesian-data-analysis-1.html#cb12-2" tabindex="-1"></a>fit.lm <span class="ot">=</span> <span class="fu">lm</span>(<span class="at">formula =</span> rating <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> complaints, </span>
+<span id="cb12-3"><a href="bayesian-data-analysis-1.html#cb12-3" tabindex="-1"></a>            <span class="at">data =</span> df.attitude)</span>
+<span id="cb12-4"><a href="bayesian-data-analysis-1.html#cb12-4" tabindex="-1"></a></span>
+<span id="cb12-5"><a href="bayesian-data-analysis-1.html#cb12-5" tabindex="-1"></a><span class="co"># print summary</span></span>
+<span id="cb12-6"><a href="bayesian-data-analysis-1.html#cb12-6" tabindex="-1"></a>fit.lm <span class="sc">%&gt;%</span> </span>
+<span id="cb12-7"><a href="bayesian-data-analysis-1.html#cb12-7" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
 <pre><code>
 Call:
 lm(formula = rating ~ 1 + complaints, data = df.attitude)
@@ -1195,178 +1175,178 @@ <h3><span class="header-section-number">22.4.2</span> Frequentist analysis<a hre
 Multiple R-squared:  0.6813,    Adjusted R-squared:  0.6699 
 F-statistic: 59.86 on 1 and 28 DF,  p-value: 1.988e-08</code></pre>
 <p>Visualize the model’s predictions</p>
-<div class="sourceCode" id="cb1241"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1241-1"><a href="bayesian-data-analysis-1.html#cb1241-1" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.attitude,</span>
-<span id="cb1241-2"><a href="bayesian-data-analysis-1.html#cb1241-2" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> complaints,</span>
-<span id="cb1241-3"><a href="bayesian-data-analysis-1.html#cb1241-3" tabindex="-1"></a>                     <span class="at">y =</span> rating)) <span class="sc">+</span></span>
-<span id="cb1241-4"><a href="bayesian-data-analysis-1.html#cb1241-4" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">method =</span> <span class="st">&quot;lm&quot;</span>,</span>
-<span id="cb1241-5"><a href="bayesian-data-analysis-1.html#cb1241-5" tabindex="-1"></a>              <span class="at">formula =</span> <span class="st">&quot;y ~ x&quot;</span>, </span>
-<span id="cb1241-6"><a href="bayesian-data-analysis-1.html#cb1241-6" tabindex="-1"></a>              <span class="at">color =</span> <span class="st">&quot;black&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1241-7"><a href="bayesian-data-analysis-1.html#cb1241-7" tabindex="-1"></a>  <span class="fu">geom_point</span>()</span></code></pre></div>
-<p><img src="22-bayesian_data_analysis1_files/figure-html/unnamed-chunk-13-1.png" width="672" /></p>
+<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb14-1"><a href="bayesian-data-analysis-1.html#cb14-1" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.attitude,</span>
+<span id="cb14-2"><a href="bayesian-data-analysis-1.html#cb14-2" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> complaints,</span>
+<span id="cb14-3"><a href="bayesian-data-analysis-1.html#cb14-3" tabindex="-1"></a>                     <span class="at">y =</span> rating)) <span class="sc">+</span></span>
+<span id="cb14-4"><a href="bayesian-data-analysis-1.html#cb14-4" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">method =</span> <span class="st">&quot;lm&quot;</span>,</span>
+<span id="cb14-5"><a href="bayesian-data-analysis-1.html#cb14-5" tabindex="-1"></a>              <span class="at">formula =</span> <span class="st">&quot;y ~ x&quot;</span>, </span>
+<span id="cb14-6"><a href="bayesian-data-analysis-1.html#cb14-6" tabindex="-1"></a>              <span class="at">color =</span> <span class="st">&quot;black&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb14-7"><a href="bayesian-data-analysis-1.html#cb14-7" tabindex="-1"></a>  <span class="fu">geom_point</span>()</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-13-1.png" width="672" /></p>
 </div>
 <div id="bayesian-regression" class="section level3 hasAnchor" number="22.4.3">
 <h3><span class="header-section-number">22.4.3</span> Bayesian regression<a href="bayesian-data-analysis-1.html#bayesian-regression" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div id="fit-the-model" class="section level4 hasAnchor" number="22.4.3.1">
-<h4><span class="header-section-number">22.4.3.1</span> Fit the model<a href="bayesian-data-analysis-1.html#fit-the-model" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1242"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1242-1"><a href="bayesian-data-analysis-1.html#cb1242-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
-<span id="cb1242-2"><a href="bayesian-data-analysis-1.html#cb1242-2" tabindex="-1"></a></span>
-<span id="cb1242-3"><a href="bayesian-data-analysis-1.html#cb1242-3" tabindex="-1"></a><span class="co"># variables &amp; priors</span></span>
-<span id="cb1242-4"><a href="bayesian-data-analysis-1.html#cb1242-4" tabindex="-1"></a>b0 <span class="ot">=</span> <span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>)</span>
-<span id="cb1242-5"><a href="bayesian-data-analysis-1.html#cb1242-5" tabindex="-1"></a>b1 <span class="ot">=</span> <span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>)</span>
-<span id="cb1242-6"><a href="bayesian-data-analysis-1.html#cb1242-6" tabindex="-1"></a>sd <span class="ot">=</span> <span class="fu">cauchy</span>(<span class="dv">0</span>, <span class="dv">3</span>, <span class="at">truncation =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="cn">Inf</span>))</span>
-<span id="cb1242-7"><a href="bayesian-data-analysis-1.html#cb1242-7" tabindex="-1"></a></span>
-<span id="cb1242-8"><a href="bayesian-data-analysis-1.html#cb1242-8" tabindex="-1"></a><span class="co"># linear predictor</span></span>
-<span id="cb1242-9"><a href="bayesian-data-analysis-1.html#cb1242-9" tabindex="-1"></a>mu <span class="ot">=</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> df.attitude<span class="sc">$</span>complaints</span>
-<span id="cb1242-10"><a href="bayesian-data-analysis-1.html#cb1242-10" tabindex="-1"></a></span>
-<span id="cb1242-11"><a href="bayesian-data-analysis-1.html#cb1242-11" tabindex="-1"></a><span class="co"># observation model (likelihood)</span></span>
-<span id="cb1242-12"><a href="bayesian-data-analysis-1.html#cb1242-12" tabindex="-1"></a><span class="fu">distribution</span>(df.attitude<span class="sc">$</span>rating) <span class="ot">=</span> <span class="fu">normal</span>(mu, sd)</span>
-<span id="cb1242-13"><a href="bayesian-data-analysis-1.html#cb1242-13" tabindex="-1"></a></span>
-<span id="cb1242-14"><a href="bayesian-data-analysis-1.html#cb1242-14" tabindex="-1"></a><span class="co"># define the model</span></span>
-<span id="cb1242-15"><a href="bayesian-data-analysis-1.html#cb1242-15" tabindex="-1"></a>m <span class="ot">=</span> <span class="fu">model</span>(b0, b1, sd)</span></code></pre></div>
+<div id="fit-the-model-3" class="section level4 hasAnchor" number="22.4.3.1">
+<h4><span class="header-section-number">22.4.3.1</span> Fit the model<a href="bayesian-data-analysis-1.html#fit-the-model-3" class="anchor-section" aria-label="Anchor link to header"></a></h4>
+<div class="sourceCode" id="cb15"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb15-1"><a href="bayesian-data-analysis-1.html#cb15-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
+<span id="cb15-2"><a href="bayesian-data-analysis-1.html#cb15-2" tabindex="-1"></a></span>
+<span id="cb15-3"><a href="bayesian-data-analysis-1.html#cb15-3" tabindex="-1"></a><span class="co"># variables &amp; priors</span></span>
+<span id="cb15-4"><a href="bayesian-data-analysis-1.html#cb15-4" tabindex="-1"></a>b0 <span class="ot">=</span> <span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>)</span>
+<span id="cb15-5"><a href="bayesian-data-analysis-1.html#cb15-5" tabindex="-1"></a>b1 <span class="ot">=</span> <span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>)</span>
+<span id="cb15-6"><a href="bayesian-data-analysis-1.html#cb15-6" tabindex="-1"></a>sd <span class="ot">=</span> <span class="fu">cauchy</span>(<span class="dv">0</span>, <span class="dv">3</span>, <span class="at">truncation =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="cn">Inf</span>))</span>
+<span id="cb15-7"><a href="bayesian-data-analysis-1.html#cb15-7" tabindex="-1"></a></span>
+<span id="cb15-8"><a href="bayesian-data-analysis-1.html#cb15-8" tabindex="-1"></a><span class="co"># linear predictor</span></span>
+<span id="cb15-9"><a href="bayesian-data-analysis-1.html#cb15-9" tabindex="-1"></a>mu <span class="ot">=</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> df.attitude<span class="sc">$</span>complaints</span>
+<span id="cb15-10"><a href="bayesian-data-analysis-1.html#cb15-10" tabindex="-1"></a></span>
+<span id="cb15-11"><a href="bayesian-data-analysis-1.html#cb15-11" tabindex="-1"></a><span class="co"># observation model (likelihood)</span></span>
+<span id="cb15-12"><a href="bayesian-data-analysis-1.html#cb15-12" tabindex="-1"></a><span class="fu">distribution</span>(df.attitude<span class="sc">$</span>rating) <span class="ot">=</span> <span class="fu">normal</span>(mu, sd)</span>
+<span id="cb15-13"><a href="bayesian-data-analysis-1.html#cb15-13" tabindex="-1"></a></span>
+<span id="cb15-14"><a href="bayesian-data-analysis-1.html#cb15-14" tabindex="-1"></a><span class="co"># define the model</span></span>
+<span id="cb15-15"><a href="bayesian-data-analysis-1.html#cb15-15" tabindex="-1"></a>m <span class="ot">=</span> <span class="fu">model</span>(b0, b1, sd)</span></code></pre></div>
 <p>Visualize the model as graph:</p>
-<div class="sourceCode" id="cb1243"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1243-1"><a href="bayesian-data-analysis-1.html#cb1243-1" tabindex="-1"></a><span class="co"># plotting</span></span>
-<span id="cb1243-2"><a href="bayesian-data-analysis-1.html#cb1243-2" tabindex="-1"></a><span class="fu">plot</span>(m)</span></code></pre></div>
+<div class="sourceCode" id="cb16"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb16-1"><a href="bayesian-data-analysis-1.html#cb16-1" tabindex="-1"></a><span class="co"># plotting</span></span>
+<span id="cb16-2"><a href="bayesian-data-analysis-1.html#cb16-2" tabindex="-1"></a><span class="fu">plot</span>(m)</span></code></pre></div>
 <p>Draw samples from the posterior distribution:</p>
-<div class="sourceCode" id="cb1244"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1244-1"><a href="bayesian-data-analysis-1.html#cb1244-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
-<span id="cb1244-2"><a href="bayesian-data-analysis-1.html#cb1244-2" tabindex="-1"></a></span>
-<span id="cb1244-3"><a href="bayesian-data-analysis-1.html#cb1244-3" tabindex="-1"></a><span class="co"># sampling</span></span>
-<span id="cb1244-4"><a href="bayesian-data-analysis-1.html#cb1244-4" tabindex="-1"></a>draws <span class="ot">=</span> <span class="fu">mcmc</span>(m, <span class="at">n_samples =</span> <span class="dv">1000</span>)</span>
-<span id="cb1244-5"><a href="bayesian-data-analysis-1.html#cb1244-5" tabindex="-1"></a></span>
-<span id="cb1244-6"><a href="bayesian-data-analysis-1.html#cb1244-6" tabindex="-1"></a><span class="co"># tidy up the draws</span></span>
-<span id="cb1244-7"><a href="bayesian-data-analysis-1.html#cb1244-7" tabindex="-1"></a>df.draws <span class="ot">=</span> <span class="fu">tidy_draws</span>(draws) <span class="sc">%&gt;%</span> </span>
-<span id="cb1244-8"><a href="bayesian-data-analysis-1.html#cb1244-8" tabindex="-1"></a>  <span class="fu">clean_names</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb17"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb17-1"><a href="bayesian-data-analysis-1.html#cb17-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
+<span id="cb17-2"><a href="bayesian-data-analysis-1.html#cb17-2" tabindex="-1"></a></span>
+<span id="cb17-3"><a href="bayesian-data-analysis-1.html#cb17-3" tabindex="-1"></a><span class="co"># sampling</span></span>
+<span id="cb17-4"><a href="bayesian-data-analysis-1.html#cb17-4" tabindex="-1"></a>draws <span class="ot">=</span> <span class="fu">mcmc</span>(m, <span class="at">n_samples =</span> <span class="dv">1000</span>)</span>
+<span id="cb17-5"><a href="bayesian-data-analysis-1.html#cb17-5" tabindex="-1"></a></span>
+<span id="cb17-6"><a href="bayesian-data-analysis-1.html#cb17-6" tabindex="-1"></a><span class="co"># tidy up the draws</span></span>
+<span id="cb17-7"><a href="bayesian-data-analysis-1.html#cb17-7" tabindex="-1"></a>df.draws <span class="ot">=</span> <span class="fu">tidy_draws</span>(draws) <span class="sc">%&gt;%</span> </span>
+<span id="cb17-8"><a href="bayesian-data-analysis-1.html#cb17-8" tabindex="-1"></a>  <span class="fu">clean_names</span>()</span></code></pre></div>
 </div>
 <div id="visualize-the-priors" class="section level4 hasAnchor" number="22.4.3.2">
 <h4><span class="header-section-number">22.4.3.2</span> Visualize the priors<a href="bayesian-data-analysis-1.html#visualize-the-priors" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>These are the priors I used for the intercept, regression weights, and the standard deviation of the Gaussian likelihood function:</p>
-<div class="sourceCode" id="cb1245"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1245-1"><a href="bayesian-data-analysis-1.html#cb1245-1" tabindex="-1"></a><span class="co"># Gaussian</span></span>
-<span id="cb1245-2"><a href="bayesian-data-analysis-1.html#cb1245-2" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="fu">tibble</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">30</span>, <span class="dv">30</span>)),</span>
-<span id="cb1245-3"><a href="bayesian-data-analysis-1.html#cb1245-3" tabindex="-1"></a>       <span class="fu">aes</span>(<span class="at">x =</span> x)) <span class="sc">+</span></span>
-<span id="cb1245-4"><a href="bayesian-data-analysis-1.html#cb1245-4" tabindex="-1"></a>  <span class="fu">stat_function</span>(<span class="at">fun =</span> <span class="st">&quot;dnorm&quot;</span>, </span>
-<span id="cb1245-5"><a href="bayesian-data-analysis-1.html#cb1245-5" tabindex="-1"></a>                <span class="at">size =</span> <span class="dv">2</span>,</span>
-<span id="cb1245-6"><a href="bayesian-data-analysis-1.html#cb1245-6" tabindex="-1"></a>                <span class="at">args =</span> <span class="fu">list</span>(<span class="at">sd =</span> <span class="dv">10</span>))</span>
-<span id="cb1245-7"><a href="bayesian-data-analysis-1.html#cb1245-7" tabindex="-1"></a></span>
-<span id="cb1245-8"><a href="bayesian-data-analysis-1.html#cb1245-8" tabindex="-1"></a><span class="co"># Cauchy</span></span>
-<span id="cb1245-9"><a href="bayesian-data-analysis-1.html#cb1245-9" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="fu">tibble</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">30</span>)),</span>
-<span id="cb1245-10"><a href="bayesian-data-analysis-1.html#cb1245-10" tabindex="-1"></a>       <span class="fu">aes</span>(<span class="at">x =</span> x)) <span class="sc">+</span></span>
-<span id="cb1245-11"><a href="bayesian-data-analysis-1.html#cb1245-11" tabindex="-1"></a>  <span class="fu">stat_function</span>(<span class="at">fun =</span> <span class="st">&quot;dcauchy&quot;</span>, </span>
-<span id="cb1245-12"><a href="bayesian-data-analysis-1.html#cb1245-12" tabindex="-1"></a>                <span class="at">size =</span> <span class="dv">2</span>,</span>
-<span id="cb1245-13"><a href="bayesian-data-analysis-1.html#cb1245-13" tabindex="-1"></a>                <span class="at">args =</span> <span class="fu">list</span>(<span class="at">location =</span> <span class="dv">0</span>,</span>
-<span id="cb1245-14"><a href="bayesian-data-analysis-1.html#cb1245-14" tabindex="-1"></a>                            <span class="at">scale =</span> <span class="dv">3</span>))</span></code></pre></div>
+<div class="sourceCode" id="cb18"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb18-1"><a href="bayesian-data-analysis-1.html#cb18-1" tabindex="-1"></a><span class="co"># Gaussian</span></span>
+<span id="cb18-2"><a href="bayesian-data-analysis-1.html#cb18-2" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="fu">tibble</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">30</span>, <span class="dv">30</span>)),</span>
+<span id="cb18-3"><a href="bayesian-data-analysis-1.html#cb18-3" tabindex="-1"></a>       <span class="fu">aes</span>(<span class="at">x =</span> x)) <span class="sc">+</span></span>
+<span id="cb18-4"><a href="bayesian-data-analysis-1.html#cb18-4" tabindex="-1"></a>  <span class="fu">stat_function</span>(<span class="at">fun =</span> <span class="st">&quot;dnorm&quot;</span>, </span>
+<span id="cb18-5"><a href="bayesian-data-analysis-1.html#cb18-5" tabindex="-1"></a>                <span class="at">size =</span> <span class="dv">2</span>,</span>
+<span id="cb18-6"><a href="bayesian-data-analysis-1.html#cb18-6" tabindex="-1"></a>                <span class="at">args =</span> <span class="fu">list</span>(<span class="at">sd =</span> <span class="dv">10</span>))</span>
+<span id="cb18-7"><a href="bayesian-data-analysis-1.html#cb18-7" tabindex="-1"></a></span>
+<span id="cb18-8"><a href="bayesian-data-analysis-1.html#cb18-8" tabindex="-1"></a><span class="co"># Cauchy</span></span>
+<span id="cb18-9"><a href="bayesian-data-analysis-1.html#cb18-9" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="fu">tibble</span>(<span class="at">x =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">30</span>)),</span>
+<span id="cb18-10"><a href="bayesian-data-analysis-1.html#cb18-10" tabindex="-1"></a>       <span class="fu">aes</span>(<span class="at">x =</span> x)) <span class="sc">+</span></span>
+<span id="cb18-11"><a href="bayesian-data-analysis-1.html#cb18-11" tabindex="-1"></a>  <span class="fu">stat_function</span>(<span class="at">fun =</span> <span class="st">&quot;dcauchy&quot;</span>, </span>
+<span id="cb18-12"><a href="bayesian-data-analysis-1.html#cb18-12" tabindex="-1"></a>                <span class="at">size =</span> <span class="dv">2</span>,</span>
+<span id="cb18-13"><a href="bayesian-data-analysis-1.html#cb18-13" tabindex="-1"></a>                <span class="at">args =</span> <span class="fu">list</span>(<span class="at">location =</span> <span class="dv">0</span>,</span>
+<span id="cb18-14"><a href="bayesian-data-analysis-1.html#cb18-14" tabindex="-1"></a>                            <span class="at">scale =</span> <span class="dv">3</span>))</span></code></pre></div>
 </div>
 <div id="visualize-the-posteriors" class="section level4 hasAnchor" number="22.4.3.3">
 <h4><span class="header-section-number">22.4.3.3</span> Visualize the posteriors<a href="bayesian-data-analysis-1.html#visualize-the-posteriors" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>This is what the posterior looks like for the three parameters in the model:</p>
-<div class="sourceCode" id="cb1246"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1246-1"><a href="bayesian-data-analysis-1.html#cb1246-1" tabindex="-1"></a>df.draws <span class="sc">%&gt;%</span> </span>
-<span id="cb1246-2"><a href="bayesian-data-analysis-1.html#cb1246-2" tabindex="-1"></a>  <span class="fu">select</span>(draw<span class="sc">:</span>sd) <span class="sc">%&gt;%</span> </span>
-<span id="cb1246-3"><a href="bayesian-data-analysis-1.html#cb1246-3" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span>draw,</span>
-<span id="cb1246-4"><a href="bayesian-data-analysis-1.html#cb1246-4" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
-<span id="cb1246-5"><a href="bayesian-data-analysis-1.html#cb1246-5" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1246-6"><a href="bayesian-data-analysis-1.html#cb1246-6" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1246-7"><a href="bayesian-data-analysis-1.html#cb1246-7" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> value)) <span class="sc">+</span> </span>
-<span id="cb1246-8"><a href="bayesian-data-analysis-1.html#cb1246-8" tabindex="-1"></a>  <span class="fu">stat_density</span>(<span class="at">geom =</span> <span class="st">&quot;line&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1246-9"><a href="bayesian-data-analysis-1.html#cb1246-9" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
-<span id="cb1246-10"><a href="bayesian-data-analysis-1.html#cb1246-10" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free_y&quot;</span>,</span>
-<span id="cb1246-11"><a href="bayesian-data-analysis-1.html#cb1246-11" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1246-12"><a href="bayesian-data-analysis-1.html#cb1246-12" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1246-13"><a href="bayesian-data-analysis-1.html#cb1246-13" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
-<span id="cb1246-14"><a href="bayesian-data-analysis-1.html#cb1246-14" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
-<span id="cb1246-15"><a href="bayesian-data-analysis-1.html#cb1246-15" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1246-16"><a href="bayesian-data-analysis-1.html#cb1246-16" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1246-17"><a href="bayesian-data-analysis-1.html#cb1246-17" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1246-18"><a href="bayesian-data-analysis-1.html#cb1246-18" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1246-19"><a href="bayesian-data-analysis-1.html#cb1246-19" tabindex="-1"></a>        <span class="at">axis.text.x =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">10</span>),</span>
-<span id="cb1246-20"><a href="bayesian-data-analysis-1.html#cb1246-20" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>(),</span>
-<span id="cb1246-21"><a href="bayesian-data-analysis-1.html#cb1246-21" tabindex="-1"></a>        <span class="at">strip.text.x =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
+<div class="sourceCode" id="cb19"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb19-1"><a href="bayesian-data-analysis-1.html#cb19-1" tabindex="-1"></a>df.draws <span class="sc">%&gt;%</span> </span>
+<span id="cb19-2"><a href="bayesian-data-analysis-1.html#cb19-2" tabindex="-1"></a>  <span class="fu">select</span>(draw<span class="sc">:</span>sd) <span class="sc">%&gt;%</span> </span>
+<span id="cb19-3"><a href="bayesian-data-analysis-1.html#cb19-3" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span>draw,</span>
+<span id="cb19-4"><a href="bayesian-data-analysis-1.html#cb19-4" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
+<span id="cb19-5"><a href="bayesian-data-analysis-1.html#cb19-5" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb19-6"><a href="bayesian-data-analysis-1.html#cb19-6" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb19-7"><a href="bayesian-data-analysis-1.html#cb19-7" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> value)) <span class="sc">+</span> </span>
+<span id="cb19-8"><a href="bayesian-data-analysis-1.html#cb19-8" tabindex="-1"></a>  <span class="fu">stat_density</span>(<span class="at">geom =</span> <span class="st">&quot;line&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb19-9"><a href="bayesian-data-analysis-1.html#cb19-9" tabindex="-1"></a>  <span class="fu">facet_grid</span>(<span class="at">rows =</span> <span class="fu">vars</span>(index),</span>
+<span id="cb19-10"><a href="bayesian-data-analysis-1.html#cb19-10" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free_y&quot;</span>,</span>
+<span id="cb19-11"><a href="bayesian-data-analysis-1.html#cb19-11" tabindex="-1"></a>             <span class="at">switch =</span> <span class="st">&quot;y&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb19-12"><a href="bayesian-data-analysis-1.html#cb19-12" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="sc">-</span><span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb19-13"><a href="bayesian-data-analysis-1.html#cb19-13" tabindex="-1"></a>  <span class="fu">annotate</span>(<span class="st">&quot;segment&quot;</span>, <span class="at">x =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">xend =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">y =</span> <span class="sc">-</span><span class="cn">Inf</span>, <span class="at">yend =</span> <span class="cn">Inf</span>) <span class="sc">+</span> </span>
+<span id="cb19-14"><a href="bayesian-data-analysis-1.html#cb19-14" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">legend.position =</span> <span class="st">&quot;none&quot;</span>,</span>
+<span id="cb19-15"><a href="bayesian-data-analysis-1.html#cb19-15" tabindex="-1"></a>        <span class="at">strip.background =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb19-16"><a href="bayesian-data-analysis-1.html#cb19-16" tabindex="-1"></a>        <span class="at">axis.title.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb19-17"><a href="bayesian-data-analysis-1.html#cb19-17" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb19-18"><a href="bayesian-data-analysis-1.html#cb19-18" tabindex="-1"></a>        <span class="at">axis.ticks.y =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb19-19"><a href="bayesian-data-analysis-1.html#cb19-19" tabindex="-1"></a>        <span class="at">axis.text.x =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">10</span>),</span>
+<span id="cb19-20"><a href="bayesian-data-analysis-1.html#cb19-20" tabindex="-1"></a>        <span class="at">axis.line =</span> <span class="fu">element_blank</span>(),</span>
+<span id="cb19-21"><a href="bayesian-data-analysis-1.html#cb19-21" tabindex="-1"></a>        <span class="at">strip.text.x =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
 </div>
 <div id="credible-interval-vs.-confidence-interval" class="section level4 hasAnchor" number="22.4.3.4">
 <h4><span class="header-section-number">22.4.3.4</span> Credible interval vs. confidence interval<a href="bayesian-data-analysis-1.html#credible-interval-vs.-confidence-interval" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1247"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1247-1"><a href="bayesian-data-analysis-1.html#cb1247-1" tabindex="-1"></a>fit.lm <span class="sc">%&gt;%</span> </span>
-<span id="cb1247-2"><a href="bayesian-data-analysis-1.html#cb1247-2" tabindex="-1"></a>  <span class="fu">tidy</span>(<span class="at">conf.int =</span> T) <span class="sc">%&gt;%</span> </span>
-<span id="cb1247-3"><a href="bayesian-data-analysis-1.html#cb1247-3" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> term,</span>
-<span id="cb1247-4"><a href="bayesian-data-analysis-1.html#cb1247-4" tabindex="-1"></a>                       <span class="at">x =</span> estimate,</span>
-<span id="cb1247-5"><a href="bayesian-data-analysis-1.html#cb1247-5" tabindex="-1"></a>                       <span class="at">xmin =</span> conf.low,</span>
-<span id="cb1247-6"><a href="bayesian-data-analysis-1.html#cb1247-6" tabindex="-1"></a>                       <span class="at">xmax =</span> conf.high)) <span class="sc">+</span></span>
-<span id="cb1247-7"><a href="bayesian-data-analysis-1.html#cb1247-7" tabindex="-1"></a>  <span class="fu">geom_pointrange</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb20"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb20-1"><a href="bayesian-data-analysis-1.html#cb20-1" tabindex="-1"></a>fit.lm <span class="sc">%&gt;%</span> </span>
+<span id="cb20-2"><a href="bayesian-data-analysis-1.html#cb20-2" tabindex="-1"></a>  <span class="fu">tidy</span>(<span class="at">conf.int =</span> T) <span class="sc">%&gt;%</span> </span>
+<span id="cb20-3"><a href="bayesian-data-analysis-1.html#cb20-3" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> term,</span>
+<span id="cb20-4"><a href="bayesian-data-analysis-1.html#cb20-4" tabindex="-1"></a>                       <span class="at">x =</span> estimate,</span>
+<span id="cb20-5"><a href="bayesian-data-analysis-1.html#cb20-5" tabindex="-1"></a>                       <span class="at">xmin =</span> conf.low,</span>
+<span id="cb20-6"><a href="bayesian-data-analysis-1.html#cb20-6" tabindex="-1"></a>                       <span class="at">xmax =</span> conf.high)) <span class="sc">+</span></span>
+<span id="cb20-7"><a href="bayesian-data-analysis-1.html#cb20-7" tabindex="-1"></a>  <span class="fu">geom_pointrange</span>()</span></code></pre></div>
 </div>
 <div id="visualize-model-predictions-2" class="section level4 hasAnchor" number="22.4.3.5">
 <h4><span class="header-section-number">22.4.3.5</span> Visualize model predictions<a href="bayesian-data-analysis-1.html#visualize-model-predictions-2" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>Let’s take some samples from the posterior to visualize the model predictions:</p>
-<div class="sourceCode" id="cb1248"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1248-1"><a href="bayesian-data-analysis-1.html#cb1248-1" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.attitude,</span>
-<span id="cb1248-2"><a href="bayesian-data-analysis-1.html#cb1248-2" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> complaints, </span>
-<span id="cb1248-3"><a href="bayesian-data-analysis-1.html#cb1248-3" tabindex="-1"></a>                     <span class="at">y =</span> rating)) <span class="sc">+</span> </span>
-<span id="cb1248-4"><a href="bayesian-data-analysis-1.html#cb1248-4" tabindex="-1"></a>  <span class="fu">geom_abline</span>(<span class="at">data =</span> df.draws <span class="sc">%&gt;%</span> </span>
-<span id="cb1248-5"><a href="bayesian-data-analysis-1.html#cb1248-5" tabindex="-1"></a>                <span class="fu">slice_sample</span>(<span class="at">n =</span> <span class="dv">50</span>),</span>
-<span id="cb1248-6"><a href="bayesian-data-analysis-1.html#cb1248-6" tabindex="-1"></a>              <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">intercept =</span> b0, </span>
-<span id="cb1248-7"><a href="bayesian-data-analysis-1.html#cb1248-7" tabindex="-1"></a>                            <span class="at">slope =</span> b1),</span>
-<span id="cb1248-8"><a href="bayesian-data-analysis-1.html#cb1248-8" tabindex="-1"></a>              <span class="at">alpha =</span> <span class="fl">0.3</span>,</span>
-<span id="cb1248-9"><a href="bayesian-data-analysis-1.html#cb1248-9" tabindex="-1"></a>              <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1248-10"><a href="bayesian-data-analysis-1.html#cb1248-10" tabindex="-1"></a>  <span class="fu">geom_point</span>() </span></code></pre></div>
+<div class="sourceCode" id="cb21"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb21-1"><a href="bayesian-data-analysis-1.html#cb21-1" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.attitude,</span>
+<span id="cb21-2"><a href="bayesian-data-analysis-1.html#cb21-2" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> complaints, </span>
+<span id="cb21-3"><a href="bayesian-data-analysis-1.html#cb21-3" tabindex="-1"></a>                     <span class="at">y =</span> rating)) <span class="sc">+</span> </span>
+<span id="cb21-4"><a href="bayesian-data-analysis-1.html#cb21-4" tabindex="-1"></a>  <span class="fu">geom_abline</span>(<span class="at">data =</span> df.draws <span class="sc">%&gt;%</span> </span>
+<span id="cb21-5"><a href="bayesian-data-analysis-1.html#cb21-5" tabindex="-1"></a>                <span class="fu">slice_sample</span>(<span class="at">n =</span> <span class="dv">50</span>),</span>
+<span id="cb21-6"><a href="bayesian-data-analysis-1.html#cb21-6" tabindex="-1"></a>              <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">intercept =</span> b0, </span>
+<span id="cb21-7"><a href="bayesian-data-analysis-1.html#cb21-7" tabindex="-1"></a>                            <span class="at">slope =</span> b1),</span>
+<span id="cb21-8"><a href="bayesian-data-analysis-1.html#cb21-8" tabindex="-1"></a>              <span class="at">alpha =</span> <span class="fl">0.3</span>,</span>
+<span id="cb21-9"><a href="bayesian-data-analysis-1.html#cb21-9" tabindex="-1"></a>              <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb21-10"><a href="bayesian-data-analysis-1.html#cb21-10" tabindex="-1"></a>  <span class="fu">geom_point</span>() </span></code></pre></div>
 </div>
 <div id="posterior-predictive-check" class="section level4 hasAnchor" number="22.4.3.6">
 <h4><span class="header-section-number">22.4.3.6</span> Posterior predictive check<a href="bayesian-data-analysis-1.html#posterior-predictive-check" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>Let’s make an animation that illustrates what predicted data sets (based on samples from the posterior) would look like:</p>
-<div class="sourceCode" id="cb1249"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1249-1"><a href="bayesian-data-analysis-1.html#cb1249-1" tabindex="-1"></a>p <span class="ot">=</span> df.draws <span class="sc">%&gt;%</span> </span>
-<span id="cb1249-2"><a href="bayesian-data-analysis-1.html#cb1249-2" tabindex="-1"></a>  <span class="fu">slice_sample</span>(<span class="at">n =</span> <span class="dv">10</span>) <span class="sc">%&gt;%</span>  </span>
-<span id="cb1249-3"><a href="bayesian-data-analysis-1.html#cb1249-3" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">complaints =</span> <span class="fu">list</span>(<span class="fu">seq</span>(<span class="fu">min</span>(df.attitude<span class="sc">$</span>complaints),</span>
-<span id="cb1249-4"><a href="bayesian-data-analysis-1.html#cb1249-4" tabindex="-1"></a>                               <span class="fu">max</span>(df.attitude<span class="sc">$</span>complaints),</span>
-<span id="cb1249-5"><a href="bayesian-data-analysis-1.html#cb1249-5" tabindex="-1"></a>                               <span class="at">length.out =</span> <span class="fu">nrow</span>(df.attitude)))) <span class="sc">%&gt;%</span> </span>
-<span id="cb1249-6"><a href="bayesian-data-analysis-1.html#cb1249-6" tabindex="-1"></a>  <span class="fu">unnest</span>(<span class="fu">c</span>(complaints)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1249-7"><a href="bayesian-data-analysis-1.html#cb1249-7" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">prediction =</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> complaints <span class="sc">+</span> <span class="fu">rnorm</span>(<span class="fu">n</span>(), <span class="at">sd =</span> sd)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1249-8"><a href="bayesian-data-analysis-1.html#cb1249-8" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> complaints, <span class="at">y =</span> prediction)) <span class="sc">+</span> </span>
-<span id="cb1249-9"><a href="bayesian-data-analysis-1.html#cb1249-9" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.8</span>,</span>
-<span id="cb1249-10"><a href="bayesian-data-analysis-1.html#cb1249-10" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1249-11"><a href="bayesian-data-analysis-1.html#cb1249-11" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> df.attitude,</span>
-<span id="cb1249-12"><a href="bayesian-data-analysis-1.html#cb1249-12" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">y =</span> rating,</span>
-<span id="cb1249-13"><a href="bayesian-data-analysis-1.html#cb1249-13" tabindex="-1"></a>                 <span class="at">x =</span> complaints)) <span class="sc">+</span></span>
-<span id="cb1249-14"><a href="bayesian-data-analysis-1.html#cb1249-14" tabindex="-1"></a>  <span class="fu">coord_cartesian</span>(<span class="at">xlim =</span> <span class="fu">c</span>(<span class="dv">20</span>, <span class="dv">100</span>),</span>
-<span id="cb1249-15"><a href="bayesian-data-analysis-1.html#cb1249-15" tabindex="-1"></a>                  <span class="at">ylim =</span> <span class="fu">c</span>(<span class="dv">20</span>, <span class="dv">100</span>)) <span class="sc">+</span></span>
-<span id="cb1249-16"><a href="bayesian-data-analysis-1.html#cb1249-16" tabindex="-1"></a>  <span class="fu">transition_manual</span>(draw)</span>
-<span id="cb1249-17"><a href="bayesian-data-analysis-1.html#cb1249-17" tabindex="-1"></a></span>
-<span id="cb1249-18"><a href="bayesian-data-analysis-1.html#cb1249-18" tabindex="-1"></a><span class="fu">animate</span>(p,</span>
-<span id="cb1249-19"><a href="bayesian-data-analysis-1.html#cb1249-19" tabindex="-1"></a>        <span class="at">nframes =</span> <span class="dv">60</span>,</span>
-<span id="cb1249-20"><a href="bayesian-data-analysis-1.html#cb1249-20" tabindex="-1"></a>        <span class="at">width =</span> <span class="dv">800</span>,</span>
-<span id="cb1249-21"><a href="bayesian-data-analysis-1.html#cb1249-21" tabindex="-1"></a>        <span class="at">height =</span> <span class="dv">600</span>,</span>
-<span id="cb1249-22"><a href="bayesian-data-analysis-1.html#cb1249-22" tabindex="-1"></a>        <span class="at">res =</span> <span class="dv">96</span>,</span>
-<span id="cb1249-23"><a href="bayesian-data-analysis-1.html#cb1249-23" tabindex="-1"></a>        <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span>
-<span id="cb1249-24"><a href="bayesian-data-analysis-1.html#cb1249-24" tabindex="-1"></a></span>
-<span id="cb1249-25"><a href="bayesian-data-analysis-1.html#cb1249-25" tabindex="-1"></a><span class="co"># anim_save(&quot;posterior_predictive.gif&quot;)</span></span></code></pre></div>
+<div class="sourceCode" id="cb22"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb22-1"><a href="bayesian-data-analysis-1.html#cb22-1" tabindex="-1"></a>p <span class="ot">=</span> df.draws <span class="sc">%&gt;%</span> </span>
+<span id="cb22-2"><a href="bayesian-data-analysis-1.html#cb22-2" tabindex="-1"></a>  <span class="fu">slice_sample</span>(<span class="at">n =</span> <span class="dv">10</span>) <span class="sc">%&gt;%</span>  </span>
+<span id="cb22-3"><a href="bayesian-data-analysis-1.html#cb22-3" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">complaints =</span> <span class="fu">list</span>(<span class="fu">seq</span>(<span class="fu">min</span>(df.attitude<span class="sc">$</span>complaints),</span>
+<span id="cb22-4"><a href="bayesian-data-analysis-1.html#cb22-4" tabindex="-1"></a>                               <span class="fu">max</span>(df.attitude<span class="sc">$</span>complaints),</span>
+<span id="cb22-5"><a href="bayesian-data-analysis-1.html#cb22-5" tabindex="-1"></a>                               <span class="at">length.out =</span> <span class="fu">nrow</span>(df.attitude)))) <span class="sc">%&gt;%</span> </span>
+<span id="cb22-6"><a href="bayesian-data-analysis-1.html#cb22-6" tabindex="-1"></a>  <span class="fu">unnest</span>(<span class="fu">c</span>(complaints)) <span class="sc">%&gt;%</span> </span>
+<span id="cb22-7"><a href="bayesian-data-analysis-1.html#cb22-7" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">prediction =</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> complaints <span class="sc">+</span> <span class="fu">rnorm</span>(<span class="fu">n</span>(), <span class="at">sd =</span> sd)) <span class="sc">%&gt;%</span> </span>
+<span id="cb22-8"><a href="bayesian-data-analysis-1.html#cb22-8" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> complaints, <span class="at">y =</span> prediction)) <span class="sc">+</span> </span>
+<span id="cb22-9"><a href="bayesian-data-analysis-1.html#cb22-9" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.8</span>,</span>
+<span id="cb22-10"><a href="bayesian-data-analysis-1.html#cb22-10" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
+<span id="cb22-11"><a href="bayesian-data-analysis-1.html#cb22-11" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> df.attitude,</span>
+<span id="cb22-12"><a href="bayesian-data-analysis-1.html#cb22-12" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">y =</span> rating,</span>
+<span id="cb22-13"><a href="bayesian-data-analysis-1.html#cb22-13" tabindex="-1"></a>                 <span class="at">x =</span> complaints)) <span class="sc">+</span></span>
+<span id="cb22-14"><a href="bayesian-data-analysis-1.html#cb22-14" tabindex="-1"></a>  <span class="fu">coord_cartesian</span>(<span class="at">xlim =</span> <span class="fu">c</span>(<span class="dv">20</span>, <span class="dv">100</span>),</span>
+<span id="cb22-15"><a href="bayesian-data-analysis-1.html#cb22-15" tabindex="-1"></a>                  <span class="at">ylim =</span> <span class="fu">c</span>(<span class="dv">20</span>, <span class="dv">100</span>)) <span class="sc">+</span></span>
+<span id="cb22-16"><a href="bayesian-data-analysis-1.html#cb22-16" tabindex="-1"></a>  <span class="fu">transition_manual</span>(draw)</span>
+<span id="cb22-17"><a href="bayesian-data-analysis-1.html#cb22-17" tabindex="-1"></a></span>
+<span id="cb22-18"><a href="bayesian-data-analysis-1.html#cb22-18" tabindex="-1"></a><span class="fu">animate</span>(p,</span>
+<span id="cb22-19"><a href="bayesian-data-analysis-1.html#cb22-19" tabindex="-1"></a>        <span class="at">nframes =</span> <span class="dv">60</span>,</span>
+<span id="cb22-20"><a href="bayesian-data-analysis-1.html#cb22-20" tabindex="-1"></a>        <span class="at">width =</span> <span class="dv">800</span>,</span>
+<span id="cb22-21"><a href="bayesian-data-analysis-1.html#cb22-21" tabindex="-1"></a>        <span class="at">height =</span> <span class="dv">600</span>,</span>
+<span id="cb22-22"><a href="bayesian-data-analysis-1.html#cb22-22" tabindex="-1"></a>        <span class="at">res =</span> <span class="dv">96</span>,</span>
+<span id="cb22-23"><a href="bayesian-data-analysis-1.html#cb22-23" tabindex="-1"></a>        <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span>
+<span id="cb22-24"><a href="bayesian-data-analysis-1.html#cb22-24" tabindex="-1"></a></span>
+<span id="cb22-25"><a href="bayesian-data-analysis-1.html#cb22-25" tabindex="-1"></a><span class="co"># anim_save(&quot;posterior_predictive.gif&quot;)</span></span></code></pre></div>
 </div>
 <div id="prior-predictive-check" class="section level4 hasAnchor" number="22.4.3.7">
 <h4><span class="header-section-number">22.4.3.7</span> Prior predictive check<a href="bayesian-data-analysis-1.html#prior-predictive-check" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>And let’s illustrate what data we would have expected to see just based on the information that we encoded in our priors.</p>
-<div class="sourceCode" id="cb1250"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1250-1"><a href="bayesian-data-analysis-1.html#cb1250-1" tabindex="-1"></a>sample_size <span class="ot">=</span> <span class="dv">10</span></span>
-<span id="cb1250-2"><a href="bayesian-data-analysis-1.html#cb1250-2" tabindex="-1"></a></span>
-<span id="cb1250-3"><a href="bayesian-data-analysis-1.html#cb1250-3" tabindex="-1"></a>p <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">b0 =</span> <span class="fu">rnorm</span>(sample_size, <span class="at">mean =</span> <span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">10</span>),</span>
-<span id="cb1250-4"><a href="bayesian-data-analysis-1.html#cb1250-4" tabindex="-1"></a>           <span class="at">b1 =</span> <span class="fu">rnorm</span>(sample_size, <span class="at">mean =</span> <span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">10</span>),</span>
-<span id="cb1250-5"><a href="bayesian-data-analysis-1.html#cb1250-5" tabindex="-1"></a>           <span class="at">sd =</span> <span class="fu">rhcauchy</span>(sample_size, <span class="at">sigma =</span> <span class="dv">3</span>),</span>
-<span id="cb1250-6"><a href="bayesian-data-analysis-1.html#cb1250-6" tabindex="-1"></a>           <span class="at">draw =</span> <span class="dv">1</span><span class="sc">:</span>sample_size) <span class="sc">%&gt;%</span> </span>
-<span id="cb1250-7"><a href="bayesian-data-analysis-1.html#cb1250-7" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">complaints =</span> <span class="fu">list</span>(<span class="fu">runif</span>(<span class="fu">nrow</span>(df.attitude),</span>
-<span id="cb1250-8"><a href="bayesian-data-analysis-1.html#cb1250-8" tabindex="-1"></a>                                 <span class="at">min =</span> <span class="fu">min</span>(df.attitude<span class="sc">$</span>complaints),</span>
-<span id="cb1250-9"><a href="bayesian-data-analysis-1.html#cb1250-9" tabindex="-1"></a>                                 <span class="at">max =</span> <span class="fu">max</span>(df.attitude<span class="sc">$</span>complaints)))) <span class="sc">%&gt;%</span> </span>
-<span id="cb1250-10"><a href="bayesian-data-analysis-1.html#cb1250-10" tabindex="-1"></a>  <span class="fu">unnest</span>(<span class="fu">c</span>(complaints)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1250-11"><a href="bayesian-data-analysis-1.html#cb1250-11" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">prediction =</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> complaints <span class="sc">+</span> <span class="fu">rnorm</span>(<span class="fu">n</span>(), <span class="at">sd =</span> sd)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1250-12"><a href="bayesian-data-analysis-1.html#cb1250-12" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> complaints, <span class="at">y =</span> prediction)) <span class="sc">+</span> </span>
-<span id="cb1250-13"><a href="bayesian-data-analysis-1.html#cb1250-13" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.8</span>,</span>
-<span id="cb1250-14"><a href="bayesian-data-analysis-1.html#cb1250-14" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1250-15"><a href="bayesian-data-analysis-1.html#cb1250-15" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> df.attitude,</span>
-<span id="cb1250-16"><a href="bayesian-data-analysis-1.html#cb1250-16" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">y =</span> rating,</span>
-<span id="cb1250-17"><a href="bayesian-data-analysis-1.html#cb1250-17" tabindex="-1"></a>                 <span class="at">x =</span> complaints)) <span class="sc">+</span></span>
-<span id="cb1250-18"><a href="bayesian-data-analysis-1.html#cb1250-18" tabindex="-1"></a>  <span class="fu">transition_manual</span>(draw)</span>
-<span id="cb1250-19"><a href="bayesian-data-analysis-1.html#cb1250-19" tabindex="-1"></a></span>
-<span id="cb1250-20"><a href="bayesian-data-analysis-1.html#cb1250-20" tabindex="-1"></a><span class="fu">animate</span>(p,</span>
-<span id="cb1250-21"><a href="bayesian-data-analysis-1.html#cb1250-21" tabindex="-1"></a>        <span class="at">nframes =</span> <span class="dv">60</span>,</span>
-<span id="cb1250-22"><a href="bayesian-data-analysis-1.html#cb1250-22" tabindex="-1"></a>        <span class="at">width =</span> <span class="dv">800</span>,</span>
-<span id="cb1250-23"><a href="bayesian-data-analysis-1.html#cb1250-23" tabindex="-1"></a>        <span class="at">height =</span> <span class="dv">600</span>,</span>
-<span id="cb1250-24"><a href="bayesian-data-analysis-1.html#cb1250-24" tabindex="-1"></a>        <span class="at">res =</span> <span class="dv">96</span>,</span>
-<span id="cb1250-25"><a href="bayesian-data-analysis-1.html#cb1250-25" tabindex="-1"></a>        <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span>
-<span id="cb1250-26"><a href="bayesian-data-analysis-1.html#cb1250-26" tabindex="-1"></a></span>
-<span id="cb1250-27"><a href="bayesian-data-analysis-1.html#cb1250-27" tabindex="-1"></a><span class="co"># anim_save(&quot;prior_predictive.gif&quot;)</span></span></code></pre></div>
+<div class="sourceCode" id="cb23"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb23-1"><a href="bayesian-data-analysis-1.html#cb23-1" tabindex="-1"></a>sample_size <span class="ot">=</span> <span class="dv">10</span></span>
+<span id="cb23-2"><a href="bayesian-data-analysis-1.html#cb23-2" tabindex="-1"></a></span>
+<span id="cb23-3"><a href="bayesian-data-analysis-1.html#cb23-3" tabindex="-1"></a>p <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">b0 =</span> <span class="fu">rnorm</span>(sample_size, <span class="at">mean =</span> <span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">10</span>),</span>
+<span id="cb23-4"><a href="bayesian-data-analysis-1.html#cb23-4" tabindex="-1"></a>           <span class="at">b1 =</span> <span class="fu">rnorm</span>(sample_size, <span class="at">mean =</span> <span class="dv">0</span>, <span class="at">sd =</span> <span class="dv">10</span>),</span>
+<span id="cb23-5"><a href="bayesian-data-analysis-1.html#cb23-5" tabindex="-1"></a>           <span class="at">sd =</span> <span class="fu">rhcauchy</span>(sample_size, <span class="at">sigma =</span> <span class="dv">3</span>),</span>
+<span id="cb23-6"><a href="bayesian-data-analysis-1.html#cb23-6" tabindex="-1"></a>           <span class="at">draw =</span> <span class="dv">1</span><span class="sc">:</span>sample_size) <span class="sc">%&gt;%</span> </span>
+<span id="cb23-7"><a href="bayesian-data-analysis-1.html#cb23-7" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">complaints =</span> <span class="fu">list</span>(<span class="fu">runif</span>(<span class="fu">nrow</span>(df.attitude),</span>
+<span id="cb23-8"><a href="bayesian-data-analysis-1.html#cb23-8" tabindex="-1"></a>                                 <span class="at">min =</span> <span class="fu">min</span>(df.attitude<span class="sc">$</span>complaints),</span>
+<span id="cb23-9"><a href="bayesian-data-analysis-1.html#cb23-9" tabindex="-1"></a>                                 <span class="at">max =</span> <span class="fu">max</span>(df.attitude<span class="sc">$</span>complaints)))) <span class="sc">%&gt;%</span> </span>
+<span id="cb23-10"><a href="bayesian-data-analysis-1.html#cb23-10" tabindex="-1"></a>  <span class="fu">unnest</span>(<span class="fu">c</span>(complaints)) <span class="sc">%&gt;%</span> </span>
+<span id="cb23-11"><a href="bayesian-data-analysis-1.html#cb23-11" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">prediction =</span> b0 <span class="sc">+</span> b1 <span class="sc">*</span> complaints <span class="sc">+</span> <span class="fu">rnorm</span>(<span class="fu">n</span>(), <span class="at">sd =</span> sd)) <span class="sc">%&gt;%</span> </span>
+<span id="cb23-12"><a href="bayesian-data-analysis-1.html#cb23-12" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> complaints, <span class="at">y =</span> prediction)) <span class="sc">+</span> </span>
+<span id="cb23-13"><a href="bayesian-data-analysis-1.html#cb23-13" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.8</span>,</span>
+<span id="cb23-14"><a href="bayesian-data-analysis-1.html#cb23-14" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
+<span id="cb23-15"><a href="bayesian-data-analysis-1.html#cb23-15" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">data =</span> df.attitude,</span>
+<span id="cb23-16"><a href="bayesian-data-analysis-1.html#cb23-16" tabindex="-1"></a>             <span class="fu">aes</span>(<span class="at">y =</span> rating,</span>
+<span id="cb23-17"><a href="bayesian-data-analysis-1.html#cb23-17" tabindex="-1"></a>                 <span class="at">x =</span> complaints)) <span class="sc">+</span></span>
+<span id="cb23-18"><a href="bayesian-data-analysis-1.html#cb23-18" tabindex="-1"></a>  <span class="fu">transition_manual</span>(draw)</span>
+<span id="cb23-19"><a href="bayesian-data-analysis-1.html#cb23-19" tabindex="-1"></a></span>
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+<span id="cb23-21"><a href="bayesian-data-analysis-1.html#cb23-21" tabindex="-1"></a>        <span class="at">nframes =</span> <span class="dv">60</span>,</span>
+<span id="cb23-22"><a href="bayesian-data-analysis-1.html#cb23-22" tabindex="-1"></a>        <span class="at">width =</span> <span class="dv">800</span>,</span>
+<span id="cb23-23"><a href="bayesian-data-analysis-1.html#cb23-23" tabindex="-1"></a>        <span class="at">height =</span> <span class="dv">600</span>,</span>
+<span id="cb23-24"><a href="bayesian-data-analysis-1.html#cb23-24" tabindex="-1"></a>        <span class="at">res =</span> <span class="dv">96</span>,</span>
+<span id="cb23-25"><a href="bayesian-data-analysis-1.html#cb23-25" tabindex="-1"></a>        <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span>
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+<span id="cb23-27"><a href="bayesian-data-analysis-1.html#cb23-27" tabindex="-1"></a><span class="co"># anim_save(&quot;prior_predictive.gif&quot;)</span></span></code></pre></div>
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 <pre><code>R version 4.3.2 (2023-10-31)
 Platform: aarch64-apple-darwin20 (64-bit)
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-[61] htmltools_0.5.7      R6_2.5.1             evaluate_0.23       
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-[67] backports_1.4.1      snakecase_0.11.1     bslib_0.6.1         
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-[73] checkmate_2.3.1      mgcv_1.9-1           whisker_0.4.1       
-[76] xfun_0.41            pkgconfig_2.0.3     </code></pre>
+[34] crayon_1.5.2         generics_0.1.3       rstudioapi_0.15.0   
+[37] tzdb_0.4.0           tfruns_1.5.2         cachem_1.0.8        
+[40] splines_4.3.2        parallel_4.3.2       base64enc_0.1-3     
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+[46] bookdown_0.37        callr_3.7.5          hms_1.1.3           
+[49] arrayhelpers_1.1-0   listenv_0.9.1        ggdist_3.3.1        
+[52] jquerylib_0.1.4      glue_1.7.0           parallelly_1.37.0   
+[55] gifski_1.12.0-2      codetools_0.2-19     ps_1.7.6            
+[58] distributional_0.3.2 stringi_1.8.3        gtable_0.3.4        
+[61] munsell_0.5.0        pillar_1.9.0         htmltools_0.5.7     
+[64] R6_2.5.1             evaluate_0.23        lattice_0.22-5      
+[67] highr_0.10           png_0.1-8            backports_1.4.1     
+[70] snakecase_0.11.1     bslib_0.6.1          Rcpp_1.0.12         
+[73] nlme_3.1-164         coda_0.19-4.1        checkmate_2.3.1     
+[76] mgcv_1.9-1           whisker_0.4.1        xfun_0.41           
+[79] pkgconfig_2.0.3     </code></pre>
 
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+<li class="chapter" data-level="7.4" data-path="simulation-1.html"><a href="simulation-1.html#pinguin-exercise"><i class="fa fa-check"></i><b>7.4</b> Pinguin exercise</a>
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-<li class="chapter" data-level="7.4.1" data-path="simulation-1.html"><a href="simulation-1.html#analytic-solution"><i class="fa fa-check"></i><b>7.4.1</b> Analytic solution</a></li>
-<li class="chapter" data-level="7.4.2" data-path="simulation-1.html"><a href="simulation-1.html#solution-via-sampling"><i class="fa fa-check"></i><b>7.4.2</b> Solution via sampling</a></li>
+<li class="chapter" data-level="7.4.1" data-path="simulation-1.html"><a href="simulation-1.html#make-the-plot"><i class="fa fa-check"></i><b>7.4.1</b> Make the plot</a></li>
+<li class="chapter" data-level="7.4.2" data-path="simulation-1.html"><a href="simulation-1.html#analytic-solutions"><i class="fa fa-check"></i><b>7.4.2</b> Analytic solutions</a></li>
+<li class="chapter" data-level="7.4.3" data-path="simulation-1.html"><a href="simulation-1.html#sampling-solution"><i class="fa fa-check"></i><b>7.4.3</b> Sampling solution</a></li>
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-<li class="chapter" data-level="7.5" data-path="simulation-1.html"><a href="simulation-1.html#additional-resources-5"><i class="fa fa-check"></i><b>7.5</b> Additional resources</a>
+<li class="chapter" data-level="7.5" data-path="simulation-1.html"><a href="simulation-1.html#bayesian-inference-with-the-normal-distribution"><i class="fa fa-check"></i><b>7.5</b> Bayesian inference with the normal distribution</a>
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-<li class="chapter" data-level="7.5.1" data-path="simulation-1.html"><a href="simulation-1.html#datacamp"><i class="fa fa-check"></i><b>7.5.1</b> Datacamp</a></li>
+<li class="chapter" data-level="7.5.1" data-path="simulation-1.html"><a href="simulation-1.html#analytic-solution"><i class="fa fa-check"></i><b>7.5.1</b> Analytic solution</a></li>
+<li class="chapter" data-level="7.5.2" data-path="simulation-1.html"><a href="simulation-1.html#solution-via-sampling"><i class="fa fa-check"></i><b>7.5.2</b> Solution via sampling</a></li>
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+<li class="chapter" data-level="7.6" data-path="simulation-1.html"><a href="simulation-1.html#additional-resources-5"><i class="fa fa-check"></i><b>7.6</b> Additional resources</a>
+<ul>
+<li class="chapter" data-level="7.6.1" data-path="simulation-1.html"><a href="simulation-1.html#datacamp"><i class="fa fa-check"></i><b>7.6.1</b> Datacamp</a></li>
+</ul></li>
+<li class="chapter" data-level="7.7" data-path="simulation-1.html"><a href="simulation-1.html#session-info-6"><i class="fa fa-check"></i><b>7.7</b> Session info</a></li>
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+<li class="chapter" data-level="8.5.1" data-path="simulation-2.html"><a href="simulation-2.html#misc-3"><i class="fa fa-check"></i><b>8.5.1</b> Misc</a></li>
+<li class="chapter" data-level="8.5.2" data-path="simulation-2.html"><a href="simulation-2.html#datacamp-1"><i class="fa fa-check"></i><b>8.5.2</b> Datacamp</a></li>
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+<li class="chapter" data-level="9.5" data-path="modeling-data.html"><a href="modeling-data.html#misc-4"><i class="fa fa-check"></i><b>9.5</b> Misc</a></li>
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@@ -446,7 +436,7 @@
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+<li class="chapter" data-level="10.5.2" data-path="linear-model-1.html"><a href="linear-model-1.html#misc-5"><i class="fa fa-check"></i><b>10.5.2</b> Misc</a></li>
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@@ -478,7 +468,7 @@
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+<li class="chapter" data-level="11.8.2" data-path="linear-model-2.html"><a href="linear-model-2.html#misc-6"><i class="fa fa-check"></i><b>11.8.2</b> Misc</a></li>
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 <li class="chapter" data-level="11.9" data-path="linear-model-2.html"><a href="linear-model-2.html#session-info-10"><i class="fa fa-check"></i><b>11.9</b> Session info</a></li>
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@@ -509,7 +499,7 @@
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+<li class="chapter" data-level="12.7.2" data-path="linear-model-3.html"><a href="linear-model-3.html#misc-7"><i class="fa fa-check"></i><b>12.7.2</b> Misc</a></li>
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 <li class="chapter" data-level="12.8" data-path="linear-model-3.html"><a href="linear-model-3.html#session-info-11"><i class="fa fa-check"></i><b>12.8</b> Session info</a></li>
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@@ -529,7 +519,7 @@
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 <li class="chapter" data-level="13.6" data-path="linear-model-4.html"><a href="linear-model-4.html#additional-resources-11"><i class="fa fa-check"></i><b>13.6</b> Additional resources</a>
 <ul>
-<li class="chapter" data-level="13.6.1" data-path="linear-model-4.html"><a href="linear-model-4.html#misc-7"><i class="fa fa-check"></i><b>13.6.1</b> Misc</a></li>
+<li class="chapter" data-level="13.6.1" data-path="linear-model-4.html"><a href="linear-model-4.html#misc-8"><i class="fa fa-check"></i><b>13.6.1</b> Misc</a></li>
 </ul></li>
 <li class="chapter" data-level="13.7" data-path="linear-model-4.html"><a href="linear-model-4.html#session-info-12"><i class="fa fa-check"></i><b>13.7</b> Session info</a></li>
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@@ -552,7 +542,7 @@
 <ul>
 <li class="chapter" data-level="14.7.1" data-path="power-analysis.html"><a href="power-analysis.html#datacamp-6"><i class="fa fa-check"></i><b>14.7.1</b> Datacamp</a></li>
 <li class="chapter" data-level="14.7.2" data-path="power-analysis.html"><a href="power-analysis.html#cheatsheets-5"><i class="fa fa-check"></i><b>14.7.2</b> Cheatsheets</a></li>
-<li class="chapter" data-level="14.7.3" data-path="power-analysis.html"><a href="power-analysis.html#misc-8"><i class="fa fa-check"></i><b>14.7.3</b> Misc</a></li>
+<li class="chapter" data-level="14.7.3" data-path="power-analysis.html"><a href="power-analysis.html#misc-9"><i class="fa fa-check"></i><b>14.7.3</b> Misc</a></li>
 </ul></li>
 <li class="chapter" data-level="14.8" data-path="power-analysis.html"><a href="power-analysis.html#session-info-13"><i class="fa fa-check"></i><b>14.8</b> Session info</a></li>
 </ul></li>
@@ -607,7 +597,7 @@
 <ul>
 <li class="chapter" data-level="16.8.1" data-path="causation.html"><a href="causation.html#books"><i class="fa fa-check"></i><b>16.8.1</b> Books</a></li>
 <li class="chapter" data-level="16.8.2" data-path="causation.html"><a href="causation.html#tutorials-1"><i class="fa fa-check"></i><b>16.8.2</b> Tutorials</a></li>
-<li class="chapter" data-level="16.8.3" data-path="causation.html"><a href="causation.html#misc-9"><i class="fa fa-check"></i><b>16.8.3</b> Misc</a></li>
+<li class="chapter" data-level="16.8.3" data-path="causation.html"><a href="causation.html#misc-10"><i class="fa fa-check"></i><b>16.8.3</b> Misc</a></li>
 </ul></li>
 <li class="chapter" data-level="16.9" data-path="causation.html"><a href="causation.html#session-info-15"><i class="fa fa-check"></i><b>16.9</b> Session info</a></li>
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@@ -653,19 +643,20 @@
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 <li class="chapter" data-level="19.3" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#load-data-sets-4"><i class="fa fa-check"></i><b>19.3</b> Load data sets</a>
 <ul>
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+<li class="chapter" data-level="19.3.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#reasoning-data"><i class="fa fa-check"></i><b>19.3.1</b> Reasoning data</a></li>
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 <li class="chapter" data-level="19.4" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#understanding-the-lmer-syntax"><i class="fa fa-check"></i><b>19.4</b> Understanding the lmer() syntax</a></li>
-<li class="chapter" data-level="19.5" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#anova-vs.-lmer"><i class="fa fa-check"></i><b>19.5</b> ANOVA vs. Lmer</a>
+<li class="chapter" data-level="19.5" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#anova-vs.-lmer"><i class="fa fa-check"></i><b>19.5</b> ANOVA vs. lmer</a>
 <ul>
 <li class="chapter" data-level="19.5.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#between-subjects-anova"><i class="fa fa-check"></i><b>19.5.1</b> Between subjects ANOVA</a></li>
 <li class="chapter" data-level="19.5.2" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#repeated-measures-anova"><i class="fa fa-check"></i><b>19.5.2</b> Repeated-measures ANOVA</a></li>
 <li class="chapter" data-level="19.5.3" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#mixed-anova"><i class="fa fa-check"></i><b>19.5.3</b> Mixed ANOVA</a></li>
 </ul></li>
-<li class="chapter" data-level="19.6" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#session-info-18"><i class="fa fa-check"></i><b>19.6</b> Session info</a></li>
+<li class="chapter" data-level="19.6" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#additional-resources-17"><i class="fa fa-check"></i><b>19.6</b> Additional resources</a>
+<ul>
+<li class="chapter" data-level="19.6.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#readings-2"><i class="fa fa-check"></i><b>19.6.1</b> Readings</a></li>
+</ul></li>
+<li class="chapter" data-level="19.7" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#session-info-18"><i class="fa fa-check"></i><b>19.7</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="20" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html"><i class="fa fa-check"></i><b>20</b> Linear mixed effects models 4</a>
 <ul>
@@ -673,10 +664,10 @@
 <li class="chapter" data-level="20.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#load-packages-and-set-plotting-theme-13"><i class="fa fa-check"></i><b>20.2</b> Load packages and set plotting theme</a></li>
 <li class="chapter" data-level="20.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#load-data-sets-5"><i class="fa fa-check"></i><b>20.3</b> Load data sets</a>
 <ul>
-<li class="chapter" data-level="20.3.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#sleep-data-1"><i class="fa fa-check"></i><b>20.3.1</b> Sleep data</a></li>
+<li class="chapter" data-level="20.3.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#sleep-data"><i class="fa fa-check"></i><b>20.3.1</b> Sleep data</a></li>
 <li class="chapter" data-level="20.3.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#reasoning-data-1"><i class="fa fa-check"></i><b>20.3.2</b> Reasoning data</a></li>
-<li class="chapter" data-level="20.3.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#weight-loss-data-1"><i class="fa fa-check"></i><b>20.3.3</b> Weight loss data</a></li>
-<li class="chapter" data-level="20.3.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politness-data-1"><i class="fa fa-check"></i><b>20.3.4</b> Politness data</a></li>
+<li class="chapter" data-level="20.3.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#weight-loss-data"><i class="fa fa-check"></i><b>20.3.3</b> Weight loss data</a></li>
+<li class="chapter" data-level="20.3.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politness-data"><i class="fa fa-check"></i><b>20.3.4</b> Politness data</a></li>
 </ul></li>
 <li class="chapter" data-level="20.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#follow-up-tests-with-emmeans"><i class="fa fa-check"></i><b>20.4</b> Follow-up tests with emmeans</a>
 <ul>
@@ -685,11 +676,17 @@
 <li class="chapter" data-level="20.4.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politeness-study"><i class="fa fa-check"></i><b>20.4.3</b> Politeness study</a></li>
 </ul></li>
 <li class="chapter" data-level="20.5" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#mixtures-of-participants"><i class="fa fa-check"></i><b>20.5</b> Mixtures of participants</a></li>
-<li class="chapter" data-level="20.6" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#bootstrapping"><i class="fa fa-check"></i><b>20.6</b> Bootstrapping</a>
+<li class="chapter" data-level="20.6" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#simulating-different-random-effects-structures"><i class="fa fa-check"></i><b>20.6</b> Simulating different random effects structures</a>
+<ul>
+<li class="chapter" data-level="20.6.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#two-level-model"><i class="fa fa-check"></i><b>20.6.1</b> Two-level model</a></li>
+<li class="chapter" data-level="20.6.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#three-level-model"><i class="fa fa-check"></i><b>20.6.2</b> Three-level model</a></li>
+</ul></li>
+<li class="chapter" data-level="20.7" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#bootstrapping"><i class="fa fa-check"></i><b>20.7</b> Bootstrapping</a>
 <ul>
-<li class="chapter" data-level="20.6.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-model"><i class="fa fa-check"></i><b>20.6.1</b> Linear model</a></li>
+<li class="chapter" data-level="20.7.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-model"><i class="fa fa-check"></i><b>20.7.1</b> Linear model</a></li>
+<li class="chapter" data-level="20.7.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-mixed-effects-model"><i class="fa fa-check"></i><b>20.7.2</b> Linear mixed effects model</a></li>
 </ul></li>
-<li class="chapter" data-level="20.7" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#session-info-19"><i class="fa fa-check"></i><b>20.7</b> Session info</a></li>
+<li class="chapter" data-level="20.8" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#session-info-19"><i class="fa fa-check"></i><b>20.8</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="21" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html"><i class="fa fa-check"></i><b>21</b> Generalized linear model</a>
 <ul>
@@ -712,7 +709,7 @@
 <li class="chapter" data-level="21.7" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#logistic-mixed-effects-model"><i class="fa fa-check"></i><b>21.7</b> Logistic mixed effects model</a></li>
 <li class="chapter" data-level="21.8" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#additional-information"><i class="fa fa-check"></i><b>21.8</b> Additional information</a>
 <ul>
-<li class="chapter" data-level="21.8.1" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#misc-10"><i class="fa fa-check"></i><b>21.8.1</b> Misc</a></li>
+<li class="chapter" data-level="21.8.1" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#misc-11"><i class="fa fa-check"></i><b>21.8.1</b> Misc</a></li>
 <li class="chapter" data-level="21.8.2" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#datacamp-7"><i class="fa fa-check"></i><b>21.8.2</b> Datacamp</a></li>
 </ul></li>
 <li class="chapter" data-level="21.9" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html#session-info-20"><i class="fa fa-check"></i><b>21.9</b> Session info</a></li>
@@ -733,7 +730,7 @@
 <li class="chapter" data-level="22.4.2" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#frequentist-analysis"><i class="fa fa-check"></i><b>22.4.2</b> Frequentist analysis</a></li>
 <li class="chapter" data-level="22.4.3" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#bayesian-regression"><i class="fa fa-check"></i><b>22.4.3</b> Bayesian regression</a></li>
 </ul></li>
-<li class="chapter" data-level="22.5" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#additional-resources-17"><i class="fa fa-check"></i><b>22.5</b> Additional resources</a>
+<li class="chapter" data-level="22.5" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#additional-resources-18"><i class="fa fa-check"></i><b>22.5</b> Additional resources</a>
 <ul>
 <li class="chapter" data-level="22.5.1" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#books-and-chapters-5"><i class="fa fa-check"></i><b>22.5.1</b> Books and chapters</a></li>
 </ul></li>
@@ -776,7 +773,7 @@
 <li class="chapter" data-level="23.7.2" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#specify-and-fit-the-model-3"><i class="fa fa-check"></i><b>23.7.2</b> 2. Specify and fit the model</a></li>
 <li class="chapter" data-level="23.7.3" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#test-specific-hypotheses-3"><i class="fa fa-check"></i><b>23.7.3</b> 5. Test specific hypotheses</a></li>
 </ul></li>
-<li class="chapter" data-level="23.8" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#additional-resources-18"><i class="fa fa-check"></i><b>23.8</b> Additional resources</a></li>
+<li class="chapter" data-level="23.8" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#additional-resources-19"><i class="fa fa-check"></i><b>23.8</b> Additional resources</a></li>
 <li class="chapter" data-level="23.9" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#session-info-22"><i class="fa fa-check"></i><b>23.9</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="24" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html"><i class="fa fa-check"></i><b>24</b> Bayesian data analysis 3</a>
@@ -803,7 +800,7 @@
 <li class="chapter" data-level="24.6.4" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#gaussian-regression-unequal-variance"><i class="fa fa-check"></i><b>24.6.4</b> Gaussian regression (unequal variance)</a></li>
 <li class="chapter" data-level="24.6.5" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#model-comparison-2"><i class="fa fa-check"></i><b>24.6.5</b> Model comparison</a></li>
 </ul></li>
-<li class="chapter" data-level="24.7" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#additional-resources-19"><i class="fa fa-check"></i><b>24.7</b> Additional resources</a></li>
+<li class="chapter" data-level="24.7" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#additional-resources-20"><i class="fa fa-check"></i><b>24.7</b> Additional resources</a></li>
 <li class="chapter" data-level="24.8" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#session-info-23"><i class="fa fa-check"></i><b>24.8</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="25" data-path="model-assumptions.html"><a href="model-assumptions.html"><i class="fa fa-check"></i><b>25</b> Model assumptions</a>
@@ -818,7 +815,7 @@
 <li class="chapter" data-level="25.3.4" data-path="model-assumptions.html"><a href="model-assumptions.html#non-parametric-tests"><i class="fa fa-check"></i><b>25.3.4</b> Non-parametric tests</a></li>
 <li class="chapter" data-level="25.3.5" data-path="model-assumptions.html"><a href="model-assumptions.html#bootstrapping-regressions"><i class="fa fa-check"></i><b>25.3.5</b> Bootstrapping regressions</a></li>
 </ul></li>
-<li class="chapter" data-level="25.4" data-path="model-assumptions.html"><a href="model-assumptions.html#additional-resources-20"><i class="fa fa-check"></i><b>25.4</b> Additional resources</a></li>
+<li class="chapter" data-level="25.4" data-path="model-assumptions.html"><a href="model-assumptions.html#additional-resources-21"><i class="fa fa-check"></i><b>25.4</b> Additional resources</a></li>
 <li class="chapter" data-level="25.5" data-path="model-assumptions.html"><a href="model-assumptions.html#session-info-24"><i class="fa fa-check"></i><b>25.5</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="26" data-path="reporting-statistics.html"><a href="reporting-statistics.html"><i class="fa fa-check"></i><b>26</b> Reporting statistics</a>
@@ -835,7 +832,11 @@
 <ul>
 <li class="chapter" data-level="26.2.1" data-path="reporting-statistics.html"><a href="reporting-statistics.html#frequentist-statistics"><i class="fa fa-check"></i><b>26.2.1</b> Frequentist statistics</a></li>
 </ul></li>
-<li class="chapter" data-level="26.3" data-path="reporting-statistics.html"><a href="reporting-statistics.html#session-info-25"><i class="fa fa-check"></i><b>26.3</b> Session info</a></li>
+<li class="chapter" data-level="26.3" data-path="reporting-statistics.html"><a href="reporting-statistics.html#additional-resources-22"><i class="fa fa-check"></i><b>26.3</b> Additional resources</a>
+<ul>
+<li class="chapter" data-level="26.3.1" data-path="reporting-statistics.html"><a href="reporting-statistics.html#misc-12"><i class="fa fa-check"></i><b>26.3.1</b> Misc</a></li>
+</ul></li>
+<li class="chapter" data-level="26.4" data-path="reporting-statistics.html"><a href="reporting-statistics.html#session-info-25"><i class="fa fa-check"></i><b>26.4</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="27" data-path="cheatsheets-6.html"><a href="cheatsheets-6.html"><i class="fa fa-check"></i><b>27</b> Cheatsheets</a>
 <ul>
@@ -881,114 +882,114 @@ <h2><span class="header-section-number">23.1</span> Learning goals<a href="bayes
 </div>
 <div id="load-packages-and-set-plotting-theme-16" class="section level2 hasAnchor" number="23.2">
 <h2><span class="header-section-number">23.2</span> Load packages and set plotting theme<a href="bayesian-data-analysis-2.html#load-packages-and-set-plotting-theme-16" class="anchor-section" aria-label="Anchor link to header"></a></h2>
-<div class="sourceCode" id="cb1253"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1253-1"><a href="bayesian-data-analysis-2.html#cb1253-1" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;knitr&quot;</span>)       <span class="co"># for knitting RMarkdown </span></span>
-<span id="cb1253-2"><a href="bayesian-data-analysis-2.html#cb1253-2" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;kableExtra&quot;</span>)  <span class="co"># for making nice tables</span></span>
-<span id="cb1253-3"><a href="bayesian-data-analysis-2.html#cb1253-3" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;janitor&quot;</span>)     <span class="co"># for cleaning column names</span></span>
-<span id="cb1253-4"><a href="bayesian-data-analysis-2.html#cb1253-4" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;tidybayes&quot;</span>)   <span class="co"># tidying up results from Bayesian models</span></span>
-<span id="cb1253-5"><a href="bayesian-data-analysis-2.html#cb1253-5" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;brms&quot;</span>)        <span class="co"># Bayesian regression models with Stan</span></span>
-<span id="cb1253-6"><a href="bayesian-data-analysis-2.html#cb1253-6" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;patchwork&quot;</span>)   <span class="co"># for making figure panels</span></span>
-<span id="cb1253-7"><a href="bayesian-data-analysis-2.html#cb1253-7" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;GGally&quot;</span>)      <span class="co"># for pairs plot</span></span>
-<span id="cb1253-8"><a href="bayesian-data-analysis-2.html#cb1253-8" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;broom.mixed&quot;</span>) <span class="co"># for tidy lmer results</span></span>
-<span id="cb1253-9"><a href="bayesian-data-analysis-2.html#cb1253-9" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;bayesplot&quot;</span>)   <span class="co"># for visualization of Bayesian model fits </span></span>
-<span id="cb1253-10"><a href="bayesian-data-analysis-2.html#cb1253-10" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;modelr&quot;</span>)      <span class="co"># for modeling functions</span></span>
-<span id="cb1253-11"><a href="bayesian-data-analysis-2.html#cb1253-11" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;lme4&quot;</span>)        <span class="co"># for linear mixed effects models </span></span>
-<span id="cb1253-12"><a href="bayesian-data-analysis-2.html#cb1253-12" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;afex&quot;</span>)        <span class="co"># for ANOVAs</span></span>
-<span id="cb1253-13"><a href="bayesian-data-analysis-2.html#cb1253-13" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;car&quot;</span>)         <span class="co"># for ANOVAs</span></span>
-<span id="cb1253-14"><a href="bayesian-data-analysis-2.html#cb1253-14" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;emmeans&quot;</span>)     <span class="co"># for linear contrasts</span></span>
-<span id="cb1253-15"><a href="bayesian-data-analysis-2.html#cb1253-15" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;ggeffects&quot;</span>)   <span class="co"># for help with logistic regressions</span></span>
-<span id="cb1253-16"><a href="bayesian-data-analysis-2.html#cb1253-16" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;titanic&quot;</span>)     <span class="co"># titanic dataset</span></span>
-<span id="cb1253-17"><a href="bayesian-data-analysis-2.html#cb1253-17" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;gganimate&quot;</span>)   <span class="co"># for animations</span></span>
-<span id="cb1253-18"><a href="bayesian-data-analysis-2.html#cb1253-18" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;parameters&quot;</span>)  <span class="co"># for getting parameters</span></span>
-<span id="cb1253-19"><a href="bayesian-data-analysis-2.html#cb1253-19" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;transformr&quot;</span>)  <span class="co"># for gganimate</span></span>
-<span id="cb1253-20"><a href="bayesian-data-analysis-2.html#cb1253-20" tabindex="-1"></a><span class="co"># install via: devtools::install_github(&quot;thomasp85/transformr&quot;)</span></span>
-<span id="cb1253-21"><a href="bayesian-data-analysis-2.html#cb1253-21" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;tidyverse&quot;</span>)   <span class="co"># for wrangling, plotting, etc. </span></span></code></pre></div>
-<div class="sourceCode" id="cb1254"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1254-1"><a href="bayesian-data-analysis-2.html#cb1254-1" tabindex="-1"></a><span class="fu">theme_set</span>(<span class="fu">theme_classic</span>() <span class="sc">+</span> <span class="co"># set the theme </span></span>
-<span id="cb1254-2"><a href="bayesian-data-analysis-2.html#cb1254-2" tabindex="-1"></a>            <span class="fu">theme</span>(<span class="at">text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">20</span>))) <span class="co"># set the default text size</span></span>
-<span id="cb1254-3"><a href="bayesian-data-analysis-2.html#cb1254-3" tabindex="-1"></a></span>
-<span id="cb1254-4"><a href="bayesian-data-analysis-2.html#cb1254-4" tabindex="-1"></a>opts_chunk<span class="sc">$</span><span class="fu">set</span>(<span class="at">comment =</span> <span class="st">&quot;&quot;</span>,</span>
-<span id="cb1254-5"><a href="bayesian-data-analysis-2.html#cb1254-5" tabindex="-1"></a>               <span class="at">fig.show =</span> <span class="st">&quot;hold&quot;</span>)</span>
-<span id="cb1254-6"><a href="bayesian-data-analysis-2.html#cb1254-6" tabindex="-1"></a></span>
-<span id="cb1254-7"><a href="bayesian-data-analysis-2.html#cb1254-7" tabindex="-1"></a><span class="fu">options</span>(<span class="at">dplyr.summarise.inform =</span> F)</span>
-<span id="cb1254-8"><a href="bayesian-data-analysis-2.html#cb1254-8" tabindex="-1"></a></span>
-<span id="cb1254-9"><a href="bayesian-data-analysis-2.html#cb1254-9" tabindex="-1"></a><span class="co"># set default color scheme in ggplot </span></span>
-<span id="cb1254-10"><a href="bayesian-data-analysis-2.html#cb1254-10" tabindex="-1"></a><span class="fu">options</span>(<span class="at">ggplot2.discrete.color =</span> RColorBrewer<span class="sc">::</span><span class="fu">brewer.pal</span>(<span class="dv">9</span>,<span class="st">&quot;Set1&quot;</span>))</span></code></pre></div>
+<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="bayesian-data-analysis-2.html#cb1-1" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;knitr&quot;</span>)       <span class="co"># for knitting RMarkdown </span></span>
+<span id="cb1-2"><a href="bayesian-data-analysis-2.html#cb1-2" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;kableExtra&quot;</span>)  <span class="co"># for making nice tables</span></span>
+<span id="cb1-3"><a href="bayesian-data-analysis-2.html#cb1-3" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;janitor&quot;</span>)     <span class="co"># for cleaning column names</span></span>
+<span id="cb1-4"><a href="bayesian-data-analysis-2.html#cb1-4" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;tidybayes&quot;</span>)   <span class="co"># tidying up results from Bayesian models</span></span>
+<span id="cb1-5"><a href="bayesian-data-analysis-2.html#cb1-5" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;brms&quot;</span>)        <span class="co"># Bayesian regression models with Stan</span></span>
+<span id="cb1-6"><a href="bayesian-data-analysis-2.html#cb1-6" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;patchwork&quot;</span>)   <span class="co"># for making figure panels</span></span>
+<span id="cb1-7"><a href="bayesian-data-analysis-2.html#cb1-7" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;GGally&quot;</span>)      <span class="co"># for pairs plot</span></span>
+<span id="cb1-8"><a href="bayesian-data-analysis-2.html#cb1-8" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;broom.mixed&quot;</span>) <span class="co"># for tidy lmer results</span></span>
+<span id="cb1-9"><a href="bayesian-data-analysis-2.html#cb1-9" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;bayesplot&quot;</span>)   <span class="co"># for visualization of Bayesian model fits </span></span>
+<span id="cb1-10"><a href="bayesian-data-analysis-2.html#cb1-10" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;modelr&quot;</span>)      <span class="co"># for modeling functions</span></span>
+<span id="cb1-11"><a href="bayesian-data-analysis-2.html#cb1-11" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;lme4&quot;</span>)        <span class="co"># for linear mixed effects models </span></span>
+<span id="cb1-12"><a href="bayesian-data-analysis-2.html#cb1-12" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;afex&quot;</span>)        <span class="co"># for ANOVAs</span></span>
+<span id="cb1-13"><a href="bayesian-data-analysis-2.html#cb1-13" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;car&quot;</span>)         <span class="co"># for ANOVAs</span></span>
+<span id="cb1-14"><a href="bayesian-data-analysis-2.html#cb1-14" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;emmeans&quot;</span>)     <span class="co"># for linear contrasts</span></span>
+<span id="cb1-15"><a href="bayesian-data-analysis-2.html#cb1-15" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;ggeffects&quot;</span>)   <span class="co"># for help with logistic regressions</span></span>
+<span id="cb1-16"><a href="bayesian-data-analysis-2.html#cb1-16" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;titanic&quot;</span>)     <span class="co"># titanic dataset</span></span>
+<span id="cb1-17"><a href="bayesian-data-analysis-2.html#cb1-17" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;gganimate&quot;</span>)   <span class="co"># for animations</span></span>
+<span id="cb1-18"><a href="bayesian-data-analysis-2.html#cb1-18" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;parameters&quot;</span>)  <span class="co"># for getting parameters</span></span>
+<span id="cb1-19"><a href="bayesian-data-analysis-2.html#cb1-19" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;transformr&quot;</span>)  <span class="co"># for gganimate</span></span>
+<span id="cb1-20"><a href="bayesian-data-analysis-2.html#cb1-20" tabindex="-1"></a><span class="co"># install via: devtools::install_github(&quot;thomasp85/transformr&quot;)</span></span>
+<span id="cb1-21"><a href="bayesian-data-analysis-2.html#cb1-21" tabindex="-1"></a><span class="fu">library</span>(<span class="st">&quot;tidyverse&quot;</span>)   <span class="co"># for wrangling, plotting, etc. </span></span></code></pre></div>
+<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="bayesian-data-analysis-2.html#cb2-1" tabindex="-1"></a><span class="fu">theme_set</span>(<span class="fu">theme_classic</span>() <span class="sc">+</span> <span class="co"># set the theme </span></span>
+<span id="cb2-2"><a href="bayesian-data-analysis-2.html#cb2-2" tabindex="-1"></a>            <span class="fu">theme</span>(<span class="at">text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">20</span>))) <span class="co"># set the default text size</span></span>
+<span id="cb2-3"><a href="bayesian-data-analysis-2.html#cb2-3" tabindex="-1"></a></span>
+<span id="cb2-4"><a href="bayesian-data-analysis-2.html#cb2-4" tabindex="-1"></a>opts_chunk<span class="sc">$</span><span class="fu">set</span>(<span class="at">comment =</span> <span class="st">&quot;&quot;</span>,</span>
+<span id="cb2-5"><a href="bayesian-data-analysis-2.html#cb2-5" tabindex="-1"></a>               <span class="at">fig.show =</span> <span class="st">&quot;hold&quot;</span>)</span>
+<span id="cb2-6"><a href="bayesian-data-analysis-2.html#cb2-6" tabindex="-1"></a></span>
+<span id="cb2-7"><a href="bayesian-data-analysis-2.html#cb2-7" tabindex="-1"></a><span class="fu">options</span>(<span class="at">dplyr.summarise.inform =</span> F)</span>
+<span id="cb2-8"><a href="bayesian-data-analysis-2.html#cb2-8" tabindex="-1"></a></span>
+<span id="cb2-9"><a href="bayesian-data-analysis-2.html#cb2-9" tabindex="-1"></a><span class="co"># set default color scheme in ggplot </span></span>
+<span id="cb2-10"><a href="bayesian-data-analysis-2.html#cb2-10" tabindex="-1"></a><span class="fu">options</span>(<span class="at">ggplot2.discrete.color =</span> RColorBrewer<span class="sc">::</span><span class="fu">brewer.pal</span>(<span class="dv">9</span>,<span class="st">&quot;Set1&quot;</span>))</span></code></pre></div>
 </div>
 <div id="load-data-sets-6" class="section level2 hasAnchor" number="23.3">
 <h2><span class="header-section-number">23.3</span> Load data sets<a href="bayesian-data-analysis-2.html#load-data-sets-6" class="anchor-section" aria-label="Anchor link to header"></a></h2>
-<div class="sourceCode" id="cb1255"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1255-1"><a href="bayesian-data-analysis-2.html#cb1255-1" tabindex="-1"></a><span class="co"># poker </span></span>
-<span id="cb1255-2"><a href="bayesian-data-analysis-2.html#cb1255-2" tabindex="-1"></a>df.poker <span class="ot">=</span> <span class="fu">read_csv</span>(<span class="st">&quot;data/poker.csv&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-3"><a href="bayesian-data-analysis-2.html#cb1255-3" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">skill =</span> <span class="fu">factor</span>(skill,</span>
-<span id="cb1255-4"><a href="bayesian-data-analysis-2.html#cb1255-4" tabindex="-1"></a>                        <span class="at">levels =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">2</span>,</span>
-<span id="cb1255-5"><a href="bayesian-data-analysis-2.html#cb1255-5" tabindex="-1"></a>                        <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;expert&quot;</span>, <span class="st">&quot;average&quot;</span>)),</span>
-<span id="cb1255-6"><a href="bayesian-data-analysis-2.html#cb1255-6" tabindex="-1"></a>         <span class="at">skill =</span> <span class="fu">fct_relevel</span>(skill, <span class="st">&quot;average&quot;</span>, <span class="st">&quot;expert&quot;</span>),</span>
-<span id="cb1255-7"><a href="bayesian-data-analysis-2.html#cb1255-7" tabindex="-1"></a>         <span class="at">hand =</span> <span class="fu">factor</span>(hand,</span>
-<span id="cb1255-8"><a href="bayesian-data-analysis-2.html#cb1255-8" tabindex="-1"></a>                       <span class="at">levels =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>,</span>
-<span id="cb1255-9"><a href="bayesian-data-analysis-2.html#cb1255-9" tabindex="-1"></a>                       <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;bad&quot;</span>, <span class="st">&quot;neutral&quot;</span>, <span class="st">&quot;good&quot;</span>)),</span>
-<span id="cb1255-10"><a href="bayesian-data-analysis-2.html#cb1255-10" tabindex="-1"></a>         <span class="at">limit =</span> <span class="fu">factor</span>(limit,</span>
-<span id="cb1255-11"><a href="bayesian-data-analysis-2.html#cb1255-11" tabindex="-1"></a>                        <span class="at">levels =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">2</span>,</span>
-<span id="cb1255-12"><a href="bayesian-data-analysis-2.html#cb1255-12" tabindex="-1"></a>                        <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;fixed&quot;</span>, <span class="st">&quot;none&quot;</span>)),</span>
-<span id="cb1255-13"><a href="bayesian-data-analysis-2.html#cb1255-13" tabindex="-1"></a>         <span class="at">participant =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">n</span>()) <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-14"><a href="bayesian-data-analysis-2.html#cb1255-14" tabindex="-1"></a>  <span class="fu">select</span>(participant, <span class="fu">everything</span>())</span>
-<span id="cb1255-15"><a href="bayesian-data-analysis-2.html#cb1255-15" tabindex="-1"></a></span>
-<span id="cb1255-16"><a href="bayesian-data-analysis-2.html#cb1255-16" tabindex="-1"></a><span class="co"># sleep</span></span>
-<span id="cb1255-17"><a href="bayesian-data-analysis-2.html#cb1255-17" tabindex="-1"></a>df.sleep <span class="ot">=</span> sleepstudy <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-18"><a href="bayesian-data-analysis-2.html#cb1255-18" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-19"><a href="bayesian-data-analysis-2.html#cb1255-19" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-20"><a href="bayesian-data-analysis-2.html#cb1255-20" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">subject =</span> <span class="fu">as.character</span>(subject)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-21"><a href="bayesian-data-analysis-2.html#cb1255-21" tabindex="-1"></a>  <span class="fu">select</span>(subject, days, reaction) <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-22"><a href="bayesian-data-analysis-2.html#cb1255-22" tabindex="-1"></a>  <span class="fu">bind_rows</span>(<span class="fu">tibble</span>(<span class="at">subject =</span> <span class="st">&quot;374&quot;</span>,</span>
-<span id="cb1255-23"><a href="bayesian-data-analysis-2.html#cb1255-23" tabindex="-1"></a>                   <span class="at">days =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">1</span>,</span>
-<span id="cb1255-24"><a href="bayesian-data-analysis-2.html#cb1255-24" tabindex="-1"></a>                   <span class="at">reaction =</span> <span class="fu">c</span>(<span class="dv">286</span>, <span class="dv">288</span>)),</span>
-<span id="cb1255-25"><a href="bayesian-data-analysis-2.html#cb1255-25" tabindex="-1"></a>            <span class="fu">tibble</span>(<span class="at">subject =</span> <span class="st">&quot;373&quot;</span>,</span>
-<span id="cb1255-26"><a href="bayesian-data-analysis-2.html#cb1255-26" tabindex="-1"></a>                   <span class="at">days =</span> <span class="dv">0</span>,</span>
-<span id="cb1255-27"><a href="bayesian-data-analysis-2.html#cb1255-27" tabindex="-1"></a>                   <span class="at">reaction =</span> <span class="dv">245</span>))</span>
-<span id="cb1255-28"><a href="bayesian-data-analysis-2.html#cb1255-28" tabindex="-1"></a></span>
-<span id="cb1255-29"><a href="bayesian-data-analysis-2.html#cb1255-29" tabindex="-1"></a><span class="co"># titanic </span></span>
-<span id="cb1255-30"><a href="bayesian-data-analysis-2.html#cb1255-30" tabindex="-1"></a>df.titanic <span class="ot">=</span> titanic_train <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-31"><a href="bayesian-data-analysis-2.html#cb1255-31" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-32"><a href="bayesian-data-analysis-2.html#cb1255-32" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">sex =</span> <span class="fu">as.factor</span>(sex))</span>
-<span id="cb1255-33"><a href="bayesian-data-analysis-2.html#cb1255-33" tabindex="-1"></a></span>
-<span id="cb1255-34"><a href="bayesian-data-analysis-2.html#cb1255-34" tabindex="-1"></a><span class="co"># politeness</span></span>
-<span id="cb1255-35"><a href="bayesian-data-analysis-2.html#cb1255-35" tabindex="-1"></a>df.politeness <span class="ot">=</span> <span class="fu">read_csv</span>(<span class="st">&quot;data/politeness_data.csv&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1255-36"><a href="bayesian-data-analysis-2.html#cb1255-36" tabindex="-1"></a>  <span class="fu">rename</span>(<span class="at">pitch =</span> frequency)</span></code></pre></div>
+<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="bayesian-data-analysis-2.html#cb3-1" tabindex="-1"></a><span class="co"># poker </span></span>
+<span id="cb3-2"><a href="bayesian-data-analysis-2.html#cb3-2" tabindex="-1"></a>df.poker <span class="ot">=</span> <span class="fu">read_csv</span>(<span class="st">&quot;data/poker.csv&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb3-3"><a href="bayesian-data-analysis-2.html#cb3-3" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">skill =</span> <span class="fu">factor</span>(skill,</span>
+<span id="cb3-4"><a href="bayesian-data-analysis-2.html#cb3-4" tabindex="-1"></a>                        <span class="at">levels =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">2</span>,</span>
+<span id="cb3-5"><a href="bayesian-data-analysis-2.html#cb3-5" tabindex="-1"></a>                        <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;expert&quot;</span>, <span class="st">&quot;average&quot;</span>)),</span>
+<span id="cb3-6"><a href="bayesian-data-analysis-2.html#cb3-6" tabindex="-1"></a>         <span class="at">skill =</span> <span class="fu">fct_relevel</span>(skill, <span class="st">&quot;average&quot;</span>, <span class="st">&quot;expert&quot;</span>),</span>
+<span id="cb3-7"><a href="bayesian-data-analysis-2.html#cb3-7" tabindex="-1"></a>         <span class="at">hand =</span> <span class="fu">factor</span>(hand,</span>
+<span id="cb3-8"><a href="bayesian-data-analysis-2.html#cb3-8" tabindex="-1"></a>                       <span class="at">levels =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>,</span>
+<span id="cb3-9"><a href="bayesian-data-analysis-2.html#cb3-9" tabindex="-1"></a>                       <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;bad&quot;</span>, <span class="st">&quot;neutral&quot;</span>, <span class="st">&quot;good&quot;</span>)),</span>
+<span id="cb3-10"><a href="bayesian-data-analysis-2.html#cb3-10" tabindex="-1"></a>         <span class="at">limit =</span> <span class="fu">factor</span>(limit,</span>
+<span id="cb3-11"><a href="bayesian-data-analysis-2.html#cb3-11" tabindex="-1"></a>                        <span class="at">levels =</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">2</span>,</span>
+<span id="cb3-12"><a href="bayesian-data-analysis-2.html#cb3-12" tabindex="-1"></a>                        <span class="at">labels =</span> <span class="fu">c</span>(<span class="st">&quot;fixed&quot;</span>, <span class="st">&quot;none&quot;</span>)),</span>
+<span id="cb3-13"><a href="bayesian-data-analysis-2.html#cb3-13" tabindex="-1"></a>         <span class="at">participant =</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">n</span>()) <span class="sc">%&gt;%</span> </span>
+<span id="cb3-14"><a href="bayesian-data-analysis-2.html#cb3-14" tabindex="-1"></a>  <span class="fu">select</span>(participant, <span class="fu">everything</span>())</span>
+<span id="cb3-15"><a href="bayesian-data-analysis-2.html#cb3-15" tabindex="-1"></a></span>
+<span id="cb3-16"><a href="bayesian-data-analysis-2.html#cb3-16" tabindex="-1"></a><span class="co"># sleep</span></span>
+<span id="cb3-17"><a href="bayesian-data-analysis-2.html#cb3-17" tabindex="-1"></a>df.sleep <span class="ot">=</span> sleepstudy <span class="sc">%&gt;%</span> </span>
+<span id="cb3-18"><a href="bayesian-data-analysis-2.html#cb3-18" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb3-19"><a href="bayesian-data-analysis-2.html#cb3-19" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb3-20"><a href="bayesian-data-analysis-2.html#cb3-20" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">subject =</span> <span class="fu">as.character</span>(subject)) <span class="sc">%&gt;%</span> </span>
+<span id="cb3-21"><a href="bayesian-data-analysis-2.html#cb3-21" tabindex="-1"></a>  <span class="fu">select</span>(subject, days, reaction) <span class="sc">%&gt;%</span> </span>
+<span id="cb3-22"><a href="bayesian-data-analysis-2.html#cb3-22" tabindex="-1"></a>  <span class="fu">bind_rows</span>(<span class="fu">tibble</span>(<span class="at">subject =</span> <span class="st">&quot;374&quot;</span>,</span>
+<span id="cb3-23"><a href="bayesian-data-analysis-2.html#cb3-23" tabindex="-1"></a>                   <span class="at">days =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">1</span>,</span>
+<span id="cb3-24"><a href="bayesian-data-analysis-2.html#cb3-24" tabindex="-1"></a>                   <span class="at">reaction =</span> <span class="fu">c</span>(<span class="dv">286</span>, <span class="dv">288</span>)),</span>
+<span id="cb3-25"><a href="bayesian-data-analysis-2.html#cb3-25" tabindex="-1"></a>            <span class="fu">tibble</span>(<span class="at">subject =</span> <span class="st">&quot;373&quot;</span>,</span>
+<span id="cb3-26"><a href="bayesian-data-analysis-2.html#cb3-26" tabindex="-1"></a>                   <span class="at">days =</span> <span class="dv">0</span>,</span>
+<span id="cb3-27"><a href="bayesian-data-analysis-2.html#cb3-27" tabindex="-1"></a>                   <span class="at">reaction =</span> <span class="dv">245</span>))</span>
+<span id="cb3-28"><a href="bayesian-data-analysis-2.html#cb3-28" tabindex="-1"></a></span>
+<span id="cb3-29"><a href="bayesian-data-analysis-2.html#cb3-29" tabindex="-1"></a><span class="co"># titanic </span></span>
+<span id="cb3-30"><a href="bayesian-data-analysis-2.html#cb3-30" tabindex="-1"></a>df.titanic <span class="ot">=</span> titanic_train <span class="sc">%&gt;%</span> </span>
+<span id="cb3-31"><a href="bayesian-data-analysis-2.html#cb3-31" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb3-32"><a href="bayesian-data-analysis-2.html#cb3-32" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">sex =</span> <span class="fu">as.factor</span>(sex))</span>
+<span id="cb3-33"><a href="bayesian-data-analysis-2.html#cb3-33" tabindex="-1"></a></span>
+<span id="cb3-34"><a href="bayesian-data-analysis-2.html#cb3-34" tabindex="-1"></a><span class="co"># politeness</span></span>
+<span id="cb3-35"><a href="bayesian-data-analysis-2.html#cb3-35" tabindex="-1"></a>df.politeness <span class="ot">=</span> <span class="fu">read_csv</span>(<span class="st">&quot;data/politeness_data.csv&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb3-36"><a href="bayesian-data-analysis-2.html#cb3-36" tabindex="-1"></a>  <span class="fu">rename</span>(<span class="at">pitch =</span> frequency)</span></code></pre></div>
 </div>
 <div id="poker" class="section level2 hasAnchor" number="23.4">
 <h2><span class="header-section-number">23.4</span> Poker<a href="bayesian-data-analysis-2.html#poker" class="anchor-section" aria-label="Anchor link to header"></a></h2>
 <div id="visualize-the-data" class="section level3 hasAnchor" number="23.4.1">
 <h3><span class="header-section-number">23.4.1</span> 1. Visualize the data<a href="bayesian-data-analysis-2.html#visualize-the-data" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <p>Let’s visualize the data first.</p>
-<div class="sourceCode" id="cb1256"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1256-1"><a href="bayesian-data-analysis-2.html#cb1256-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
-<span id="cb1256-2"><a href="bayesian-data-analysis-2.html#cb1256-2" tabindex="-1"></a></span>
-<span id="cb1256-3"><a href="bayesian-data-analysis-2.html#cb1256-3" tabindex="-1"></a>df.poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1256-4"><a href="bayesian-data-analysis-2.html#cb1256-4" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> hand,</span>
-<span id="cb1256-5"><a href="bayesian-data-analysis-2.html#cb1256-5" tabindex="-1"></a>                       <span class="at">y =</span> balance,</span>
-<span id="cb1256-6"><a href="bayesian-data-analysis-2.html#cb1256-6" tabindex="-1"></a>                       <span class="at">fill =</span> hand,</span>
-<span id="cb1256-7"><a href="bayesian-data-analysis-2.html#cb1256-7" tabindex="-1"></a>                       <span class="at">group =</span> skill,</span>
-<span id="cb1256-8"><a href="bayesian-data-analysis-2.html#cb1256-8" tabindex="-1"></a>                       <span class="at">shape =</span> skill)) <span class="sc">+</span> </span>
-<span id="cb1256-9"><a href="bayesian-data-analysis-2.html#cb1256-9" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
-<span id="cb1256-10"><a href="bayesian-data-analysis-2.html#cb1256-10" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitterdodge</span>(<span class="at">dodge.width =</span> <span class="fl">0.5</span>,</span>
-<span id="cb1256-11"><a href="bayesian-data-analysis-2.html#cb1256-11" tabindex="-1"></a>                                             <span class="at">jitter.height =</span> <span class="dv">0</span>, </span>
-<span id="cb1256-12"><a href="bayesian-data-analysis-2.html#cb1256-12" tabindex="-1"></a>                                             <span class="at">jitter.width =</span> <span class="fl">0.2</span>)) <span class="sc">+</span> </span>
-<span id="cb1256-13"><a href="bayesian-data-analysis-2.html#cb1256-13" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>,</span>
-<span id="cb1256-14"><a href="bayesian-data-analysis-2.html#cb1256-14" tabindex="-1"></a>               <span class="at">position =</span> <span class="fu">position_dodge</span>(<span class="at">width =</span> <span class="fl">0.5</span>),</span>
-<span id="cb1256-15"><a href="bayesian-data-analysis-2.html#cb1256-15" tabindex="-1"></a>               <span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
-<span id="cb1256-16"><a href="bayesian-data-analysis-2.html#cb1256-16" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">&quot;final balance (in Euros)&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1256-17"><a href="bayesian-data-analysis-2.html#cb1256-17" tabindex="-1"></a>  <span class="fu">scale_shape_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="dv">21</span>, <span class="dv">22</span>)) <span class="sc">+</span> </span>
-<span id="cb1256-18"><a href="bayesian-data-analysis-2.html#cb1256-18" tabindex="-1"></a>  <span class="fu">guides</span>(<span class="at">fill =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">shape =</span> <span class="dv">21</span>,</span>
-<span id="cb1256-19"><a href="bayesian-data-analysis-2.html#cb1256-19" tabindex="-1"></a>                                                 <span class="at">fill =</span> RColorBrewer<span class="sc">::</span><span class="fu">brewer.pal</span>(<span class="dv">3</span>, <span class="st">&quot;Set1&quot;</span>))),</span>
-<span id="cb1256-20"><a href="bayesian-data-analysis-2.html#cb1256-20" tabindex="-1"></a>         <span class="at">shape =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">alpha =</span> <span class="dv">1</span>, <span class="at">fill =</span> <span class="st">&quot;black&quot;</span>)))</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
+<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="bayesian-data-analysis-2.html#cb4-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
+<span id="cb4-2"><a href="bayesian-data-analysis-2.html#cb4-2" tabindex="-1"></a></span>
+<span id="cb4-3"><a href="bayesian-data-analysis-2.html#cb4-3" tabindex="-1"></a>df.poker <span class="sc">%&gt;%</span> </span>
+<span id="cb4-4"><a href="bayesian-data-analysis-2.html#cb4-4" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> hand,</span>
+<span id="cb4-5"><a href="bayesian-data-analysis-2.html#cb4-5" tabindex="-1"></a>                       <span class="at">y =</span> balance,</span>
+<span id="cb4-6"><a href="bayesian-data-analysis-2.html#cb4-6" tabindex="-1"></a>                       <span class="at">fill =</span> hand,</span>
+<span id="cb4-7"><a href="bayesian-data-analysis-2.html#cb4-7" tabindex="-1"></a>                       <span class="at">group =</span> skill,</span>
+<span id="cb4-8"><a href="bayesian-data-analysis-2.html#cb4-8" tabindex="-1"></a>                       <span class="at">shape =</span> skill)) <span class="sc">+</span> </span>
+<span id="cb4-9"><a href="bayesian-data-analysis-2.html#cb4-9" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
+<span id="cb4-10"><a href="bayesian-data-analysis-2.html#cb4-10" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitterdodge</span>(<span class="at">dodge.width =</span> <span class="fl">0.5</span>,</span>
+<span id="cb4-11"><a href="bayesian-data-analysis-2.html#cb4-11" tabindex="-1"></a>                                             <span class="at">jitter.height =</span> <span class="dv">0</span>, </span>
+<span id="cb4-12"><a href="bayesian-data-analysis-2.html#cb4-12" tabindex="-1"></a>                                             <span class="at">jitter.width =</span> <span class="fl">0.2</span>)) <span class="sc">+</span> </span>
+<span id="cb4-13"><a href="bayesian-data-analysis-2.html#cb4-13" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>,</span>
+<span id="cb4-14"><a href="bayesian-data-analysis-2.html#cb4-14" tabindex="-1"></a>               <span class="at">position =</span> <span class="fu">position_dodge</span>(<span class="at">width =</span> <span class="fl">0.5</span>),</span>
+<span id="cb4-15"><a href="bayesian-data-analysis-2.html#cb4-15" tabindex="-1"></a>               <span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
+<span id="cb4-16"><a href="bayesian-data-analysis-2.html#cb4-16" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">&quot;final balance (in Euros)&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb4-17"><a href="bayesian-data-analysis-2.html#cb4-17" tabindex="-1"></a>  <span class="fu">scale_shape_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="dv">21</span>, <span class="dv">22</span>)) <span class="sc">+</span> </span>
+<span id="cb4-18"><a href="bayesian-data-analysis-2.html#cb4-18" tabindex="-1"></a>  <span class="fu">guides</span>(<span class="at">fill =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">shape =</span> <span class="dv">21</span>,</span>
+<span id="cb4-19"><a href="bayesian-data-analysis-2.html#cb4-19" tabindex="-1"></a>                                                 <span class="at">fill =</span> RColorBrewer<span class="sc">::</span><span class="fu">brewer.pal</span>(<span class="dv">3</span>, <span class="st">&quot;Set1&quot;</span>))),</span>
+<span id="cb4-20"><a href="bayesian-data-analysis-2.html#cb4-20" tabindex="-1"></a>         <span class="at">shape =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">alpha =</span> <span class="dv">1</span>, <span class="at">fill =</span> <span class="st">&quot;black&quot;</span>)))</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
 </div>
 <div id="specify-and-fit-the-model" class="section level3 hasAnchor" number="23.4.2">
 <h3><span class="header-section-number">23.4.2</span> 2. Specify and fit the model<a href="bayesian-data-analysis-2.html#specify-and-fit-the-model" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="frequentist-model" class="section level4 hasAnchor" number="23.4.2.1">
 <h4><span class="header-section-number">23.4.2.1</span> Frequentist model<a href="bayesian-data-analysis-2.html#frequentist-model" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>And let’s now fit a simple (frequentist) ANOVA model. You have multiple options to do so:</p>
-<div class="sourceCode" id="cb1257"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1257-1"><a href="bayesian-data-analysis-2.html#cb1257-1" tabindex="-1"></a><span class="co"># Option 1: Using the &quot;afex&quot; package</span></span>
-<span id="cb1257-2"><a href="bayesian-data-analysis-2.html#cb1257-2" tabindex="-1"></a><span class="fu">aov_ez</span>(<span class="at">id =</span> <span class="st">&quot;participant&quot;</span>,</span>
-<span id="cb1257-3"><a href="bayesian-data-analysis-2.html#cb1257-3" tabindex="-1"></a>       <span class="at">dv =</span> <span class="st">&quot;balance&quot;</span>,</span>
-<span id="cb1257-4"><a href="bayesian-data-analysis-2.html#cb1257-4" tabindex="-1"></a>       <span class="at">between =</span> <span class="fu">c</span>(<span class="st">&quot;hand&quot;</span>, <span class="st">&quot;skill&quot;</span>),</span>
-<span id="cb1257-5"><a href="bayesian-data-analysis-2.html#cb1257-5" tabindex="-1"></a>       <span class="at">data =</span> df.poker)</span></code></pre></div>
+<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="bayesian-data-analysis-2.html#cb5-1" tabindex="-1"></a><span class="co"># Option 1: Using the &quot;afex&quot; package</span></span>
+<span id="cb5-2"><a href="bayesian-data-analysis-2.html#cb5-2" tabindex="-1"></a><span class="fu">aov_ez</span>(<span class="at">id =</span> <span class="st">&quot;participant&quot;</span>,</span>
+<span id="cb5-3"><a href="bayesian-data-analysis-2.html#cb5-3" tabindex="-1"></a>       <span class="at">dv =</span> <span class="st">&quot;balance&quot;</span>,</span>
+<span id="cb5-4"><a href="bayesian-data-analysis-2.html#cb5-4" tabindex="-1"></a>       <span class="at">between =</span> <span class="fu">c</span>(<span class="st">&quot;hand&quot;</span>, <span class="st">&quot;skill&quot;</span>),</span>
+<span id="cb5-5"><a href="bayesian-data-analysis-2.html#cb5-5" tabindex="-1"></a>       <span class="at">data =</span> df.poker)</span></code></pre></div>
 <pre><code>Contrasts set to contr.sum for the following variables: hand, skill</code></pre>
 <pre><code>Anova Table (Type 3 tests)
 
@@ -999,13 +1000,13 @@ <h4><span class="header-section-number">23.4.2.1</span> Frequentist model<a href
 3 hand:skill 2, 294 16.16  7.08 *** .046   &lt;.001
 ---
 Signif. codes:  0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;+&#39; 0.1 &#39; &#39; 1</code></pre>
-<div class="sourceCode" id="cb1260"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1260-1"><a href="bayesian-data-analysis-2.html#cb1260-1" tabindex="-1"></a><span class="co"># Option 2: Using the car package (here we have to remember to set the contrasts to sum</span></span>
-<span id="cb1260-2"><a href="bayesian-data-analysis-2.html#cb1260-2" tabindex="-1"></a><span class="co"># contrasts!)</span></span>
-<span id="cb1260-3"><a href="bayesian-data-analysis-2.html#cb1260-3" tabindex="-1"></a><span class="fu">lm</span>(balance <span class="sc">~</span> hand <span class="sc">*</span> skill,</span>
-<span id="cb1260-4"><a href="bayesian-data-analysis-2.html#cb1260-4" tabindex="-1"></a>   <span class="at">contrasts =</span> <span class="fu">list</span>(<span class="at">hand =</span> <span class="st">&quot;contr.sum&quot;</span>,</span>
-<span id="cb1260-5"><a href="bayesian-data-analysis-2.html#cb1260-5" tabindex="-1"></a>                    <span class="at">skill =</span> <span class="st">&quot;contr.sum&quot;</span>),</span>
-<span id="cb1260-6"><a href="bayesian-data-analysis-2.html#cb1260-6" tabindex="-1"></a>   <span class="at">data =</span> df.poker) <span class="sc">%&gt;%</span> </span>
-<span id="cb1260-7"><a href="bayesian-data-analysis-2.html#cb1260-7" tabindex="-1"></a>  car<span class="sc">::</span><span class="fu">Anova</span>(<span class="at">type =</span> <span class="dv">3</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb8"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb8-1"><a href="bayesian-data-analysis-2.html#cb8-1" tabindex="-1"></a><span class="co"># Option 2: Using the car package (here we have to remember to set the contrasts to sum</span></span>
+<span id="cb8-2"><a href="bayesian-data-analysis-2.html#cb8-2" tabindex="-1"></a><span class="co"># contrasts!)</span></span>
+<span id="cb8-3"><a href="bayesian-data-analysis-2.html#cb8-3" tabindex="-1"></a><span class="fu">lm</span>(balance <span class="sc">~</span> hand <span class="sc">*</span> skill,</span>
+<span id="cb8-4"><a href="bayesian-data-analysis-2.html#cb8-4" tabindex="-1"></a>   <span class="at">contrasts =</span> <span class="fu">list</span>(<span class="at">hand =</span> <span class="st">&quot;contr.sum&quot;</span>,</span>
+<span id="cb8-5"><a href="bayesian-data-analysis-2.html#cb8-5" tabindex="-1"></a>                    <span class="at">skill =</span> <span class="st">&quot;contr.sum&quot;</span>),</span>
+<span id="cb8-6"><a href="bayesian-data-analysis-2.html#cb8-6" tabindex="-1"></a>   <span class="at">data =</span> df.poker) <span class="sc">%&gt;%</span> </span>
+<span id="cb8-7"><a href="bayesian-data-analysis-2.html#cb8-7" tabindex="-1"></a>  car<span class="sc">::</span><span class="fu">Anova</span>(<span class="at">type =</span> <span class="dv">3</span>)</span></code></pre></div>
 <pre><code>Anova Table (Type III tests)
 
 Response: balance
@@ -1017,13 +1018,13 @@ <h4><span class="header-section-number">23.4.2.1</span> Frequentist model<a href
 Residuals    4752.3 294                        
 ---
 Signif. codes:  0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1</code></pre>
-<div class="sourceCode" id="cb1262"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1262-1"><a href="bayesian-data-analysis-2.html#cb1262-1" tabindex="-1"></a><span class="co"># Option 3: Using the emmeans package (I like this one the best! It let&#39;s us use the </span></span>
-<span id="cb1262-2"><a href="bayesian-data-analysis-2.html#cb1262-2" tabindex="-1"></a><span class="co"># general lm() syntax and we don&#39;t have to remember to set the contrast)</span></span>
-<span id="cb1262-3"><a href="bayesian-data-analysis-2.html#cb1262-3" tabindex="-1"></a>fit.lm_poker <span class="ot">=</span> <span class="fu">lm</span>(balance <span class="sc">~</span> hand <span class="sc">*</span> skill,</span>
-<span id="cb1262-4"><a href="bayesian-data-analysis-2.html#cb1262-4" tabindex="-1"></a>                  <span class="at">data =</span> df.poker) </span>
-<span id="cb1262-5"><a href="bayesian-data-analysis-2.html#cb1262-5" tabindex="-1"></a></span>
-<span id="cb1262-6"><a href="bayesian-data-analysis-2.html#cb1262-6" tabindex="-1"></a>fit.lm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1262-7"><a href="bayesian-data-analysis-2.html#cb1262-7" tabindex="-1"></a>  <span class="fu">joint_tests</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="bayesian-data-analysis-2.html#cb10-1" tabindex="-1"></a><span class="co"># Option 3: Using the emmeans package (I like this one the best! It let&#39;s us use the </span></span>
+<span id="cb10-2"><a href="bayesian-data-analysis-2.html#cb10-2" tabindex="-1"></a><span class="co"># general lm() syntax and we don&#39;t have to remember to set the contrast)</span></span>
+<span id="cb10-3"><a href="bayesian-data-analysis-2.html#cb10-3" tabindex="-1"></a>fit.lm_poker <span class="ot">=</span> <span class="fu">lm</span>(balance <span class="sc">~</span> hand <span class="sc">*</span> skill,</span>
+<span id="cb10-4"><a href="bayesian-data-analysis-2.html#cb10-4" tabindex="-1"></a>                  <span class="at">data =</span> df.poker) </span>
+<span id="cb10-5"><a href="bayesian-data-analysis-2.html#cb10-5" tabindex="-1"></a></span>
+<span id="cb10-6"><a href="bayesian-data-analysis-2.html#cb10-6" tabindex="-1"></a>fit.lm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb10-7"><a href="bayesian-data-analysis-2.html#cb10-7" tabindex="-1"></a>  <span class="fu">joint_tests</span>()</span></code></pre></div>
 <pre><code> model term df1 df2 F.ratio p.value
  hand         2 294  79.169  &lt;.0001
  skill        1 294   2.434  0.1198
@@ -1033,19 +1034,19 @@ <h4><span class="header-section-number">23.4.2.1</span> Frequentist model<a href
 <div id="bayesian-model" class="section level4 hasAnchor" number="23.4.2.2">
 <h4><span class="header-section-number">23.4.2.2</span> Bayesian model<a href="bayesian-data-analysis-2.html#bayesian-model" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>Now, let’s fit a Bayesian regression model using the <code>brm()</code> function (starting with a simple model that only considers <code>hand</code> as a predictor):</p>
-<div class="sourceCode" id="cb1264"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1264-1"><a href="bayesian-data-analysis-2.html#cb1264-1" tabindex="-1"></a>fit.brm_poker <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand,</span>
-<span id="cb1264-2"><a href="bayesian-data-analysis-2.html#cb1264-2" tabindex="-1"></a>                    <span class="at">data =</span> df.poker,</span>
-<span id="cb1264-3"><a href="bayesian-data-analysis-2.html#cb1264-3" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>, </span>
-<span id="cb1264-4"><a href="bayesian-data-analysis-2.html#cb1264-4" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_poker&quot;</span>)</span>
-<span id="cb1264-5"><a href="bayesian-data-analysis-2.html#cb1264-5" tabindex="-1"></a></span>
-<span id="cb1264-6"><a href="bayesian-data-analysis-2.html#cb1264-6" tabindex="-1"></a><span class="co"># we&#39;ll use this model here later </span></span>
-<span id="cb1264-7"><a href="bayesian-data-analysis-2.html#cb1264-7" tabindex="-1"></a>fit.brm_poker2 <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand <span class="sc">*</span> skill,</span>
-<span id="cb1264-8"><a href="bayesian-data-analysis-2.html#cb1264-8" tabindex="-1"></a>                    <span class="at">data =</span> df.poker,</span>
-<span id="cb1264-9"><a href="bayesian-data-analysis-2.html#cb1264-9" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>, </span>
-<span id="cb1264-10"><a href="bayesian-data-analysis-2.html#cb1264-10" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_poker2&quot;</span>)</span>
-<span id="cb1264-11"><a href="bayesian-data-analysis-2.html#cb1264-11" tabindex="-1"></a></span>
-<span id="cb1264-12"><a href="bayesian-data-analysis-2.html#cb1264-12" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span></span>
-<span id="cb1264-13"><a href="bayesian-data-analysis-2.html#cb1264-13" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb12"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb12-1"><a href="bayesian-data-analysis-2.html#cb12-1" tabindex="-1"></a>fit.brm_poker <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand,</span>
+<span id="cb12-2"><a href="bayesian-data-analysis-2.html#cb12-2" tabindex="-1"></a>                    <span class="at">data =</span> df.poker,</span>
+<span id="cb12-3"><a href="bayesian-data-analysis-2.html#cb12-3" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>, </span>
+<span id="cb12-4"><a href="bayesian-data-analysis-2.html#cb12-4" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_poker&quot;</span>)</span>
+<span id="cb12-5"><a href="bayesian-data-analysis-2.html#cb12-5" tabindex="-1"></a></span>
+<span id="cb12-6"><a href="bayesian-data-analysis-2.html#cb12-6" tabindex="-1"></a><span class="co"># we&#39;ll use this model here later </span></span>
+<span id="cb12-7"><a href="bayesian-data-analysis-2.html#cb12-7" tabindex="-1"></a>fit.brm_poker2 <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand <span class="sc">*</span> skill,</span>
+<span id="cb12-8"><a href="bayesian-data-analysis-2.html#cb12-8" tabindex="-1"></a>                    <span class="at">data =</span> df.poker,</span>
+<span id="cb12-9"><a href="bayesian-data-analysis-2.html#cb12-9" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>, </span>
+<span id="cb12-10"><a href="bayesian-data-analysis-2.html#cb12-10" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_poker2&quot;</span>)</span>
+<span id="cb12-11"><a href="bayesian-data-analysis-2.html#cb12-11" tabindex="-1"></a></span>
+<span id="cb12-12"><a href="bayesian-data-analysis-2.html#cb12-12" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span></span>
+<span id="cb12-13"><a href="bayesian-data-analysis-2.html#cb12-13" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
 <pre><code> Family: gaussian 
   Links: mu = identity; sigma = identity 
 Formula: balance ~ 1 + hand 
@@ -1071,8 +1072,8 @@ <h4><span class="header-section-number">23.4.2.2</span> Bayesian model<a href="b
 <h5><span class="header-section-number">23.4.2.2.1</span> Full specification<a href="bayesian-data-analysis-2.html#full-specification" class="anchor-section" aria-label="Anchor link to header"></a></h5>
 <p>So far, we have used the defaults that <code>brm()</code> comes with and not bothered about specifiying the priors, etc.</p>
 <p>Notice that we didn’t specify any priors in the model. By default, “brms” assigns weakly informative priors to the parameters in the model. We can see what these are by running the following command:</p>
-<div class="sourceCode" id="cb1266"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1266-1"><a href="bayesian-data-analysis-2.html#cb1266-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1266-2"><a href="bayesian-data-analysis-2.html#cb1266-2" tabindex="-1"></a>  <span class="fu">prior_summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb14"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb14-1"><a href="bayesian-data-analysis-2.html#cb14-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb14-2"><a href="bayesian-data-analysis-2.html#cb14-2" tabindex="-1"></a>  <span class="fu">prior_summary</span>()</span></code></pre></div>
 <pre><code>                  prior     class        coef group resp dpar nlpar lb ub
                  (flat)         b                                        
                  (flat)         b    handgood                            
@@ -1086,9 +1087,9 @@ <h5><span class="header-section-number">23.4.2.2.1</span> Full specification<a h
       default
       default</code></pre>
 <p>We can also get information about which priors need to be specified before fitting a model:</p>
-<div class="sourceCode" id="cb1268"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1268-1"><a href="bayesian-data-analysis-2.html#cb1268-1" tabindex="-1"></a><span class="fu">get_prior</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand,</span>
-<span id="cb1268-2"><a href="bayesian-data-analysis-2.html#cb1268-2" tabindex="-1"></a>          <span class="at">family =</span> <span class="st">&quot;gaussian&quot;</span>,</span>
-<span id="cb1268-3"><a href="bayesian-data-analysis-2.html#cb1268-3" tabindex="-1"></a>          <span class="at">data =</span> df.poker)</span></code></pre></div>
+<div class="sourceCode" id="cb16"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb16-1"><a href="bayesian-data-analysis-2.html#cb16-1" tabindex="-1"></a><span class="fu">get_prior</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand,</span>
+<span id="cb16-2"><a href="bayesian-data-analysis-2.html#cb16-2" tabindex="-1"></a>          <span class="at">family =</span> <span class="st">&quot;gaussian&quot;</span>,</span>
+<span id="cb16-3"><a href="bayesian-data-analysis-2.html#cb16-3" tabindex="-1"></a>          <span class="at">data =</span> df.poker)</span></code></pre></div>
 <pre><code>                  prior     class        coef group resp dpar nlpar lb ub
                  (flat)         b                                        
                  (flat)         b    handgood                            
@@ -1102,43 +1103,43 @@ <h5><span class="header-section-number">23.4.2.2.1</span> Full specification<a h
       default
       default</code></pre>
 <p>Here is an example for what a more complete model specification could look like:</p>
-<div class="sourceCode" id="cb1270"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1270-1"><a href="bayesian-data-analysis-2.html#cb1270-1" tabindex="-1"></a>fit.brm_poker_full <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand,</span>
-<span id="cb1270-2"><a href="bayesian-data-analysis-2.html#cb1270-2" tabindex="-1"></a>                         <span class="at">family =</span> <span class="st">&quot;gaussian&quot;</span>,</span>
-<span id="cb1270-3"><a href="bayesian-data-analysis-2.html#cb1270-3" tabindex="-1"></a>                         <span class="at">data =</span> df.poker,</span>
-<span id="cb1270-4"><a href="bayesian-data-analysis-2.html#cb1270-4" tabindex="-1"></a>                         <span class="at">prior =</span> <span class="fu">c</span>(<span class="fu">prior</span>(<span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>),</span>
-<span id="cb1270-5"><a href="bayesian-data-analysis-2.html#cb1270-5" tabindex="-1"></a>                                         <span class="at">class =</span> <span class="st">&quot;b&quot;</span>,</span>
-<span id="cb1270-6"><a href="bayesian-data-analysis-2.html#cb1270-6" tabindex="-1"></a>                                         <span class="at">coef =</span> <span class="st">&quot;handgood&quot;</span>),</span>
-<span id="cb1270-7"><a href="bayesian-data-analysis-2.html#cb1270-7" tabindex="-1"></a>                                   <span class="fu">prior</span>(<span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>),</span>
-<span id="cb1270-8"><a href="bayesian-data-analysis-2.html#cb1270-8" tabindex="-1"></a>                                         <span class="at">class =</span> <span class="st">&quot;b&quot;</span>,</span>
-<span id="cb1270-9"><a href="bayesian-data-analysis-2.html#cb1270-9" tabindex="-1"></a>                                         <span class="at">coef =</span> <span class="st">&quot;handneutral&quot;</span>),</span>
-<span id="cb1270-10"><a href="bayesian-data-analysis-2.html#cb1270-10" tabindex="-1"></a>                                   <span class="fu">prior</span>(<span class="fu">student_t</span>(<span class="dv">3</span>, <span class="dv">3</span>, <span class="dv">10</span>),</span>
-<span id="cb1270-11"><a href="bayesian-data-analysis-2.html#cb1270-11" tabindex="-1"></a>                                         <span class="at">class =</span> <span class="st">&quot;Intercept&quot;</span>),</span>
-<span id="cb1270-12"><a href="bayesian-data-analysis-2.html#cb1270-12" tabindex="-1"></a>                                   <span class="fu">prior</span>(<span class="fu">student_t</span>(<span class="dv">3</span>, <span class="dv">0</span>, <span class="dv">10</span>),</span>
-<span id="cb1270-13"><a href="bayesian-data-analysis-2.html#cb1270-13" tabindex="-1"></a>                                         <span class="at">class =</span> <span class="st">&quot;sigma&quot;</span>)),</span>
-<span id="cb1270-14"><a href="bayesian-data-analysis-2.html#cb1270-14" tabindex="-1"></a>                         <span class="at">inits =</span> <span class="fu">list</span>(<span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">0</span>,</span>
-<span id="cb1270-15"><a href="bayesian-data-analysis-2.html#cb1270-15" tabindex="-1"></a>                                           <span class="at">sigma =</span> <span class="dv">1</span>,</span>
-<span id="cb1270-16"><a href="bayesian-data-analysis-2.html#cb1270-16" tabindex="-1"></a>                                           <span class="at">handgood =</span> <span class="dv">5</span>,</span>
-<span id="cb1270-17"><a href="bayesian-data-analysis-2.html#cb1270-17" tabindex="-1"></a>                                           <span class="at">handneutral =</span> <span class="dv">5</span>),</span>
-<span id="cb1270-18"><a href="bayesian-data-analysis-2.html#cb1270-18" tabindex="-1"></a>                                      <span class="fu">list</span>(<span class="at">Intercept =</span> <span class="sc">-</span><span class="dv">5</span>,</span>
-<span id="cb1270-19"><a href="bayesian-data-analysis-2.html#cb1270-19" tabindex="-1"></a>                                           <span class="at">sigma =</span> <span class="dv">3</span>,</span>
-<span id="cb1270-20"><a href="bayesian-data-analysis-2.html#cb1270-20" tabindex="-1"></a>                                           <span class="at">handgood =</span> <span class="dv">2</span>,</span>
-<span id="cb1270-21"><a href="bayesian-data-analysis-2.html#cb1270-21" tabindex="-1"></a>                                           <span class="at">handneutral =</span> <span class="dv">2</span>),</span>
-<span id="cb1270-22"><a href="bayesian-data-analysis-2.html#cb1270-22" tabindex="-1"></a>                                      <span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">2</span>,</span>
-<span id="cb1270-23"><a href="bayesian-data-analysis-2.html#cb1270-23" tabindex="-1"></a>                                           <span class="at">sigma =</span> <span class="dv">1</span>,</span>
-<span id="cb1270-24"><a href="bayesian-data-analysis-2.html#cb1270-24" tabindex="-1"></a>                                           <span class="at">handgood =</span> <span class="sc">-</span><span class="dv">1</span>,</span>
-<span id="cb1270-25"><a href="bayesian-data-analysis-2.html#cb1270-25" tabindex="-1"></a>                                           <span class="at">handneutral =</span> <span class="dv">1</span>),</span>
-<span id="cb1270-26"><a href="bayesian-data-analysis-2.html#cb1270-26" tabindex="-1"></a>                                      <span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">1</span>,</span>
-<span id="cb1270-27"><a href="bayesian-data-analysis-2.html#cb1270-27" tabindex="-1"></a>                                           <span class="at">sigma =</span> <span class="dv">2</span>,</span>
-<span id="cb1270-28"><a href="bayesian-data-analysis-2.html#cb1270-28" tabindex="-1"></a>                                           <span class="at">handgood =</span> <span class="dv">2</span>,</span>
-<span id="cb1270-29"><a href="bayesian-data-analysis-2.html#cb1270-29" tabindex="-1"></a>                                           <span class="at">handneutral =</span> <span class="sc">-</span><span class="dv">2</span>)),</span>
-<span id="cb1270-30"><a href="bayesian-data-analysis-2.html#cb1270-30" tabindex="-1"></a>                         <span class="at">iter =</span> <span class="dv">4000</span>,</span>
-<span id="cb1270-31"><a href="bayesian-data-analysis-2.html#cb1270-31" tabindex="-1"></a>                         <span class="at">warmup =</span> <span class="dv">1000</span>,</span>
-<span id="cb1270-32"><a href="bayesian-data-analysis-2.html#cb1270-32" tabindex="-1"></a>                         <span class="at">chains =</span> <span class="dv">4</span>,</span>
-<span id="cb1270-33"><a href="bayesian-data-analysis-2.html#cb1270-33" tabindex="-1"></a>                         <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_full&quot;</span>,</span>
-<span id="cb1270-34"><a href="bayesian-data-analysis-2.html#cb1270-34" tabindex="-1"></a>                         <span class="at">seed =</span> <span class="dv">1</span>)</span>
-<span id="cb1270-35"><a href="bayesian-data-analysis-2.html#cb1270-35" tabindex="-1"></a></span>
-<span id="cb1270-36"><a href="bayesian-data-analysis-2.html#cb1270-36" tabindex="-1"></a>fit.brm_poker_full <span class="sc">%&gt;%</span></span>
-<span id="cb1270-37"><a href="bayesian-data-analysis-2.html#cb1270-37" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb18"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb18-1"><a href="bayesian-data-analysis-2.html#cb18-1" tabindex="-1"></a>fit.brm_poker_full <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand,</span>
+<span id="cb18-2"><a href="bayesian-data-analysis-2.html#cb18-2" tabindex="-1"></a>                         <span class="at">family =</span> <span class="st">&quot;gaussian&quot;</span>,</span>
+<span id="cb18-3"><a href="bayesian-data-analysis-2.html#cb18-3" tabindex="-1"></a>                         <span class="at">data =</span> df.poker,</span>
+<span id="cb18-4"><a href="bayesian-data-analysis-2.html#cb18-4" tabindex="-1"></a>                         <span class="at">prior =</span> <span class="fu">c</span>(<span class="fu">prior</span>(<span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>),</span>
+<span id="cb18-5"><a href="bayesian-data-analysis-2.html#cb18-5" tabindex="-1"></a>                                         <span class="at">class =</span> <span class="st">&quot;b&quot;</span>,</span>
+<span id="cb18-6"><a href="bayesian-data-analysis-2.html#cb18-6" tabindex="-1"></a>                                         <span class="at">coef =</span> <span class="st">&quot;handgood&quot;</span>),</span>
+<span id="cb18-7"><a href="bayesian-data-analysis-2.html#cb18-7" tabindex="-1"></a>                                   <span class="fu">prior</span>(<span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>),</span>
+<span id="cb18-8"><a href="bayesian-data-analysis-2.html#cb18-8" tabindex="-1"></a>                                         <span class="at">class =</span> <span class="st">&quot;b&quot;</span>,</span>
+<span id="cb18-9"><a href="bayesian-data-analysis-2.html#cb18-9" tabindex="-1"></a>                                         <span class="at">coef =</span> <span class="st">&quot;handneutral&quot;</span>),</span>
+<span id="cb18-10"><a href="bayesian-data-analysis-2.html#cb18-10" tabindex="-1"></a>                                   <span class="fu">prior</span>(<span class="fu">student_t</span>(<span class="dv">3</span>, <span class="dv">3</span>, <span class="dv">10</span>),</span>
+<span id="cb18-11"><a href="bayesian-data-analysis-2.html#cb18-11" tabindex="-1"></a>                                         <span class="at">class =</span> <span class="st">&quot;Intercept&quot;</span>),</span>
+<span id="cb18-12"><a href="bayesian-data-analysis-2.html#cb18-12" tabindex="-1"></a>                                   <span class="fu">prior</span>(<span class="fu">student_t</span>(<span class="dv">3</span>, <span class="dv">0</span>, <span class="dv">10</span>),</span>
+<span id="cb18-13"><a href="bayesian-data-analysis-2.html#cb18-13" tabindex="-1"></a>                                         <span class="at">class =</span> <span class="st">&quot;sigma&quot;</span>)),</span>
+<span id="cb18-14"><a href="bayesian-data-analysis-2.html#cb18-14" tabindex="-1"></a>                         <span class="at">inits =</span> <span class="fu">list</span>(<span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">0</span>,</span>
+<span id="cb18-15"><a href="bayesian-data-analysis-2.html#cb18-15" tabindex="-1"></a>                                           <span class="at">sigma =</span> <span class="dv">1</span>,</span>
+<span id="cb18-16"><a href="bayesian-data-analysis-2.html#cb18-16" tabindex="-1"></a>                                           <span class="at">handgood =</span> <span class="dv">5</span>,</span>
+<span id="cb18-17"><a href="bayesian-data-analysis-2.html#cb18-17" tabindex="-1"></a>                                           <span class="at">handneutral =</span> <span class="dv">5</span>),</span>
+<span id="cb18-18"><a href="bayesian-data-analysis-2.html#cb18-18" tabindex="-1"></a>                                      <span class="fu">list</span>(<span class="at">Intercept =</span> <span class="sc">-</span><span class="dv">5</span>,</span>
+<span id="cb18-19"><a href="bayesian-data-analysis-2.html#cb18-19" tabindex="-1"></a>                                           <span class="at">sigma =</span> <span class="dv">3</span>,</span>
+<span id="cb18-20"><a href="bayesian-data-analysis-2.html#cb18-20" tabindex="-1"></a>                                           <span class="at">handgood =</span> <span class="dv">2</span>,</span>
+<span id="cb18-21"><a href="bayesian-data-analysis-2.html#cb18-21" tabindex="-1"></a>                                           <span class="at">handneutral =</span> <span class="dv">2</span>),</span>
+<span id="cb18-22"><a href="bayesian-data-analysis-2.html#cb18-22" tabindex="-1"></a>                                      <span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">2</span>,</span>
+<span id="cb18-23"><a href="bayesian-data-analysis-2.html#cb18-23" tabindex="-1"></a>                                           <span class="at">sigma =</span> <span class="dv">1</span>,</span>
+<span id="cb18-24"><a href="bayesian-data-analysis-2.html#cb18-24" tabindex="-1"></a>                                           <span class="at">handgood =</span> <span class="sc">-</span><span class="dv">1</span>,</span>
+<span id="cb18-25"><a href="bayesian-data-analysis-2.html#cb18-25" tabindex="-1"></a>                                           <span class="at">handneutral =</span> <span class="dv">1</span>),</span>
+<span id="cb18-26"><a href="bayesian-data-analysis-2.html#cb18-26" tabindex="-1"></a>                                      <span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">1</span>,</span>
+<span id="cb18-27"><a href="bayesian-data-analysis-2.html#cb18-27" tabindex="-1"></a>                                           <span class="at">sigma =</span> <span class="dv">2</span>,</span>
+<span id="cb18-28"><a href="bayesian-data-analysis-2.html#cb18-28" tabindex="-1"></a>                                           <span class="at">handgood =</span> <span class="dv">2</span>,</span>
+<span id="cb18-29"><a href="bayesian-data-analysis-2.html#cb18-29" tabindex="-1"></a>                                           <span class="at">handneutral =</span> <span class="sc">-</span><span class="dv">2</span>)),</span>
+<span id="cb18-30"><a href="bayesian-data-analysis-2.html#cb18-30" tabindex="-1"></a>                         <span class="at">iter =</span> <span class="dv">4000</span>,</span>
+<span id="cb18-31"><a href="bayesian-data-analysis-2.html#cb18-31" tabindex="-1"></a>                         <span class="at">warmup =</span> <span class="dv">1000</span>,</span>
+<span id="cb18-32"><a href="bayesian-data-analysis-2.html#cb18-32" tabindex="-1"></a>                         <span class="at">chains =</span> <span class="dv">4</span>,</span>
+<span id="cb18-33"><a href="bayesian-data-analysis-2.html#cb18-33" tabindex="-1"></a>                         <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_full&quot;</span>,</span>
+<span id="cb18-34"><a href="bayesian-data-analysis-2.html#cb18-34" tabindex="-1"></a>                         <span class="at">seed =</span> <span class="dv">1</span>)</span>
+<span id="cb18-35"><a href="bayesian-data-analysis-2.html#cb18-35" tabindex="-1"></a></span>
+<span id="cb18-36"><a href="bayesian-data-analysis-2.html#cb18-36" tabindex="-1"></a>fit.brm_poker_full <span class="sc">%&gt;%</span></span>
+<span id="cb18-37"><a href="bayesian-data-analysis-2.html#cb18-37" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
 <pre><code> Family: gaussian 
   Links: mu = identity; sigma = identity 
 Formula: balance ~ 1 + hand 
@@ -1160,8 +1161,8 @@ <h5><span class="header-section-number">23.4.2.2.1</span> Full specification<a h
 and Tail_ESS are effective sample size measures, and Rhat is the potential
 scale reduction factor on split chains (at convergence, Rhat = 1).</code></pre>
 <p>We can also take a look at the Stan code that the <code>brm()</code> function creates:</p>
-<div class="sourceCode" id="cb1272"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1272-1"><a href="bayesian-data-analysis-2.html#cb1272-1" tabindex="-1"></a>fit.brm_poker_full <span class="sc">%&gt;%</span></span>
-<span id="cb1272-2"><a href="bayesian-data-analysis-2.html#cb1272-2" tabindex="-1"></a>  <span class="fu">stancode</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb20"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb20-1"><a href="bayesian-data-analysis-2.html#cb20-1" tabindex="-1"></a>fit.brm_poker_full <span class="sc">%&gt;%</span></span>
+<span id="cb20-2"><a href="bayesian-data-analysis-2.html#cb20-2" tabindex="-1"></a>  <span class="fu">stancode</span>()</span></code></pre></div>
 <pre><code>// generated with brms 2.20.4
 functions {
 }
@@ -1215,59 +1216,60 @@ <h3><span class="header-section-number">23.4.3</span> 3. Model evaluation<a href
 <div id="a-did-the-inference-work" class="section level4 hasAnchor" number="23.4.3.1">
 <h4><span class="header-section-number">23.4.3.1</span> a) Did the inference work?<a href="bayesian-data-analysis-2.html#a-did-the-inference-work" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>So far, we’ve assumed that the inference has worked out. We can check this by running <code>plot()</code> on our brm object:</p>
-<div class="sourceCode" id="cb1274"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1274-1"><a href="bayesian-data-analysis-2.html#cb1274-1" tabindex="-1"></a><span class="fu">plot</span>(fit.brm_poker,</span>
-<span id="cb1274-2"><a href="bayesian-data-analysis-2.html#cb1274-2" tabindex="-1"></a>     <span class="at">N =</span> <span class="dv">7</span>,</span>
-<span id="cb1274-3"><a href="bayesian-data-analysis-2.html#cb1274-3" tabindex="-1"></a>     <span class="at">ask =</span> F)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-11-1.png" width="960" /></p>
+<div class="sourceCode" id="cb22"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb22-1"><a href="bayesian-data-analysis-2.html#cb22-1" tabindex="-1"></a><span class="fu">plot</span>(fit.brm_poker,</span>
+<span id="cb22-2"><a href="bayesian-data-analysis-2.html#cb22-2" tabindex="-1"></a>     <span class="at">N =</span> <span class="dv">7</span>,</span>
+<span id="cb22-3"><a href="bayesian-data-analysis-2.html#cb22-3" tabindex="-1"></a>     <span class="at">ask =</span> F)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-11-1.png" width="960" /></p>
 <p>The posterior distributions (left hand side), and the trace plots of the samples from the posterior (right hand side) look good.</p>
 <p>Let’s make our own version of a trace plot for one parameter in the model:</p>
-<div class="sourceCode" id="cb1275"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1275-1"><a href="bayesian-data-analysis-2.html#cb1275-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1275-2"><a href="bayesian-data-analysis-2.html#cb1275-2" tabindex="-1"></a>  <span class="fu">spread_draws</span>(b_Intercept) <span class="sc">%&gt;%</span> </span>
-<span id="cb1275-3"><a href="bayesian-data-analysis-2.html#cb1275-3" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1275-4"><a href="bayesian-data-analysis-2.html#cb1275-4" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">chain =</span> <span class="fu">as.factor</span>(chain)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1275-5"><a href="bayesian-data-analysis-2.html#cb1275-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> iteration,</span>
-<span id="cb1275-6"><a href="bayesian-data-analysis-2.html#cb1275-6" tabindex="-1"></a>             <span class="at">y =</span> b_intercept,</span>
-<span id="cb1275-7"><a href="bayesian-data-analysis-2.html#cb1275-7" tabindex="-1"></a>             <span class="at">group =</span> chain,</span>
-<span id="cb1275-8"><a href="bayesian-data-analysis-2.html#cb1275-8" tabindex="-1"></a>             <span class="at">color =</span> chain)) <span class="sc">+</span> </span>
-<span id="cb1275-9"><a href="bayesian-data-analysis-2.html#cb1275-9" tabindex="-1"></a>  <span class="fu">geom_line</span>() <span class="sc">+</span> </span>
-<span id="cb1275-10"><a href="bayesian-data-analysis-2.html#cb1275-10" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">type =</span> <span class="st">&quot;seq&quot;</span>,</span>
-<span id="cb1275-11"><a href="bayesian-data-analysis-2.html#cb1275-11" tabindex="-1"></a>                     <span class="at">direction =</span> <span class="sc">-</span><span class="dv">1</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-12-1.png" width="672" /></p>
+<div class="sourceCode" id="cb23"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb23-1"><a href="bayesian-data-analysis-2.html#cb23-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb23-2"><a href="bayesian-data-analysis-2.html#cb23-2" tabindex="-1"></a>  <span class="fu">spread_draws</span>(b_Intercept) <span class="sc">%&gt;%</span> </span>
+<span id="cb23-3"><a href="bayesian-data-analysis-2.html#cb23-3" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb23-4"><a href="bayesian-data-analysis-2.html#cb23-4" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">chain =</span> <span class="fu">as.factor</span>(chain)) <span class="sc">%&gt;%</span> </span>
+<span id="cb23-5"><a href="bayesian-data-analysis-2.html#cb23-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> iteration,</span>
+<span id="cb23-6"><a href="bayesian-data-analysis-2.html#cb23-6" tabindex="-1"></a>             <span class="at">y =</span> b_intercept,</span>
+<span id="cb23-7"><a href="bayesian-data-analysis-2.html#cb23-7" tabindex="-1"></a>             <span class="at">group =</span> chain,</span>
+<span id="cb23-8"><a href="bayesian-data-analysis-2.html#cb23-8" tabindex="-1"></a>             <span class="at">color =</span> chain)) <span class="sc">+</span> </span>
+<span id="cb23-9"><a href="bayesian-data-analysis-2.html#cb23-9" tabindex="-1"></a>  <span class="fu">geom_line</span>() <span class="sc">+</span> </span>
+<span id="cb23-10"><a href="bayesian-data-analysis-2.html#cb23-10" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">type =</span> <span class="st">&quot;seq&quot;</span>,</span>
+<span id="cb23-11"><a href="bayesian-data-analysis-2.html#cb23-11" tabindex="-1"></a>                     <span class="at">direction =</span> <span class="sc">-</span><span class="dv">1</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-12-1.png" width="672" /></p>
 <p>We can also take a look at the auto-correlation plot. Ideally, we want to generate independent samples from the posterior. So we don’t want subsequent samples to be strongly correlated with each other. Let’s take a look:</p>
-<div class="sourceCode" id="cb1276"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1276-1"><a href="bayesian-data-analysis-2.html#cb1276-1" tabindex="-1"></a>variables <span class="ot">=</span> fit.brm_poker <span class="sc">%&gt;%</span></span>
-<span id="cb1276-2"><a href="bayesian-data-analysis-2.html#cb1276-2" tabindex="-1"></a>  <span class="fu">get_variables</span>() <span class="sc">%&gt;%</span></span>
-<span id="cb1276-3"><a href="bayesian-data-analysis-2.html#cb1276-3" tabindex="-1"></a>  .[<span class="dv">1</span><span class="sc">:</span><span class="dv">4</span>]</span>
-<span id="cb1276-4"><a href="bayesian-data-analysis-2.html#cb1276-4" tabindex="-1"></a></span>
-<span id="cb1276-5"><a href="bayesian-data-analysis-2.html#cb1276-5" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1276-6"><a href="bayesian-data-analysis-2.html#cb1276-6" tabindex="-1"></a>  <span class="fu">as_draws</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1276-7"><a href="bayesian-data-analysis-2.html#cb1276-7" tabindex="-1"></a>  <span class="fu">mcmc_acf</span>(<span class="at">pars =</span> variables,</span>
-<span id="cb1276-8"><a href="bayesian-data-analysis-2.html#cb1276-8" tabindex="-1"></a>           <span class="at">lags =</span> <span class="dv">4</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb24"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb24-1"><a href="bayesian-data-analysis-2.html#cb24-1" tabindex="-1"></a>variables <span class="ot">=</span> fit.brm_poker <span class="sc">%&gt;%</span></span>
+<span id="cb24-2"><a href="bayesian-data-analysis-2.html#cb24-2" tabindex="-1"></a>  <span class="fu">get_variables</span>() <span class="sc">%&gt;%</span></span>
+<span id="cb24-3"><a href="bayesian-data-analysis-2.html#cb24-3" tabindex="-1"></a>  .[<span class="dv">1</span><span class="sc">:</span><span class="dv">4</span>]</span>
+<span id="cb24-4"><a href="bayesian-data-analysis-2.html#cb24-4" tabindex="-1"></a></span>
+<span id="cb24-5"><a href="bayesian-data-analysis-2.html#cb24-5" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb24-6"><a href="bayesian-data-analysis-2.html#cb24-6" tabindex="-1"></a>  <span class="fu">as_draws</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb24-7"><a href="bayesian-data-analysis-2.html#cb24-7" tabindex="-1"></a>  <span class="fu">mcmc_acf</span>(<span class="at">pars =</span> variables,</span>
+<span id="cb24-8"><a href="bayesian-data-analysis-2.html#cb24-8" tabindex="-1"></a>           <span class="at">lags =</span> <span class="dv">4</span>)</span></code></pre></div>
 <pre><code>Warning: The `facets` argument of `facet_grid()` is deprecated as of ggplot2 2.2.0.
 ℹ Please use the `rows` argument instead.
 ℹ The deprecated feature was likely used in the bayesplot package.
   Please report the issue at &lt;https://github.com/stan-dev/bayesplot/issues/&gt;.
 This warning is displayed once every 8 hours.
-Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.</code></pre>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-13-1.png" width="672" /></p>
+Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
+generated.</code></pre>
+<p><img src="psych252_files/figure-html/unnamed-chunk-13-1.png" width="672" /></p>
 <p>Looking good! The autocorrelation should become very small as the lag increases (indicating that we are getting independent samples from the posterior).</p>
 <div id="when-things-go-wrong" class="section level6 hasAnchor" number="23.4.3.1.0.1">
 <h6><span class="header-section-number">23.4.3.1.0.1</span> When things go wrong<a href="bayesian-data-analysis-2.html#when-things-go-wrong" class="anchor-section" aria-label="Anchor link to header"></a></h6>
 <p>Let’s try to fit a model to very little data (just two observations) with extremely uninformative priors:</p>
-<div class="sourceCode" id="cb1278"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1278-1"><a href="bayesian-data-analysis-2.html#cb1278-1" tabindex="-1"></a>df.data <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">y =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">1</span>, <span class="dv">1</span>))</span>
-<span id="cb1278-2"><a href="bayesian-data-analysis-2.html#cb1278-2" tabindex="-1"></a></span>
-<span id="cb1278-3"><a href="bayesian-data-analysis-2.html#cb1278-3" tabindex="-1"></a>fit.brm_wrong <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">data =</span> df.data,</span>
-<span id="cb1278-4"><a href="bayesian-data-analysis-2.html#cb1278-4" tabindex="-1"></a>                    <span class="at">family =</span> gaussian,</span>
-<span id="cb1278-5"><a href="bayesian-data-analysis-2.html#cb1278-5" tabindex="-1"></a>                    <span class="at">formula =</span> y <span class="sc">~</span> <span class="dv">1</span>,</span>
-<span id="cb1278-6"><a href="bayesian-data-analysis-2.html#cb1278-6" tabindex="-1"></a>                    <span class="at">prior =</span> <span class="fu">c</span>(<span class="fu">prior</span>(<span class="fu">uniform</span>(<span class="sc">-</span><span class="fl">1e10</span>, <span class="fl">1e10</span>), <span class="at">class =</span> Intercept),</span>
-<span id="cb1278-7"><a href="bayesian-data-analysis-2.html#cb1278-7" tabindex="-1"></a>                              <span class="fu">prior</span>(<span class="fu">uniform</span>(<span class="dv">0</span>, <span class="fl">1e10</span>), <span class="at">class =</span> sigma)),</span>
-<span id="cb1278-8"><a href="bayesian-data-analysis-2.html#cb1278-8" tabindex="-1"></a>                    <span class="at">inits =</span> <span class="fu">list</span>(<span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">0</span>, <span class="at">sigma =</span> <span class="dv">1</span>),</span>
-<span id="cb1278-9"><a href="bayesian-data-analysis-2.html#cb1278-9" tabindex="-1"></a>                                 <span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">0</span>, <span class="at">sigma =</span> <span class="dv">1</span>)),</span>
-<span id="cb1278-10"><a href="bayesian-data-analysis-2.html#cb1278-10" tabindex="-1"></a>                    <span class="at">iter =</span> <span class="dv">4000</span>,</span>
-<span id="cb1278-11"><a href="bayesian-data-analysis-2.html#cb1278-11" tabindex="-1"></a>                    <span class="at">warmup =</span> <span class="dv">1000</span>,</span>
-<span id="cb1278-12"><a href="bayesian-data-analysis-2.html#cb1278-12" tabindex="-1"></a>                    <span class="at">chains =</span> <span class="dv">2</span>,</span>
-<span id="cb1278-13"><a href="bayesian-data-analysis-2.html#cb1278-13" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_wrong&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb26"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb26-1"><a href="bayesian-data-analysis-2.html#cb26-1" tabindex="-1"></a>df.data <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">y =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">1</span>, <span class="dv">1</span>))</span>
+<span id="cb26-2"><a href="bayesian-data-analysis-2.html#cb26-2" tabindex="-1"></a></span>
+<span id="cb26-3"><a href="bayesian-data-analysis-2.html#cb26-3" tabindex="-1"></a>fit.brm_wrong <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">data =</span> df.data,</span>
+<span id="cb26-4"><a href="bayesian-data-analysis-2.html#cb26-4" tabindex="-1"></a>                    <span class="at">family =</span> gaussian,</span>
+<span id="cb26-5"><a href="bayesian-data-analysis-2.html#cb26-5" tabindex="-1"></a>                    <span class="at">formula =</span> y <span class="sc">~</span> <span class="dv">1</span>,</span>
+<span id="cb26-6"><a href="bayesian-data-analysis-2.html#cb26-6" tabindex="-1"></a>                    <span class="at">prior =</span> <span class="fu">c</span>(<span class="fu">prior</span>(<span class="fu">uniform</span>(<span class="sc">-</span><span class="fl">1e10</span>, <span class="fl">1e10</span>), <span class="at">class =</span> Intercept),</span>
+<span id="cb26-7"><a href="bayesian-data-analysis-2.html#cb26-7" tabindex="-1"></a>                              <span class="fu">prior</span>(<span class="fu">uniform</span>(<span class="dv">0</span>, <span class="fl">1e10</span>), <span class="at">class =</span> sigma)),</span>
+<span id="cb26-8"><a href="bayesian-data-analysis-2.html#cb26-8" tabindex="-1"></a>                    <span class="at">inits =</span> <span class="fu">list</span>(<span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">0</span>, <span class="at">sigma =</span> <span class="dv">1</span>),</span>
+<span id="cb26-9"><a href="bayesian-data-analysis-2.html#cb26-9" tabindex="-1"></a>                                 <span class="fu">list</span>(<span class="at">Intercept =</span> <span class="dv">0</span>, <span class="at">sigma =</span> <span class="dv">1</span>)),</span>
+<span id="cb26-10"><a href="bayesian-data-analysis-2.html#cb26-10" tabindex="-1"></a>                    <span class="at">iter =</span> <span class="dv">4000</span>,</span>
+<span id="cb26-11"><a href="bayesian-data-analysis-2.html#cb26-11" tabindex="-1"></a>                    <span class="at">warmup =</span> <span class="dv">1000</span>,</span>
+<span id="cb26-12"><a href="bayesian-data-analysis-2.html#cb26-12" tabindex="-1"></a>                    <span class="at">chains =</span> <span class="dv">2</span>,</span>
+<span id="cb26-13"><a href="bayesian-data-analysis-2.html#cb26-13" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_wrong&quot;</span>)</span></code></pre></div>
 <p>Let’s take a look at the posterior distributions of the model parameters:</p>
-<div class="sourceCode" id="cb1279"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1279-1"><a href="bayesian-data-analysis-2.html#cb1279-1" tabindex="-1"></a><span class="fu">summary</span>(fit.brm_wrong)</span></code></pre></div>
+<div class="sourceCode" id="cb27"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb27-1"><a href="bayesian-data-analysis-2.html#cb27-1" tabindex="-1"></a><span class="fu">summary</span>(fit.brm_wrong)</span></code></pre></div>
 <pre><code>Warning: There were 1396 divergent transitions after warmup. Increasing
 adapt_delta above 0.8 may help. See
 http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup</code></pre>
@@ -1293,34 +1295,34 @@ <h6><span class="header-section-number">23.4.3.1.0.1</span> When things go wrong
 scale reduction factor on split chains (at convergence, Rhat = 1).</code></pre>
 <p>Not looking good – The estimates and credible intervals are off the charts. And the effective samples sizes in the chains are very small.</p>
 <p>Let’s visualize the trace plots:</p>
-<div class="sourceCode" id="cb1282"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1282-1"><a href="bayesian-data-analysis-2.html#cb1282-1" tabindex="-1"></a><span class="fu">plot</span>(fit.brm_wrong,</span>
-<span id="cb1282-2"><a href="bayesian-data-analysis-2.html#cb1282-2" tabindex="-1"></a>     <span class="at">N =</span> <span class="dv">2</span>, </span>
-<span id="cb1282-3"><a href="bayesian-data-analysis-2.html#cb1282-3" tabindex="-1"></a>     <span class="at">ask =</span> F)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-16-1.png" width="1152" /></p>
-<div class="sourceCode" id="cb1283"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1283-1"><a href="bayesian-data-analysis-2.html#cb1283-1" tabindex="-1"></a>fit.brm_wrong <span class="sc">%&gt;%</span> </span>
-<span id="cb1283-2"><a href="bayesian-data-analysis-2.html#cb1283-2" tabindex="-1"></a>  <span class="fu">spread_draws</span>(b_Intercept) <span class="sc">%&gt;%</span> </span>
-<span id="cb1283-3"><a href="bayesian-data-analysis-2.html#cb1283-3" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1283-4"><a href="bayesian-data-analysis-2.html#cb1283-4" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">chain =</span> <span class="fu">as.factor</span>(chain)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1283-5"><a href="bayesian-data-analysis-2.html#cb1283-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> iteration,</span>
-<span id="cb1283-6"><a href="bayesian-data-analysis-2.html#cb1283-6" tabindex="-1"></a>             <span class="at">y =</span> b_intercept,</span>
-<span id="cb1283-7"><a href="bayesian-data-analysis-2.html#cb1283-7" tabindex="-1"></a>             <span class="at">group =</span> chain,</span>
-<span id="cb1283-8"><a href="bayesian-data-analysis-2.html#cb1283-8" tabindex="-1"></a>             <span class="at">color =</span> chain)) <span class="sc">+</span> </span>
-<span id="cb1283-9"><a href="bayesian-data-analysis-2.html#cb1283-9" tabindex="-1"></a>  <span class="fu">geom_line</span>() <span class="sc">+</span> </span>
-<span id="cb1283-10"><a href="bayesian-data-analysis-2.html#cb1283-10" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">direction =</span> <span class="sc">-</span><span class="dv">1</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-17-1.png" width="672" /></p>
+<div class="sourceCode" id="cb30"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb30-1"><a href="bayesian-data-analysis-2.html#cb30-1" tabindex="-1"></a><span class="fu">plot</span>(fit.brm_wrong,</span>
+<span id="cb30-2"><a href="bayesian-data-analysis-2.html#cb30-2" tabindex="-1"></a>     <span class="at">N =</span> <span class="dv">2</span>, </span>
+<span id="cb30-3"><a href="bayesian-data-analysis-2.html#cb30-3" tabindex="-1"></a>     <span class="at">ask =</span> F)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-16-1.png" width="1152" /></p>
+<div class="sourceCode" id="cb31"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb31-1"><a href="bayesian-data-analysis-2.html#cb31-1" tabindex="-1"></a>fit.brm_wrong <span class="sc">%&gt;%</span> </span>
+<span id="cb31-2"><a href="bayesian-data-analysis-2.html#cb31-2" tabindex="-1"></a>  <span class="fu">spread_draws</span>(b_Intercept) <span class="sc">%&gt;%</span> </span>
+<span id="cb31-3"><a href="bayesian-data-analysis-2.html#cb31-3" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb31-4"><a href="bayesian-data-analysis-2.html#cb31-4" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">chain =</span> <span class="fu">as.factor</span>(chain)) <span class="sc">%&gt;%</span> </span>
+<span id="cb31-5"><a href="bayesian-data-analysis-2.html#cb31-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> iteration,</span>
+<span id="cb31-6"><a href="bayesian-data-analysis-2.html#cb31-6" tabindex="-1"></a>             <span class="at">y =</span> b_intercept,</span>
+<span id="cb31-7"><a href="bayesian-data-analysis-2.html#cb31-7" tabindex="-1"></a>             <span class="at">group =</span> chain,</span>
+<span id="cb31-8"><a href="bayesian-data-analysis-2.html#cb31-8" tabindex="-1"></a>             <span class="at">color =</span> chain)) <span class="sc">+</span> </span>
+<span id="cb31-9"><a href="bayesian-data-analysis-2.html#cb31-9" tabindex="-1"></a>  <span class="fu">geom_line</span>() <span class="sc">+</span> </span>
+<span id="cb31-10"><a href="bayesian-data-analysis-2.html#cb31-10" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">direction =</span> <span class="sc">-</span><span class="dv">1</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-17-1.png" width="672" /></p>
 <p>Given that we have so little data in this case, we need to help the model a little bit by providing some slighlty more specific priors.</p>
-<div class="sourceCode" id="cb1284"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1284-1"><a href="bayesian-data-analysis-2.html#cb1284-1" tabindex="-1"></a>fit.brm_right <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">data =</span> df.data,</span>
-<span id="cb1284-2"><a href="bayesian-data-analysis-2.html#cb1284-2" tabindex="-1"></a>                    <span class="at">family =</span> gaussian,</span>
-<span id="cb1284-3"><a href="bayesian-data-analysis-2.html#cb1284-3" tabindex="-1"></a>                    <span class="at">formula =</span> y <span class="sc">~</span> <span class="dv">1</span>,</span>
-<span id="cb1284-4"><a href="bayesian-data-analysis-2.html#cb1284-4" tabindex="-1"></a>                    <span class="at">prior =</span> <span class="fu">c</span>(<span class="fu">prior</span>(<span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>), <span class="at">class =</span> Intercept), <span class="co"># more reasonable priors</span></span>
-<span id="cb1284-5"><a href="bayesian-data-analysis-2.html#cb1284-5" tabindex="-1"></a>                              <span class="fu">prior</span>(<span class="fu">cauchy</span>(<span class="dv">0</span>, <span class="dv">1</span>), <span class="at">class =</span> sigma)),</span>
-<span id="cb1284-6"><a href="bayesian-data-analysis-2.html#cb1284-6" tabindex="-1"></a>                    <span class="at">iter =</span> <span class="dv">4000</span>,</span>
-<span id="cb1284-7"><a href="bayesian-data-analysis-2.html#cb1284-7" tabindex="-1"></a>                    <span class="at">warmup =</span> <span class="dv">1000</span>,</span>
-<span id="cb1284-8"><a href="bayesian-data-analysis-2.html#cb1284-8" tabindex="-1"></a>                    <span class="at">chains =</span> <span class="dv">2</span>,</span>
-<span id="cb1284-9"><a href="bayesian-data-analysis-2.html#cb1284-9" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>,</span>
-<span id="cb1284-10"><a href="bayesian-data-analysis-2.html#cb1284-10" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_right&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb32"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb32-1"><a href="bayesian-data-analysis-2.html#cb32-1" tabindex="-1"></a>fit.brm_right <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">data =</span> df.data,</span>
+<span id="cb32-2"><a href="bayesian-data-analysis-2.html#cb32-2" tabindex="-1"></a>                    <span class="at">family =</span> gaussian,</span>
+<span id="cb32-3"><a href="bayesian-data-analysis-2.html#cb32-3" tabindex="-1"></a>                    <span class="at">formula =</span> y <span class="sc">~</span> <span class="dv">1</span>,</span>
+<span id="cb32-4"><a href="bayesian-data-analysis-2.html#cb32-4" tabindex="-1"></a>                    <span class="at">prior =</span> <span class="fu">c</span>(<span class="fu">prior</span>(<span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>), <span class="at">class =</span> Intercept), <span class="co"># more reasonable priors</span></span>
+<span id="cb32-5"><a href="bayesian-data-analysis-2.html#cb32-5" tabindex="-1"></a>                              <span class="fu">prior</span>(<span class="fu">cauchy</span>(<span class="dv">0</span>, <span class="dv">1</span>), <span class="at">class =</span> sigma)),</span>
+<span id="cb32-6"><a href="bayesian-data-analysis-2.html#cb32-6" tabindex="-1"></a>                    <span class="at">iter =</span> <span class="dv">4000</span>,</span>
+<span id="cb32-7"><a href="bayesian-data-analysis-2.html#cb32-7" tabindex="-1"></a>                    <span class="at">warmup =</span> <span class="dv">1000</span>,</span>
+<span id="cb32-8"><a href="bayesian-data-analysis-2.html#cb32-8" tabindex="-1"></a>                    <span class="at">chains =</span> <span class="dv">2</span>,</span>
+<span id="cb32-9"><a href="bayesian-data-analysis-2.html#cb32-9" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>,</span>
+<span id="cb32-10"><a href="bayesian-data-analysis-2.html#cb32-10" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_right&quot;</span>)</span></code></pre></div>
 <p>Let’s take a look at the posterior distributions of the model parameters:</p>
-<div class="sourceCode" id="cb1285"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1285-1"><a href="bayesian-data-analysis-2.html#cb1285-1" tabindex="-1"></a><span class="fu">summary</span>(fit.brm_right)</span></code></pre></div>
+<div class="sourceCode" id="cb33"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb33-1"><a href="bayesian-data-analysis-2.html#cb33-1" tabindex="-1"></a><span class="fu">summary</span>(fit.brm_right)</span></code></pre></div>
 <pre><code> Family: gaussian 
   Links: mu = identity; sigma = identity 
 Formula: y ~ 1 
@@ -1341,23 +1343,23 @@ <h6><span class="header-section-number">23.4.3.1.0.1</span> When things go wrong
 scale reduction factor on split chains (at convergence, Rhat = 1).</code></pre>
 <p>This looks much better. There is still quite a bit of uncertainty in our paremeter estimates, but it has reduced dramatically.</p>
 <p>Let’s visualize the trace plots:</p>
-<div class="sourceCode" id="cb1287"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1287-1"><a href="bayesian-data-analysis-2.html#cb1287-1" tabindex="-1"></a><span class="fu">plot</span>(fit.brm_right,</span>
-<span id="cb1287-2"><a href="bayesian-data-analysis-2.html#cb1287-2" tabindex="-1"></a>     <span class="at">N =</span> <span class="dv">2</span>, </span>
-<span id="cb1287-3"><a href="bayesian-data-analysis-2.html#cb1287-3" tabindex="-1"></a>     <span class="at">ask =</span> F)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-20-1.png" width="672" /></p>
-<div class="sourceCode" id="cb1288"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1288-1"><a href="bayesian-data-analysis-2.html#cb1288-1" tabindex="-1"></a>fit.brm_right <span class="sc">%&gt;%</span> </span>
-<span id="cb1288-2"><a href="bayesian-data-analysis-2.html#cb1288-2" tabindex="-1"></a>  <span class="fu">spread_draws</span>(b_Intercept, sigma) <span class="sc">%&gt;%</span> </span>
-<span id="cb1288-3"><a href="bayesian-data-analysis-2.html#cb1288-3" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1288-4"><a href="bayesian-data-analysis-2.html#cb1288-4" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">chain =</span> <span class="fu">as.factor</span>(chain)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1288-5"><a href="bayesian-data-analysis-2.html#cb1288-5" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">c</span>(b_intercept, sigma)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1288-6"><a href="bayesian-data-analysis-2.html#cb1288-6" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> iteration,</span>
-<span id="cb1288-7"><a href="bayesian-data-analysis-2.html#cb1288-7" tabindex="-1"></a>             <span class="at">y =</span> value,</span>
-<span id="cb1288-8"><a href="bayesian-data-analysis-2.html#cb1288-8" tabindex="-1"></a>             <span class="at">group =</span> chain,</span>
-<span id="cb1288-9"><a href="bayesian-data-analysis-2.html#cb1288-9" tabindex="-1"></a>             <span class="at">color =</span> chain)) <span class="sc">+</span> </span>
-<span id="cb1288-10"><a href="bayesian-data-analysis-2.html#cb1288-10" tabindex="-1"></a>  <span class="fu">geom_line</span>() <span class="sc">+</span> </span>
-<span id="cb1288-11"><a href="bayesian-data-analysis-2.html#cb1288-11" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="fu">vars</span>(name), <span class="at">ncol =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
-<span id="cb1288-12"><a href="bayesian-data-analysis-2.html#cb1288-12" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">direction =</span> <span class="sc">-</span><span class="dv">1</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-21-1.png" width="672" /></p>
+<div class="sourceCode" id="cb35"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb35-1"><a href="bayesian-data-analysis-2.html#cb35-1" tabindex="-1"></a><span class="fu">plot</span>(fit.brm_right,</span>
+<span id="cb35-2"><a href="bayesian-data-analysis-2.html#cb35-2" tabindex="-1"></a>     <span class="at">N =</span> <span class="dv">2</span>, </span>
+<span id="cb35-3"><a href="bayesian-data-analysis-2.html#cb35-3" tabindex="-1"></a>     <span class="at">ask =</span> F)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-20-1.png" width="672" /></p>
+<div class="sourceCode" id="cb36"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb36-1"><a href="bayesian-data-analysis-2.html#cb36-1" tabindex="-1"></a>fit.brm_right <span class="sc">%&gt;%</span> </span>
+<span id="cb36-2"><a href="bayesian-data-analysis-2.html#cb36-2" tabindex="-1"></a>  <span class="fu">spread_draws</span>(b_Intercept, sigma) <span class="sc">%&gt;%</span> </span>
+<span id="cb36-3"><a href="bayesian-data-analysis-2.html#cb36-3" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb36-4"><a href="bayesian-data-analysis-2.html#cb36-4" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">chain =</span> <span class="fu">as.factor</span>(chain)) <span class="sc">%&gt;%</span> </span>
+<span id="cb36-5"><a href="bayesian-data-analysis-2.html#cb36-5" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">c</span>(b_intercept, sigma)) <span class="sc">%&gt;%</span> </span>
+<span id="cb36-6"><a href="bayesian-data-analysis-2.html#cb36-6" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> iteration,</span>
+<span id="cb36-7"><a href="bayesian-data-analysis-2.html#cb36-7" tabindex="-1"></a>             <span class="at">y =</span> value,</span>
+<span id="cb36-8"><a href="bayesian-data-analysis-2.html#cb36-8" tabindex="-1"></a>             <span class="at">group =</span> chain,</span>
+<span id="cb36-9"><a href="bayesian-data-analysis-2.html#cb36-9" tabindex="-1"></a>             <span class="at">color =</span> chain)) <span class="sc">+</span> </span>
+<span id="cb36-10"><a href="bayesian-data-analysis-2.html#cb36-10" tabindex="-1"></a>  <span class="fu">geom_line</span>() <span class="sc">+</span> </span>
+<span id="cb36-11"><a href="bayesian-data-analysis-2.html#cb36-11" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="fu">vars</span>(name), <span class="at">ncol =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
+<span id="cb36-12"><a href="bayesian-data-analysis-2.html#cb36-12" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">direction =</span> <span class="sc">-</span><span class="dv">1</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-21-1.png" width="672" /></p>
 <p>Looking mostly good!</p>
 </div>
 </div>
@@ -1366,83 +1368,83 @@ <h4><span class="header-section-number">23.4.3.2</span> b) Visualize model predi
 <div id="posterior-predictive-check-1" class="section level5 hasAnchor" number="23.4.3.2.1">
 <h5><span class="header-section-number">23.4.3.2.1</span> Posterior predictive check<a href="bayesian-data-analysis-2.html#posterior-predictive-check-1" class="anchor-section" aria-label="Anchor link to header"></a></h5>
 <p>To check whether the model did a good job capturing the data, we can simulate what future data the Bayesian model predicts, now that it has learned from the data we feed into it.</p>
-<div class="sourceCode" id="cb1289"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1289-1"><a href="bayesian-data-analysis-2.html#cb1289-1" tabindex="-1"></a><span class="fu">pp_check</span>(fit.brm_poker, <span class="at">ndraws =</span> <span class="dv">100</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-22-1.png" width="672" /></p>
+<div class="sourceCode" id="cb37"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb37-1"><a href="bayesian-data-analysis-2.html#cb37-1" tabindex="-1"></a><span class="fu">pp_check</span>(fit.brm_poker, <span class="at">ndraws =</span> <span class="dv">100</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-22-1.png" width="672" /></p>
 <p>This looks good! The predicted shaped of the data based on samples from the posterior distribution looks very similar to the shape of the actual data.</p>
 <p>Let’s make a hypothetical outcome plot that shows what concrete data sets the model would predict. The <code>add_predicted_draws()</code> function from the “tidybayes” package is helpful for generating predictions from the posterior.</p>
-<div class="sourceCode" id="cb1290"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1290-1"><a href="bayesian-data-analysis-2.html#cb1290-1" tabindex="-1"></a>df.predictive_samples <span class="ot">=</span> df.poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1290-2"><a href="bayesian-data-analysis-2.html#cb1290-2" tabindex="-1"></a>  <span class="fu">add_predicted_draws</span>(<span class="at">newdata =</span> .,</span>
-<span id="cb1290-3"><a href="bayesian-data-analysis-2.html#cb1290-3" tabindex="-1"></a>                      <span class="at">object =</span> fit.brm_poker2,</span>
-<span id="cb1290-4"><a href="bayesian-data-analysis-2.html#cb1290-4" tabindex="-1"></a>                      <span class="at">ndraws =</span> <span class="dv">10</span>)</span>
-<span id="cb1290-5"><a href="bayesian-data-analysis-2.html#cb1290-5" tabindex="-1"></a></span>
-<span id="cb1290-6"><a href="bayesian-data-analysis-2.html#cb1290-6" tabindex="-1"></a>p <span class="ot">=</span> <span class="fu">ggplot</span>(<span class="at">data =</span> df.predictive_samples,</span>
-<span id="cb1290-7"><a href="bayesian-data-analysis-2.html#cb1290-7" tabindex="-1"></a>           <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> hand,</span>
-<span id="cb1290-8"><a href="bayesian-data-analysis-2.html#cb1290-8" tabindex="-1"></a>                         <span class="at">y =</span> .prediction,</span>
-<span id="cb1290-9"><a href="bayesian-data-analysis-2.html#cb1290-9" tabindex="-1"></a>                         <span class="at">fill =</span> hand,</span>
-<span id="cb1290-10"><a href="bayesian-data-analysis-2.html#cb1290-10" tabindex="-1"></a>                         <span class="at">group =</span> skill,</span>
-<span id="cb1290-11"><a href="bayesian-data-analysis-2.html#cb1290-11" tabindex="-1"></a>                         <span class="at">shape =</span> skill)) <span class="sc">+</span> </span>
-<span id="cb1290-12"><a href="bayesian-data-analysis-2.html#cb1290-12" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
-<span id="cb1290-13"><a href="bayesian-data-analysis-2.html#cb1290-13" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitterdodge</span>(<span class="at">dodge.width =</span> <span class="fl">0.5</span>,</span>
-<span id="cb1290-14"><a href="bayesian-data-analysis-2.html#cb1290-14" tabindex="-1"></a>                                             <span class="at">jitter.height =</span> <span class="dv">0</span>, </span>
-<span id="cb1290-15"><a href="bayesian-data-analysis-2.html#cb1290-15" tabindex="-1"></a>                                             <span class="at">jitter.width =</span> <span class="fl">0.2</span>)) <span class="sc">+</span> </span>
-<span id="cb1290-16"><a href="bayesian-data-analysis-2.html#cb1290-16" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>,</span>
-<span id="cb1290-17"><a href="bayesian-data-analysis-2.html#cb1290-17" tabindex="-1"></a>               <span class="at">position =</span> <span class="fu">position_dodge</span>(<span class="at">width =</span> <span class="fl">0.5</span>),</span>
-<span id="cb1290-18"><a href="bayesian-data-analysis-2.html#cb1290-18" tabindex="-1"></a>               <span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
-<span id="cb1290-19"><a href="bayesian-data-analysis-2.html#cb1290-19" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">&quot;final balance (in Euros)&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1290-20"><a href="bayesian-data-analysis-2.html#cb1290-20" tabindex="-1"></a>  <span class="fu">scale_shape_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="dv">21</span>, <span class="dv">22</span>)) <span class="sc">+</span> </span>
-<span id="cb1290-21"><a href="bayesian-data-analysis-2.html#cb1290-21" tabindex="-1"></a>  <span class="fu">guides</span>(<span class="at">fill =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">shape =</span> <span class="dv">21</span>)),</span>
-<span id="cb1290-22"><a href="bayesian-data-analysis-2.html#cb1290-22" tabindex="-1"></a>         <span class="at">shape =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">alpha =</span> <span class="dv">1</span>, <span class="at">fill =</span> <span class="st">&quot;black&quot;</span>))) <span class="sc">+</span> </span>
-<span id="cb1290-23"><a href="bayesian-data-analysis-2.html#cb1290-23" tabindex="-1"></a>  <span class="fu">transition_manual</span>(.draw)</span>
-<span id="cb1290-24"><a href="bayesian-data-analysis-2.html#cb1290-24" tabindex="-1"></a></span>
-<span id="cb1290-25"><a href="bayesian-data-analysis-2.html#cb1290-25" tabindex="-1"></a><span class="fu">animate</span>(p, <span class="at">nframes =</span> <span class="dv">120</span>, <span class="at">width =</span> <span class="dv">800</span>, <span class="at">height =</span> <span class="dv">600</span>, <span class="at">res =</span> <span class="dv">96</span>, <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span></code></pre></div>
-<pre><code>Warning: No renderer available. Please install the gifski, av, or magick
-package to create animated output</code></pre>
+<div class="sourceCode" id="cb38"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb38-1"><a href="bayesian-data-analysis-2.html#cb38-1" tabindex="-1"></a>df.predictive_samples <span class="ot">=</span> df.poker <span class="sc">%&gt;%</span> </span>
+<span id="cb38-2"><a href="bayesian-data-analysis-2.html#cb38-2" tabindex="-1"></a>  <span class="fu">add_predicted_draws</span>(<span class="at">newdata =</span> .,</span>
+<span id="cb38-3"><a href="bayesian-data-analysis-2.html#cb38-3" tabindex="-1"></a>                      <span class="at">object =</span> fit.brm_poker2,</span>
+<span id="cb38-4"><a href="bayesian-data-analysis-2.html#cb38-4" tabindex="-1"></a>                      <span class="at">ndraws =</span> <span class="dv">10</span>)</span>
+<span id="cb38-5"><a href="bayesian-data-analysis-2.html#cb38-5" tabindex="-1"></a></span>
+<span id="cb38-6"><a href="bayesian-data-analysis-2.html#cb38-6" tabindex="-1"></a>p <span class="ot">=</span> <span class="fu">ggplot</span>(<span class="at">data =</span> df.predictive_samples,</span>
+<span id="cb38-7"><a href="bayesian-data-analysis-2.html#cb38-7" tabindex="-1"></a>           <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> hand,</span>
+<span id="cb38-8"><a href="bayesian-data-analysis-2.html#cb38-8" tabindex="-1"></a>                         <span class="at">y =</span> .prediction,</span>
+<span id="cb38-9"><a href="bayesian-data-analysis-2.html#cb38-9" tabindex="-1"></a>                         <span class="at">fill =</span> hand,</span>
+<span id="cb38-10"><a href="bayesian-data-analysis-2.html#cb38-10" tabindex="-1"></a>                         <span class="at">group =</span> skill,</span>
+<span id="cb38-11"><a href="bayesian-data-analysis-2.html#cb38-11" tabindex="-1"></a>                         <span class="at">shape =</span> skill)) <span class="sc">+</span> </span>
+<span id="cb38-12"><a href="bayesian-data-analysis-2.html#cb38-12" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
+<span id="cb38-13"><a href="bayesian-data-analysis-2.html#cb38-13" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitterdodge</span>(<span class="at">dodge.width =</span> <span class="fl">0.5</span>,</span>
+<span id="cb38-14"><a href="bayesian-data-analysis-2.html#cb38-14" tabindex="-1"></a>                                             <span class="at">jitter.height =</span> <span class="dv">0</span>, </span>
+<span id="cb38-15"><a href="bayesian-data-analysis-2.html#cb38-15" tabindex="-1"></a>                                             <span class="at">jitter.width =</span> <span class="fl">0.2</span>)) <span class="sc">+</span> </span>
+<span id="cb38-16"><a href="bayesian-data-analysis-2.html#cb38-16" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>,</span>
+<span id="cb38-17"><a href="bayesian-data-analysis-2.html#cb38-17" tabindex="-1"></a>               <span class="at">position =</span> <span class="fu">position_dodge</span>(<span class="at">width =</span> <span class="fl">0.5</span>),</span>
+<span id="cb38-18"><a href="bayesian-data-analysis-2.html#cb38-18" tabindex="-1"></a>               <span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
+<span id="cb38-19"><a href="bayesian-data-analysis-2.html#cb38-19" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">&quot;final balance (in Euros)&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb38-20"><a href="bayesian-data-analysis-2.html#cb38-20" tabindex="-1"></a>  <span class="fu">scale_shape_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="dv">21</span>, <span class="dv">22</span>)) <span class="sc">+</span> </span>
+<span id="cb38-21"><a href="bayesian-data-analysis-2.html#cb38-21" tabindex="-1"></a>  <span class="fu">guides</span>(<span class="at">fill =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">shape =</span> <span class="dv">21</span>)),</span>
+<span id="cb38-22"><a href="bayesian-data-analysis-2.html#cb38-22" tabindex="-1"></a>         <span class="at">shape =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">alpha =</span> <span class="dv">1</span>, <span class="at">fill =</span> <span class="st">&quot;black&quot;</span>))) <span class="sc">+</span> </span>
+<span id="cb38-23"><a href="bayesian-data-analysis-2.html#cb38-23" tabindex="-1"></a>  <span class="fu">transition_manual</span>(.draw)</span>
+<span id="cb38-24"><a href="bayesian-data-analysis-2.html#cb38-24" tabindex="-1"></a></span>
+<span id="cb38-25"><a href="bayesian-data-analysis-2.html#cb38-25" tabindex="-1"></a><span class="fu">animate</span>(p, <span class="at">nframes =</span> <span class="dv">120</span>, <span class="at">width =</span> <span class="dv">800</span>, <span class="at">height =</span> <span class="dv">600</span>, <span class="at">res =</span> <span class="dv">96</span>, <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span></code></pre></div>
+<pre><code>nframes and fps adjusted to match transition</code></pre>
+<p><img src="psych252_files/figure-html/unnamed-chunk-23-1.gif" /><!-- --></p>
 </div>
 <div id="prior-predictive-check-1" class="section level5 hasAnchor" number="23.4.3.2.2">
 <h5><span class="header-section-number">23.4.3.2.2</span> Prior predictive check<a href="bayesian-data-analysis-2.html#prior-predictive-check-1" class="anchor-section" aria-label="Anchor link to header"></a></h5>
-<div class="sourceCode" id="cb1292"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1292-1"><a href="bayesian-data-analysis-2.html#cb1292-1" tabindex="-1"></a>fit.brm_poker_prior <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">0</span> <span class="sc">+</span> Intercept <span class="sc">+</span> hand <span class="sc">*</span> skill,</span>
-<span id="cb1292-2"><a href="bayesian-data-analysis-2.html#cb1292-2" tabindex="-1"></a>                          <span class="at">family =</span> <span class="st">&quot;gaussian&quot;</span>,</span>
-<span id="cb1292-3"><a href="bayesian-data-analysis-2.html#cb1292-3" tabindex="-1"></a>                          <span class="at">data =</span> df.poker,</span>
-<span id="cb1292-4"><a href="bayesian-data-analysis-2.html#cb1292-4" tabindex="-1"></a>                          <span class="at">prior =</span> <span class="fu">c</span>(<span class="fu">prior</span>(<span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>), <span class="at">class =</span> <span class="st">&quot;b&quot;</span>),</span>
-<span id="cb1292-5"><a href="bayesian-data-analysis-2.html#cb1292-5" tabindex="-1"></a>                                    <span class="fu">prior</span>(<span class="fu">student_t</span>(<span class="dv">3</span>, <span class="dv">0</span>, <span class="dv">10</span>), <span class="at">class =</span> <span class="st">&quot;sigma&quot;</span>)),</span>
-<span id="cb1292-6"><a href="bayesian-data-analysis-2.html#cb1292-6" tabindex="-1"></a>                          <span class="at">iter =</span> <span class="dv">4000</span>,</span>
-<span id="cb1292-7"><a href="bayesian-data-analysis-2.html#cb1292-7" tabindex="-1"></a>                          <span class="at">warmup =</span> <span class="dv">1000</span>,</span>
-<span id="cb1292-8"><a href="bayesian-data-analysis-2.html#cb1292-8" tabindex="-1"></a>                          <span class="at">chains =</span> <span class="dv">4</span>,</span>
-<span id="cb1292-9"><a href="bayesian-data-analysis-2.html#cb1292-9" tabindex="-1"></a>                          <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_prior&quot;</span>,</span>
-<span id="cb1292-10"><a href="bayesian-data-analysis-2.html#cb1292-10" tabindex="-1"></a>                          <span class="at">sample_prior =</span> <span class="st">&quot;only&quot;</span>,</span>
-<span id="cb1292-11"><a href="bayesian-data-analysis-2.html#cb1292-11" tabindex="-1"></a>                          <span class="at">seed =</span> <span class="dv">1</span>)</span>
-<span id="cb1292-12"><a href="bayesian-data-analysis-2.html#cb1292-12" tabindex="-1"></a></span>
-<span id="cb1292-13"><a href="bayesian-data-analysis-2.html#cb1292-13" tabindex="-1"></a><span class="co"># generate prior samples </span></span>
-<span id="cb1292-14"><a href="bayesian-data-analysis-2.html#cb1292-14" tabindex="-1"></a>df.prior_samples <span class="ot">=</span> df.poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1292-15"><a href="bayesian-data-analysis-2.html#cb1292-15" tabindex="-1"></a>  <span class="fu">add_predicted_draws</span>(<span class="at">newdata =</span> .,</span>
-<span id="cb1292-16"><a href="bayesian-data-analysis-2.html#cb1292-16" tabindex="-1"></a>                      <span class="at">object =</span> fit.brm_poker_prior,</span>
-<span id="cb1292-17"><a href="bayesian-data-analysis-2.html#cb1292-17" tabindex="-1"></a>                      <span class="at">ndraws =</span> <span class="dv">10</span>)</span>
-<span id="cb1292-18"><a href="bayesian-data-analysis-2.html#cb1292-18" tabindex="-1"></a></span>
-<span id="cb1292-19"><a href="bayesian-data-analysis-2.html#cb1292-19" tabindex="-1"></a><span class="co"># plot the results as an animation</span></span>
-<span id="cb1292-20"><a href="bayesian-data-analysis-2.html#cb1292-20" tabindex="-1"></a>p <span class="ot">=</span> <span class="fu">ggplot</span>(<span class="at">data =</span> df.prior_samples,</span>
-<span id="cb1292-21"><a href="bayesian-data-analysis-2.html#cb1292-21" tabindex="-1"></a>           <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> hand,</span>
-<span id="cb1292-22"><a href="bayesian-data-analysis-2.html#cb1292-22" tabindex="-1"></a>                         <span class="at">y =</span> .prediction,</span>
-<span id="cb1292-23"><a href="bayesian-data-analysis-2.html#cb1292-23" tabindex="-1"></a>                         <span class="at">fill =</span> hand,</span>
-<span id="cb1292-24"><a href="bayesian-data-analysis-2.html#cb1292-24" tabindex="-1"></a>                         <span class="at">group =</span> skill,</span>
-<span id="cb1292-25"><a href="bayesian-data-analysis-2.html#cb1292-25" tabindex="-1"></a>                         <span class="at">shape =</span> skill)) <span class="sc">+</span> </span>
-<span id="cb1292-26"><a href="bayesian-data-analysis-2.html#cb1292-26" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
-<span id="cb1292-27"><a href="bayesian-data-analysis-2.html#cb1292-27" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitterdodge</span>(<span class="at">dodge.width =</span> <span class="fl">0.5</span>,</span>
-<span id="cb1292-28"><a href="bayesian-data-analysis-2.html#cb1292-28" tabindex="-1"></a>                                             <span class="at">jitter.height =</span> <span class="dv">0</span>, </span>
-<span id="cb1292-29"><a href="bayesian-data-analysis-2.html#cb1292-29" tabindex="-1"></a>                                             <span class="at">jitter.width =</span> <span class="fl">0.2</span>)) <span class="sc">+</span> </span>
-<span id="cb1292-30"><a href="bayesian-data-analysis-2.html#cb1292-30" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>,</span>
-<span id="cb1292-31"><a href="bayesian-data-analysis-2.html#cb1292-31" tabindex="-1"></a>               <span class="at">position =</span> <span class="fu">position_dodge</span>(<span class="at">width =</span> <span class="fl">0.5</span>),</span>
-<span id="cb1292-32"><a href="bayesian-data-analysis-2.html#cb1292-32" tabindex="-1"></a>               <span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
-<span id="cb1292-33"><a href="bayesian-data-analysis-2.html#cb1292-33" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">&quot;final balance (in Euros)&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1292-34"><a href="bayesian-data-analysis-2.html#cb1292-34" tabindex="-1"></a>  <span class="fu">scale_shape_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="dv">21</span>, <span class="dv">22</span>)) <span class="sc">+</span> </span>
-<span id="cb1292-35"><a href="bayesian-data-analysis-2.html#cb1292-35" tabindex="-1"></a>  <span class="fu">guides</span>(<span class="at">fill =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">shape =</span> <span class="dv">21</span>,</span>
-<span id="cb1292-36"><a href="bayesian-data-analysis-2.html#cb1292-36" tabindex="-1"></a>                                                 <span class="at">fill =</span> RColorBrewer<span class="sc">::</span><span class="fu">brewer.pal</span>(<span class="dv">3</span>, <span class="st">&quot;Set1&quot;</span>))),</span>
-<span id="cb1292-37"><a href="bayesian-data-analysis-2.html#cb1292-37" tabindex="-1"></a>         <span class="at">shape =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">alpha =</span> <span class="dv">1</span>, <span class="at">fill =</span> <span class="st">&quot;black&quot;</span>))) <span class="sc">+</span> </span>
-<span id="cb1292-38"><a href="bayesian-data-analysis-2.html#cb1292-38" tabindex="-1"></a>  <span class="fu">transition_manual</span>(.draw)</span>
-<span id="cb1292-39"><a href="bayesian-data-analysis-2.html#cb1292-39" tabindex="-1"></a></span>
-<span id="cb1292-40"><a href="bayesian-data-analysis-2.html#cb1292-40" tabindex="-1"></a><span class="fu">animate</span>(p, <span class="at">nframes =</span> <span class="dv">120</span>, <span class="at">width =</span> <span class="dv">800</span>, <span class="at">height =</span> <span class="dv">600</span>, <span class="at">res =</span> <span class="dv">96</span>, <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span></code></pre></div>
-<pre><code>Warning: No renderer available. Please install the gifski, av, or magick
-package to create animated output</code></pre>
-<div class="sourceCode" id="cb1294"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1294-1"><a href="bayesian-data-analysis-2.html#cb1294-1" tabindex="-1"></a><span class="co"># anim_save(&quot;poker_prior_predictive.gif&quot;)</span></span></code></pre></div>
+<div class="sourceCode" id="cb40"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb40-1"><a href="bayesian-data-analysis-2.html#cb40-1" tabindex="-1"></a>fit.brm_poker_prior <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">0</span> <span class="sc">+</span> Intercept <span class="sc">+</span> hand <span class="sc">*</span> skill,</span>
+<span id="cb40-2"><a href="bayesian-data-analysis-2.html#cb40-2" tabindex="-1"></a>                          <span class="at">family =</span> <span class="st">&quot;gaussian&quot;</span>,</span>
+<span id="cb40-3"><a href="bayesian-data-analysis-2.html#cb40-3" tabindex="-1"></a>                          <span class="at">data =</span> df.poker,</span>
+<span id="cb40-4"><a href="bayesian-data-analysis-2.html#cb40-4" tabindex="-1"></a>                          <span class="at">prior =</span> <span class="fu">c</span>(<span class="fu">prior</span>(<span class="fu">normal</span>(<span class="dv">0</span>, <span class="dv">10</span>), <span class="at">class =</span> <span class="st">&quot;b&quot;</span>),</span>
+<span id="cb40-5"><a href="bayesian-data-analysis-2.html#cb40-5" tabindex="-1"></a>                                    <span class="fu">prior</span>(<span class="fu">student_t</span>(<span class="dv">3</span>, <span class="dv">0</span>, <span class="dv">10</span>), <span class="at">class =</span> <span class="st">&quot;sigma&quot;</span>)),</span>
+<span id="cb40-6"><a href="bayesian-data-analysis-2.html#cb40-6" tabindex="-1"></a>                          <span class="at">iter =</span> <span class="dv">4000</span>,</span>
+<span id="cb40-7"><a href="bayesian-data-analysis-2.html#cb40-7" tabindex="-1"></a>                          <span class="at">warmup =</span> <span class="dv">1000</span>,</span>
+<span id="cb40-8"><a href="bayesian-data-analysis-2.html#cb40-8" tabindex="-1"></a>                          <span class="at">chains =</span> <span class="dv">4</span>,</span>
+<span id="cb40-9"><a href="bayesian-data-analysis-2.html#cb40-9" tabindex="-1"></a>                          <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_prior&quot;</span>,</span>
+<span id="cb40-10"><a href="bayesian-data-analysis-2.html#cb40-10" tabindex="-1"></a>                          <span class="at">sample_prior =</span> <span class="st">&quot;only&quot;</span>,</span>
+<span id="cb40-11"><a href="bayesian-data-analysis-2.html#cb40-11" tabindex="-1"></a>                          <span class="at">seed =</span> <span class="dv">1</span>)</span>
+<span id="cb40-12"><a href="bayesian-data-analysis-2.html#cb40-12" tabindex="-1"></a></span>
+<span id="cb40-13"><a href="bayesian-data-analysis-2.html#cb40-13" tabindex="-1"></a><span class="co"># generate prior samples </span></span>
+<span id="cb40-14"><a href="bayesian-data-analysis-2.html#cb40-14" tabindex="-1"></a>df.prior_samples <span class="ot">=</span> df.poker <span class="sc">%&gt;%</span> </span>
+<span id="cb40-15"><a href="bayesian-data-analysis-2.html#cb40-15" tabindex="-1"></a>  <span class="fu">add_predicted_draws</span>(<span class="at">newdata =</span> .,</span>
+<span id="cb40-16"><a href="bayesian-data-analysis-2.html#cb40-16" tabindex="-1"></a>                      <span class="at">object =</span> fit.brm_poker_prior,</span>
+<span id="cb40-17"><a href="bayesian-data-analysis-2.html#cb40-17" tabindex="-1"></a>                      <span class="at">ndraws =</span> <span class="dv">10</span>)</span>
+<span id="cb40-18"><a href="bayesian-data-analysis-2.html#cb40-18" tabindex="-1"></a></span>
+<span id="cb40-19"><a href="bayesian-data-analysis-2.html#cb40-19" tabindex="-1"></a><span class="co"># plot the results as an animation</span></span>
+<span id="cb40-20"><a href="bayesian-data-analysis-2.html#cb40-20" tabindex="-1"></a>p <span class="ot">=</span> <span class="fu">ggplot</span>(<span class="at">data =</span> df.prior_samples,</span>
+<span id="cb40-21"><a href="bayesian-data-analysis-2.html#cb40-21" tabindex="-1"></a>           <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> hand,</span>
+<span id="cb40-22"><a href="bayesian-data-analysis-2.html#cb40-22" tabindex="-1"></a>                         <span class="at">y =</span> .prediction,</span>
+<span id="cb40-23"><a href="bayesian-data-analysis-2.html#cb40-23" tabindex="-1"></a>                         <span class="at">fill =</span> hand,</span>
+<span id="cb40-24"><a href="bayesian-data-analysis-2.html#cb40-24" tabindex="-1"></a>                         <span class="at">group =</span> skill,</span>
+<span id="cb40-25"><a href="bayesian-data-analysis-2.html#cb40-25" tabindex="-1"></a>                         <span class="at">shape =</span> skill)) <span class="sc">+</span> </span>
+<span id="cb40-26"><a href="bayesian-data-analysis-2.html#cb40-26" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
+<span id="cb40-27"><a href="bayesian-data-analysis-2.html#cb40-27" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitterdodge</span>(<span class="at">dodge.width =</span> <span class="fl">0.5</span>,</span>
+<span id="cb40-28"><a href="bayesian-data-analysis-2.html#cb40-28" tabindex="-1"></a>                                             <span class="at">jitter.height =</span> <span class="dv">0</span>, </span>
+<span id="cb40-29"><a href="bayesian-data-analysis-2.html#cb40-29" tabindex="-1"></a>                                             <span class="at">jitter.width =</span> <span class="fl">0.2</span>)) <span class="sc">+</span> </span>
+<span id="cb40-30"><a href="bayesian-data-analysis-2.html#cb40-30" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>,</span>
+<span id="cb40-31"><a href="bayesian-data-analysis-2.html#cb40-31" tabindex="-1"></a>               <span class="at">position =</span> <span class="fu">position_dodge</span>(<span class="at">width =</span> <span class="fl">0.5</span>),</span>
+<span id="cb40-32"><a href="bayesian-data-analysis-2.html#cb40-32" tabindex="-1"></a>               <span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span> </span>
+<span id="cb40-33"><a href="bayesian-data-analysis-2.html#cb40-33" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">y =</span> <span class="st">&quot;final balance (in Euros)&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb40-34"><a href="bayesian-data-analysis-2.html#cb40-34" tabindex="-1"></a>  <span class="fu">scale_shape_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="dv">21</span>, <span class="dv">22</span>)) <span class="sc">+</span> </span>
+<span id="cb40-35"><a href="bayesian-data-analysis-2.html#cb40-35" tabindex="-1"></a>  <span class="fu">guides</span>(<span class="at">fill =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">shape =</span> <span class="dv">21</span>,</span>
+<span id="cb40-36"><a href="bayesian-data-analysis-2.html#cb40-36" tabindex="-1"></a>                                                 <span class="at">fill =</span> RColorBrewer<span class="sc">::</span><span class="fu">brewer.pal</span>(<span class="dv">3</span>, <span class="st">&quot;Set1&quot;</span>))),</span>
+<span id="cb40-37"><a href="bayesian-data-analysis-2.html#cb40-37" tabindex="-1"></a>         <span class="at">shape =</span> <span class="fu">guide_legend</span>(<span class="at">override.aes =</span> <span class="fu">list</span>(<span class="at">alpha =</span> <span class="dv">1</span>, <span class="at">fill =</span> <span class="st">&quot;black&quot;</span>))) <span class="sc">+</span> </span>
+<span id="cb40-38"><a href="bayesian-data-analysis-2.html#cb40-38" tabindex="-1"></a>  <span class="fu">transition_manual</span>(.draw)</span>
+<span id="cb40-39"><a href="bayesian-data-analysis-2.html#cb40-39" tabindex="-1"></a></span>
+<span id="cb40-40"><a href="bayesian-data-analysis-2.html#cb40-40" tabindex="-1"></a><span class="fu">animate</span>(p, <span class="at">nframes =</span> <span class="dv">120</span>, <span class="at">width =</span> <span class="dv">800</span>, <span class="at">height =</span> <span class="dv">600</span>, <span class="at">res =</span> <span class="dv">96</span>, <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span></code></pre></div>
+<pre><code>nframes and fps adjusted to match transition</code></pre>
+<div class="sourceCode" id="cb42"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb42-1"><a href="bayesian-data-analysis-2.html#cb42-1" tabindex="-1"></a><span class="co"># anim_save(&quot;poker_prior_predictive.gif&quot;)</span></span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-24-1.gif" /><!-- --></p>
 </div>
 </div>
 </div>
@@ -1451,34 +1453,34 @@ <h3><span class="header-section-number">23.4.4</span> 4. Interpret the model par
 <div id="visualize-the-posteriors-1" class="section level4 hasAnchor" number="23.4.4.1">
 <h4><span class="header-section-number">23.4.4.1</span> Visualize the posteriors<a href="bayesian-data-analysis-2.html#visualize-the-posteriors-1" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>Let’s visualize what the posterior for the different parameters looks like. We use the <code>stat_halfeye()</code> function from the “tidybayes” package to do so:</p>
-<div class="sourceCode" id="cb1295"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1295-1"><a href="bayesian-data-analysis-2.html#cb1295-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1295-2"><a href="bayesian-data-analysis-2.html#cb1295-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span></span>
-<span id="cb1295-3"><a href="bayesian-data-analysis-2.html#cb1295-3" tabindex="-1"></a>  <span class="fu">select</span>(<span class="fu">starts_with</span>(<span class="st">&quot;b_&quot;</span>), sigma) <span class="sc">%&gt;%</span></span>
-<span id="cb1295-4"><a href="bayesian-data-analysis-2.html#cb1295-4" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">everything</span>(),</span>
-<span id="cb1295-5"><a href="bayesian-data-analysis-2.html#cb1295-5" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;variable&quot;</span>,</span>
-<span id="cb1295-6"><a href="bayesian-data-analysis-2.html#cb1295-6" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1295-7"><a href="bayesian-data-analysis-2.html#cb1295-7" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1295-8"><a href="bayesian-data-analysis-2.html#cb1295-8" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> <span class="fu">fct_rev</span>(variable),</span>
-<span id="cb1295-9"><a href="bayesian-data-analysis-2.html#cb1295-9" tabindex="-1"></a>                       <span class="at">x =</span> value)) <span class="sc">+</span></span>
-<span id="cb1295-10"><a href="bayesian-data-analysis-2.html#cb1295-10" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1295-11"><a href="bayesian-data-analysis-2.html#cb1295-11" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">axis.title.y =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-25-1.png" width="672" /></p>
+<div class="sourceCode" id="cb43"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb43-1"><a href="bayesian-data-analysis-2.html#cb43-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb43-2"><a href="bayesian-data-analysis-2.html#cb43-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span></span>
+<span id="cb43-3"><a href="bayesian-data-analysis-2.html#cb43-3" tabindex="-1"></a>  <span class="fu">select</span>(<span class="fu">starts_with</span>(<span class="st">&quot;b_&quot;</span>), sigma) <span class="sc">%&gt;%</span></span>
+<span id="cb43-4"><a href="bayesian-data-analysis-2.html#cb43-4" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">everything</span>(),</span>
+<span id="cb43-5"><a href="bayesian-data-analysis-2.html#cb43-5" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;variable&quot;</span>,</span>
+<span id="cb43-6"><a href="bayesian-data-analysis-2.html#cb43-6" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb43-7"><a href="bayesian-data-analysis-2.html#cb43-7" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb43-8"><a href="bayesian-data-analysis-2.html#cb43-8" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> <span class="fu">fct_rev</span>(variable),</span>
+<span id="cb43-9"><a href="bayesian-data-analysis-2.html#cb43-9" tabindex="-1"></a>                       <span class="at">x =</span> value)) <span class="sc">+</span></span>
+<span id="cb43-10"><a href="bayesian-data-analysis-2.html#cb43-10" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb43-11"><a href="bayesian-data-analysis-2.html#cb43-11" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">axis.title.y =</span> <span class="fu">element_blank</span>())</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-25-1.png" width="672" /></p>
 </div>
 <div id="compute-highest-density-intervals" class="section level4 hasAnchor" number="23.4.4.2">
 <h4><span class="header-section-number">23.4.4.2</span> Compute highest density intervals<a href="bayesian-data-analysis-2.html#compute-highest-density-intervals" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>To compute the MAP (maximum a posteriori probability) estimate and highest density interval, we use the <code>mean_hdi()</code> function that comes with the “tidybayes” package.</p>
-<div class="sourceCode" id="cb1296"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1296-1"><a href="bayesian-data-analysis-2.html#cb1296-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1296-2"><a href="bayesian-data-analysis-2.html#cb1296-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span></span>
-<span id="cb1296-3"><a href="bayesian-data-analysis-2.html#cb1296-3" tabindex="-1"></a>  <span class="fu">select</span>(<span class="fu">starts_with</span>(<span class="st">&quot;b_&quot;</span>), sigma) <span class="sc">%&gt;%</span> </span>
-<span id="cb1296-4"><a href="bayesian-data-analysis-2.html#cb1296-4" tabindex="-1"></a>  <span class="fu">mean_hdi</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1296-5"><a href="bayesian-data-analysis-2.html#cb1296-5" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(.width<span class="sc">:</span>.interval),</span>
-<span id="cb1296-6"><a href="bayesian-data-analysis-2.html#cb1296-6" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
-<span id="cb1296-7"><a href="bayesian-data-analysis-2.html#cb1296-7" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1296-8"><a href="bayesian-data-analysis-2.html#cb1296-8" tabindex="-1"></a>  <span class="fu">select</span>(index, value) <span class="sc">%&gt;%</span> </span>
-<span id="cb1296-9"><a href="bayesian-data-analysis-2.html#cb1296-9" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">ifelse</span>(<span class="fu">str_detect</span>(index, <span class="fu">fixed</span>(<span class="st">&quot;.&quot;</span>)), index, <span class="fu">str_c</span>(index, <span class="st">&quot;.mean&quot;</span>))) <span class="sc">%&gt;%</span> </span>
-<span id="cb1296-10"><a href="bayesian-data-analysis-2.html#cb1296-10" tabindex="-1"></a>  <span class="fu">separate</span>(index, <span class="at">into =</span> <span class="fu">c</span>(<span class="st">&quot;parameter&quot;</span>, <span class="st">&quot;type&quot;</span>), <span class="at">sep =</span> <span class="st">&quot;</span><span class="sc">\\</span><span class="st">.&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1296-11"><a href="bayesian-data-analysis-2.html#cb1296-11" tabindex="-1"></a>  <span class="fu">pivot_wider</span>(<span class="at">names_from =</span> type, </span>
-<span id="cb1296-12"><a href="bayesian-data-analysis-2.html#cb1296-12" tabindex="-1"></a>              <span class="at">values_from =</span> value)</span></code></pre></div>
+<div class="sourceCode" id="cb44"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb44-1"><a href="bayesian-data-analysis-2.html#cb44-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb44-2"><a href="bayesian-data-analysis-2.html#cb44-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span></span>
+<span id="cb44-3"><a href="bayesian-data-analysis-2.html#cb44-3" tabindex="-1"></a>  <span class="fu">select</span>(<span class="fu">starts_with</span>(<span class="st">&quot;b_&quot;</span>), sigma) <span class="sc">%&gt;%</span> </span>
+<span id="cb44-4"><a href="bayesian-data-analysis-2.html#cb44-4" tabindex="-1"></a>  <span class="fu">mean_hdi</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb44-5"><a href="bayesian-data-analysis-2.html#cb44-5" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="sc">-</span><span class="fu">c</span>(.width<span class="sc">:</span>.interval),</span>
+<span id="cb44-6"><a href="bayesian-data-analysis-2.html#cb44-6" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;index&quot;</span>,</span>
+<span id="cb44-7"><a href="bayesian-data-analysis-2.html#cb44-7" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb44-8"><a href="bayesian-data-analysis-2.html#cb44-8" tabindex="-1"></a>  <span class="fu">select</span>(index, value) <span class="sc">%&gt;%</span> </span>
+<span id="cb44-9"><a href="bayesian-data-analysis-2.html#cb44-9" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">index =</span> <span class="fu">ifelse</span>(<span class="fu">str_detect</span>(index, <span class="fu">fixed</span>(<span class="st">&quot;.&quot;</span>)), index, <span class="fu">str_c</span>(index, <span class="st">&quot;.mean&quot;</span>))) <span class="sc">%&gt;%</span> </span>
+<span id="cb44-10"><a href="bayesian-data-analysis-2.html#cb44-10" tabindex="-1"></a>  <span class="fu">separate</span>(index, <span class="at">into =</span> <span class="fu">c</span>(<span class="st">&quot;parameter&quot;</span>, <span class="st">&quot;type&quot;</span>), <span class="at">sep =</span> <span class="st">&quot;</span><span class="sc">\\</span><span class="st">.&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb44-11"><a href="bayesian-data-analysis-2.html#cb44-11" tabindex="-1"></a>  <span class="fu">pivot_wider</span>(<span class="at">names_from =</span> type, </span>
+<span id="cb44-12"><a href="bayesian-data-analysis-2.html#cb44-12" tabindex="-1"></a>              <span class="at">values_from =</span> value)</span></code></pre></div>
 <pre><code># A tibble: 4 × 4
   parameter      mean lower upper
   &lt;chr&gt;         &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt;
@@ -1494,23 +1496,23 @@ <h3><span class="header-section-number">23.4.5</span> 5. Test specific hypothese
 <h4><span class="header-section-number">23.4.5.1</span> with <code>hypothesis()</code><a href="bayesian-data-analysis-2.html#with-hypothesis" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>One key advantage of Bayesian over frequentist analysis is that we can test hypothesis in a very flexible manner by directly probing our posterior samples in different ways.</p>
 <p>We may ask, for example, what the probability is that the parameter for the difference between a bad hand and a neutral hand (<code>b_handneutral</code>) is greater than 0. Let’s plot the posterior distribution together with the criterion:</p>
-<div class="sourceCode" id="cb1298"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1298-1"><a href="bayesian-data-analysis-2.html#cb1298-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1298-2"><a href="bayesian-data-analysis-2.html#cb1298-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1298-3"><a href="bayesian-data-analysis-2.html#cb1298-3" tabindex="-1"></a>  <span class="fu">select</span>(b_handneutral) <span class="sc">%&gt;%</span> </span>
-<span id="cb1298-4"><a href="bayesian-data-analysis-2.html#cb1298-4" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">everything</span>(),</span>
-<span id="cb1298-5"><a href="bayesian-data-analysis-2.html#cb1298-5" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;variable&quot;</span>,</span>
-<span id="cb1298-6"><a href="bayesian-data-analysis-2.html#cb1298-6" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1298-7"><a href="bayesian-data-analysis-2.html#cb1298-7" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1298-8"><a href="bayesian-data-analysis-2.html#cb1298-8" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> variable, <span class="at">x =</span> value)) <span class="sc">+</span></span>
-<span id="cb1298-9"><a href="bayesian-data-analysis-2.html#cb1298-9" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1298-10"><a href="bayesian-data-analysis-2.html#cb1298-10" tabindex="-1"></a>  <span class="fu">geom_vline</span>(<span class="at">xintercept =</span> <span class="dv">0</span>,</span>
-<span id="cb1298-11"><a href="bayesian-data-analysis-2.html#cb1298-11" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">&quot;red&quot;</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-27-1.png" width="672" /></p>
+<div class="sourceCode" id="cb46"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb46-1"><a href="bayesian-data-analysis-2.html#cb46-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb46-2"><a href="bayesian-data-analysis-2.html#cb46-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb46-3"><a href="bayesian-data-analysis-2.html#cb46-3" tabindex="-1"></a>  <span class="fu">select</span>(b_handneutral) <span class="sc">%&gt;%</span> </span>
+<span id="cb46-4"><a href="bayesian-data-analysis-2.html#cb46-4" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">everything</span>(),</span>
+<span id="cb46-5"><a href="bayesian-data-analysis-2.html#cb46-5" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;variable&quot;</span>,</span>
+<span id="cb46-6"><a href="bayesian-data-analysis-2.html#cb46-6" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb46-7"><a href="bayesian-data-analysis-2.html#cb46-7" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb46-8"><a href="bayesian-data-analysis-2.html#cb46-8" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> variable, <span class="at">x =</span> value)) <span class="sc">+</span></span>
+<span id="cb46-9"><a href="bayesian-data-analysis-2.html#cb46-9" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb46-10"><a href="bayesian-data-analysis-2.html#cb46-10" tabindex="-1"></a>  <span class="fu">geom_vline</span>(<span class="at">xintercept =</span> <span class="dv">0</span>,</span>
+<span id="cb46-11"><a href="bayesian-data-analysis-2.html#cb46-11" tabindex="-1"></a>             <span class="at">color =</span> <span class="st">&quot;red&quot;</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-27-1.png" width="672" /></p>
 <p>We see that the posterior is definitely greater than 0.</p>
 <p>We can ask many different kinds of questions about the data by doing basic arithmetic on our posterior samples. The <code>hypothesis()</code> function makes this even easier. Here are some examples:</p>
-<div class="sourceCode" id="cb1299"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1299-1"><a href="bayesian-data-analysis-2.html#cb1299-1" tabindex="-1"></a><span class="co"># the probability that the posterior for handneutral is less than 0</span></span>
-<span id="cb1299-2"><a href="bayesian-data-analysis-2.html#cb1299-2" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
-<span id="cb1299-3"><a href="bayesian-data-analysis-2.html#cb1299-3" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;handneutral &lt; 0&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb47"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb47-1"><a href="bayesian-data-analysis-2.html#cb47-1" tabindex="-1"></a><span class="co"># the probability that the posterior for handneutral is less than 0</span></span>
+<span id="cb47-2"><a href="bayesian-data-analysis-2.html#cb47-2" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
+<span id="cb47-3"><a href="bayesian-data-analysis-2.html#cb47-3" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;handneutral &lt; 0&quot;</span>)</span></code></pre></div>
 <pre><code>Hypothesis Tests for class b:
          Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
 1 (handneutral) &lt; 0     4.41       0.6      3.4     5.37          0         0
@@ -1521,14 +1523,14 @@ <h4><span class="header-section-number">23.4.5.1</span> with <code>hypothesis()<
 &#39;*&#39;: For one-sided hypotheses, the posterior probability exceeds 95%;
 for two-sided hypotheses, the value tested against lies outside the 95%-CI.
 Posterior probabilities of point hypotheses assume equal prior probabilities.</code></pre>
-<div class="sourceCode" id="cb1301"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1301-1"><a href="bayesian-data-analysis-2.html#cb1301-1" tabindex="-1"></a><span class="co"># the probability that the posterior for handneutral is greater than 4</span></span>
-<span id="cb1301-2"><a href="bayesian-data-analysis-2.html#cb1301-2" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
-<span id="cb1301-3"><a href="bayesian-data-analysis-2.html#cb1301-3" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;handneutral &gt; 4&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1301-4"><a href="bayesian-data-analysis-2.html#cb1301-4" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-29-1.png" width="672" /></p>
-<div class="sourceCode" id="cb1302"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1302-1"><a href="bayesian-data-analysis-2.html#cb1302-1" tabindex="-1"></a><span class="co"># the probability that good hands make twice as much as bad hands</span></span>
-<span id="cb1302-2"><a href="bayesian-data-analysis-2.html#cb1302-2" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
-<span id="cb1302-3"><a href="bayesian-data-analysis-2.html#cb1302-3" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;Intercept + handgood &gt; 2 * Intercept&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb49"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb49-1"><a href="bayesian-data-analysis-2.html#cb49-1" tabindex="-1"></a><span class="co"># the probability that the posterior for handneutral is greater than 4</span></span>
+<span id="cb49-2"><a href="bayesian-data-analysis-2.html#cb49-2" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
+<span id="cb49-3"><a href="bayesian-data-analysis-2.html#cb49-3" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;handneutral &gt; 4&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb49-4"><a href="bayesian-data-analysis-2.html#cb49-4" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-29-1.png" width="672" /></p>
+<div class="sourceCode" id="cb50"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb50-1"><a href="bayesian-data-analysis-2.html#cb50-1" tabindex="-1"></a><span class="co"># the probability that good hands make twice as much as bad hands</span></span>
+<span id="cb50-2"><a href="bayesian-data-analysis-2.html#cb50-2" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
+<span id="cb50-3"><a href="bayesian-data-analysis-2.html#cb50-3" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;Intercept + handgood &gt; 2 * Intercept&quot;</span>)</span></code></pre></div>
 <pre><code>Hypothesis Tests for class b:
                 Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio
 1 (Intercept+handgo... &gt; 0     1.15      0.96     -0.4     2.73       7.93
@@ -1540,13 +1542,13 @@ <h4><span class="header-section-number">23.4.5.1</span> with <code>hypothesis()<
 for two-sided hypotheses, the value tested against lies outside the 95%-CI.
 Posterior probabilities of point hypotheses assume equal prior probabilities.</code></pre>
 <p>We can also make a plot of what the posterior distribution of the hypothesis looks like:</p>
-<div class="sourceCode" id="cb1304"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1304-1"><a href="bayesian-data-analysis-2.html#cb1304-1" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
-<span id="cb1304-2"><a href="bayesian-data-analysis-2.html#cb1304-2" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;Intercept + handgood &gt; 2 * Intercept&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1304-3"><a href="bayesian-data-analysis-2.html#cb1304-3" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-31-1.png" width="672" /></p>
-<div class="sourceCode" id="cb1305"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1305-1"><a href="bayesian-data-analysis-2.html#cb1305-1" tabindex="-1"></a><span class="co"># the probability that neutral hands make less than the average of bad and good hands</span></span>
-<span id="cb1305-2"><a href="bayesian-data-analysis-2.html#cb1305-2" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
-<span id="cb1305-3"><a href="bayesian-data-analysis-2.html#cb1305-3" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;Intercept + handneutral &lt; (Intercept + Intercept + handgood) / 2&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb52"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb52-1"><a href="bayesian-data-analysis-2.html#cb52-1" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
+<span id="cb52-2"><a href="bayesian-data-analysis-2.html#cb52-2" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;Intercept + handgood &gt; 2 * Intercept&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb52-3"><a href="bayesian-data-analysis-2.html#cb52-3" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-31-1.png" width="672" /></p>
+<div class="sourceCode" id="cb53"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb53-1"><a href="bayesian-data-analysis-2.html#cb53-1" tabindex="-1"></a><span class="co"># the probability that neutral hands make less than the average of bad and good hands</span></span>
+<span id="cb53-2"><a href="bayesian-data-analysis-2.html#cb53-2" tabindex="-1"></a><span class="fu">hypothesis</span>(fit.brm_poker,</span>
+<span id="cb53-3"><a href="bayesian-data-analysis-2.html#cb53-3" tabindex="-1"></a>           <span class="at">hypothesis =</span> <span class="st">&quot;Intercept + handneutral &lt; (Intercept + Intercept + handgood) / 2&quot;</span>)</span></code></pre></div>
 <pre><code>Hypothesis Tests for class b:
                 Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio
 1 (Intercept+handne... &lt; 0     0.86      0.52     0.01     1.72       0.05
@@ -1558,16 +1560,16 @@ <h4><span class="header-section-number">23.4.5.1</span> with <code>hypothesis()<
 for two-sided hypotheses, the value tested against lies outside the 95%-CI.
 Posterior probabilities of point hypotheses assume equal prior probabilities.</code></pre>
 <p>Let’s double check one example, and calculate the result directly based on the posterior samples:</p>
-<div class="sourceCode" id="cb1307"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1307-1"><a href="bayesian-data-analysis-2.html#cb1307-1" tabindex="-1"></a>df.hypothesis <span class="ot">=</span> fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1307-2"><a href="bayesian-data-analysis-2.html#cb1307-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1307-3"><a href="bayesian-data-analysis-2.html#cb1307-3" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1307-4"><a href="bayesian-data-analysis-2.html#cb1307-4" tabindex="-1"></a>  <span class="fu">select</span>(<span class="fu">starts_with</span>(<span class="st">&quot;b_&quot;</span>)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1307-5"><a href="bayesian-data-analysis-2.html#cb1307-5" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">neutral =</span> b_intercept <span class="sc">+</span> b_handneutral,</span>
-<span id="cb1307-6"><a href="bayesian-data-analysis-2.html#cb1307-6" tabindex="-1"></a>         <span class="at">bad_good_average =</span> (b_intercept <span class="sc">+</span> b_intercept <span class="sc">+</span> b_handgood)<span class="sc">/</span><span class="dv">2</span>,</span>
-<span id="cb1307-7"><a href="bayesian-data-analysis-2.html#cb1307-7" tabindex="-1"></a>         <span class="at">hypothesis =</span> neutral <span class="sc">&lt;</span> bad_good_average)</span></code></pre></div>
+<div class="sourceCode" id="cb55"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb55-1"><a href="bayesian-data-analysis-2.html#cb55-1" tabindex="-1"></a>df.hypothesis <span class="ot">=</span> fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb55-2"><a href="bayesian-data-analysis-2.html#cb55-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb55-3"><a href="bayesian-data-analysis-2.html#cb55-3" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb55-4"><a href="bayesian-data-analysis-2.html#cb55-4" tabindex="-1"></a>  <span class="fu">select</span>(<span class="fu">starts_with</span>(<span class="st">&quot;b_&quot;</span>)) <span class="sc">%&gt;%</span> </span>
+<span id="cb55-5"><a href="bayesian-data-analysis-2.html#cb55-5" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">neutral =</span> b_intercept <span class="sc">+</span> b_handneutral,</span>
+<span id="cb55-6"><a href="bayesian-data-analysis-2.html#cb55-6" tabindex="-1"></a>         <span class="at">bad_good_average =</span> (b_intercept <span class="sc">+</span> b_intercept <span class="sc">+</span> b_handgood)<span class="sc">/</span><span class="dv">2</span>,</span>
+<span id="cb55-7"><a href="bayesian-data-analysis-2.html#cb55-7" tabindex="-1"></a>         <span class="at">hypothesis =</span> neutral <span class="sc">&lt;</span> bad_good_average)</span></code></pre></div>
 <pre><code>Warning: Dropping &#39;draws_df&#39; class as required metadata was removed.</code></pre>
-<div class="sourceCode" id="cb1309"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1309-1"><a href="bayesian-data-analysis-2.html#cb1309-1" tabindex="-1"></a>df.hypothesis <span class="sc">%&gt;%</span> </span>
-<span id="cb1309-2"><a href="bayesian-data-analysis-2.html#cb1309-2" tabindex="-1"></a>  <span class="fu">summarize</span>(<span class="at">p =</span> <span class="fu">sum</span>(hypothesis)<span class="sc">/</span><span class="fu">n</span>())</span></code></pre></div>
+<div class="sourceCode" id="cb57"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb57-1"><a href="bayesian-data-analysis-2.html#cb57-1" tabindex="-1"></a>df.hypothesis <span class="sc">%&gt;%</span> </span>
+<span id="cb57-2"><a href="bayesian-data-analysis-2.html#cb57-2" tabindex="-1"></a>  <span class="fu">summarize</span>(<span class="at">p =</span> <span class="fu">sum</span>(hypothesis)<span class="sc">/</span><span class="fu">n</span>())</span></code></pre></div>
 <pre><code># A tibble: 1 × 1
        p
    &lt;dbl&gt;
@@ -1576,8 +1578,8 @@ <h4><span class="header-section-number">23.4.5.1</span> with <code>hypothesis()<
 <div id="with-emmeans" class="section level4 hasAnchor" number="23.4.5.2">
 <h4><span class="header-section-number">23.4.5.2</span> with <code>emmeans()</code><a href="bayesian-data-analysis-2.html#with-emmeans" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>We can also use the <code>emmeans()</code> function to compute contrasts.</p>
-<div class="sourceCode" id="cb1311"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1311-1"><a href="bayesian-data-analysis-2.html#cb1311-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1311-2"><a href="bayesian-data-analysis-2.html#cb1311-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> consec <span class="sc">~</span> hand)</span></code></pre></div>
+<div class="sourceCode" id="cb59"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb59-1"><a href="bayesian-data-analysis-2.html#cb59-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb59-2"><a href="bayesian-data-analysis-2.html#cb59-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> consec <span class="sc">~</span> hand)</span></code></pre></div>
 <pre><code>Loading required namespace: rstanarm</code></pre>
 <pre><code>$emmeans
  hand    emmean lower.HPD upper.HPD
@@ -1597,40 +1599,40 @@ <h4><span class="header-section-number">23.4.5.2</span> with <code>emmeans()</co
 HPD interval probability: 0.95 </code></pre>
 <p>Here, it computed the estimated means for each group for us, as well as the consecutive contrasts between each group.</p>
 <p>Let’s visualize the contrasts. First, let’s just use the <code>plot()</code> function as it’s been adapted by the emmeans package:</p>
-<div class="sourceCode" id="cb1314"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1314-1"><a href="bayesian-data-analysis-2.html#cb1314-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1314-2"><a href="bayesian-data-analysis-2.html#cb1314-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> consec <span class="sc">~</span> hand) <span class="sc">%&gt;%</span> </span>
-<span id="cb1314-3"><a href="bayesian-data-analysis-2.html#cb1314-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1314-4"><a href="bayesian-data-analysis-2.html#cb1314-4" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-35-1.png" width="672" /></p>
+<div class="sourceCode" id="cb62"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb62-1"><a href="bayesian-data-analysis-2.html#cb62-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb62-2"><a href="bayesian-data-analysis-2.html#cb62-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> consec <span class="sc">~</span> hand) <span class="sc">%&gt;%</span> </span>
+<span id="cb62-3"><a href="bayesian-data-analysis-2.html#cb62-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb62-4"><a href="bayesian-data-analysis-2.html#cb62-4" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-35-1.png" width="672" /></p>
 <p>To get full posterior distributions instead of summaries, we can use the “tidybayes” package like so:</p>
-<div class="sourceCode" id="cb1315"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1315-1"><a href="bayesian-data-analysis-2.html#cb1315-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1315-2"><a href="bayesian-data-analysis-2.html#cb1315-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> consec <span class="sc">~</span> hand) <span class="sc">%&gt;%</span> </span>
-<span id="cb1315-3"><a href="bayesian-data-analysis-2.html#cb1315-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1315-4"><a href="bayesian-data-analysis-2.html#cb1315-4" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1315-5"><a href="bayesian-data-analysis-2.html#cb1315-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> contrast,</span>
-<span id="cb1315-6"><a href="bayesian-data-analysis-2.html#cb1315-6" tabindex="-1"></a>                       <span class="at">x =</span> .value)) <span class="sc">+</span> </span>
-<span id="cb1315-7"><a href="bayesian-data-analysis-2.html#cb1315-7" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>,</span>
-<span id="cb1315-8"><a href="bayesian-data-analysis-2.html#cb1315-8" tabindex="-1"></a>               <span class="at">point_interval =</span> mean_hdi,</span>
-<span id="cb1315-9"><a href="bayesian-data-analysis-2.html#cb1315-9" tabindex="-1"></a>               <span class="at">.width =</span> <span class="fu">c</span>(<span class="fl">0.5</span>, <span class="fl">0.75</span>, <span class="fl">0.95</span>))</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-36-1.png" width="672" /></p>
+<div class="sourceCode" id="cb63"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb63-1"><a href="bayesian-data-analysis-2.html#cb63-1" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb63-2"><a href="bayesian-data-analysis-2.html#cb63-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> consec <span class="sc">~</span> hand) <span class="sc">%&gt;%</span> </span>
+<span id="cb63-3"><a href="bayesian-data-analysis-2.html#cb63-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb63-4"><a href="bayesian-data-analysis-2.html#cb63-4" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb63-5"><a href="bayesian-data-analysis-2.html#cb63-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> contrast,</span>
+<span id="cb63-6"><a href="bayesian-data-analysis-2.html#cb63-6" tabindex="-1"></a>                       <span class="at">x =</span> .value)) <span class="sc">+</span> </span>
+<span id="cb63-7"><a href="bayesian-data-analysis-2.html#cb63-7" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>,</span>
+<span id="cb63-8"><a href="bayesian-data-analysis-2.html#cb63-8" tabindex="-1"></a>               <span class="at">point_interval =</span> mean_hdi,</span>
+<span id="cb63-9"><a href="bayesian-data-analysis-2.html#cb63-9" tabindex="-1"></a>               <span class="at">.width =</span> <span class="fu">c</span>(<span class="fl">0.5</span>, <span class="fl">0.75</span>, <span class="fl">0.95</span>))</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-36-1.png" width="672" /></p>
 <p>To see whether neutral hands did differently from bad and good hands (combined), we can define the following contrast.</p>
-<div class="sourceCode" id="cb1316"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1316-1"><a href="bayesian-data-analysis-2.html#cb1316-1" tabindex="-1"></a>contrasts <span class="ot">=</span> <span class="fu">list</span>(<span class="at">neutral_vs_rest =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">1</span>, <span class="dv">2</span>, <span class="sc">-</span><span class="dv">1</span>))</span>
-<span id="cb1316-2"><a href="bayesian-data-analysis-2.html#cb1316-2" tabindex="-1"></a></span>
-<span id="cb1316-3"><a href="bayesian-data-analysis-2.html#cb1316-3" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1316-4"><a href="bayesian-data-analysis-2.html#cb1316-4" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> <span class="st">&quot;hand&quot;</span>,</span>
-<span id="cb1316-5"><a href="bayesian-data-analysis-2.html#cb1316-5" tabindex="-1"></a>          <span class="at">contr =</span> contrasts) <span class="sc">%&gt;%</span> </span>
-<span id="cb1316-6"><a href="bayesian-data-analysis-2.html#cb1316-6" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1316-7"><a href="bayesian-data-analysis-2.html#cb1316-7" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1316-8"><a href="bayesian-data-analysis-2.html#cb1316-8" tabindex="-1"></a>  <span class="fu">mean_hdi</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb64"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb64-1"><a href="bayesian-data-analysis-2.html#cb64-1" tabindex="-1"></a>contrasts <span class="ot">=</span> <span class="fu">list</span>(<span class="at">neutral_vs_rest =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">1</span>, <span class="dv">2</span>, <span class="sc">-</span><span class="dv">1</span>))</span>
+<span id="cb64-2"><a href="bayesian-data-analysis-2.html#cb64-2" tabindex="-1"></a></span>
+<span id="cb64-3"><a href="bayesian-data-analysis-2.html#cb64-3" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb64-4"><a href="bayesian-data-analysis-2.html#cb64-4" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> <span class="st">&quot;hand&quot;</span>,</span>
+<span id="cb64-5"><a href="bayesian-data-analysis-2.html#cb64-5" tabindex="-1"></a>          <span class="at">contr =</span> contrasts) <span class="sc">%&gt;%</span> </span>
+<span id="cb64-6"><a href="bayesian-data-analysis-2.html#cb64-6" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb64-7"><a href="bayesian-data-analysis-2.html#cb64-7" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb64-8"><a href="bayesian-data-analysis-2.html#cb64-8" tabindex="-1"></a>  <span class="fu">mean_hdi</span>()</span></code></pre></div>
 <pre><code># A tibble: 1 × 7
   contrast        .value .lower .upper .width .point .interval
   &lt;chr&gt;            &lt;dbl&gt;  &lt;dbl&gt;  &lt;dbl&gt;  &lt;dbl&gt; &lt;chr&gt;  &lt;chr&gt;    
 1 neutral_vs_rest   1.72 -0.247   3.82   0.95 mean   hdi      </code></pre>
 <p>Here, the HDP does not exclude 0.</p>
 <p>Let’s double check that we get the same result using the <code>hypothesis()</code> function, or by directly computing from the posterior samples.</p>
-<div class="sourceCode" id="cb1318"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1318-1"><a href="bayesian-data-analysis-2.html#cb1318-1" tabindex="-1"></a><span class="co"># using hypothesis()</span></span>
-<span id="cb1318-2"><a href="bayesian-data-analysis-2.html#cb1318-2" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1318-3"><a href="bayesian-data-analysis-2.html#cb1318-3" tabindex="-1"></a>  <span class="fu">hypothesis</span>(<span class="st">&quot;(Intercept + handneutral)*2 &lt; (Intercept + Intercept + handgood)&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb66"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb66-1"><a href="bayesian-data-analysis-2.html#cb66-1" tabindex="-1"></a><span class="co"># using hypothesis()</span></span>
+<span id="cb66-2"><a href="bayesian-data-analysis-2.html#cb66-2" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb66-3"><a href="bayesian-data-analysis-2.html#cb66-3" tabindex="-1"></a>  <span class="fu">hypothesis</span>(<span class="st">&quot;(Intercept + handneutral)*2 &lt; (Intercept + Intercept + handgood)&quot;</span>)</span></code></pre></div>
 <pre><code>Hypothesis Tests for class b:
                 Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio
 1 ((Intercept+handn... &lt; 0     1.72      1.03     0.01     3.43       0.05
@@ -1641,25 +1643,25 @@ <h4><span class="header-section-number">23.4.5.2</span> with <code>emmeans()</co
 &#39;*&#39;: For one-sided hypotheses, the posterior probability exceeds 95%;
 for two-sided hypotheses, the value tested against lies outside the 95%-CI.
 Posterior probabilities of point hypotheses assume equal prior probabilities.</code></pre>
-<div class="sourceCode" id="cb1320"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1320-1"><a href="bayesian-data-analysis-2.html#cb1320-1" tabindex="-1"></a><span class="co"># directly computing from the posterior</span></span>
-<span id="cb1320-2"><a href="bayesian-data-analysis-2.html#cb1320-2" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
-<span id="cb1320-3"><a href="bayesian-data-analysis-2.html#cb1320-3" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1320-4"><a href="bayesian-data-analysis-2.html#cb1320-4" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1320-5"><a href="bayesian-data-analysis-2.html#cb1320-5" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">contrast =</span> (b_intercept <span class="sc">+</span> b_handneutral) <span class="sc">*</span> <span class="dv">2</span> <span class="sc">-</span> (b_intercept <span class="sc">+</span> b_intercept <span class="sc">+</span> b_handgood)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1320-6"><a href="bayesian-data-analysis-2.html#cb1320-6" tabindex="-1"></a>  <span class="fu">summarize</span>(<span class="at">contrast =</span> <span class="fu">mean</span>(contrast))</span></code></pre></div>
+<div class="sourceCode" id="cb68"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb68-1"><a href="bayesian-data-analysis-2.html#cb68-1" tabindex="-1"></a><span class="co"># directly computing from the posterior</span></span>
+<span id="cb68-2"><a href="bayesian-data-analysis-2.html#cb68-2" tabindex="-1"></a>fit.brm_poker <span class="sc">%&gt;%</span> </span>
+<span id="cb68-3"><a href="bayesian-data-analysis-2.html#cb68-3" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb68-4"><a href="bayesian-data-analysis-2.html#cb68-4" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb68-5"><a href="bayesian-data-analysis-2.html#cb68-5" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">contrast =</span> (b_intercept <span class="sc">+</span> b_handneutral) <span class="sc">*</span> <span class="dv">2</span> <span class="sc">-</span> (b_intercept <span class="sc">+</span> b_intercept <span class="sc">+</span> b_handgood)) <span class="sc">%&gt;%</span> </span>
+<span id="cb68-6"><a href="bayesian-data-analysis-2.html#cb68-6" tabindex="-1"></a>  <span class="fu">summarize</span>(<span class="at">contrast =</span> <span class="fu">mean</span>(contrast))</span></code></pre></div>
 <pre><code># A tibble: 1 × 1
   contrast
      &lt;dbl&gt;
 1     1.72</code></pre>
 <p>The <code>emmeans()</code> function becomes particularly useful when our model has several categorical predictors, and we are interested in comparing differences along one predictor while marginalizing over the values of the other predictor.</p>
 <p>Let’s take a look for a model that considers both <code>skill</code> and <code>hand</code> as predictors (as well as the interaction).</p>
-<div class="sourceCode" id="cb1322"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1322-1"><a href="bayesian-data-analysis-2.html#cb1322-1" tabindex="-1"></a>fit.brm_poker2 <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> hand <span class="sc">*</span> skill,</span>
-<span id="cb1322-2"><a href="bayesian-data-analysis-2.html#cb1322-2" tabindex="-1"></a>                     <span class="at">data =</span> df.poker,</span>
-<span id="cb1322-3"><a href="bayesian-data-analysis-2.html#cb1322-3" tabindex="-1"></a>                     <span class="at">seed =</span> <span class="dv">1</span>, </span>
-<span id="cb1322-4"><a href="bayesian-data-analysis-2.html#cb1322-4" tabindex="-1"></a>                     <span class="at">file =</span> <span class="st">&quot;cache/brm_poker2&quot;</span>)</span>
-<span id="cb1322-5"><a href="bayesian-data-analysis-2.html#cb1322-5" tabindex="-1"></a></span>
-<span id="cb1322-6"><a href="bayesian-data-analysis-2.html#cb1322-6" tabindex="-1"></a>fit.brm_poker2 <span class="sc">%&gt;%</span> </span>
-<span id="cb1322-7"><a href="bayesian-data-analysis-2.html#cb1322-7" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb70"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb70-1"><a href="bayesian-data-analysis-2.html#cb70-1" tabindex="-1"></a>fit.brm_poker2 <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> hand <span class="sc">*</span> skill,</span>
+<span id="cb70-2"><a href="bayesian-data-analysis-2.html#cb70-2" tabindex="-1"></a>                     <span class="at">data =</span> df.poker,</span>
+<span id="cb70-3"><a href="bayesian-data-analysis-2.html#cb70-3" tabindex="-1"></a>                     <span class="at">seed =</span> <span class="dv">1</span>, </span>
+<span id="cb70-4"><a href="bayesian-data-analysis-2.html#cb70-4" tabindex="-1"></a>                     <span class="at">file =</span> <span class="st">&quot;cache/brm_poker2&quot;</span>)</span>
+<span id="cb70-5"><a href="bayesian-data-analysis-2.html#cb70-5" tabindex="-1"></a></span>
+<span id="cb70-6"><a href="bayesian-data-analysis-2.html#cb70-6" tabindex="-1"></a>fit.brm_poker2 <span class="sc">%&gt;%</span> </span>
+<span id="cb70-7"><a href="bayesian-data-analysis-2.html#cb70-7" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
 <pre><code> Family: gaussian 
   Links: mu = identity; sigma = identity 
 Formula: balance ~ 1 + hand * skill 
@@ -1691,8 +1693,8 @@ <h4><span class="header-section-number">23.4.5.2</span> with <code>emmeans()</co
 and Tail_ESS are effective sample size measures, and Rhat is the potential
 scale reduction factor on split chains (at convergence, Rhat = 1).</code></pre>
 <p>In the summary table above, <code>skillexpert</code> captures the difference between an expert and an average player <strong>when they have a bad hand</strong>. To see whether there was a difference in expertise overall (i.e. across all three kinds of hands), we can calculate a linear contrast.</p>
-<div class="sourceCode" id="cb1324"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1324-1"><a href="bayesian-data-analysis-2.html#cb1324-1" tabindex="-1"></a>fit.brm_poker2 <span class="sc">%&gt;%</span> </span>
-<span id="cb1324-2"><a href="bayesian-data-analysis-2.html#cb1324-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(pairwise <span class="sc">~</span> skill)</span></code></pre></div>
+<div class="sourceCode" id="cb72"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb72-1"><a href="bayesian-data-analysis-2.html#cb72-1" tabindex="-1"></a>fit.brm_poker2 <span class="sc">%&gt;%</span> </span>
+<span id="cb72-2"><a href="bayesian-data-analysis-2.html#cb72-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(pairwise <span class="sc">~</span> skill)</span></code></pre></div>
 <pre><code>NOTE: Results may be misleading due to involvement in interactions</code></pre>
 <pre><code>$emmeans
  skill   emmean lower.HPD upper.HPD
@@ -1712,16 +1714,16 @@ <h4><span class="header-section-number">23.4.5.2</span> with <code>emmeans()</co
 HPD interval probability: 0.95 </code></pre>
 <p>It looks like overall, skilled players weren’t doing much better than average players.</p>
 <p>We can even do something like an equivalent of an ANOVA using <code>emmeans()</code>, like so:</p>
-<div class="sourceCode" id="cb1327"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1327-1"><a href="bayesian-data-analysis-2.html#cb1327-1" tabindex="-1"></a><span class="fu">joint_tests</span>(fit.brm_poker2)</span></code></pre></div>
+<div class="sourceCode" id="cb75"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb75-1"><a href="bayesian-data-analysis-2.html#cb75-1" tabindex="-1"></a><span class="fu">joint_tests</span>(fit.brm_poker2)</span></code></pre></div>
 <pre><code> model term df1 df2 F.ratio   Chisq p.value
  hand         2 Inf  82.446 164.892  &lt;.0001
  skill        1 Inf   2.549   2.549  0.1103
  hand:skill   2 Inf   7.347  14.694  0.0006</code></pre>
 <p>The values we get here are very similar to what we would get from a frequentist ANOVA:</p>
-<div class="sourceCode" id="cb1329"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1329-1"><a href="bayesian-data-analysis-2.html#cb1329-1" tabindex="-1"></a><span class="fu">aov_ez</span>(<span class="at">id =</span> <span class="st">&quot;participant&quot;</span>,</span>
-<span id="cb1329-2"><a href="bayesian-data-analysis-2.html#cb1329-2" tabindex="-1"></a>       <span class="at">dv =</span> <span class="st">&quot;balance&quot;</span>,</span>
-<span id="cb1329-3"><a href="bayesian-data-analysis-2.html#cb1329-3" tabindex="-1"></a>       <span class="at">between =</span> <span class="fu">c</span>(<span class="st">&quot;hand&quot;</span>, <span class="st">&quot;skill&quot;</span>),</span>
-<span id="cb1329-4"><a href="bayesian-data-analysis-2.html#cb1329-4" tabindex="-1"></a>       <span class="at">data =</span> df.poker)</span></code></pre></div>
+<div class="sourceCode" id="cb77"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb77-1"><a href="bayesian-data-analysis-2.html#cb77-1" tabindex="-1"></a><span class="fu">aov_ez</span>(<span class="at">id =</span> <span class="st">&quot;participant&quot;</span>,</span>
+<span id="cb77-2"><a href="bayesian-data-analysis-2.html#cb77-2" tabindex="-1"></a>       <span class="at">dv =</span> <span class="st">&quot;balance&quot;</span>,</span>
+<span id="cb77-3"><a href="bayesian-data-analysis-2.html#cb77-3" tabindex="-1"></a>       <span class="at">between =</span> <span class="fu">c</span>(<span class="st">&quot;hand&quot;</span>, <span class="st">&quot;skill&quot;</span>),</span>
+<span id="cb77-4"><a href="bayesian-data-analysis-2.html#cb77-4" tabindex="-1"></a>       <span class="at">data =</span> df.poker)</span></code></pre></div>
 <pre><code>Contrasts set to contr.sum for the following variables: hand, skill</code></pre>
 <pre><code>Anova Table (Type 3 tests)
 
@@ -1736,44 +1738,43 @@ <h4><span class="header-section-number">23.4.5.2</span> with <code>emmeans()</co
 <div id="bayes-factor" class="section level4 hasAnchor" number="23.4.5.3">
 <h4><span class="header-section-number">23.4.5.3</span> Bayes factor<a href="bayesian-data-analysis-2.html#bayes-factor" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>Another way of testing hypothesis is via the Bayes factor. Let’s fit the two models we are interested in comparing with each other:</p>
-<div class="sourceCode" id="cb1332"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1332-1"><a href="bayesian-data-analysis-2.html#cb1332-1" tabindex="-1"></a>fit.brm_poker_bf1 <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand,</span>
-<span id="cb1332-2"><a href="bayesian-data-analysis-2.html#cb1332-2" tabindex="-1"></a>                        <span class="at">data =</span> df.poker,</span>
-<span id="cb1332-3"><a href="bayesian-data-analysis-2.html#cb1332-3" tabindex="-1"></a>                        <span class="at">save_pars =</span> <span class="fu">save_pars</span>(<span class="at">all =</span> T),</span>
-<span id="cb1332-4"><a href="bayesian-data-analysis-2.html#cb1332-4" tabindex="-1"></a>                        <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_bf1&quot;</span>)</span>
-<span id="cb1332-5"><a href="bayesian-data-analysis-2.html#cb1332-5" tabindex="-1"></a></span>
-<span id="cb1332-6"><a href="bayesian-data-analysis-2.html#cb1332-6" tabindex="-1"></a>fit.brm_poker_bf2 <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand <span class="sc">+</span> skill,</span>
-<span id="cb1332-7"><a href="bayesian-data-analysis-2.html#cb1332-7" tabindex="-1"></a>                        <span class="at">data =</span> df.poker,</span>
-<span id="cb1332-8"><a href="bayesian-data-analysis-2.html#cb1332-8" tabindex="-1"></a>                        <span class="at">save_pars =</span> <span class="fu">save_pars</span>(<span class="at">all =</span> T),</span>
-<span id="cb1332-9"><a href="bayesian-data-analysis-2.html#cb1332-9" tabindex="-1"></a>                        <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_bf2&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb80"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb80-1"><a href="bayesian-data-analysis-2.html#cb80-1" tabindex="-1"></a>fit.brm_poker_bf1 <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand,</span>
+<span id="cb80-2"><a href="bayesian-data-analysis-2.html#cb80-2" tabindex="-1"></a>                        <span class="at">data =</span> df.poker,</span>
+<span id="cb80-3"><a href="bayesian-data-analysis-2.html#cb80-3" tabindex="-1"></a>                        <span class="at">save_pars =</span> <span class="fu">save_pars</span>(<span class="at">all =</span> T),</span>
+<span id="cb80-4"><a href="bayesian-data-analysis-2.html#cb80-4" tabindex="-1"></a>                        <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_bf1&quot;</span>)</span>
+<span id="cb80-5"><a href="bayesian-data-analysis-2.html#cb80-5" tabindex="-1"></a></span>
+<span id="cb80-6"><a href="bayesian-data-analysis-2.html#cb80-6" tabindex="-1"></a>fit.brm_poker_bf2 <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> balance <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> hand <span class="sc">+</span> skill,</span>
+<span id="cb80-7"><a href="bayesian-data-analysis-2.html#cb80-7" tabindex="-1"></a>                        <span class="at">data =</span> df.poker,</span>
+<span id="cb80-8"><a href="bayesian-data-analysis-2.html#cb80-8" tabindex="-1"></a>                        <span class="at">save_pars =</span> <span class="fu">save_pars</span>(<span class="at">all =</span> T),</span>
+<span id="cb80-9"><a href="bayesian-data-analysis-2.html#cb80-9" tabindex="-1"></a>                        <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_bf2&quot;</span>)</span></code></pre></div>
 <p>And then compare the models using the <code>bayes_factor()</code> function:</p>
-<div class="sourceCode" id="cb1333"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1333-1"><a href="bayesian-data-analysis-2.html#cb1333-1" tabindex="-1"></a><span class="fu">bayes_factor</span>(fit.brm_poker_bf2, fit.brm_poker_bf1)</span></code></pre></div>
+<div class="sourceCode" id="cb81"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb81-1"><a href="bayesian-data-analysis-2.html#cb81-1" tabindex="-1"></a><span class="fu">bayes_factor</span>(fit.brm_poker_bf2, fit.brm_poker_bf1)</span></code></pre></div>
 <pre><code>Iteration: 1
 Iteration: 2
 Iteration: 3
 Iteration: 4
-Iteration: 5
 Iteration: 1
 Iteration: 2
 Iteration: 3
 Iteration: 4
 Iteration: 5</code></pre>
-<pre><code>Estimated Bayes factor in favor of fit.brm_poker_bf2 over fit.brm_poker_bf1: 3.82034</code></pre>
+<pre><code>Estimated Bayes factor in favor of fit.brm_poker_bf2 over fit.brm_poker_bf1: 3.83098</code></pre>
 <p>Bayes factors don’t have a very good reputation (see here and here). Instead, the way to go these days appears to be via approximate leave one out cross-validation.</p>
 </div>
 <div id="approximate-leave-one-out-cross-validation" class="section level4 hasAnchor" number="23.4.5.4">
 <h4><span class="header-section-number">23.4.5.4</span> Approximate leave one out cross-validation<a href="bayesian-data-analysis-2.html#approximate-leave-one-out-cross-validation" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1336"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1336-1"><a href="bayesian-data-analysis-2.html#cb1336-1" tabindex="-1"></a>fit.brm_poker_bf1 <span class="ot">=</span> <span class="fu">add_criterion</span>(fit.brm_poker_bf1,</span>
-<span id="cb1336-2"><a href="bayesian-data-analysis-2.html#cb1336-2" tabindex="-1"></a>                                  <span class="at">criterion =</span> <span class="st">&quot;loo&quot;</span>,</span>
-<span id="cb1336-3"><a href="bayesian-data-analysis-2.html#cb1336-3" tabindex="-1"></a>                                  <span class="at">reloo =</span> T,</span>
-<span id="cb1336-4"><a href="bayesian-data-analysis-2.html#cb1336-4" tabindex="-1"></a>                                  <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_bf1&quot;</span>)</span>
-<span id="cb1336-5"><a href="bayesian-data-analysis-2.html#cb1336-5" tabindex="-1"></a></span>
-<span id="cb1336-6"><a href="bayesian-data-analysis-2.html#cb1336-6" tabindex="-1"></a>fit.brm_poker_bf2 <span class="ot">=</span> <span class="fu">add_criterion</span>(fit.brm_poker_bf2,</span>
-<span id="cb1336-7"><a href="bayesian-data-analysis-2.html#cb1336-7" tabindex="-1"></a>                                  <span class="at">criterion =</span> <span class="st">&quot;loo&quot;</span>,</span>
-<span id="cb1336-8"><a href="bayesian-data-analysis-2.html#cb1336-8" tabindex="-1"></a>                                  <span class="at">reloo =</span> T,</span>
-<span id="cb1336-9"><a href="bayesian-data-analysis-2.html#cb1336-9" tabindex="-1"></a>                                  <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_bf2&quot;</span>)</span>
-<span id="cb1336-10"><a href="bayesian-data-analysis-2.html#cb1336-10" tabindex="-1"></a></span>
-<span id="cb1336-11"><a href="bayesian-data-analysis-2.html#cb1336-11" tabindex="-1"></a><span class="fu">loo_compare</span>(fit.brm_poker_bf1,</span>
-<span id="cb1336-12"><a href="bayesian-data-analysis-2.html#cb1336-12" tabindex="-1"></a>            fit.brm_poker_bf2)</span></code></pre></div>
+<div class="sourceCode" id="cb84"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb84-1"><a href="bayesian-data-analysis-2.html#cb84-1" tabindex="-1"></a>fit.brm_poker_bf1 <span class="ot">=</span> <span class="fu">add_criterion</span>(fit.brm_poker_bf1,</span>
+<span id="cb84-2"><a href="bayesian-data-analysis-2.html#cb84-2" tabindex="-1"></a>                                  <span class="at">criterion =</span> <span class="st">&quot;loo&quot;</span>,</span>
+<span id="cb84-3"><a href="bayesian-data-analysis-2.html#cb84-3" tabindex="-1"></a>                                  <span class="at">reloo =</span> T,</span>
+<span id="cb84-4"><a href="bayesian-data-analysis-2.html#cb84-4" tabindex="-1"></a>                                  <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_bf1&quot;</span>)</span>
+<span id="cb84-5"><a href="bayesian-data-analysis-2.html#cb84-5" tabindex="-1"></a></span>
+<span id="cb84-6"><a href="bayesian-data-analysis-2.html#cb84-6" tabindex="-1"></a>fit.brm_poker_bf2 <span class="ot">=</span> <span class="fu">add_criterion</span>(fit.brm_poker_bf2,</span>
+<span id="cb84-7"><a href="bayesian-data-analysis-2.html#cb84-7" tabindex="-1"></a>                                  <span class="at">criterion =</span> <span class="st">&quot;loo&quot;</span>,</span>
+<span id="cb84-8"><a href="bayesian-data-analysis-2.html#cb84-8" tabindex="-1"></a>                                  <span class="at">reloo =</span> T,</span>
+<span id="cb84-9"><a href="bayesian-data-analysis-2.html#cb84-9" tabindex="-1"></a>                                  <span class="at">file =</span> <span class="st">&quot;cache/brm_poker_bf2&quot;</span>)</span>
+<span id="cb84-10"><a href="bayesian-data-analysis-2.html#cb84-10" tabindex="-1"></a></span>
+<span id="cb84-11"><a href="bayesian-data-analysis-2.html#cb84-11" tabindex="-1"></a><span class="fu">loo_compare</span>(fit.brm_poker_bf1,</span>
+<span id="cb84-12"><a href="bayesian-data-analysis-2.html#cb84-12" tabindex="-1"></a>            fit.brm_poker_bf2)</span></code></pre></div>
 <pre><code>                  elpd_diff se_diff
 fit.brm_poker_bf2  0.0       0.0   
 fit.brm_poker_bf1 -0.3       1.5   </code></pre>
@@ -1784,26 +1785,26 @@ <h4><span class="header-section-number">23.4.5.4</span> Approximate leave one ou
 <h2><span class="header-section-number">23.5</span> Sleep study<a href="bayesian-data-analysis-2.html#sleep-study-1" class="anchor-section" aria-label="Anchor link to header"></a></h2>
 <div id="visualize-the-data-1" class="section level3 hasAnchor" number="23.5.1">
 <h3><span class="header-section-number">23.5.1</span> 1. Visualize the data<a href="bayesian-data-analysis-2.html#visualize-the-data-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1338"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1338-1"><a href="bayesian-data-analysis-2.html#cb1338-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
-<span id="cb1338-2"><a href="bayesian-data-analysis-2.html#cb1338-2" tabindex="-1"></a></span>
-<span id="cb1338-3"><a href="bayesian-data-analysis-2.html#cb1338-3" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1338-4"><a href="bayesian-data-analysis-2.html#cb1338-4" tabindex="-1"></a>         <span class="fu">mutate</span>(<span class="at">days =</span> <span class="fu">as.factor</span>(days)),</span>
-<span id="cb1338-5"><a href="bayesian-data-analysis-2.html#cb1338-5" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
-<span id="cb1338-6"><a href="bayesian-data-analysis-2.html#cb1338-6" tabindex="-1"></a>                     <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
-<span id="cb1338-7"><a href="bayesian-data-analysis-2.html#cb1338-7" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
-<span id="cb1338-8"><a href="bayesian-data-analysis-2.html#cb1338-8" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitter</span>(<span class="at">width =</span> <span class="fl">0.1</span>)) <span class="sc">+</span> </span>
-<span id="cb1338-9"><a href="bayesian-data-analysis-2.html#cb1338-9" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>) </span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-46-1.png" width="672" /></p>
+<div class="sourceCode" id="cb86"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb86-1"><a href="bayesian-data-analysis-2.html#cb86-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
+<span id="cb86-2"><a href="bayesian-data-analysis-2.html#cb86-2" tabindex="-1"></a></span>
+<span id="cb86-3"><a href="bayesian-data-analysis-2.html#cb86-3" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb86-4"><a href="bayesian-data-analysis-2.html#cb86-4" tabindex="-1"></a>         <span class="fu">mutate</span>(<span class="at">days =</span> <span class="fu">as.factor</span>(days)),</span>
+<span id="cb86-5"><a href="bayesian-data-analysis-2.html#cb86-5" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
+<span id="cb86-6"><a href="bayesian-data-analysis-2.html#cb86-6" tabindex="-1"></a>                     <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
+<span id="cb86-7"><a href="bayesian-data-analysis-2.html#cb86-7" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
+<span id="cb86-8"><a href="bayesian-data-analysis-2.html#cb86-8" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitter</span>(<span class="at">width =</span> <span class="fl">0.1</span>)) <span class="sc">+</span> </span>
+<span id="cb86-9"><a href="bayesian-data-analysis-2.html#cb86-9" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>) </span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-46-1.png" width="672" /></p>
 </div>
 <div id="specify-and-fit-the-model-1" class="section level3 hasAnchor" number="23.5.2">
 <h3><span class="header-section-number">23.5.2</span> 2. Specify and fit the model<a href="bayesian-data-analysis-2.html#specify-and-fit-the-model-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="frequentist-analysis-1" class="section level4 hasAnchor" number="23.5.2.1">
 <h4><span class="header-section-number">23.5.2.1</span> Frequentist analysis<a href="bayesian-data-analysis-2.html#frequentist-analysis-1" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1339"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1339-1"><a href="bayesian-data-analysis-2.html#cb1339-1" tabindex="-1"></a>fit.lmer_sleep <span class="ot">=</span> <span class="fu">lmer</span>(<span class="at">formula =</span> reaction <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> days <span class="sc">+</span> (<span class="dv">1</span> <span class="sc">+</span> days <span class="sc">|</span> subject),</span>
-<span id="cb1339-2"><a href="bayesian-data-analysis-2.html#cb1339-2" tabindex="-1"></a>                      <span class="at">data =</span> df.sleep)</span>
-<span id="cb1339-3"><a href="bayesian-data-analysis-2.html#cb1339-3" tabindex="-1"></a></span>
-<span id="cb1339-4"><a href="bayesian-data-analysis-2.html#cb1339-4" tabindex="-1"></a>fit.lmer_sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1339-5"><a href="bayesian-data-analysis-2.html#cb1339-5" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb87"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb87-1"><a href="bayesian-data-analysis-2.html#cb87-1" tabindex="-1"></a>fit.lmer_sleep <span class="ot">=</span> <span class="fu">lmer</span>(<span class="at">formula =</span> reaction <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> days <span class="sc">+</span> (<span class="dv">1</span> <span class="sc">+</span> days <span class="sc">|</span> subject),</span>
+<span id="cb87-2"><a href="bayesian-data-analysis-2.html#cb87-2" tabindex="-1"></a>                      <span class="at">data =</span> df.sleep)</span>
+<span id="cb87-3"><a href="bayesian-data-analysis-2.html#cb87-3" tabindex="-1"></a></span>
+<span id="cb87-4"><a href="bayesian-data-analysis-2.html#cb87-4" tabindex="-1"></a>fit.lmer_sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb87-5"><a href="bayesian-data-analysis-2.html#cb87-5" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
 <pre><code>Linear mixed model fit by REML. t-tests use Satterthwaite&#39;s method [
 lmerModLmerTest]
 Formula: reaction ~ 1 + days + (1 + days | subject)
@@ -1835,18 +1836,18 @@ <h4><span class="header-section-number">23.5.2.1</span> Frequentist analysis<a h
 </div>
 <div id="bayesian-analysis" class="section level4 hasAnchor" number="23.5.2.2">
 <h4><span class="header-section-number">23.5.2.2</span> Bayesian analysis<a href="bayesian-data-analysis-2.html#bayesian-analysis" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1341"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1341-1"><a href="bayesian-data-analysis-2.html#cb1341-1" tabindex="-1"></a>fit.brm_sleep <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> reaction <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> days <span class="sc">+</span> (<span class="dv">1</span> <span class="sc">+</span> days <span class="sc">|</span> subject),</span>
-<span id="cb1341-2"><a href="bayesian-data-analysis-2.html#cb1341-2" tabindex="-1"></a>                    <span class="at">data =</span> df.sleep,</span>
-<span id="cb1341-3"><a href="bayesian-data-analysis-2.html#cb1341-3" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>,</span>
-<span id="cb1341-4"><a href="bayesian-data-analysis-2.html#cb1341-4" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_sleep&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb89"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb89-1"><a href="bayesian-data-analysis-2.html#cb89-1" tabindex="-1"></a>fit.brm_sleep <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> reaction <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> days <span class="sc">+</span> (<span class="dv">1</span> <span class="sc">+</span> days <span class="sc">|</span> subject),</span>
+<span id="cb89-2"><a href="bayesian-data-analysis-2.html#cb89-2" tabindex="-1"></a>                    <span class="at">data =</span> df.sleep,</span>
+<span id="cb89-3"><a href="bayesian-data-analysis-2.html#cb89-3" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>,</span>
+<span id="cb89-4"><a href="bayesian-data-analysis-2.html#cb89-4" tabindex="-1"></a>                    <span class="at">file =</span> <span class="st">&quot;cache/brm_sleep&quot;</span>)</span></code></pre></div>
 </div>
 </div>
 <div id="model-evaluation-1" class="section level3 hasAnchor" number="23.5.3">
 <h3><span class="header-section-number">23.5.3</span> 3. Model evaluation<a href="bayesian-data-analysis-2.html#model-evaluation-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="a-did-the-inference-work-1" class="section level4 hasAnchor" number="23.5.3.1">
 <h4><span class="header-section-number">23.5.3.1</span> a) Did the inference work?<a href="bayesian-data-analysis-2.html#a-did-the-inference-work-1" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1342"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1342-1"><a href="bayesian-data-analysis-2.html#cb1342-1" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1342-2"><a href="bayesian-data-analysis-2.html#cb1342-2" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb90"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb90-1"><a href="bayesian-data-analysis-2.html#cb90-1" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb90-2"><a href="bayesian-data-analysis-2.html#cb90-2" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
 <pre><code> Family: gaussian 
   Links: mu = identity; sigma = identity 
 Formula: reaction ~ 1 + days + (1 + days | subject) 
@@ -1873,21 +1874,21 @@ <h4><span class="header-section-number">23.5.3.1</span> a) Did the inference wor
 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
 and Tail_ESS are effective sample size measures, and Rhat is the potential
 scale reduction factor on split chains (at convergence, Rhat = 1).</code></pre>
-<div class="sourceCode" id="cb1344"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1344-1"><a href="bayesian-data-analysis-2.html#cb1344-1" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1344-2"><a href="bayesian-data-analysis-2.html#cb1344-2" tabindex="-1"></a>  <span class="fu">plot</span>(<span class="at">N =</span> <span class="dv">6</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-49-1.png" width="768" /></p>
+<div class="sourceCode" id="cb92"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb92-1"><a href="bayesian-data-analysis-2.html#cb92-1" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb92-2"><a href="bayesian-data-analysis-2.html#cb92-2" tabindex="-1"></a>  <span class="fu">plot</span>(<span class="at">N =</span> <span class="dv">6</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-49-1.png" width="768" /></p>
 </div>
 <div id="b-visualize-model-predictions-1" class="section level4 hasAnchor" number="23.5.3.2">
 <h4><span class="header-section-number">23.5.3.2</span> b) Visualize model predictions<a href="bayesian-data-analysis-2.html#b-visualize-model-predictions-1" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1345"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1345-1"><a href="bayesian-data-analysis-2.html#cb1345-1" tabindex="-1"></a><span class="fu">pp_check</span>(fit.brm_sleep,</span>
-<span id="cb1345-2"><a href="bayesian-data-analysis-2.html#cb1345-2" tabindex="-1"></a>         <span class="at">ndraws =</span> <span class="dv">100</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-50-1.png" width="672" /></p>
+<div class="sourceCode" id="cb93"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb93-1"><a href="bayesian-data-analysis-2.html#cb93-1" tabindex="-1"></a><span class="fu">pp_check</span>(fit.brm_sleep,</span>
+<span id="cb93-2"><a href="bayesian-data-analysis-2.html#cb93-2" tabindex="-1"></a>         <span class="at">ndraws =</span> <span class="dv">100</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-50-1.png" width="672" /></p>
 </div>
 </div>
 <div id="interpret-the-parameters" class="section level3 hasAnchor" number="23.5.4">
 <h3><span class="header-section-number">23.5.4</span> 4. Interpret the parameters<a href="bayesian-data-analysis-2.html#interpret-the-parameters" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1346"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1346-1"><a href="bayesian-data-analysis-2.html#cb1346-1" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1346-2"><a href="bayesian-data-analysis-2.html#cb1346-2" tabindex="-1"></a>  <span class="fu">tidy</span>(<span class="at">conf.method =</span> <span class="st">&quot;HPDinterval&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb94"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb94-1"><a href="bayesian-data-analysis-2.html#cb94-1" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb94-2"><a href="bayesian-data-analysis-2.html#cb94-2" tabindex="-1"></a>  <span class="fu">tidy</span>(<span class="at">conf.method =</span> <span class="st">&quot;HPDinterval&quot;</span>)</span></code></pre></div>
 <pre><code># A tibble: 6 × 8
   effect   component group    term         estimate std.error conf.low conf.high
   &lt;chr&gt;    &lt;chr&gt;     &lt;chr&gt;    &lt;chr&gt;           &lt;dbl&gt;     &lt;dbl&gt;    &lt;dbl&gt;     &lt;dbl&gt;
@@ -1899,32 +1900,32 @@ <h3><span class="header-section-number">23.5.4</span> 4. Interpret the parameter
 6 ran_pars cond      Residual sd__Observa…  25.8        1.54    22.7      28.7  </code></pre>
 <div id="summary-of-posterior-distributions" class="section level4 hasAnchor" number="23.5.4.1">
 <h4><span class="header-section-number">23.5.4.1</span> Summary of posterior distributions<a href="bayesian-data-analysis-2.html#summary-of-posterior-distributions" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1348"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1348-1"><a href="bayesian-data-analysis-2.html#cb1348-1" tabindex="-1"></a><span class="co"># all parameters</span></span>
-<span id="cb1348-2"><a href="bayesian-data-analysis-2.html#cb1348-2" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1348-3"><a href="bayesian-data-analysis-2.html#cb1348-3" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1348-4"><a href="bayesian-data-analysis-2.html#cb1348-4" tabindex="-1"></a>  <span class="fu">select</span>(<span class="sc">-</span><span class="fu">c</span>(lp__, <span class="fu">contains</span>(<span class="st">&quot;[&quot;</span>))) <span class="sc">%&gt;%</span></span>
-<span id="cb1348-5"><a href="bayesian-data-analysis-2.html#cb1348-5" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">everything</span>(),</span>
-<span id="cb1348-6"><a href="bayesian-data-analysis-2.html#cb1348-6" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;variable&quot;</span>,</span>
-<span id="cb1348-7"><a href="bayesian-data-analysis-2.html#cb1348-7" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1348-8"><a href="bayesian-data-analysis-2.html#cb1348-8" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1348-9"><a href="bayesian-data-analysis-2.html#cb1348-9" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> value)) <span class="sc">+</span></span>
-<span id="cb1348-10"><a href="bayesian-data-analysis-2.html#cb1348-10" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">point_interval =</span> mode_hdi,</span>
-<span id="cb1348-11"><a href="bayesian-data-analysis-2.html#cb1348-11" tabindex="-1"></a>               <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1348-12"><a href="bayesian-data-analysis-2.html#cb1348-12" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span> variable,</span>
-<span id="cb1348-13"><a href="bayesian-data-analysis-2.html#cb1348-13" tabindex="-1"></a>             <span class="at">ncol =</span> <span class="dv">2</span>,</span>
-<span id="cb1348-14"><a href="bayesian-data-analysis-2.html#cb1348-14" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1348-15"><a href="bayesian-data-analysis-2.html#cb1348-15" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>))</span>
-<span id="cb1348-16"><a href="bayesian-data-analysis-2.html#cb1348-16" tabindex="-1"></a></span>
-<span id="cb1348-17"><a href="bayesian-data-analysis-2.html#cb1348-17" tabindex="-1"></a><span class="co"># just the parameter of interest</span></span>
-<span id="cb1348-18"><a href="bayesian-data-analysis-2.html#cb1348-18" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1348-19"><a href="bayesian-data-analysis-2.html#cb1348-19" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1348-20"><a href="bayesian-data-analysis-2.html#cb1348-20" tabindex="-1"></a>  <span class="fu">select</span>(b_days) <span class="sc">%&gt;%</span></span>
-<span id="cb1348-21"><a href="bayesian-data-analysis-2.html#cb1348-21" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1348-22"><a href="bayesian-data-analysis-2.html#cb1348-22" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> b_days)) <span class="sc">+</span></span>
-<span id="cb1348-23"><a href="bayesian-data-analysis-2.html#cb1348-23" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">point_interval =</span> mode_hdi,</span>
-<span id="cb1348-24"><a href="bayesian-data-analysis-2.html#cb1348-24" tabindex="-1"></a>               <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1348-25"><a href="bayesian-data-analysis-2.html#cb1348-25" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>))</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-52-1.png" width="672" /><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-52-2.png" width="672" /></p>
+<div class="sourceCode" id="cb96"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb96-1"><a href="bayesian-data-analysis-2.html#cb96-1" tabindex="-1"></a><span class="co"># all parameters</span></span>
+<span id="cb96-2"><a href="bayesian-data-analysis-2.html#cb96-2" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb96-3"><a href="bayesian-data-analysis-2.html#cb96-3" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb96-4"><a href="bayesian-data-analysis-2.html#cb96-4" tabindex="-1"></a>  <span class="fu">select</span>(<span class="sc">-</span><span class="fu">c</span>(lp__, <span class="fu">contains</span>(<span class="st">&quot;[&quot;</span>))) <span class="sc">%&gt;%</span></span>
+<span id="cb96-5"><a href="bayesian-data-analysis-2.html#cb96-5" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">everything</span>(),</span>
+<span id="cb96-6"><a href="bayesian-data-analysis-2.html#cb96-6" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;variable&quot;</span>,</span>
+<span id="cb96-7"><a href="bayesian-data-analysis-2.html#cb96-7" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb96-8"><a href="bayesian-data-analysis-2.html#cb96-8" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb96-9"><a href="bayesian-data-analysis-2.html#cb96-9" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> value)) <span class="sc">+</span></span>
+<span id="cb96-10"><a href="bayesian-data-analysis-2.html#cb96-10" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">point_interval =</span> mode_hdi,</span>
+<span id="cb96-11"><a href="bayesian-data-analysis-2.html#cb96-11" tabindex="-1"></a>               <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb96-12"><a href="bayesian-data-analysis-2.html#cb96-12" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span> variable,</span>
+<span id="cb96-13"><a href="bayesian-data-analysis-2.html#cb96-13" tabindex="-1"></a>             <span class="at">ncol =</span> <span class="dv">2</span>,</span>
+<span id="cb96-14"><a href="bayesian-data-analysis-2.html#cb96-14" tabindex="-1"></a>             <span class="at">scales =</span> <span class="st">&quot;free&quot;</span>) <span class="sc">+</span></span>
+<span id="cb96-15"><a href="bayesian-data-analysis-2.html#cb96-15" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>))</span>
+<span id="cb96-16"><a href="bayesian-data-analysis-2.html#cb96-16" tabindex="-1"></a></span>
+<span id="cb96-17"><a href="bayesian-data-analysis-2.html#cb96-17" tabindex="-1"></a><span class="co"># just the parameter of interest</span></span>
+<span id="cb96-18"><a href="bayesian-data-analysis-2.html#cb96-18" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb96-19"><a href="bayesian-data-analysis-2.html#cb96-19" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb96-20"><a href="bayesian-data-analysis-2.html#cb96-20" tabindex="-1"></a>  <span class="fu">select</span>(b_days) <span class="sc">%&gt;%</span></span>
+<span id="cb96-21"><a href="bayesian-data-analysis-2.html#cb96-21" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb96-22"><a href="bayesian-data-analysis-2.html#cb96-22" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> b_days)) <span class="sc">+</span></span>
+<span id="cb96-23"><a href="bayesian-data-analysis-2.html#cb96-23" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">point_interval =</span> mode_hdi,</span>
+<span id="cb96-24"><a href="bayesian-data-analysis-2.html#cb96-24" tabindex="-1"></a>               <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb96-25"><a href="bayesian-data-analysis-2.html#cb96-25" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>))</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-52-1.png" width="672" /><img src="psych252_files/figure-html/unnamed-chunk-52-2.png" width="672" /></p>
 </div>
 </div>
 <div id="test-specific-hypotheses-1" class="section level3 hasAnchor" number="23.5.5">
@@ -1935,140 +1936,141 @@ <h3><span class="header-section-number">23.5.5</span> 5. Test specific hypothese
 <h3><span class="header-section-number">23.5.6</span> 6. Report results<a href="bayesian-data-analysis-2.html#report-results" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="model-prediction-with-posterior-draws-aggregate" class="section level4 hasAnchor" number="23.5.6.1">
 <h4><span class="header-section-number">23.5.6.1</span> Model prediction with posterior draws (aggregate)<a href="bayesian-data-analysis-2.html#model-prediction-with-posterior-draws-aggregate" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1349"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1349-1"><a href="bayesian-data-analysis-2.html#cb1349-1" tabindex="-1"></a>df.model <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">days =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1349-2"><a href="bayesian-data-analysis-2.html#cb1349-2" tabindex="-1"></a>  <span class="fu">add_linpred_draws</span>(<span class="at">newdata =</span> .,</span>
-<span id="cb1349-3"><a href="bayesian-data-analysis-2.html#cb1349-3" tabindex="-1"></a>                    <span class="at">object =</span> fit.brm_sleep,</span>
-<span id="cb1349-4"><a href="bayesian-data-analysis-2.html#cb1349-4" tabindex="-1"></a>                    <span class="at">ndraws =</span> <span class="dv">10</span>,</span>
-<span id="cb1349-5"><a href="bayesian-data-analysis-2.html#cb1349-5" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>,</span>
-<span id="cb1349-6"><a href="bayesian-data-analysis-2.html#cb1349-6" tabindex="-1"></a>                    <span class="at">re_formula =</span> <span class="cn">NA</span>)</span>
-<span id="cb1349-7"><a href="bayesian-data-analysis-2.html#cb1349-7" tabindex="-1"></a></span>
-<span id="cb1349-8"><a href="bayesian-data-analysis-2.html#cb1349-8" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.sleep,</span>
-<span id="cb1349-9"><a href="bayesian-data-analysis-2.html#cb1349-9" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
-<span id="cb1349-10"><a href="bayesian-data-analysis-2.html#cb1349-10" tabindex="-1"></a>                     <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
-<span id="cb1349-11"><a href="bayesian-data-analysis-2.html#cb1349-11" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
-<span id="cb1349-12"><a href="bayesian-data-analysis-2.html#cb1349-12" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitter</span>(<span class="at">width =</span> <span class="fl">0.1</span>)) <span class="sc">+</span> </span>
-<span id="cb1349-13"><a href="bayesian-data-analysis-2.html#cb1349-13" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model,</span>
-<span id="cb1349-14"><a href="bayesian-data-analysis-2.html#cb1349-14" tabindex="-1"></a>            <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
-<span id="cb1349-15"><a href="bayesian-data-analysis-2.html#cb1349-15" tabindex="-1"></a>                          <span class="at">group =</span> .draw),</span>
-<span id="cb1349-16"><a href="bayesian-data-analysis-2.html#cb1349-16" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1349-17"><a href="bayesian-data-analysis-2.html#cb1349-17" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1349-18"><a href="bayesian-data-analysis-2.html#cb1349-18" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-53-1.png" width="672" /></p>
+<div class="sourceCode" id="cb97"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb97-1"><a href="bayesian-data-analysis-2.html#cb97-1" tabindex="-1"></a>df.model <span class="ot">=</span> <span class="fu">tibble</span>(<span class="at">days =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb97-2"><a href="bayesian-data-analysis-2.html#cb97-2" tabindex="-1"></a>  <span class="fu">add_linpred_draws</span>(<span class="at">newdata =</span> .,</span>
+<span id="cb97-3"><a href="bayesian-data-analysis-2.html#cb97-3" tabindex="-1"></a>                    <span class="at">object =</span> fit.brm_sleep,</span>
+<span id="cb97-4"><a href="bayesian-data-analysis-2.html#cb97-4" tabindex="-1"></a>                    <span class="at">ndraws =</span> <span class="dv">10</span>,</span>
+<span id="cb97-5"><a href="bayesian-data-analysis-2.html#cb97-5" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>,</span>
+<span id="cb97-6"><a href="bayesian-data-analysis-2.html#cb97-6" tabindex="-1"></a>                    <span class="at">re_formula =</span> <span class="cn">NA</span>)</span>
+<span id="cb97-7"><a href="bayesian-data-analysis-2.html#cb97-7" tabindex="-1"></a></span>
+<span id="cb97-8"><a href="bayesian-data-analysis-2.html#cb97-8" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.sleep,</span>
+<span id="cb97-9"><a href="bayesian-data-analysis-2.html#cb97-9" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
+<span id="cb97-10"><a href="bayesian-data-analysis-2.html#cb97-10" tabindex="-1"></a>                     <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
+<span id="cb97-11"><a href="bayesian-data-analysis-2.html#cb97-11" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
+<span id="cb97-12"><a href="bayesian-data-analysis-2.html#cb97-12" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitter</span>(<span class="at">width =</span> <span class="fl">0.1</span>)) <span class="sc">+</span> </span>
+<span id="cb97-13"><a href="bayesian-data-analysis-2.html#cb97-13" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model,</span>
+<span id="cb97-14"><a href="bayesian-data-analysis-2.html#cb97-14" tabindex="-1"></a>            <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
+<span id="cb97-15"><a href="bayesian-data-analysis-2.html#cb97-15" tabindex="-1"></a>                          <span class="at">group =</span> .draw),</span>
+<span id="cb97-16"><a href="bayesian-data-analysis-2.html#cb97-16" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
+<span id="cb97-17"><a href="bayesian-data-analysis-2.html#cb97-17" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>) <span class="sc">+</span></span>
+<span id="cb97-18"><a href="bayesian-data-analysis-2.html#cb97-18" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-53-1.png" width="672" /></p>
 </div>
 <div id="model-prediction-with-credible-intervals-aggregate" class="section level4 hasAnchor" number="23.5.6.2">
 <h4><span class="header-section-number">23.5.6.2</span> Model prediction with credible intervals (aggregate)<a href="bayesian-data-analysis-2.html#model-prediction-with-credible-intervals-aggregate" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1350"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1350-1"><a href="bayesian-data-analysis-2.html#cb1350-1" tabindex="-1"></a>df.model <span class="ot">=</span> fit.brm_sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1350-2"><a href="bayesian-data-analysis-2.html#cb1350-2" tabindex="-1"></a>  <span class="fu">fitted</span>(<span class="at">re_formula =</span> <span class="cn">NA</span>,</span>
-<span id="cb1350-3"><a href="bayesian-data-analysis-2.html#cb1350-3" tabindex="-1"></a>         <span class="at">newdata =</span> <span class="fu">tibble</span>(<span class="at">days =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1350-4"><a href="bayesian-data-analysis-2.html#cb1350-4" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1350-5"><a href="bayesian-data-analysis-2.html#cb1350-5" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">days =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1350-6"><a href="bayesian-data-analysis-2.html#cb1350-6" tabindex="-1"></a>  <span class="fu">clean_names</span>()</span>
-<span id="cb1350-7"><a href="bayesian-data-analysis-2.html#cb1350-7" tabindex="-1"></a>  </span>
-<span id="cb1350-8"><a href="bayesian-data-analysis-2.html#cb1350-8" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.sleep,</span>
-<span id="cb1350-9"><a href="bayesian-data-analysis-2.html#cb1350-9" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
-<span id="cb1350-10"><a href="bayesian-data-analysis-2.html#cb1350-10" tabindex="-1"></a>                     <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
-<span id="cb1350-11"><a href="bayesian-data-analysis-2.html#cb1350-11" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
-<span id="cb1350-12"><a href="bayesian-data-analysis-2.html#cb1350-12" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitter</span>(<span class="at">width =</span> <span class="fl">0.1</span>)) <span class="sc">+</span> </span>
-<span id="cb1350-13"><a href="bayesian-data-analysis-2.html#cb1350-13" tabindex="-1"></a>  <span class="fu">geom_ribbon</span>(<span class="at">data =</span> df.model,</span>
-<span id="cb1350-14"><a href="bayesian-data-analysis-2.html#cb1350-14" tabindex="-1"></a>              <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> estimate,</span>
-<span id="cb1350-15"><a href="bayesian-data-analysis-2.html#cb1350-15" tabindex="-1"></a>                            <span class="at">ymin =</span> q2_5,</span>
-<span id="cb1350-16"><a href="bayesian-data-analysis-2.html#cb1350-16" tabindex="-1"></a>                            <span class="at">ymax =</span> q97_5),</span>
-<span id="cb1350-17"><a href="bayesian-data-analysis-2.html#cb1350-17" tabindex="-1"></a>              <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>,</span>
-<span id="cb1350-18"><a href="bayesian-data-analysis-2.html#cb1350-18" tabindex="-1"></a>              <span class="at">alpha =</span> <span class="fl">0.5</span>) <span class="sc">+</span></span>
-<span id="cb1350-19"><a href="bayesian-data-analysis-2.html#cb1350-19" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model,</span>
-<span id="cb1350-20"><a href="bayesian-data-analysis-2.html#cb1350-20" tabindex="-1"></a>            <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> estimate),</span>
-<span id="cb1350-21"><a href="bayesian-data-analysis-2.html#cb1350-21" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>,</span>
-<span id="cb1350-22"><a href="bayesian-data-analysis-2.html#cb1350-22" tabindex="-1"></a>            <span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span></span>
-<span id="cb1350-23"><a href="bayesian-data-analysis-2.html#cb1350-23" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1350-24"><a href="bayesian-data-analysis-2.html#cb1350-24" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb98"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb98-1"><a href="bayesian-data-analysis-2.html#cb98-1" tabindex="-1"></a>df.model <span class="ot">=</span> fit.brm_sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb98-2"><a href="bayesian-data-analysis-2.html#cb98-2" tabindex="-1"></a>  <span class="fu">fitted</span>(<span class="at">re_formula =</span> <span class="cn">NA</span>,</span>
+<span id="cb98-3"><a href="bayesian-data-analysis-2.html#cb98-3" tabindex="-1"></a>         <span class="at">newdata =</span> <span class="fu">tibble</span>(<span class="at">days =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>)) <span class="sc">%&gt;%</span> </span>
+<span id="cb98-4"><a href="bayesian-data-analysis-2.html#cb98-4" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb98-5"><a href="bayesian-data-analysis-2.html#cb98-5" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">days =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb98-6"><a href="bayesian-data-analysis-2.html#cb98-6" tabindex="-1"></a>  <span class="fu">clean_names</span>()</span>
+<span id="cb98-7"><a href="bayesian-data-analysis-2.html#cb98-7" tabindex="-1"></a>  </span>
+<span id="cb98-8"><a href="bayesian-data-analysis-2.html#cb98-8" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.sleep,</span>
+<span id="cb98-9"><a href="bayesian-data-analysis-2.html#cb98-9" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
+<span id="cb98-10"><a href="bayesian-data-analysis-2.html#cb98-10" tabindex="-1"></a>                     <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
+<span id="cb98-11"><a href="bayesian-data-analysis-2.html#cb98-11" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
+<span id="cb98-12"><a href="bayesian-data-analysis-2.html#cb98-12" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitter</span>(<span class="at">width =</span> <span class="fl">0.1</span>)) <span class="sc">+</span> </span>
+<span id="cb98-13"><a href="bayesian-data-analysis-2.html#cb98-13" tabindex="-1"></a>  <span class="fu">geom_ribbon</span>(<span class="at">data =</span> df.model,</span>
+<span id="cb98-14"><a href="bayesian-data-analysis-2.html#cb98-14" tabindex="-1"></a>              <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> estimate,</span>
+<span id="cb98-15"><a href="bayesian-data-analysis-2.html#cb98-15" tabindex="-1"></a>                            <span class="at">ymin =</span> q2_5,</span>
+<span id="cb98-16"><a href="bayesian-data-analysis-2.html#cb98-16" tabindex="-1"></a>                            <span class="at">ymax =</span> q97_5),</span>
+<span id="cb98-17"><a href="bayesian-data-analysis-2.html#cb98-17" tabindex="-1"></a>              <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>,</span>
+<span id="cb98-18"><a href="bayesian-data-analysis-2.html#cb98-18" tabindex="-1"></a>              <span class="at">alpha =</span> <span class="fl">0.5</span>) <span class="sc">+</span></span>
+<span id="cb98-19"><a href="bayesian-data-analysis-2.html#cb98-19" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model,</span>
+<span id="cb98-20"><a href="bayesian-data-analysis-2.html#cb98-20" tabindex="-1"></a>            <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> estimate),</span>
+<span id="cb98-21"><a href="bayesian-data-analysis-2.html#cb98-21" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>,</span>
+<span id="cb98-22"><a href="bayesian-data-analysis-2.html#cb98-22" tabindex="-1"></a>            <span class="at">size =</span> <span class="dv">1</span>) <span class="sc">+</span></span>
+<span id="cb98-23"><a href="bayesian-data-analysis-2.html#cb98-23" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>) <span class="sc">+</span></span>
+<span id="cb98-24"><a href="bayesian-data-analysis-2.html#cb98-24" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">9</span>)</span></code></pre></div>
 <pre><code>Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
 ℹ Please use `linewidth` instead.
 This warning is displayed once every 8 hours.
-Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.</code></pre>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-54-1.png" width="672" /></p>
+Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
+generated.</code></pre>
+<p><img src="psych252_files/figure-html/unnamed-chunk-54-1.png" width="672" /></p>
 </div>
 <div id="model-prediction-with-credible-intervals-individual-participants" class="section level4 hasAnchor" number="23.5.6.3">
 <h4><span class="header-section-number">23.5.6.3</span> Model prediction with credible intervals (individual participants)<a href="bayesian-data-analysis-2.html#model-prediction-with-credible-intervals-individual-participants" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1352"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1352-1"><a href="bayesian-data-analysis-2.html#cb1352-1" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1352-2"><a href="bayesian-data-analysis-2.html#cb1352-2" tabindex="-1"></a>  <span class="fu">fitted</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1352-3"><a href="bayesian-data-analysis-2.html#cb1352-3" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1352-4"><a href="bayesian-data-analysis-2.html#cb1352-4" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1352-5"><a href="bayesian-data-analysis-2.html#cb1352-5" tabindex="-1"></a>  <span class="fu">bind_cols</span>(df.sleep) <span class="sc">%&gt;%</span> </span>
-<span id="cb1352-6"><a href="bayesian-data-analysis-2.html#cb1352-6" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1352-7"><a href="bayesian-data-analysis-2.html#cb1352-7" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
-<span id="cb1352-8"><a href="bayesian-data-analysis-2.html#cb1352-8" tabindex="-1"></a>                     <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
-<span id="cb1352-9"><a href="bayesian-data-analysis-2.html#cb1352-9" tabindex="-1"></a>  <span class="fu">geom_ribbon</span>(<span class="fu">aes</span>(<span class="at">ymin =</span> q2_5,</span>
-<span id="cb1352-10"><a href="bayesian-data-analysis-2.html#cb1352-10" tabindex="-1"></a>                  <span class="at">ymax =</span> q97_5),</span>
-<span id="cb1352-11"><a href="bayesian-data-analysis-2.html#cb1352-11" tabindex="-1"></a>              <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1352-12"><a href="bayesian-data-analysis-2.html#cb1352-12" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="fu">aes</span>(<span class="at">y =</span> estimate),</span>
-<span id="cb1352-13"><a href="bayesian-data-analysis-2.html#cb1352-13" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;blue&quot;</span>) <span class="sc">+</span></span>
-<span id="cb1352-14"><a href="bayesian-data-analysis-2.html#cb1352-14" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
-<span id="cb1352-15"><a href="bayesian-data-analysis-2.html#cb1352-15" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span>subject, <span class="at">ncol =</span> <span class="dv">5</span>) <span class="sc">+</span></span>
-<span id="cb1352-16"><a href="bayesian-data-analysis-2.html#cb1352-16" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">&quot;Days of sleep deprivation&quot;</span>, </span>
-<span id="cb1352-17"><a href="bayesian-data-analysis-2.html#cb1352-17" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">&quot;Average reaction time (ms)&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1352-18"><a href="bayesian-data-analysis-2.html#cb1352-18" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">4</span> <span class="sc">*</span> <span class="dv">2</span>) <span class="sc">+</span></span>
-<span id="cb1352-19"><a href="bayesian-data-analysis-2.html#cb1352-19" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">strip.text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>),</span>
-<span id="cb1352-20"><a href="bayesian-data-analysis-2.html#cb1352-20" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>))</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-55-1.png" width="672" /></p>
+<div class="sourceCode" id="cb100"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb100-1"><a href="bayesian-data-analysis-2.html#cb100-1" tabindex="-1"></a>fit.brm_sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb100-2"><a href="bayesian-data-analysis-2.html#cb100-2" tabindex="-1"></a>  <span class="fu">fitted</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb100-3"><a href="bayesian-data-analysis-2.html#cb100-3" tabindex="-1"></a>  <span class="fu">as_tibble</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb100-4"><a href="bayesian-data-analysis-2.html#cb100-4" tabindex="-1"></a>  <span class="fu">clean_names</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb100-5"><a href="bayesian-data-analysis-2.html#cb100-5" tabindex="-1"></a>  <span class="fu">bind_cols</span>(df.sleep) <span class="sc">%&gt;%</span> </span>
+<span id="cb100-6"><a href="bayesian-data-analysis-2.html#cb100-6" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb100-7"><a href="bayesian-data-analysis-2.html#cb100-7" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
+<span id="cb100-8"><a href="bayesian-data-analysis-2.html#cb100-8" tabindex="-1"></a>                     <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
+<span id="cb100-9"><a href="bayesian-data-analysis-2.html#cb100-9" tabindex="-1"></a>  <span class="fu">geom_ribbon</span>(<span class="fu">aes</span>(<span class="at">ymin =</span> q2_5,</span>
+<span id="cb100-10"><a href="bayesian-data-analysis-2.html#cb100-10" tabindex="-1"></a>                  <span class="at">ymax =</span> q97_5),</span>
+<span id="cb100-11"><a href="bayesian-data-analysis-2.html#cb100-11" tabindex="-1"></a>              <span class="at">fill =</span> <span class="st">&quot;lightblue&quot;</span>) <span class="sc">+</span></span>
+<span id="cb100-12"><a href="bayesian-data-analysis-2.html#cb100-12" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="fu">aes</span>(<span class="at">y =</span> estimate),</span>
+<span id="cb100-13"><a href="bayesian-data-analysis-2.html#cb100-13" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;blue&quot;</span>) <span class="sc">+</span></span>
+<span id="cb100-14"><a href="bayesian-data-analysis-2.html#cb100-14" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
+<span id="cb100-15"><a href="bayesian-data-analysis-2.html#cb100-15" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span>subject, <span class="at">ncol =</span> <span class="dv">5</span>) <span class="sc">+</span></span>
+<span id="cb100-16"><a href="bayesian-data-analysis-2.html#cb100-16" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">&quot;Days of sleep deprivation&quot;</span>, </span>
+<span id="cb100-17"><a href="bayesian-data-analysis-2.html#cb100-17" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">&quot;Average reaction time (ms)&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb100-18"><a href="bayesian-data-analysis-2.html#cb100-18" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">4</span> <span class="sc">*</span> <span class="dv">2</span>) <span class="sc">+</span></span>
+<span id="cb100-19"><a href="bayesian-data-analysis-2.html#cb100-19" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">strip.text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>),</span>
+<span id="cb100-20"><a href="bayesian-data-analysis-2.html#cb100-20" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>))</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-55-1.png" width="672" /></p>
 </div>
 <div id="model-prediction-for-random-samples" class="section level4 hasAnchor" number="23.5.6.4">
 <h4><span class="header-section-number">23.5.6.4</span> Model prediction for random samples<a href="bayesian-data-analysis-2.html#model-prediction-for-random-samples" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1353"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1353-1"><a href="bayesian-data-analysis-2.html#cb1353-1" tabindex="-1"></a>df.model <span class="ot">=</span> df.sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1353-2"><a href="bayesian-data-analysis-2.html#cb1353-2" tabindex="-1"></a>  <span class="fu">complete</span>(subject, days) <span class="sc">%&gt;%</span> </span>
-<span id="cb1353-3"><a href="bayesian-data-analysis-2.html#cb1353-3" tabindex="-1"></a>  <span class="fu">add_linpred_draws</span>(<span class="at">newdata =</span> .,</span>
-<span id="cb1353-4"><a href="bayesian-data-analysis-2.html#cb1353-4" tabindex="-1"></a>                    <span class="at">object =</span> fit.brm_sleep,</span>
-<span id="cb1353-5"><a href="bayesian-data-analysis-2.html#cb1353-5" tabindex="-1"></a>                    <span class="at">ndraws =</span> <span class="dv">10</span>,</span>
-<span id="cb1353-6"><a href="bayesian-data-analysis-2.html#cb1353-6" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>)</span>
-<span id="cb1353-7"><a href="bayesian-data-analysis-2.html#cb1353-7" tabindex="-1"></a></span>
-<span id="cb1353-8"><a href="bayesian-data-analysis-2.html#cb1353-8" tabindex="-1"></a>df.sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1353-9"><a href="bayesian-data-analysis-2.html#cb1353-9" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1353-10"><a href="bayesian-data-analysis-2.html#cb1353-10" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
-<span id="cb1353-11"><a href="bayesian-data-analysis-2.html#cb1353-11" tabindex="-1"></a>                       <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
-<span id="cb1353-12"><a href="bayesian-data-analysis-2.html#cb1353-12" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model,</span>
-<span id="cb1353-13"><a href="bayesian-data-analysis-2.html#cb1353-13" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
-<span id="cb1353-14"><a href="bayesian-data-analysis-2.html#cb1353-14" tabindex="-1"></a>                <span class="at">group =</span> .draw),</span>
-<span id="cb1353-15"><a href="bayesian-data-analysis-2.html#cb1353-15" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>,</span>
-<span id="cb1353-16"><a href="bayesian-data-analysis-2.html#cb1353-16" tabindex="-1"></a>            <span class="at">alpha =</span> <span class="fl">0.5</span>) <span class="sc">+</span> </span>
-<span id="cb1353-17"><a href="bayesian-data-analysis-2.html#cb1353-17" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
-<span id="cb1353-18"><a href="bayesian-data-analysis-2.html#cb1353-18" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span>subject, <span class="at">ncol =</span> <span class="dv">5</span>) <span class="sc">+</span></span>
-<span id="cb1353-19"><a href="bayesian-data-analysis-2.html#cb1353-19" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">&quot;Days of sleep deprivation&quot;</span>, </span>
-<span id="cb1353-20"><a href="bayesian-data-analysis-2.html#cb1353-20" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">&quot;Average reaction time (ms)&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1353-21"><a href="bayesian-data-analysis-2.html#cb1353-21" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">4</span> <span class="sc">*</span> <span class="dv">2</span>) <span class="sc">+</span></span>
-<span id="cb1353-22"><a href="bayesian-data-analysis-2.html#cb1353-22" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">strip.text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>),</span>
-<span id="cb1353-23"><a href="bayesian-data-analysis-2.html#cb1353-23" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>))</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-56-1.png" width="672" /></p>
+<div class="sourceCode" id="cb101"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb101-1"><a href="bayesian-data-analysis-2.html#cb101-1" tabindex="-1"></a>df.model <span class="ot">=</span> df.sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb101-2"><a href="bayesian-data-analysis-2.html#cb101-2" tabindex="-1"></a>  <span class="fu">complete</span>(subject, days) <span class="sc">%&gt;%</span> </span>
+<span id="cb101-3"><a href="bayesian-data-analysis-2.html#cb101-3" tabindex="-1"></a>  <span class="fu">add_linpred_draws</span>(<span class="at">newdata =</span> .,</span>
+<span id="cb101-4"><a href="bayesian-data-analysis-2.html#cb101-4" tabindex="-1"></a>                    <span class="at">object =</span> fit.brm_sleep,</span>
+<span id="cb101-5"><a href="bayesian-data-analysis-2.html#cb101-5" tabindex="-1"></a>                    <span class="at">ndraws =</span> <span class="dv">10</span>,</span>
+<span id="cb101-6"><a href="bayesian-data-analysis-2.html#cb101-6" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>)</span>
+<span id="cb101-7"><a href="bayesian-data-analysis-2.html#cb101-7" tabindex="-1"></a></span>
+<span id="cb101-8"><a href="bayesian-data-analysis-2.html#cb101-8" tabindex="-1"></a>df.sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb101-9"><a href="bayesian-data-analysis-2.html#cb101-9" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb101-10"><a href="bayesian-data-analysis-2.html#cb101-10" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
+<span id="cb101-11"><a href="bayesian-data-analysis-2.html#cb101-11" tabindex="-1"></a>                       <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
+<span id="cb101-12"><a href="bayesian-data-analysis-2.html#cb101-12" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model,</span>
+<span id="cb101-13"><a href="bayesian-data-analysis-2.html#cb101-13" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
+<span id="cb101-14"><a href="bayesian-data-analysis-2.html#cb101-14" tabindex="-1"></a>                <span class="at">group =</span> .draw),</span>
+<span id="cb101-15"><a href="bayesian-data-analysis-2.html#cb101-15" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;lightblue&quot;</span>,</span>
+<span id="cb101-16"><a href="bayesian-data-analysis-2.html#cb101-16" tabindex="-1"></a>            <span class="at">alpha =</span> <span class="fl">0.5</span>) <span class="sc">+</span> </span>
+<span id="cb101-17"><a href="bayesian-data-analysis-2.html#cb101-17" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
+<span id="cb101-18"><a href="bayesian-data-analysis-2.html#cb101-18" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span>subject, <span class="at">ncol =</span> <span class="dv">5</span>) <span class="sc">+</span></span>
+<span id="cb101-19"><a href="bayesian-data-analysis-2.html#cb101-19" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">&quot;Days of sleep deprivation&quot;</span>, </span>
+<span id="cb101-20"><a href="bayesian-data-analysis-2.html#cb101-20" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">&quot;Average reaction time (ms)&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb101-21"><a href="bayesian-data-analysis-2.html#cb101-21" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">4</span> <span class="sc">*</span> <span class="dv">2</span>) <span class="sc">+</span></span>
+<span id="cb101-22"><a href="bayesian-data-analysis-2.html#cb101-22" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">strip.text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>),</span>
+<span id="cb101-23"><a href="bayesian-data-analysis-2.html#cb101-23" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>))</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-56-1.png" width="672" /></p>
 </div>
 <div id="animated-model-prediction-for-random-samples" class="section level4 hasAnchor" number="23.5.6.5">
 <h4><span class="header-section-number">23.5.6.5</span> Animated model prediction for random samples<a href="bayesian-data-analysis-2.html#animated-model-prediction-for-random-samples" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1354"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1354-1"><a href="bayesian-data-analysis-2.html#cb1354-1" tabindex="-1"></a>df.model <span class="ot">=</span> df.sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1354-2"><a href="bayesian-data-analysis-2.html#cb1354-2" tabindex="-1"></a>  <span class="fu">complete</span>(subject, days) <span class="sc">%&gt;%</span> </span>
-<span id="cb1354-3"><a href="bayesian-data-analysis-2.html#cb1354-3" tabindex="-1"></a>  <span class="fu">add_linpred_draws</span>(<span class="at">newdata =</span> .,</span>
-<span id="cb1354-4"><a href="bayesian-data-analysis-2.html#cb1354-4" tabindex="-1"></a>                    <span class="at">object =</span> fit.brm_sleep,</span>
-<span id="cb1354-5"><a href="bayesian-data-analysis-2.html#cb1354-5" tabindex="-1"></a>                    <span class="at">ndraws =</span> <span class="dv">10</span>,</span>
-<span id="cb1354-6"><a href="bayesian-data-analysis-2.html#cb1354-6" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>)</span>
-<span id="cb1354-7"><a href="bayesian-data-analysis-2.html#cb1354-7" tabindex="-1"></a></span>
-<span id="cb1354-8"><a href="bayesian-data-analysis-2.html#cb1354-8" tabindex="-1"></a>p <span class="ot">=</span> df.sleep <span class="sc">%&gt;%</span> </span>
-<span id="cb1354-9"><a href="bayesian-data-analysis-2.html#cb1354-9" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1354-10"><a href="bayesian-data-analysis-2.html#cb1354-10" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
-<span id="cb1354-11"><a href="bayesian-data-analysis-2.html#cb1354-11" tabindex="-1"></a>                       <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
-<span id="cb1354-12"><a href="bayesian-data-analysis-2.html#cb1354-12" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model,</span>
-<span id="cb1354-13"><a href="bayesian-data-analysis-2.html#cb1354-13" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
-<span id="cb1354-14"><a href="bayesian-data-analysis-2.html#cb1354-14" tabindex="-1"></a>                <span class="at">group =</span> .draw),</span>
-<span id="cb1354-15"><a href="bayesian-data-analysis-2.html#cb1354-15" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;black&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1354-16"><a href="bayesian-data-analysis-2.html#cb1354-16" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
-<span id="cb1354-17"><a href="bayesian-data-analysis-2.html#cb1354-17" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span>subject, <span class="at">ncol =</span> <span class="dv">5</span>) <span class="sc">+</span></span>
-<span id="cb1354-18"><a href="bayesian-data-analysis-2.html#cb1354-18" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">&quot;Days of sleep deprivation&quot;</span>, </span>
-<span id="cb1354-19"><a href="bayesian-data-analysis-2.html#cb1354-19" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">&quot;Average reaction time (ms)&quot;</span>) <span class="sc">+</span> </span>
-<span id="cb1354-20"><a href="bayesian-data-analysis-2.html#cb1354-20" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">4</span> <span class="sc">*</span> <span class="dv">2</span>) <span class="sc">+</span></span>
-<span id="cb1354-21"><a href="bayesian-data-analysis-2.html#cb1354-21" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">strip.text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>),</span>
-<span id="cb1354-22"><a href="bayesian-data-analysis-2.html#cb1354-22" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>)) <span class="sc">+</span> </span>
-<span id="cb1354-23"><a href="bayesian-data-analysis-2.html#cb1354-23" tabindex="-1"></a>  <span class="fu">transition_states</span>(.draw, <span class="dv">0</span>, <span class="dv">1</span>) <span class="sc">+</span></span>
-<span id="cb1354-24"><a href="bayesian-data-analysis-2.html#cb1354-24" tabindex="-1"></a>  <span class="fu">shadow_mark</span>(<span class="at">past =</span> <span class="cn">TRUE</span>, <span class="at">alpha =</span> <span class="dv">1</span><span class="sc">/</span><span class="dv">5</span>, <span class="at">color =</span> <span class="st">&quot;gray50&quot;</span>)</span>
-<span id="cb1354-25"><a href="bayesian-data-analysis-2.html#cb1354-25" tabindex="-1"></a></span>
-<span id="cb1354-26"><a href="bayesian-data-analysis-2.html#cb1354-26" tabindex="-1"></a><span class="fu">animate</span>(p, <span class="at">nframes =</span> <span class="dv">10</span>, <span class="at">fps =</span> <span class="dv">1</span>, <span class="at">width =</span> <span class="dv">800</span>, <span class="at">height =</span> <span class="dv">600</span>, <span class="at">res =</span> <span class="dv">96</span>, <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span></code></pre></div>
-<pre><code>Warning: No renderer available. Please install the gifski, av, or magick
-package to create animated output</code></pre>
-<div class="sourceCode" id="cb1356"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1356-1"><a href="bayesian-data-analysis-2.html#cb1356-1" tabindex="-1"></a><span class="co"># anim_save(&quot;sleep_posterior_predictive.gif&quot;)</span></span></code></pre></div>
+<div class="sourceCode" id="cb102"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb102-1"><a href="bayesian-data-analysis-2.html#cb102-1" tabindex="-1"></a>df.model <span class="ot">=</span> df.sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb102-2"><a href="bayesian-data-analysis-2.html#cb102-2" tabindex="-1"></a>  <span class="fu">complete</span>(subject, days) <span class="sc">%&gt;%</span> </span>
+<span id="cb102-3"><a href="bayesian-data-analysis-2.html#cb102-3" tabindex="-1"></a>  <span class="fu">add_linpred_draws</span>(<span class="at">newdata =</span> .,</span>
+<span id="cb102-4"><a href="bayesian-data-analysis-2.html#cb102-4" tabindex="-1"></a>                    <span class="at">object =</span> fit.brm_sleep,</span>
+<span id="cb102-5"><a href="bayesian-data-analysis-2.html#cb102-5" tabindex="-1"></a>                    <span class="at">ndraws =</span> <span class="dv">10</span>,</span>
+<span id="cb102-6"><a href="bayesian-data-analysis-2.html#cb102-6" tabindex="-1"></a>                    <span class="at">seed =</span> <span class="dv">1</span>)</span>
+<span id="cb102-7"><a href="bayesian-data-analysis-2.html#cb102-7" tabindex="-1"></a></span>
+<span id="cb102-8"><a href="bayesian-data-analysis-2.html#cb102-8" tabindex="-1"></a>p <span class="ot">=</span> df.sleep <span class="sc">%&gt;%</span> </span>
+<span id="cb102-9"><a href="bayesian-data-analysis-2.html#cb102-9" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb102-10"><a href="bayesian-data-analysis-2.html#cb102-10" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> days,</span>
+<span id="cb102-11"><a href="bayesian-data-analysis-2.html#cb102-11" tabindex="-1"></a>                       <span class="at">y =</span> reaction)) <span class="sc">+</span> </span>
+<span id="cb102-12"><a href="bayesian-data-analysis-2.html#cb102-12" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model,</span>
+<span id="cb102-13"><a href="bayesian-data-analysis-2.html#cb102-13" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
+<span id="cb102-14"><a href="bayesian-data-analysis-2.html#cb102-14" tabindex="-1"></a>                <span class="at">group =</span> .draw),</span>
+<span id="cb102-15"><a href="bayesian-data-analysis-2.html#cb102-15" tabindex="-1"></a>            <span class="at">color =</span> <span class="st">&quot;black&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb102-16"><a href="bayesian-data-analysis-2.html#cb102-16" tabindex="-1"></a>  <span class="fu">geom_point</span>() <span class="sc">+</span></span>
+<span id="cb102-17"><a href="bayesian-data-analysis-2.html#cb102-17" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span>subject, <span class="at">ncol =</span> <span class="dv">5</span>) <span class="sc">+</span></span>
+<span id="cb102-18"><a href="bayesian-data-analysis-2.html#cb102-18" tabindex="-1"></a>  <span class="fu">labs</span>(<span class="at">x =</span> <span class="st">&quot;Days of sleep deprivation&quot;</span>, </span>
+<span id="cb102-19"><a href="bayesian-data-analysis-2.html#cb102-19" tabindex="-1"></a>       <span class="at">y =</span> <span class="st">&quot;Average reaction time (ms)&quot;</span>) <span class="sc">+</span> </span>
+<span id="cb102-20"><a href="bayesian-data-analysis-2.html#cb102-20" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">breaks =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">4</span> <span class="sc">*</span> <span class="dv">2</span>) <span class="sc">+</span></span>
+<span id="cb102-21"><a href="bayesian-data-analysis-2.html#cb102-21" tabindex="-1"></a>  <span class="fu">theme</span>(<span class="at">strip.text =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>),</span>
+<span id="cb102-22"><a href="bayesian-data-analysis-2.html#cb102-22" tabindex="-1"></a>        <span class="at">axis.text.y =</span> <span class="fu">element_text</span>(<span class="at">size =</span> <span class="dv">12</span>)) <span class="sc">+</span> </span>
+<span id="cb102-23"><a href="bayesian-data-analysis-2.html#cb102-23" tabindex="-1"></a>  <span class="fu">transition_states</span>(.draw, <span class="dv">0</span>, <span class="dv">1</span>) <span class="sc">+</span></span>
+<span id="cb102-24"><a href="bayesian-data-analysis-2.html#cb102-24" tabindex="-1"></a>  <span class="fu">shadow_mark</span>(<span class="at">past =</span> <span class="cn">TRUE</span>, <span class="at">alpha =</span> <span class="dv">1</span><span class="sc">/</span><span class="dv">5</span>, <span class="at">color =</span> <span class="st">&quot;gray50&quot;</span>)</span>
+<span id="cb102-25"><a href="bayesian-data-analysis-2.html#cb102-25" tabindex="-1"></a></span>
+<span id="cb102-26"><a href="bayesian-data-analysis-2.html#cb102-26" tabindex="-1"></a><span class="fu">animate</span>(p, <span class="at">nframes =</span> <span class="dv">10</span>, <span class="at">fps =</span> <span class="dv">1</span>, <span class="at">width =</span> <span class="dv">800</span>, <span class="at">height =</span> <span class="dv">600</span>, <span class="at">res =</span> <span class="dv">96</span>, <span class="at">type =</span> <span class="st">&quot;cairo&quot;</span>)</span>
+<span id="cb102-27"><a href="bayesian-data-analysis-2.html#cb102-27" tabindex="-1"></a></span>
+<span id="cb102-28"><a href="bayesian-data-analysis-2.html#cb102-28" tabindex="-1"></a><span class="co"># anim_save(&quot;sleep_posterior_predictive.gif&quot;)</span></span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-57-1.gif" /><!-- --></p>
 </div>
 </div>
 </div>
@@ -2076,30 +2078,30 @@ <h4><span class="header-section-number">23.5.6.5</span> Animated model predictio
 <h2><span class="header-section-number">23.6</span> Titanic study<a href="bayesian-data-analysis-2.html#titanic-study" class="anchor-section" aria-label="Anchor link to header"></a></h2>
 <div id="visualize-the-data-2" class="section level3 hasAnchor" number="23.6.1">
 <h3><span class="header-section-number">23.6.1</span> 1. Visualize the data<a href="bayesian-data-analysis-2.html#visualize-the-data-2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1357"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1357-1"><a href="bayesian-data-analysis-2.html#cb1357-1" tabindex="-1"></a>df.titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1357-2"><a href="bayesian-data-analysis-2.html#cb1357-2" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">sex =</span> <span class="fu">as.factor</span>(sex)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1357-3"><a href="bayesian-data-analysis-2.html#cb1357-3" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1357-4"><a href="bayesian-data-analysis-2.html#cb1357-4" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> fare,</span>
-<span id="cb1357-5"><a href="bayesian-data-analysis-2.html#cb1357-5" tabindex="-1"></a>                       <span class="at">y =</span> survived,</span>
-<span id="cb1357-6"><a href="bayesian-data-analysis-2.html#cb1357-6" tabindex="-1"></a>                       <span class="at">color =</span> sex)) <span class="sc">+</span></span>
-<span id="cb1357-7"><a href="bayesian-data-analysis-2.html#cb1357-7" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.1</span>, <span class="at">size =</span> <span class="dv">2</span>) <span class="sc">+</span> </span>
-<span id="cb1357-8"><a href="bayesian-data-analysis-2.html#cb1357-8" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">method =</span> <span class="st">&quot;glm&quot;</span>,</span>
-<span id="cb1357-9"><a href="bayesian-data-analysis-2.html#cb1357-9" tabindex="-1"></a>              <span class="at">method.args =</span> <span class="fu">list</span>(<span class="at">family =</span> <span class="st">&quot;binomial&quot;</span>),</span>
-<span id="cb1357-10"><a href="bayesian-data-analysis-2.html#cb1357-10" tabindex="-1"></a>              <span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
-<span id="cb1357-11"><a href="bayesian-data-analysis-2.html#cb1357-11" tabindex="-1"></a>              <span class="fu">aes</span>(<span class="at">fill =</span> sex)) <span class="sc">+</span></span>
-<span id="cb1357-12"><a href="bayesian-data-analysis-2.html#cb1357-12" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">palette =</span> <span class="st">&quot;Set1&quot;</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-58-1.png" width="672" /></p>
+<div class="sourceCode" id="cb103"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb103-1"><a href="bayesian-data-analysis-2.html#cb103-1" tabindex="-1"></a>df.titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb103-2"><a href="bayesian-data-analysis-2.html#cb103-2" tabindex="-1"></a>  <span class="fu">mutate</span>(<span class="at">sex =</span> <span class="fu">as.factor</span>(sex)) <span class="sc">%&gt;%</span> </span>
+<span id="cb103-3"><a href="bayesian-data-analysis-2.html#cb103-3" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb103-4"><a href="bayesian-data-analysis-2.html#cb103-4" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> fare,</span>
+<span id="cb103-5"><a href="bayesian-data-analysis-2.html#cb103-5" tabindex="-1"></a>                       <span class="at">y =</span> survived,</span>
+<span id="cb103-6"><a href="bayesian-data-analysis-2.html#cb103-6" tabindex="-1"></a>                       <span class="at">color =</span> sex)) <span class="sc">+</span></span>
+<span id="cb103-7"><a href="bayesian-data-analysis-2.html#cb103-7" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.1</span>, <span class="at">size =</span> <span class="dv">2</span>) <span class="sc">+</span> </span>
+<span id="cb103-8"><a href="bayesian-data-analysis-2.html#cb103-8" tabindex="-1"></a>  <span class="fu">geom_smooth</span>(<span class="at">method =</span> <span class="st">&quot;glm&quot;</span>,</span>
+<span id="cb103-9"><a href="bayesian-data-analysis-2.html#cb103-9" tabindex="-1"></a>              <span class="at">method.args =</span> <span class="fu">list</span>(<span class="at">family =</span> <span class="st">&quot;binomial&quot;</span>),</span>
+<span id="cb103-10"><a href="bayesian-data-analysis-2.html#cb103-10" tabindex="-1"></a>              <span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
+<span id="cb103-11"><a href="bayesian-data-analysis-2.html#cb103-11" tabindex="-1"></a>              <span class="fu">aes</span>(<span class="at">fill =</span> sex)) <span class="sc">+</span></span>
+<span id="cb103-12"><a href="bayesian-data-analysis-2.html#cb103-12" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">palette =</span> <span class="st">&quot;Set1&quot;</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-58-1.png" width="672" /></p>
 </div>
 <div id="specify-and-fit-the-model-2" class="section level3 hasAnchor" number="23.6.2">
 <h3><span class="header-section-number">23.6.2</span> 2. Specify and fit the model<a href="bayesian-data-analysis-2.html#specify-and-fit-the-model-2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="frequentist-analysis-2" class="section level4 hasAnchor" number="23.6.2.1">
 <h4><span class="header-section-number">23.6.2.1</span> Frequentist analysis<a href="bayesian-data-analysis-2.html#frequentist-analysis-2" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1358"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1358-1"><a href="bayesian-data-analysis-2.html#cb1358-1" tabindex="-1"></a>fit.glm_titanic <span class="ot">=</span> <span class="fu">glm</span>(<span class="at">formula =</span> survived <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> fare <span class="sc">*</span> sex,</span>
-<span id="cb1358-2"><a href="bayesian-data-analysis-2.html#cb1358-2" tabindex="-1"></a>                      <span class="at">family =</span> <span class="st">&quot;binomial&quot;</span>,</span>
-<span id="cb1358-3"><a href="bayesian-data-analysis-2.html#cb1358-3" tabindex="-1"></a>                      <span class="at">data =</span> df.titanic)</span>
-<span id="cb1358-4"><a href="bayesian-data-analysis-2.html#cb1358-4" tabindex="-1"></a></span>
-<span id="cb1358-5"><a href="bayesian-data-analysis-2.html#cb1358-5" tabindex="-1"></a>fit.glm_titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1358-6"><a href="bayesian-data-analysis-2.html#cb1358-6" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb104"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb104-1"><a href="bayesian-data-analysis-2.html#cb104-1" tabindex="-1"></a>fit.glm_titanic <span class="ot">=</span> <span class="fu">glm</span>(<span class="at">formula =</span> survived <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> fare <span class="sc">*</span> sex,</span>
+<span id="cb104-2"><a href="bayesian-data-analysis-2.html#cb104-2" tabindex="-1"></a>                      <span class="at">family =</span> <span class="st">&quot;binomial&quot;</span>,</span>
+<span id="cb104-3"><a href="bayesian-data-analysis-2.html#cb104-3" tabindex="-1"></a>                      <span class="at">data =</span> df.titanic)</span>
+<span id="cb104-4"><a href="bayesian-data-analysis-2.html#cb104-4" tabindex="-1"></a></span>
+<span id="cb104-5"><a href="bayesian-data-analysis-2.html#cb104-5" tabindex="-1"></a>fit.glm_titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb104-6"><a href="bayesian-data-analysis-2.html#cb104-6" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
 <pre><code>
 Call:
 glm(formula = survived ~ 1 + fare * sex, family = &quot;binomial&quot;, 
@@ -2124,19 +2126,19 @@ <h4><span class="header-section-number">23.6.2.1</span> Frequentist analysis<a h
 </div>
 <div id="bayesian-analysis-1" class="section level4 hasAnchor" number="23.6.2.2">
 <h4><span class="header-section-number">23.6.2.2</span> Bayesian analysis<a href="bayesian-data-analysis-2.html#bayesian-analysis-1" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1360"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1360-1"><a href="bayesian-data-analysis-2.html#cb1360-1" tabindex="-1"></a>fit.brm_titanic <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> survived <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> fare <span class="sc">*</span> sex,</span>
-<span id="cb1360-2"><a href="bayesian-data-analysis-2.html#cb1360-2" tabindex="-1"></a>                      <span class="at">family =</span> <span class="st">&quot;bernoulli&quot;</span>,</span>
-<span id="cb1360-3"><a href="bayesian-data-analysis-2.html#cb1360-3" tabindex="-1"></a>                      <span class="at">data =</span> df.titanic,</span>
-<span id="cb1360-4"><a href="bayesian-data-analysis-2.html#cb1360-4" tabindex="-1"></a>                      <span class="at">file =</span> <span class="st">&quot;cache/brm_titanic&quot;</span>,</span>
-<span id="cb1360-5"><a href="bayesian-data-analysis-2.html#cb1360-5" tabindex="-1"></a>                      <span class="at">seed =</span> <span class="dv">1</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb106"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb106-1"><a href="bayesian-data-analysis-2.html#cb106-1" tabindex="-1"></a>fit.brm_titanic <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> survived <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> fare <span class="sc">*</span> sex,</span>
+<span id="cb106-2"><a href="bayesian-data-analysis-2.html#cb106-2" tabindex="-1"></a>                      <span class="at">family =</span> <span class="st">&quot;bernoulli&quot;</span>,</span>
+<span id="cb106-3"><a href="bayesian-data-analysis-2.html#cb106-3" tabindex="-1"></a>                      <span class="at">data =</span> df.titanic,</span>
+<span id="cb106-4"><a href="bayesian-data-analysis-2.html#cb106-4" tabindex="-1"></a>                      <span class="at">file =</span> <span class="st">&quot;cache/brm_titanic&quot;</span>,</span>
+<span id="cb106-5"><a href="bayesian-data-analysis-2.html#cb106-5" tabindex="-1"></a>                      <span class="at">seed =</span> <span class="dv">1</span>)</span></code></pre></div>
 </div>
 </div>
 <div id="model-evaluation-2" class="section level3 hasAnchor" number="23.6.3">
 <h3><span class="header-section-number">23.6.3</span> 3. Model evaluation<a href="bayesian-data-analysis-2.html#model-evaluation-2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="a-did-the-inference-work-2" class="section level4 hasAnchor" number="23.6.3.1">
 <h4><span class="header-section-number">23.6.3.1</span> a) Did the inference work?<a href="bayesian-data-analysis-2.html#a-did-the-inference-work-2" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1361"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1361-1"><a href="bayesian-data-analysis-2.html#cb1361-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1361-2"><a href="bayesian-data-analysis-2.html#cb1361-2" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb107"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb107-1"><a href="bayesian-data-analysis-2.html#cb107-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb107-2"><a href="bayesian-data-analysis-2.html#cb107-2" tabindex="-1"></a>  <span class="fu">summary</span>()</span></code></pre></div>
 <pre><code> Family: bernoulli 
   Links: mu = logit 
 Formula: survived ~ 1 + fare * sex 
@@ -2154,54 +2156,54 @@ <h4><span class="header-section-number">23.6.3.1</span> a) Did the inference wor
 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
 and Tail_ESS are effective sample size measures, and Rhat is the potential
 scale reduction factor on split chains (at convergence, Rhat = 1).</code></pre>
-<div class="sourceCode" id="cb1363"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1363-1"><a href="bayesian-data-analysis-2.html#cb1363-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1363-2"><a href="bayesian-data-analysis-2.html#cb1363-2" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-61-1.png" width="960" /></p>
+<div class="sourceCode" id="cb109"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb109-1"><a href="bayesian-data-analysis-2.html#cb109-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb109-2"><a href="bayesian-data-analysis-2.html#cb109-2" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-61-1.png" width="960" /></p>
 </div>
 <div id="b-visualize-model-predictions-2" class="section level4 hasAnchor" number="23.6.3.2">
 <h4><span class="header-section-number">23.6.3.2</span> b) Visualize model predictions<a href="bayesian-data-analysis-2.html#b-visualize-model-predictions-2" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1364"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1364-1"><a href="bayesian-data-analysis-2.html#cb1364-1" tabindex="-1"></a><span class="fu">pp_check</span>(fit.brm_titanic,</span>
-<span id="cb1364-2"><a href="bayesian-data-analysis-2.html#cb1364-2" tabindex="-1"></a>         <span class="at">ndraws =</span> <span class="dv">100</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-62-1.png" width="672" /></p>
+<div class="sourceCode" id="cb110"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb110-1"><a href="bayesian-data-analysis-2.html#cb110-1" tabindex="-1"></a><span class="fu">pp_check</span>(fit.brm_titanic,</span>
+<span id="cb110-2"><a href="bayesian-data-analysis-2.html#cb110-2" tabindex="-1"></a>         <span class="at">ndraws =</span> <span class="dv">100</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-62-1.png" width="672" /></p>
 <p>Let’s visualize what the posterior predictive would have looked like for a linear model (instead of a logistic model).</p>
-<div class="sourceCode" id="cb1365"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1365-1"><a href="bayesian-data-analysis-2.html#cb1365-1" tabindex="-1"></a>fit.brm_titanic_linear <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> survived <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> fare <span class="sc">*</span> sex,</span>
-<span id="cb1365-2"><a href="bayesian-data-analysis-2.html#cb1365-2" tabindex="-1"></a>                             <span class="at">data =</span> df.titanic,</span>
-<span id="cb1365-3"><a href="bayesian-data-analysis-2.html#cb1365-3" tabindex="-1"></a>                             <span class="at">file =</span> <span class="st">&quot;cache/brm_titanic_linear&quot;</span>,</span>
-<span id="cb1365-4"><a href="bayesian-data-analysis-2.html#cb1365-4" tabindex="-1"></a>                             <span class="at">seed =</span> <span class="dv">1</span>)</span>
-<span id="cb1365-5"><a href="bayesian-data-analysis-2.html#cb1365-5" tabindex="-1"></a></span>
-<span id="cb1365-6"><a href="bayesian-data-analysis-2.html#cb1365-6" tabindex="-1"></a><span class="fu">pp_check</span>(fit.brm_titanic_linear,</span>
-<span id="cb1365-7"><a href="bayesian-data-analysis-2.html#cb1365-7" tabindex="-1"></a>         <span class="at">ndraws =</span> <span class="dv">100</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-63-1.png" width="672" /></p>
+<div class="sourceCode" id="cb111"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb111-1"><a href="bayesian-data-analysis-2.html#cb111-1" tabindex="-1"></a>fit.brm_titanic_linear <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> survived <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> fare <span class="sc">*</span> sex,</span>
+<span id="cb111-2"><a href="bayesian-data-analysis-2.html#cb111-2" tabindex="-1"></a>                             <span class="at">data =</span> df.titanic,</span>
+<span id="cb111-3"><a href="bayesian-data-analysis-2.html#cb111-3" tabindex="-1"></a>                             <span class="at">file =</span> <span class="st">&quot;cache/brm_titanic_linear&quot;</span>,</span>
+<span id="cb111-4"><a href="bayesian-data-analysis-2.html#cb111-4" tabindex="-1"></a>                             <span class="at">seed =</span> <span class="dv">1</span>)</span>
+<span id="cb111-5"><a href="bayesian-data-analysis-2.html#cb111-5" tabindex="-1"></a></span>
+<span id="cb111-6"><a href="bayesian-data-analysis-2.html#cb111-6" tabindex="-1"></a><span class="fu">pp_check</span>(fit.brm_titanic_linear,</span>
+<span id="cb111-7"><a href="bayesian-data-analysis-2.html#cb111-7" tabindex="-1"></a>         <span class="at">ndraws =</span> <span class="dv">100</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-63-1.png" width="672" /></p>
 </div>
 </div>
 <div id="interpret-the-parameters-1" class="section level3 hasAnchor" number="23.6.4">
 <h3><span class="header-section-number">23.6.4</span> 4. Interpret the parameters<a href="bayesian-data-analysis-2.html#interpret-the-parameters-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1366"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1366-1"><a href="bayesian-data-analysis-2.html#cb1366-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1366-2"><a href="bayesian-data-analysis-2.html#cb1366-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1366-3"><a href="bayesian-data-analysis-2.html#cb1366-3" tabindex="-1"></a>  <span class="fu">select</span>(<span class="sc">-</span>lp__) <span class="sc">%&gt;%</span></span>
-<span id="cb1366-4"><a href="bayesian-data-analysis-2.html#cb1366-4" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">everything</span>(),</span>
-<span id="cb1366-5"><a href="bayesian-data-analysis-2.html#cb1366-5" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;variable&quot;</span>,</span>
-<span id="cb1366-6"><a href="bayesian-data-analysis-2.html#cb1366-6" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1366-7"><a href="bayesian-data-analysis-2.html#cb1366-7" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
-<span id="cb1366-8"><a href="bayesian-data-analysis-2.html#cb1366-8" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> variable,</span>
-<span id="cb1366-9"><a href="bayesian-data-analysis-2.html#cb1366-9" tabindex="-1"></a>                       <span class="at">x =</span> value)) <span class="sc">+</span></span>
-<span id="cb1366-10"><a href="bayesian-data-analysis-2.html#cb1366-10" tabindex="-1"></a>  <span class="fu">stat_intervalh</span>() <span class="sc">+</span> </span>
-<span id="cb1366-11"><a href="bayesian-data-analysis-2.html#cb1366-11" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>()</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-64-1.png" width="672" /></p>
-<div class="sourceCode" id="cb1367"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1367-1"><a href="bayesian-data-analysis-2.html#cb1367-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1367-2"><a href="bayesian-data-analysis-2.html#cb1367-2" tabindex="-1"></a>  <span class="fu">parameters</span>(<span class="at">centrality =</span> <span class="st">&quot;mean&quot;</span>,</span>
-<span id="cb1367-3"><a href="bayesian-data-analysis-2.html#cb1367-3" tabindex="-1"></a>             <span class="at">ci =</span> <span class="fl">0.95</span>)</span></code></pre></div>
-<div class="sourceCode" id="cb1368"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1368-1"><a href="bayesian-data-analysis-2.html#cb1368-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1368-2"><a href="bayesian-data-analysis-2.html#cb1368-2" tabindex="-1"></a>  <span class="fu">ggpredict</span>(<span class="at">terms =</span> <span class="fu">c</span>(<span class="st">&quot;fare [0:500]&quot;</span>, <span class="st">&quot;sex&quot;</span>)) <span class="sc">%&gt;%</span> </span>
-<span id="cb1368-3"><a href="bayesian-data-analysis-2.html#cb1368-3" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-66-1.png" width="672" /></p>
+<div class="sourceCode" id="cb112"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb112-1"><a href="bayesian-data-analysis-2.html#cb112-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb112-2"><a href="bayesian-data-analysis-2.html#cb112-2" tabindex="-1"></a>  <span class="fu">as_draws_df</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb112-3"><a href="bayesian-data-analysis-2.html#cb112-3" tabindex="-1"></a>  <span class="fu">select</span>(<span class="sc">-</span>lp__) <span class="sc">%&gt;%</span></span>
+<span id="cb112-4"><a href="bayesian-data-analysis-2.html#cb112-4" tabindex="-1"></a>  <span class="fu">pivot_longer</span>(<span class="at">cols =</span> <span class="fu">everything</span>(),</span>
+<span id="cb112-5"><a href="bayesian-data-analysis-2.html#cb112-5" tabindex="-1"></a>               <span class="at">names_to =</span> <span class="st">&quot;variable&quot;</span>,</span>
+<span id="cb112-6"><a href="bayesian-data-analysis-2.html#cb112-6" tabindex="-1"></a>               <span class="at">values_to =</span> <span class="st">&quot;value&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb112-7"><a href="bayesian-data-analysis-2.html#cb112-7" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">data =</span> .,</span>
+<span id="cb112-8"><a href="bayesian-data-analysis-2.html#cb112-8" tabindex="-1"></a>         <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">y =</span> variable,</span>
+<span id="cb112-9"><a href="bayesian-data-analysis-2.html#cb112-9" tabindex="-1"></a>                       <span class="at">x =</span> value)) <span class="sc">+</span></span>
+<span id="cb112-10"><a href="bayesian-data-analysis-2.html#cb112-10" tabindex="-1"></a>  <span class="fu">stat_intervalh</span>() <span class="sc">+</span> </span>
+<span id="cb112-11"><a href="bayesian-data-analysis-2.html#cb112-11" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>()</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-64-1.png" width="672" /></p>
+<div class="sourceCode" id="cb113"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb113-1"><a href="bayesian-data-analysis-2.html#cb113-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb113-2"><a href="bayesian-data-analysis-2.html#cb113-2" tabindex="-1"></a>  <span class="fu">parameters</span>(<span class="at">centrality =</span> <span class="st">&quot;mean&quot;</span>,</span>
+<span id="cb113-3"><a href="bayesian-data-analysis-2.html#cb113-3" tabindex="-1"></a>             <span class="at">ci =</span> <span class="fl">0.95</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb114"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb114-1"><a href="bayesian-data-analysis-2.html#cb114-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb114-2"><a href="bayesian-data-analysis-2.html#cb114-2" tabindex="-1"></a>  <span class="fu">ggpredict</span>(<span class="at">terms =</span> <span class="fu">c</span>(<span class="st">&quot;fare [0:500]&quot;</span>, <span class="st">&quot;sex&quot;</span>)) <span class="sc">%&gt;%</span> </span>
+<span id="cb114-3"><a href="bayesian-data-analysis-2.html#cb114-3" tabindex="-1"></a>  <span class="fu">plot</span>()</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-66-1.png" width="672" /></p>
 </div>
 <div id="test-specific-hypotheses-2" class="section level3 hasAnchor" number="23.6.5">
 <h3><span class="header-section-number">23.6.5</span> 5. Test specific hypotheses<a href="bayesian-data-analysis-2.html#test-specific-hypotheses-2" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <p>Difference between men and women in survival?</p>
-<div class="sourceCode" id="cb1369"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1369-1"><a href="bayesian-data-analysis-2.html#cb1369-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1369-2"><a href="bayesian-data-analysis-2.html#cb1369-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> sex,</span>
-<span id="cb1369-3"><a href="bayesian-data-analysis-2.html#cb1369-3" tabindex="-1"></a>          <span class="at">type =</span> <span class="st">&quot;response&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb115"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb115-1"><a href="bayesian-data-analysis-2.html#cb115-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb115-2"><a href="bayesian-data-analysis-2.html#cb115-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> sex,</span>
+<span id="cb115-3"><a href="bayesian-data-analysis-2.html#cb115-3" tabindex="-1"></a>          <span class="at">type =</span> <span class="st">&quot;response&quot;</span>)</span></code></pre></div>
 <pre><code>NOTE: Results may be misleading due to involvement in interactions</code></pre>
 <pre><code>$emmeans
  sex    response lower.HPD upper.HPD
@@ -2220,9 +2222,9 @@ <h3><span class="header-section-number">23.6.5</span> 5. Test specific hypothese
 Results are back-transformed from the log odds ratio scale 
 HPD interval probability: 0.95 </code></pre>
 <p>Difference in how fare affected the chances of survival for men and women?</p>
-<div class="sourceCode" id="cb1372"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1372-1"><a href="bayesian-data-analysis-2.html#cb1372-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
-<span id="cb1372-2"><a href="bayesian-data-analysis-2.html#cb1372-2" tabindex="-1"></a>  <span class="fu">emtrends</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> sex,</span>
-<span id="cb1372-3"><a href="bayesian-data-analysis-2.html#cb1372-3" tabindex="-1"></a>           <span class="at">var =</span> <span class="st">&quot;fare&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb118"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb118-1"><a href="bayesian-data-analysis-2.html#cb118-1" tabindex="-1"></a>fit.brm_titanic <span class="sc">%&gt;%</span> </span>
+<span id="cb118-2"><a href="bayesian-data-analysis-2.html#cb118-2" tabindex="-1"></a>  <span class="fu">emtrends</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> sex,</span>
+<span id="cb118-3"><a href="bayesian-data-analysis-2.html#cb118-3" tabindex="-1"></a>           <span class="at">var =</span> <span class="st">&quot;fare&quot;</span>)</span></code></pre></div>
 <pre><code>$emtrends
  sex    fare.trend lower.HPD upper.HPD
  female     0.0206   0.01065    0.0311
@@ -2240,28 +2242,28 @@ <h3><span class="header-section-number">23.6.5</span> 5. Test specific hypothese
 </div>
 <div id="report-results-1" class="section level3 hasAnchor" number="23.6.6">
 <h3><span class="header-section-number">23.6.6</span> 6. Report results<a href="bayesian-data-analysis-2.html#report-results-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1374"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1374-1"><a href="bayesian-data-analysis-2.html#cb1374-1" tabindex="-1"></a>df.model <span class="ot">=</span> <span class="fu">add_linpred_draws</span>(<span class="at">newdata =</span> <span class="fu">expand_grid</span>(<span class="at">sex =</span> <span class="fu">c</span>(<span class="st">&quot;female&quot;</span>, <span class="st">&quot;male&quot;</span>),</span>
-<span id="cb1374-2"><a href="bayesian-data-analysis-2.html#cb1374-2" tabindex="-1"></a>                                                   <span class="at">fare =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">500</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1374-3"><a href="bayesian-data-analysis-2.html#cb1374-3" tabindex="-1"></a>                               <span class="fu">mutate</span>(<span class="at">sex =</span> <span class="fu">factor</span>(sex, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;female&quot;</span>, <span class="st">&quot;male&quot;</span>))),</span>
-<span id="cb1374-4"><a href="bayesian-data-analysis-2.html#cb1374-4" tabindex="-1"></a>                             <span class="at">object =</span> fit.brm_titanic,</span>
-<span id="cb1374-5"><a href="bayesian-data-analysis-2.html#cb1374-5" tabindex="-1"></a>                             <span class="at">ndraws =</span> <span class="dv">10</span>)</span>
-<span id="cb1374-6"><a href="bayesian-data-analysis-2.html#cb1374-6" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.titanic,</span>
-<span id="cb1374-7"><a href="bayesian-data-analysis-2.html#cb1374-7" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> fare,</span>
-<span id="cb1374-8"><a href="bayesian-data-analysis-2.html#cb1374-8" tabindex="-1"></a>                     <span class="at">y =</span> survived,</span>
-<span id="cb1374-9"><a href="bayesian-data-analysis-2.html#cb1374-9" tabindex="-1"></a>                     <span class="at">color =</span> sex)) <span class="sc">+</span></span>
-<span id="cb1374-10"><a href="bayesian-data-analysis-2.html#cb1374-10" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.1</span>, <span class="at">size =</span> <span class="dv">2</span>) <span class="sc">+</span> </span>
-<span id="cb1374-11"><a href="bayesian-data-analysis-2.html#cb1374-11" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model <span class="sc">%&gt;%</span> </span>
-<span id="cb1374-12"><a href="bayesian-data-analysis-2.html#cb1374-12" tabindex="-1"></a>              <span class="fu">filter</span>(sex <span class="sc">==</span> <span class="st">&quot;male&quot;</span>),</span>
-<span id="cb1374-13"><a href="bayesian-data-analysis-2.html#cb1374-13" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
-<span id="cb1374-14"><a href="bayesian-data-analysis-2.html#cb1374-14" tabindex="-1"></a>                <span class="at">group =</span> .draw,</span>
-<span id="cb1374-15"><a href="bayesian-data-analysis-2.html#cb1374-15" tabindex="-1"></a>                <span class="at">color =</span> sex)) <span class="sc">+</span> </span>
-<span id="cb1374-16"><a href="bayesian-data-analysis-2.html#cb1374-16" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model <span class="sc">%&gt;%</span> </span>
-<span id="cb1374-17"><a href="bayesian-data-analysis-2.html#cb1374-17" tabindex="-1"></a>              <span class="fu">filter</span>(sex <span class="sc">==</span> <span class="st">&quot;female&quot;</span>),</span>
-<span id="cb1374-18"><a href="bayesian-data-analysis-2.html#cb1374-18" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
-<span id="cb1374-19"><a href="bayesian-data-analysis-2.html#cb1374-19" tabindex="-1"></a>                <span class="at">group =</span> .draw,</span>
-<span id="cb1374-20"><a href="bayesian-data-analysis-2.html#cb1374-20" tabindex="-1"></a>                <span class="at">color =</span> sex)) <span class="sc">+</span> </span>
-<span id="cb1374-21"><a href="bayesian-data-analysis-2.html#cb1374-21" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">palette =</span> <span class="st">&quot;Set1&quot;</span>)</span></code></pre></div>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-69-1.png" width="672" /></p>
+<div class="sourceCode" id="cb120"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb120-1"><a href="bayesian-data-analysis-2.html#cb120-1" tabindex="-1"></a>df.model <span class="ot">=</span> <span class="fu">add_linpred_draws</span>(<span class="at">newdata =</span> <span class="fu">expand_grid</span>(<span class="at">sex =</span> <span class="fu">c</span>(<span class="st">&quot;female&quot;</span>, <span class="st">&quot;male&quot;</span>),</span>
+<span id="cb120-2"><a href="bayesian-data-analysis-2.html#cb120-2" tabindex="-1"></a>                                                   <span class="at">fare =</span> <span class="dv">0</span><span class="sc">:</span><span class="dv">500</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb120-3"><a href="bayesian-data-analysis-2.html#cb120-3" tabindex="-1"></a>                               <span class="fu">mutate</span>(<span class="at">sex =</span> <span class="fu">factor</span>(sex, <span class="at">levels =</span> <span class="fu">c</span>(<span class="st">&quot;female&quot;</span>, <span class="st">&quot;male&quot;</span>))),</span>
+<span id="cb120-4"><a href="bayesian-data-analysis-2.html#cb120-4" tabindex="-1"></a>                             <span class="at">object =</span> fit.brm_titanic,</span>
+<span id="cb120-5"><a href="bayesian-data-analysis-2.html#cb120-5" tabindex="-1"></a>                             <span class="at">ndraws =</span> <span class="dv">10</span>)</span>
+<span id="cb120-6"><a href="bayesian-data-analysis-2.html#cb120-6" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.titanic,</span>
+<span id="cb120-7"><a href="bayesian-data-analysis-2.html#cb120-7" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> fare,</span>
+<span id="cb120-8"><a href="bayesian-data-analysis-2.html#cb120-8" tabindex="-1"></a>                     <span class="at">y =</span> survived,</span>
+<span id="cb120-9"><a href="bayesian-data-analysis-2.html#cb120-9" tabindex="-1"></a>                     <span class="at">color =</span> sex)) <span class="sc">+</span></span>
+<span id="cb120-10"><a href="bayesian-data-analysis-2.html#cb120-10" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.1</span>, <span class="at">size =</span> <span class="dv">2</span>) <span class="sc">+</span> </span>
+<span id="cb120-11"><a href="bayesian-data-analysis-2.html#cb120-11" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model <span class="sc">%&gt;%</span> </span>
+<span id="cb120-12"><a href="bayesian-data-analysis-2.html#cb120-12" tabindex="-1"></a>              <span class="fu">filter</span>(sex <span class="sc">==</span> <span class="st">&quot;male&quot;</span>),</span>
+<span id="cb120-13"><a href="bayesian-data-analysis-2.html#cb120-13" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
+<span id="cb120-14"><a href="bayesian-data-analysis-2.html#cb120-14" tabindex="-1"></a>                <span class="at">group =</span> .draw,</span>
+<span id="cb120-15"><a href="bayesian-data-analysis-2.html#cb120-15" tabindex="-1"></a>                <span class="at">color =</span> sex)) <span class="sc">+</span> </span>
+<span id="cb120-16"><a href="bayesian-data-analysis-2.html#cb120-16" tabindex="-1"></a>  <span class="fu">geom_line</span>(<span class="at">data =</span> df.model <span class="sc">%&gt;%</span> </span>
+<span id="cb120-17"><a href="bayesian-data-analysis-2.html#cb120-17" tabindex="-1"></a>              <span class="fu">filter</span>(sex <span class="sc">==</span> <span class="st">&quot;female&quot;</span>),</span>
+<span id="cb120-18"><a href="bayesian-data-analysis-2.html#cb120-18" tabindex="-1"></a>            <span class="fu">aes</span>(<span class="at">y =</span> .linpred,</span>
+<span id="cb120-19"><a href="bayesian-data-analysis-2.html#cb120-19" tabindex="-1"></a>                <span class="at">group =</span> .draw,</span>
+<span id="cb120-20"><a href="bayesian-data-analysis-2.html#cb120-20" tabindex="-1"></a>                <span class="at">color =</span> sex)) <span class="sc">+</span> </span>
+<span id="cb120-21"><a href="bayesian-data-analysis-2.html#cb120-21" tabindex="-1"></a>  <span class="fu">scale_color_brewer</span>(<span class="at">palette =</span> <span class="st">&quot;Set1&quot;</span>)</span></code></pre></div>
+<p><img src="psych252_files/figure-html/unnamed-chunk-69-1.png" width="672" /></p>
 </div>
 </div>
 <div id="politeness-data" class="section level2 hasAnchor" number="23.7">
@@ -2270,51 +2272,51 @@ <h2><span class="header-section-number">23.7</span> Politeness data<a href="baye
 <p>(I’m skipping some of the steps of our recipe for Bayesian data analysis here.)</p>
 <div id="visualize-the-data-3" class="section level3 hasAnchor" number="23.7.1">
 <h3><span class="header-section-number">23.7.1</span> 1. Visualize the data<a href="bayesian-data-analysis-2.html#visualize-the-data-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div class="sourceCode" id="cb1375"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1375-1"><a href="bayesian-data-analysis-2.html#cb1375-1" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.politeness,</span>
-<span id="cb1375-2"><a href="bayesian-data-analysis-2.html#cb1375-2" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> attitude,</span>
-<span id="cb1375-3"><a href="bayesian-data-analysis-2.html#cb1375-3" tabindex="-1"></a>                     <span class="at">y =</span> pitch,</span>
-<span id="cb1375-4"><a href="bayesian-data-analysis-2.html#cb1375-4" tabindex="-1"></a>                     <span class="at">fill =</span> gender,</span>
-<span id="cb1375-5"><a href="bayesian-data-analysis-2.html#cb1375-5" tabindex="-1"></a>                     <span class="at">color =</span> gender)) <span class="sc">+</span> </span>
-<span id="cb1375-6"><a href="bayesian-data-analysis-2.html#cb1375-6" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
-<span id="cb1375-7"><a href="bayesian-data-analysis-2.html#cb1375-7" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitter</span>(<span class="at">width =</span> <span class="fl">0.1</span>, <span class="at">height =</span> <span class="dv">0</span>)) <span class="sc">+</span> </span>
-<span id="cb1375-8"><a href="bayesian-data-analysis-2.html#cb1375-8" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>,</span>
-<span id="cb1375-9"><a href="bayesian-data-analysis-2.html#cb1375-9" tabindex="-1"></a>               <span class="at">shape =</span> <span class="dv">21</span>,</span>
-<span id="cb1375-10"><a href="bayesian-data-analysis-2.html#cb1375-10" tabindex="-1"></a>               <span class="at">size =</span> <span class="dv">1</span>,</span>
-<span id="cb1375-11"><a href="bayesian-data-analysis-2.html#cb1375-11" tabindex="-1"></a>               <span class="at">color =</span> <span class="st">&quot;black&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb121"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb121-1"><a href="bayesian-data-analysis-2.html#cb121-1" tabindex="-1"></a><span class="fu">ggplot</span>(<span class="at">data =</span> df.politeness,</span>
+<span id="cb121-2"><a href="bayesian-data-analysis-2.html#cb121-2" tabindex="-1"></a>       <span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> attitude,</span>
+<span id="cb121-3"><a href="bayesian-data-analysis-2.html#cb121-3" tabindex="-1"></a>                     <span class="at">y =</span> pitch,</span>
+<span id="cb121-4"><a href="bayesian-data-analysis-2.html#cb121-4" tabindex="-1"></a>                     <span class="at">fill =</span> gender,</span>
+<span id="cb121-5"><a href="bayesian-data-analysis-2.html#cb121-5" tabindex="-1"></a>                     <span class="at">color =</span> gender)) <span class="sc">+</span> </span>
+<span id="cb121-6"><a href="bayesian-data-analysis-2.html#cb121-6" tabindex="-1"></a>  <span class="fu">geom_point</span>(<span class="at">alpha =</span> <span class="fl">0.2</span>,</span>
+<span id="cb121-7"><a href="bayesian-data-analysis-2.html#cb121-7" tabindex="-1"></a>             <span class="at">position =</span> <span class="fu">position_jitter</span>(<span class="at">width =</span> <span class="fl">0.1</span>, <span class="at">height =</span> <span class="dv">0</span>)) <span class="sc">+</span> </span>
+<span id="cb121-8"><a href="bayesian-data-analysis-2.html#cb121-8" tabindex="-1"></a>  <span class="fu">stat_summary</span>(<span class="at">fun.data =</span> <span class="st">&quot;mean_cl_boot&quot;</span>,</span>
+<span id="cb121-9"><a href="bayesian-data-analysis-2.html#cb121-9" tabindex="-1"></a>               <span class="at">shape =</span> <span class="dv">21</span>,</span>
+<span id="cb121-10"><a href="bayesian-data-analysis-2.html#cb121-10" tabindex="-1"></a>               <span class="at">size =</span> <span class="dv">1</span>,</span>
+<span id="cb121-11"><a href="bayesian-data-analysis-2.html#cb121-11" tabindex="-1"></a>               <span class="at">color =</span> <span class="st">&quot;black&quot;</span>)</span></code></pre></div>
 <pre><code>Warning: Removed 1 rows containing non-finite values (`stat_summary()`).</code></pre>
 <pre><code>Warning: Removed 1 rows containing missing values (`geom_point()`).</code></pre>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-70-1.png" width="672" /></p>
+<p><img src="psych252_files/figure-html/unnamed-chunk-70-1.png" width="672" /></p>
 </div>
 <div id="specify-and-fit-the-model-3" class="section level3 hasAnchor" number="23.7.2">
 <h3><span class="header-section-number">23.7.2</span> 2. Specify and fit the model<a href="bayesian-data-analysis-2.html#specify-and-fit-the-model-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="frequentist-analysis-3" class="section level4 hasAnchor" number="23.7.2.1">
 <h4><span class="header-section-number">23.7.2.1</span> Frequentist analysis<a href="bayesian-data-analysis-2.html#frequentist-analysis-3" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1378"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1378-1"><a href="bayesian-data-analysis-2.html#cb1378-1" tabindex="-1"></a>fit.lm_polite <span class="ot">=</span> <span class="fu">lm</span>(<span class="at">formula =</span> pitch <span class="sc">~</span> gender <span class="sc">*</span> attitude,</span>
-<span id="cb1378-2"><a href="bayesian-data-analysis-2.html#cb1378-2" tabindex="-1"></a>                   <span class="at">data =</span> df.politeness)</span></code></pre></div>
+<div class="sourceCode" id="cb124"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb124-1"><a href="bayesian-data-analysis-2.html#cb124-1" tabindex="-1"></a>fit.lm_polite <span class="ot">=</span> <span class="fu">lm</span>(<span class="at">formula =</span> pitch <span class="sc">~</span> gender <span class="sc">*</span> attitude,</span>
+<span id="cb124-2"><a href="bayesian-data-analysis-2.html#cb124-2" tabindex="-1"></a>                   <span class="at">data =</span> df.politeness)</span></code></pre></div>
 </div>
 <div id="bayesian-analysis-2" class="section level4 hasAnchor" number="23.7.2.2">
 <h4><span class="header-section-number">23.7.2.2</span> Bayesian analysis<a href="bayesian-data-analysis-2.html#bayesian-analysis-2" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1379"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1379-1"><a href="bayesian-data-analysis-2.html#cb1379-1" tabindex="-1"></a>fit.brm_polite <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> pitch <span class="sc">~</span> gender <span class="sc">*</span> attitude, </span>
-<span id="cb1379-2"><a href="bayesian-data-analysis-2.html#cb1379-2" tabindex="-1"></a>                     <span class="at">data =</span> df.politeness, </span>
-<span id="cb1379-3"><a href="bayesian-data-analysis-2.html#cb1379-3" tabindex="-1"></a>                     <span class="at">file =</span> <span class="st">&quot;cache/brm_polite&quot;</span>,</span>
-<span id="cb1379-4"><a href="bayesian-data-analysis-2.html#cb1379-4" tabindex="-1"></a>                     <span class="at">seed =</span> <span class="dv">1</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb125"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb125-1"><a href="bayesian-data-analysis-2.html#cb125-1" tabindex="-1"></a>fit.brm_polite <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> pitch <span class="sc">~</span> gender <span class="sc">*</span> attitude, </span>
+<span id="cb125-2"><a href="bayesian-data-analysis-2.html#cb125-2" tabindex="-1"></a>                     <span class="at">data =</span> df.politeness, </span>
+<span id="cb125-3"><a href="bayesian-data-analysis-2.html#cb125-3" tabindex="-1"></a>                     <span class="at">file =</span> <span class="st">&quot;cache/brm_polite&quot;</span>,</span>
+<span id="cb125-4"><a href="bayesian-data-analysis-2.html#cb125-4" tabindex="-1"></a>                     <span class="at">seed =</span> <span class="dv">1</span>)</span></code></pre></div>
 </div>
 </div>
 <div id="test-specific-hypotheses-3" class="section level3 hasAnchor" number="23.7.3">
 <h3><span class="header-section-number">23.7.3</span> 5. Test specific hypotheses<a href="bayesian-data-analysis-2.html#test-specific-hypotheses-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
 <div id="frequentist" class="section level4 hasAnchor" number="23.7.3.1">
 <h4><span class="header-section-number">23.7.3.1</span> Frequentist<a href="bayesian-data-analysis-2.html#frequentist" class="anchor-section" aria-label="Anchor link to header"></a></h4>
-<div class="sourceCode" id="cb1380"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1380-1"><a href="bayesian-data-analysis-2.html#cb1380-1" tabindex="-1"></a>fit.lm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1380-2"><a href="bayesian-data-analysis-2.html#cb1380-2" tabindex="-1"></a>  <span class="fu">joint_tests</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb126"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb126-1"><a href="bayesian-data-analysis-2.html#cb126-1" tabindex="-1"></a>fit.lm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb126-2"><a href="bayesian-data-analysis-2.html#cb126-2" tabindex="-1"></a>  <span class="fu">joint_tests</span>()</span></code></pre></div>
 <pre><code> model term      df1 df2 F.ratio p.value
  gender            1  79 191.028  &lt;.0001
  attitude          1  79   6.170  0.0151
  gender:attitude   1  79   1.029  0.3135</code></pre>
 <p>It looks like there are significant main effects of gender and attitude, but no interaction effect.</p>
 <p>Let’s check whether there is a difference in attitude separately for each gender:</p>
-<div class="sourceCode" id="cb1382"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1382-1"><a href="bayesian-data-analysis-2.html#cb1382-1" tabindex="-1"></a>fit.lm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1382-2"><a href="bayesian-data-analysis-2.html#cb1382-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
-<span id="cb1382-3"><a href="bayesian-data-analysis-2.html#cb1382-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb128"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb128-1"><a href="bayesian-data-analysis-2.html#cb128-1" tabindex="-1"></a>fit.lm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb128-2"><a href="bayesian-data-analysis-2.html#cb128-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
+<span id="cb128-3"><a href="bayesian-data-analysis-2.html#cb128-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>)</span></code></pre></div>
 <pre><code>gender = F:
  contrast  estimate   SE df t.ratio p.value
  inf - pol     27.4 11.0 79   2.489  0.0149
@@ -2327,10 +2329,10 @@ <h4><span class="header-section-number">23.7.3.1</span> Frequentist<a href="baye
 <div id="bayesian" class="section level4 hasAnchor" number="23.7.3.2">
 <h4><span class="header-section-number">23.7.3.2</span> Bayesian<a href="bayesian-data-analysis-2.html#bayesian" class="anchor-section" aria-label="Anchor link to header"></a></h4>
 <p>Let’s whether there was a main effect of gender.</p>
-<div class="sourceCode" id="cb1384"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1384-1"><a href="bayesian-data-analysis-2.html#cb1384-1" tabindex="-1"></a><span class="co"># main effect of gender</span></span>
-<span id="cb1384-2"><a href="bayesian-data-analysis-2.html#cb1384-2" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1384-3"><a href="bayesian-data-analysis-2.html#cb1384-3" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> gender) <span class="sc">%&gt;%</span> </span>
-<span id="cb1384-4"><a href="bayesian-data-analysis-2.html#cb1384-4" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb130"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb130-1"><a href="bayesian-data-analysis-2.html#cb130-1" tabindex="-1"></a><span class="co"># main effect of gender</span></span>
+<span id="cb130-2"><a href="bayesian-data-analysis-2.html#cb130-2" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb130-3"><a href="bayesian-data-analysis-2.html#cb130-3" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> gender) <span class="sc">%&gt;%</span> </span>
+<span id="cb130-4"><a href="bayesian-data-analysis-2.html#cb130-4" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>)</span></code></pre></div>
 <pre><code>NOTE: Results may be misleading due to involvement in interactions</code></pre>
 <pre><code> contrast estimate lower.HPD upper.HPD
  F - M         108      91.7       124
@@ -2339,45 +2341,45 @@ <h4><span class="header-section-number">23.7.3.2</span> Bayesian<a href="bayesia
 Point estimate displayed: median 
 HPD interval probability: 0.95 </code></pre>
 <p>Let’s take a look what the full posterior distribution over this contrast looks like:</p>
-<div class="sourceCode" id="cb1387"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1387-1"><a href="bayesian-data-analysis-2.html#cb1387-1" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1387-2"><a href="bayesian-data-analysis-2.html#cb1387-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> gender) <span class="sc">%&gt;%</span> </span>
-<span id="cb1387-3"><a href="bayesian-data-analysis-2.html#cb1387-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1387-4"><a href="bayesian-data-analysis-2.html#cb1387-4" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1387-5"><a href="bayesian-data-analysis-2.html#cb1387-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> .value)) <span class="sc">+</span> </span>
-<span id="cb1387-6"><a href="bayesian-data-analysis-2.html#cb1387-6" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb133"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb133-1"><a href="bayesian-data-analysis-2.html#cb133-1" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb133-2"><a href="bayesian-data-analysis-2.html#cb133-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> gender) <span class="sc">%&gt;%</span> </span>
+<span id="cb133-3"><a href="bayesian-data-analysis-2.html#cb133-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb133-4"><a href="bayesian-data-analysis-2.html#cb133-4" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb133-5"><a href="bayesian-data-analysis-2.html#cb133-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="at">mapping =</span> <span class="fu">aes</span>(<span class="at">x =</span> .value)) <span class="sc">+</span> </span>
+<span id="cb133-6"><a href="bayesian-data-analysis-2.html#cb133-6" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>()</span></code></pre></div>
 <pre><code>NOTE: Results may be misleading due to involvement in interactions</code></pre>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-76-1.png" width="672" /></p>
+<p><img src="psych252_files/figure-html/unnamed-chunk-76-1.png" width="672" /></p>
 <p>Looks neat!</p>
 <p>And let’s confirm that we really estimated the main effect here. Let’s fit a model that only has gender as a predictor, and then compare:</p>
-<div class="sourceCode" id="cb1389"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1389-1"><a href="bayesian-data-analysis-2.html#cb1389-1" tabindex="-1"></a>fit.brm_polite_gender <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> pitch <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> gender, </span>
-<span id="cb1389-2"><a href="bayesian-data-analysis-2.html#cb1389-2" tabindex="-1"></a>                            <span class="at">data =</span> df.politeness, </span>
-<span id="cb1389-3"><a href="bayesian-data-analysis-2.html#cb1389-3" tabindex="-1"></a>                            <span class="at">file =</span> <span class="st">&quot;cache/brm_polite_gender&quot;</span>,</span>
-<span id="cb1389-4"><a href="bayesian-data-analysis-2.html#cb1389-4" tabindex="-1"></a>                            <span class="at">seed =</span> <span class="dv">1</span>)</span>
-<span id="cb1389-5"><a href="bayesian-data-analysis-2.html#cb1389-5" tabindex="-1"></a></span>
-<span id="cb1389-6"><a href="bayesian-data-analysis-2.html#cb1389-6" tabindex="-1"></a><span class="co"># using the gather_emmeans_draws to get means rather than medians </span></span>
-<span id="cb1389-7"><a href="bayesian-data-analysis-2.html#cb1389-7" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1389-8"><a href="bayesian-data-analysis-2.html#cb1389-8" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">spec =</span> pairwise <span class="sc">~</span> gender) <span class="sc">%&gt;%</span> </span>
-<span id="cb1389-9"><a href="bayesian-data-analysis-2.html#cb1389-9" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1389-10"><a href="bayesian-data-analysis-2.html#cb1389-10" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1389-11"><a href="bayesian-data-analysis-2.html#cb1389-11" tabindex="-1"></a>  <span class="fu">mean_hdi</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb135"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb135-1"><a href="bayesian-data-analysis-2.html#cb135-1" tabindex="-1"></a>fit.brm_polite_gender <span class="ot">=</span> <span class="fu">brm</span>(<span class="at">formula =</span> pitch <span class="sc">~</span> <span class="dv">1</span> <span class="sc">+</span> gender, </span>
+<span id="cb135-2"><a href="bayesian-data-analysis-2.html#cb135-2" tabindex="-1"></a>                            <span class="at">data =</span> df.politeness, </span>
+<span id="cb135-3"><a href="bayesian-data-analysis-2.html#cb135-3" tabindex="-1"></a>                            <span class="at">file =</span> <span class="st">&quot;cache/brm_polite_gender&quot;</span>,</span>
+<span id="cb135-4"><a href="bayesian-data-analysis-2.html#cb135-4" tabindex="-1"></a>                            <span class="at">seed =</span> <span class="dv">1</span>)</span>
+<span id="cb135-5"><a href="bayesian-data-analysis-2.html#cb135-5" tabindex="-1"></a></span>
+<span id="cb135-6"><a href="bayesian-data-analysis-2.html#cb135-6" tabindex="-1"></a><span class="co"># using the gather_emmeans_draws to get means rather than medians </span></span>
+<span id="cb135-7"><a href="bayesian-data-analysis-2.html#cb135-7" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb135-8"><a href="bayesian-data-analysis-2.html#cb135-8" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">spec =</span> pairwise <span class="sc">~</span> gender) <span class="sc">%&gt;%</span> </span>
+<span id="cb135-9"><a href="bayesian-data-analysis-2.html#cb135-9" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb135-10"><a href="bayesian-data-analysis-2.html#cb135-10" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb135-11"><a href="bayesian-data-analysis-2.html#cb135-11" tabindex="-1"></a>  <span class="fu">mean_hdi</span>()</span></code></pre></div>
 <pre><code>NOTE: Results may be misleading due to involvement in interactions</code></pre>
 <pre><code># A tibble: 1 × 7
   contrast .value .lower .upper .width .point .interval
   &lt;chr&gt;     &lt;dbl&gt;  &lt;dbl&gt;  &lt;dbl&gt;  &lt;dbl&gt; &lt;chr&gt;  &lt;chr&gt;    
 1 F - M      108.   91.9   124.   0.95 mean   hdi      </code></pre>
-<div class="sourceCode" id="cb1392"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1392-1"><a href="bayesian-data-analysis-2.html#cb1392-1" tabindex="-1"></a>fit.brm_polite_gender <span class="sc">%&gt;%</span> </span>
-<span id="cb1392-2"><a href="bayesian-data-analysis-2.html#cb1392-2" tabindex="-1"></a>  <span class="fu">fixef</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1392-3"><a href="bayesian-data-analysis-2.html#cb1392-3" tabindex="-1"></a>  <span class="fu">as_tibble</span>(<span class="at">rownames =</span> <span class="st">&quot;term&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb138"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb138-1"><a href="bayesian-data-analysis-2.html#cb138-1" tabindex="-1"></a>fit.brm_polite_gender <span class="sc">%&gt;%</span> </span>
+<span id="cb138-2"><a href="bayesian-data-analysis-2.html#cb138-2" tabindex="-1"></a>  <span class="fu">fixef</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb138-3"><a href="bayesian-data-analysis-2.html#cb138-3" tabindex="-1"></a>  <span class="fu">as_tibble</span>(<span class="at">rownames =</span> <span class="st">&quot;term&quot;</span>)</span></code></pre></div>
 <pre><code># A tibble: 2 × 5
   term      Estimate Est.Error  Q2.5 Q97.5
   &lt;chr&gt;        &lt;dbl&gt;     &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt;
 1 Intercept     247.      5.72  236. 258. 
 2 genderM      -108.      8.16 -124. -92.3</code></pre>
 <p>Yip, both of these methods give us the same result (the sign is flipped but that’s just because emmeans computed F-M, whereas the other method computed M-F)! Again, the <code>emmeans()</code> route is more convenient because we can more easily check for several main effects (and take a look at specific contrast, too).</p>
-<div class="sourceCode" id="cb1394"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1394-1"><a href="bayesian-data-analysis-2.html#cb1394-1" tabindex="-1"></a><span class="co"># main effect attitude</span></span>
-<span id="cb1394-2"><a href="bayesian-data-analysis-2.html#cb1394-2" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1394-3"><a href="bayesian-data-analysis-2.html#cb1394-3" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude) <span class="sc">%&gt;%</span> </span>
-<span id="cb1394-4"><a href="bayesian-data-analysis-2.html#cb1394-4" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb140"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb140-1"><a href="bayesian-data-analysis-2.html#cb140-1" tabindex="-1"></a><span class="co"># main effect attitude</span></span>
+<span id="cb140-2"><a href="bayesian-data-analysis-2.html#cb140-2" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb140-3"><a href="bayesian-data-analysis-2.html#cb140-3" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude) <span class="sc">%&gt;%</span> </span>
+<span id="cb140-4"><a href="bayesian-data-analysis-2.html#cb140-4" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>)</span></code></pre></div>
 <pre><code>NOTE: Results may be misleading due to involvement in interactions</code></pre>
 <pre><code> contrast  estimate lower.HPD upper.HPD
  inf - pol     19.8      5.16      35.9
@@ -2385,10 +2387,10 @@ <h4><span class="header-section-number">23.7.3.2</span> Bayesian<a href="bayesia
 Results are averaged over the levels of: gender 
 Point estimate displayed: median 
 HPD interval probability: 0.95 </code></pre>
-<div class="sourceCode" id="cb1397"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1397-1"><a href="bayesian-data-analysis-2.html#cb1397-1" tabindex="-1"></a><span class="co"># effect of attitude separately for each gender</span></span>
-<span id="cb1397-2"><a href="bayesian-data-analysis-2.html#cb1397-2" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1397-3"><a href="bayesian-data-analysis-2.html#cb1397-3" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
-<span id="cb1397-4"><a href="bayesian-data-analysis-2.html#cb1397-4" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb143"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb143-1"><a href="bayesian-data-analysis-2.html#cb143-1" tabindex="-1"></a><span class="co"># effect of attitude separately for each gender</span></span>
+<span id="cb143-2"><a href="bayesian-data-analysis-2.html#cb143-2" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb143-3"><a href="bayesian-data-analysis-2.html#cb143-3" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
+<span id="cb143-4"><a href="bayesian-data-analysis-2.html#cb143-4" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>)</span></code></pre></div>
 <pre><code>gender = F:
  contrast  estimate lower.HPD upper.HPD
  inf - pol     27.9      5.96      49.5
@@ -2399,41 +2401,42 @@ <h4><span class="header-section-number">23.7.3.2</span> Bayesian<a href="bayesia
 
 Point estimate displayed: median 
 HPD interval probability: 0.95 </code></pre>
-<div class="sourceCode" id="cb1399"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1399-1"><a href="bayesian-data-analysis-2.html#cb1399-1" tabindex="-1"></a><span class="co"># in case you want the means instead of medians </span></span>
-<span id="cb1399-2"><a href="bayesian-data-analysis-2.html#cb1399-2" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1399-3"><a href="bayesian-data-analysis-2.html#cb1399-3" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
-<span id="cb1399-4"><a href="bayesian-data-analysis-2.html#cb1399-4" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1399-5"><a href="bayesian-data-analysis-2.html#cb1399-5" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1399-6"><a href="bayesian-data-analysis-2.html#cb1399-6" tabindex="-1"></a>  <span class="fu">mean_hdi</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb145"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb145-1"><a href="bayesian-data-analysis-2.html#cb145-1" tabindex="-1"></a><span class="co"># in case you want the means instead of medians </span></span>
+<span id="cb145-2"><a href="bayesian-data-analysis-2.html#cb145-2" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb145-3"><a href="bayesian-data-analysis-2.html#cb145-3" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
+<span id="cb145-4"><a href="bayesian-data-analysis-2.html#cb145-4" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb145-5"><a href="bayesian-data-analysis-2.html#cb145-5" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb145-6"><a href="bayesian-data-analysis-2.html#cb145-6" tabindex="-1"></a>  <span class="fu">mean_hdi</span>()</span></code></pre></div>
 <pre><code># A tibble: 2 × 8
   contrast  gender .value .lower .upper .width .point .interval
   &lt;fct&gt;     &lt;fct&gt;   &lt;dbl&gt;  &lt;dbl&gt;  &lt;dbl&gt;  &lt;dbl&gt; &lt;chr&gt;  &lt;chr&gt;    
 1 inf - pol F        27.8   5.96   49.5   0.95 mean   hdi      
 2 inf - pol M        11.9  -9.29   34.2   0.95 mean   hdi      </code></pre>
 <p>Here is a way to visualize the contrasts:</p>
-<div class="sourceCode" id="cb1401"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1401-1"><a href="bayesian-data-analysis-2.html#cb1401-1" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1401-2"><a href="bayesian-data-analysis-2.html#cb1401-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
-<span id="cb1401-3"><a href="bayesian-data-analysis-2.html#cb1401-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1401-4"><a href="bayesian-data-analysis-2.html#cb1401-4" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
-<span id="cb1401-5"><a href="bayesian-data-analysis-2.html#cb1401-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> .value,</span>
-<span id="cb1401-6"><a href="bayesian-data-analysis-2.html#cb1401-6" tabindex="-1"></a>             <span class="at">y =</span> gender,</span>
-<span id="cb1401-7"><a href="bayesian-data-analysis-2.html#cb1401-7" tabindex="-1"></a>             <span class="at">fill =</span> <span class="fu">stat</span>(x <span class="sc">&gt;</span> <span class="dv">0</span>))) <span class="sc">+</span> </span>
-<span id="cb1401-8"><a href="bayesian-data-analysis-2.html#cb1401-8" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span> contrast) <span class="sc">+</span></span>
-<span id="cb1401-9"><a href="bayesian-data-analysis-2.html#cb1401-9" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">show.legend =</span> F) <span class="sc">+</span> </span>
-<span id="cb1401-10"><a href="bayesian-data-analysis-2.html#cb1401-10" tabindex="-1"></a>  <span class="fu">geom_vline</span>(<span class="at">xintercept =</span> <span class="dv">0</span>, </span>
-<span id="cb1401-11"><a href="bayesian-data-analysis-2.html#cb1401-11" tabindex="-1"></a>             <span class="at">linetype =</span> <span class="dv">2</span>) <span class="sc">+</span> </span>
-<span id="cb1401-12"><a href="bayesian-data-analysis-2.html#cb1401-12" tabindex="-1"></a>  <span class="fu">scale_fill_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">&quot;gray80&quot;</span>, <span class="st">&quot;skyblue&quot;</span>))</span></code></pre></div>
+<div class="sourceCode" id="cb147"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb147-1"><a href="bayesian-data-analysis-2.html#cb147-1" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb147-2"><a href="bayesian-data-analysis-2.html#cb147-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(<span class="at">specs =</span> pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
+<span id="cb147-3"><a href="bayesian-data-analysis-2.html#cb147-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;contrasts&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb147-4"><a href="bayesian-data-analysis-2.html#cb147-4" tabindex="-1"></a>  <span class="fu">gather_emmeans_draws</span>() <span class="sc">%&gt;%</span> </span>
+<span id="cb147-5"><a href="bayesian-data-analysis-2.html#cb147-5" tabindex="-1"></a>  <span class="fu">ggplot</span>(<span class="fu">aes</span>(<span class="at">x =</span> .value,</span>
+<span id="cb147-6"><a href="bayesian-data-analysis-2.html#cb147-6" tabindex="-1"></a>             <span class="at">y =</span> gender,</span>
+<span id="cb147-7"><a href="bayesian-data-analysis-2.html#cb147-7" tabindex="-1"></a>             <span class="at">fill =</span> <span class="fu">stat</span>(x <span class="sc">&gt;</span> <span class="dv">0</span>))) <span class="sc">+</span> </span>
+<span id="cb147-8"><a href="bayesian-data-analysis-2.html#cb147-8" tabindex="-1"></a>  <span class="fu">facet_wrap</span>(<span class="sc">~</span> contrast) <span class="sc">+</span></span>
+<span id="cb147-9"><a href="bayesian-data-analysis-2.html#cb147-9" tabindex="-1"></a>  <span class="fu">stat_halfeye</span>(<span class="at">show.legend =</span> F) <span class="sc">+</span> </span>
+<span id="cb147-10"><a href="bayesian-data-analysis-2.html#cb147-10" tabindex="-1"></a>  <span class="fu">geom_vline</span>(<span class="at">xintercept =</span> <span class="dv">0</span>, </span>
+<span id="cb147-11"><a href="bayesian-data-analysis-2.html#cb147-11" tabindex="-1"></a>             <span class="at">linetype =</span> <span class="dv">2</span>) <span class="sc">+</span> </span>
+<span id="cb147-12"><a href="bayesian-data-analysis-2.html#cb147-12" tabindex="-1"></a>  <span class="fu">scale_fill_manual</span>(<span class="at">values =</span> <span class="fu">c</span>(<span class="st">&quot;gray80&quot;</span>, <span class="st">&quot;skyblue&quot;</span>))</span></code></pre></div>
 <pre><code>Warning: `stat(x &gt; 0)` was deprecated in ggplot2 3.4.0.
 ℹ Please use `after_stat(x &gt; 0)` instead.
 This warning is displayed once every 8 hours.
-Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.</code></pre>
-<p><img src="23-bayesian_data_analysis2_files/figure-html/unnamed-chunk-79-1.png" width="672" /></p>
+Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
+generated.</code></pre>
+<p><img src="psych252_files/figure-html/unnamed-chunk-79-1.png" width="672" /></p>
 <p>Here is one way to check whether there was an interaction between attitude and gender (see <a href="https://cran.r-project.org/web/packages/emmeans/vignettes/interactions.html">this vignette</a> for more info).</p>
-<div class="sourceCode" id="cb1403"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1403-1"><a href="bayesian-data-analysis-2.html#cb1403-1" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
-<span id="cb1403-2"><a href="bayesian-data-analysis-2.html#cb1403-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
-<span id="cb1403-3"><a href="bayesian-data-analysis-2.html#cb1403-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;emmeans&quot;</span>) <span class="sc">%&gt;%</span> </span>
-<span id="cb1403-4"><a href="bayesian-data-analysis-2.html#cb1403-4" tabindex="-1"></a>  <span class="fu">contrast</span>(<span class="at">interaction =</span> <span class="fu">c</span>(<span class="st">&quot;consec&quot;</span>),</span>
-<span id="cb1403-5"><a href="bayesian-data-analysis-2.html#cb1403-5" tabindex="-1"></a>           <span class="at">by =</span> <span class="cn">NULL</span>)</span></code></pre></div>
+<div class="sourceCode" id="cb149"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb149-1"><a href="bayesian-data-analysis-2.html#cb149-1" tabindex="-1"></a>fit.brm_polite <span class="sc">%&gt;%</span> </span>
+<span id="cb149-2"><a href="bayesian-data-analysis-2.html#cb149-2" tabindex="-1"></a>  <span class="fu">emmeans</span>(pairwise <span class="sc">~</span> attitude <span class="sc">|</span> gender) <span class="sc">%&gt;%</span> </span>
+<span id="cb149-3"><a href="bayesian-data-analysis-2.html#cb149-3" tabindex="-1"></a>  <span class="fu">pluck</span>(<span class="st">&quot;emmeans&quot;</span>) <span class="sc">%&gt;%</span> </span>
+<span id="cb149-4"><a href="bayesian-data-analysis-2.html#cb149-4" tabindex="-1"></a>  <span class="fu">contrast</span>(<span class="at">interaction =</span> <span class="fu">c</span>(<span class="st">&quot;consec&quot;</span>),</span>
+<span id="cb149-5"><a href="bayesian-data-analysis-2.html#cb149-5" tabindex="-1"></a>           <span class="at">by =</span> <span class="cn">NULL</span>)</span></code></pre></div>
 <pre><code> attitude_consec gender_consec estimate lower.HPD upper.HPD
  pol - inf       M - F             16.1     -15.1      47.2
 
@@ -2442,8 +2445,8 @@ <h4><span class="header-section-number">23.7.3.2</span> Bayesian<a href="bayesia
 </div>
 </div>
 </div>
-<div id="additional-resources-18" class="section level2 hasAnchor" number="23.8">
-<h2><span class="header-section-number">23.8</span> Additional resources<a href="bayesian-data-analysis-2.html#additional-resources-18" class="anchor-section" aria-label="Anchor link to header"></a></h2>
+<div id="additional-resources-19" class="section level2 hasAnchor" number="23.8">
+<h2><span class="header-section-number">23.8</span> Additional resources<a href="bayesian-data-analysis-2.html#additional-resources-19" class="anchor-section" aria-label="Anchor link to header"></a></h2>
 <ul>
 <li><a href="https://michael-franke.github.io/Bayesian-Regression/">Bayesian regression: Theory &amp; Practice</a></li>
 <li><a href="https://mjskay.github.io/tidybayes/articles/tidy-brms.html">Tutorial on visualizing brms posteriors with tidybayes</a></li>
@@ -2456,7 +2459,7 @@ <h2><span class="header-section-number">23.8</span> Additional resources<a href=
 <div id="session-info-22" class="section level2 hasAnchor" number="23.9">
 <h2><span class="header-section-number">23.9</span> Session info<a href="bayesian-data-analysis-2.html#session-info-22" class="anchor-section" aria-label="Anchor link to header"></a></h2>
 <p>Information about this R session including which version of R was used, and what packages were loaded.</p>
-<div class="sourceCode" id="cb1405"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1405-1"><a href="bayesian-data-analysis-2.html#cb1405-1" tabindex="-1"></a><span class="fu">sessionInfo</span>()</span></code></pre></div>
+<div class="sourceCode" id="cb151"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb151-1"><a href="bayesian-data-analysis-2.html#cb151-1" tabindex="-1"></a><span class="fu">sessionInfo</span>()</span></code></pre></div>
 <pre><code>R version 4.3.2 (2023-10-31)
 Platform: aarch64-apple-darwin20 (64-bit)
 Running under: macOS Sonoma 14.1.2
@@ -2483,8 +2486,8 @@ <h2><span class="header-section-number">23.9</span> Session info<a href="bayesia
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-[28] Rcpp_1.0.11           tidybayes_3.0.6       janitor_2.2.0        
+[25] ggplot2_3.4.4         patchwork_1.2.0       brms_2.20.4          
+[28] Rcpp_1.0.12           tidybayes_3.0.6       janitor_2.2.0        
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- [91] gtable_0.3.4         furrr_0.3.1          htmltools_0.5.7     
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-[124] mvtnorm_1.2-4        cli_3.6.2            compiler_4.3.2      
-[127] rlang_1.1.2          crayon_1.5.2         rstantools_2.3.1.1  
-[130] labeling_0.4.3       classInt_0.4-10      plyr_1.8.9          
-[133] stringi_1.8.3        rstan_2.32.3         viridisLite_0.4.2   
-[136] QuickJSR_1.0.9       lmerTest_3.1-3       munsell_0.5.0       
-[139] colourpicker_1.3.0   Brobdingnag_1.2-9    bayestestR_0.13.1   
-[142] V8_4.4.1             hms_1.1.3            bit64_4.0.5         
-[145] future_1.33.1        shiny_1.8.0          haven_2.5.4         
-[148] highr_0.10           igraph_1.6.0         broom_1.0.5         
-[151] RcppParallel_5.1.7   bslib_0.6.1          bit_4.0.5           </code></pre>
+ [25] httpuv_1.6.13        gifski_1.12.0-2      pkgbuild_1.4.3      
+ [28] DBI_1.2.0            minqa_1.2.6          RColorBrewer_1.1-3  
+ [31] abind_1.4-5          rvest_1.0.3          nnet_7.3-19         
+ [34] tensorA_0.36.2.1     tweenr_2.0.2         inline_0.3.19       
+ [37] listenv_0.9.1        units_0.8-5          bridgesampling_1.1-2
+ [40] parallelly_1.37.0    svglite_2.1.3        codetools_0.2-19    
+ [43] DT_0.31              xml2_1.3.6           tidyselect_1.2.0    
+ [46] rstanarm_2.26.1      farver_2.1.1         matrixStats_1.2.0   
+ [49] stats4_4.3.2         base64enc_0.1-3      webshot_0.5.5       
+ [52] jsonlite_1.8.8       e1071_1.7-14         Formula_1.2-5       
+ [55] ellipsis_0.3.2       survival_3.5-7       systemfonts_1.0.5   
+ [58] tools_4.3.2          progress_1.2.3       glue_1.7.0          
+ [61] gridExtra_2.3        mgcv_1.9-1           xfun_0.41           
+ [64] distributional_0.3.2 loo_2.6.0            withr_3.0.0         
+ [67] numDeriv_2016.8-1.1  fastmap_1.1.1        boot_1.3-28.1       
+ [70] fansi_1.0.6          shinyjs_2.1.0        digest_0.6.34       
+ [73] timechange_0.2.0     R6_2.5.1             mime_0.12           
+ [76] estimability_1.4.1   colorspace_2.1-0     lpSolve_5.6.20      
+ [79] gtools_3.9.5         markdown_1.12        threejs_0.3.3       
+ [82] utf8_1.2.4           generics_0.1.3       data.table_1.14.10  
+ [85] class_7.3-22         prettyunits_1.2.0    httr_1.4.7          
+ [88] htmlwidgets_1.6.4    ggstats_0.5.1        pkgconfig_2.0.3     
+ [91] dygraphs_1.1.1.6     gtable_0.3.4         furrr_0.3.1         
+ [94] htmltools_0.5.7      bookdown_0.37        scales_1.3.0        
+ [97] posterior_1.5.0      snakecase_0.11.1     rstudioapi_0.15.0   
+[100] tzdb_0.4.0           reshape2_1.4.4       coda_0.19-4.1       
+[103] checkmate_2.3.1      nlme_3.1-164         curl_5.2.0          
+[106] nloptr_2.0.3         proxy_0.4-27         cachem_1.0.8        
+[109] zoo_1.8-12           sjlabelled_1.2.0     KernSmooth_2.23-22  
+[112] parallel_4.3.2       miniUI_0.1.1.1       foreign_0.8-86      
+[115] pillar_1.9.0         grid_4.3.2           vctrs_0.6.5         
+[118] shinystan_2.6.0      promises_1.2.1       arrayhelpers_1.1-0  
+[121] cluster_2.1.6        xtable_1.8-4         htmlTable_2.4.2     
+[124] evaluate_0.23        mvtnorm_1.2-4        cli_3.6.2           
+[127] compiler_4.3.2       rlang_1.1.3          crayon_1.5.2        
+[130] rstantools_2.3.1.1   labeling_0.4.3       classInt_0.4-10     
+[133] plyr_1.8.9           stringi_1.8.3        rstan_2.32.3        
+[136] viridisLite_0.4.2    QuickJSR_1.0.9       lmerTest_3.1-3      
+[139] munsell_0.5.0        colourpicker_1.3.0   Brobdingnag_1.2-9   
+[142] bayestestR_0.13.1    V8_4.4.1             hms_1.1.3           
+[145] bit64_4.0.5          future_1.33.1        shiny_1.8.0         
+[148] haven_2.5.4          highr_0.10           igraph_1.6.0        
+[151] broom_1.0.5          RcppParallel_5.1.7   bslib_0.6.1         
+[154] bit_4.0.5           </code></pre>
 
 </div>
 </div>
 <h3>References<a href="references-1.html#references-1" class="anchor-section" aria-label="Anchor link to header"></a></h3>
-<div id="refs" class="references csl-bib-body hanging-indent" entry-spacing="0">
+<div id="refs" class="references csl-bib-body hanging-indent">
 <div id="ref-franke2019bayesian" class="csl-entry">
 Franke, Michael, and Timo B Roettger. 2019. <span>“Bayesian Regression Modeling (for Factorial Designs): A Tutorial.”</span> <em>PsyArXiv</em>. <a href="https://psyarxiv.com/cdxv3">https://psyarxiv.com/cdxv3</a>.
 </div>
@@ -2611,20 +2615,6 @@ <h3>References<a href="references-1.html#references-1" class="anchor-section" ar
 });
 </script>
 
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diff --git a/docs/linear-mixed-effects-models-2.html b/docs/linear-mixed-effects-models-2.html
index 7b1515d..1730bbc 100644
--- a/docs/linear-mixed-effects-models-2.html
+++ b/docs/linear-mixed-effects-models-2.html
@@ -23,7 +23,7 @@
 <meta name="author" content="Tobias Gerstenberg" />
 
 
-<meta name="date" content="2024-02-22" />
+<meta name="date" content="2024-03-06" />
 
   <meta name="viewport" content="width=device-width, initial-scale=1" />
   <meta name="apple-mobile-web-app-capable" content="yes" />
@@ -621,19 +621,20 @@
 <li class="chapter" data-level="19.2" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#load-packages-and-set-plotting-theme-12"><i class="fa fa-check"></i><b>19.2</b> Load packages and set plotting theme</a></li>
 <li class="chapter" data-level="19.3" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#load-data-sets-4"><i class="fa fa-check"></i><b>19.3</b> Load data sets</a>
 <ul>
-<li class="chapter" data-level="19.3.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#sleep-data"><i class="fa fa-check"></i><b>19.3.1</b> Sleep data</a></li>
-<li class="chapter" data-level="19.3.2" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#reasoning-data"><i class="fa fa-check"></i><b>19.3.2</b> Reasoning data</a></li>
-<li class="chapter" data-level="19.3.3" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#weight-loss-data"><i class="fa fa-check"></i><b>19.3.3</b> Weight loss data</a></li>
-<li class="chapter" data-level="19.3.4" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#politness-data"><i class="fa fa-check"></i><b>19.3.4</b> Politness data</a></li>
+<li class="chapter" data-level="19.3.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#reasoning-data"><i class="fa fa-check"></i><b>19.3.1</b> Reasoning data</a></li>
 </ul></li>
 <li class="chapter" data-level="19.4" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#understanding-the-lmer-syntax"><i class="fa fa-check"></i><b>19.4</b> Understanding the lmer() syntax</a></li>
-<li class="chapter" data-level="19.5" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#anova-vs.-lmer"><i class="fa fa-check"></i><b>19.5</b> ANOVA vs. Lmer</a>
+<li class="chapter" data-level="19.5" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#anova-vs.-lmer"><i class="fa fa-check"></i><b>19.5</b> ANOVA vs. lmer</a>
 <ul>
 <li class="chapter" data-level="19.5.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#between-subjects-anova"><i class="fa fa-check"></i><b>19.5.1</b> Between subjects ANOVA</a></li>
 <li class="chapter" data-level="19.5.2" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#repeated-measures-anova"><i class="fa fa-check"></i><b>19.5.2</b> Repeated-measures ANOVA</a></li>
 <li class="chapter" data-level="19.5.3" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#mixed-anova"><i class="fa fa-check"></i><b>19.5.3</b> Mixed ANOVA</a></li>
 </ul></li>
-<li class="chapter" data-level="19.6" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#session-info-18"><i class="fa fa-check"></i><b>19.6</b> Session info</a></li>
+<li class="chapter" data-level="19.6" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#additional-resources-17"><i class="fa fa-check"></i><b>19.6</b> Additional resources</a>
+<ul>
+<li class="chapter" data-level="19.6.1" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#readings-2"><i class="fa fa-check"></i><b>19.6.1</b> Readings</a></li>
+</ul></li>
+<li class="chapter" data-level="19.7" data-path="linear-mixed-effects-models-3.html"><a href="linear-mixed-effects-models-3.html#session-info-18"><i class="fa fa-check"></i><b>19.7</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="20" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html"><i class="fa fa-check"></i><b>20</b> Linear mixed effects models 4</a>
 <ul>
@@ -641,10 +642,10 @@
 <li class="chapter" data-level="20.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#load-packages-and-set-plotting-theme-13"><i class="fa fa-check"></i><b>20.2</b> Load packages and set plotting theme</a></li>
 <li class="chapter" data-level="20.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#load-data-sets-5"><i class="fa fa-check"></i><b>20.3</b> Load data sets</a>
 <ul>
-<li class="chapter" data-level="20.3.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#sleep-data-1"><i class="fa fa-check"></i><b>20.3.1</b> Sleep data</a></li>
+<li class="chapter" data-level="20.3.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#sleep-data"><i class="fa fa-check"></i><b>20.3.1</b> Sleep data</a></li>
 <li class="chapter" data-level="20.3.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#reasoning-data-1"><i class="fa fa-check"></i><b>20.3.2</b> Reasoning data</a></li>
-<li class="chapter" data-level="20.3.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#weight-loss-data-1"><i class="fa fa-check"></i><b>20.3.3</b> Weight loss data</a></li>
-<li class="chapter" data-level="20.3.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politness-data-1"><i class="fa fa-check"></i><b>20.3.4</b> Politness data</a></li>
+<li class="chapter" data-level="20.3.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#weight-loss-data"><i class="fa fa-check"></i><b>20.3.3</b> Weight loss data</a></li>
+<li class="chapter" data-level="20.3.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politness-data"><i class="fa fa-check"></i><b>20.3.4</b> Politness data</a></li>
 </ul></li>
 <li class="chapter" data-level="20.4" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#follow-up-tests-with-emmeans"><i class="fa fa-check"></i><b>20.4</b> Follow-up tests with emmeans</a>
 <ul>
@@ -653,11 +654,17 @@
 <li class="chapter" data-level="20.4.3" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#politeness-study"><i class="fa fa-check"></i><b>20.4.3</b> Politeness study</a></li>
 </ul></li>
 <li class="chapter" data-level="20.5" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#mixtures-of-participants"><i class="fa fa-check"></i><b>20.5</b> Mixtures of participants</a></li>
-<li class="chapter" data-level="20.6" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#bootstrapping"><i class="fa fa-check"></i><b>20.6</b> Bootstrapping</a>
+<li class="chapter" data-level="20.6" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#simulating-different-random-effects-structures"><i class="fa fa-check"></i><b>20.6</b> Simulating different random effects structures</a>
+<ul>
+<li class="chapter" data-level="20.6.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#two-level-model"><i class="fa fa-check"></i><b>20.6.1</b> Two-level model</a></li>
+<li class="chapter" data-level="20.6.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#three-level-model"><i class="fa fa-check"></i><b>20.6.2</b> Three-level model</a></li>
+</ul></li>
+<li class="chapter" data-level="20.7" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#bootstrapping"><i class="fa fa-check"></i><b>20.7</b> Bootstrapping</a>
 <ul>
-<li class="chapter" data-level="20.6.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-model"><i class="fa fa-check"></i><b>20.6.1</b> Linear model</a></li>
+<li class="chapter" data-level="20.7.1" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-model"><i class="fa fa-check"></i><b>20.7.1</b> Linear model</a></li>
+<li class="chapter" data-level="20.7.2" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#linear-mixed-effects-model"><i class="fa fa-check"></i><b>20.7.2</b> Linear mixed effects model</a></li>
 </ul></li>
-<li class="chapter" data-level="20.7" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#session-info-19"><i class="fa fa-check"></i><b>20.7</b> Session info</a></li>
+<li class="chapter" data-level="20.8" data-path="linear-mixed-effects-models-4.html"><a href="linear-mixed-effects-models-4.html#session-info-19"><i class="fa fa-check"></i><b>20.8</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="21" data-path="generalized-linear-model.html"><a href="generalized-linear-model.html"><i class="fa fa-check"></i><b>21</b> Generalized linear model</a>
 <ul>
@@ -701,7 +708,7 @@
 <li class="chapter" data-level="22.4.2" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#frequentist-analysis"><i class="fa fa-check"></i><b>22.4.2</b> Frequentist analysis</a></li>
 <li class="chapter" data-level="22.4.3" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#bayesian-regression"><i class="fa fa-check"></i><b>22.4.3</b> Bayesian regression</a></li>
 </ul></li>
-<li class="chapter" data-level="22.5" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#additional-resources-17"><i class="fa fa-check"></i><b>22.5</b> Additional resources</a>
+<li class="chapter" data-level="22.5" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#additional-resources-18"><i class="fa fa-check"></i><b>22.5</b> Additional resources</a>
 <ul>
 <li class="chapter" data-level="22.5.1" data-path="bayesian-data-analysis-1.html"><a href="bayesian-data-analysis-1.html#books-and-chapters-5"><i class="fa fa-check"></i><b>22.5.1</b> Books and chapters</a></li>
 </ul></li>
@@ -744,7 +751,7 @@
 <li class="chapter" data-level="23.7.2" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#specify-and-fit-the-model-3"><i class="fa fa-check"></i><b>23.7.2</b> 2. Specify and fit the model</a></li>
 <li class="chapter" data-level="23.7.3" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#test-specific-hypotheses-3"><i class="fa fa-check"></i><b>23.7.3</b> 5. Test specific hypotheses</a></li>
 </ul></li>
-<li class="chapter" data-level="23.8" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#additional-resources-18"><i class="fa fa-check"></i><b>23.8</b> Additional resources</a></li>
+<li class="chapter" data-level="23.8" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#additional-resources-19"><i class="fa fa-check"></i><b>23.8</b> Additional resources</a></li>
 <li class="chapter" data-level="23.9" data-path="bayesian-data-analysis-2.html"><a href="bayesian-data-analysis-2.html#session-info-22"><i class="fa fa-check"></i><b>23.9</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="24" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html"><i class="fa fa-check"></i><b>24</b> Bayesian data analysis 3</a>
@@ -771,7 +778,7 @@
 <li class="chapter" data-level="24.6.4" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#gaussian-regression-unequal-variance"><i class="fa fa-check"></i><b>24.6.4</b> Gaussian regression (unequal variance)</a></li>
 <li class="chapter" data-level="24.6.5" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#model-comparison-2"><i class="fa fa-check"></i><b>24.6.5</b> Model comparison</a></li>
 </ul></li>
-<li class="chapter" data-level="24.7" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#additional-resources-19"><i class="fa fa-check"></i><b>24.7</b> Additional resources</a></li>
+<li class="chapter" data-level="24.7" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#additional-resources-20"><i class="fa fa-check"></i><b>24.7</b> Additional resources</a></li>
 <li class="chapter" data-level="24.8" data-path="bayesian-data-analysis-3.html"><a href="bayesian-data-analysis-3.html#session-info-23"><i class="fa fa-check"></i><b>24.8</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="25" data-path="model-assumptions.html"><a href="model-assumptions.html"><i class="fa fa-check"></i><b>25</b> Model assumptions</a>
@@ -786,7 +793,7 @@
 <li class="chapter" data-level="25.3.4" data-path="model-assumptions.html"><a href="model-assumptions.html#non-parametric-tests"><i class="fa fa-check"></i><b>25.3.4</b> Non-parametric tests</a></li>
 <li class="chapter" data-level="25.3.5" data-path="model-assumptions.html"><a href="model-assumptions.html#bootstrapping-regressions"><i class="fa fa-check"></i><b>25.3.5</b> Bootstrapping regressions</a></li>
 </ul></li>
-<li class="chapter" data-level="25.4" data-path="model-assumptions.html"><a href="model-assumptions.html#additional-resources-20"><i class="fa fa-check"></i><b>25.4</b> Additional resources</a></li>
+<li class="chapter" data-level="25.4" data-path="model-assumptions.html"><a href="model-assumptions.html#additional-resources-21"><i class="fa fa-check"></i><b>25.4</b> Additional resources</a></li>
 <li class="chapter" data-level="25.5" data-path="model-assumptions.html"><a href="model-assumptions.html#session-info-24"><i class="fa fa-check"></i><b>25.5</b> Session info</a></li>
 </ul></li>
 <li class="chapter" data-level="26" data-path="reporting-statistics.html"><a href="reporting-statistics.html"><i class="fa fa-check"></i><b>26</b> Reporting statistics</a>
@@ -803,7 +810,7 @@
 <ul>
 <li class="chapter" data-level="26.2.1" data-path="reporting-statistics.html"><a href="reporting-statistics.html#frequentist-statistics"><i class="fa fa-check"></i><b>26.2.1</b> Frequentist statistics</a></li>
 </ul></li>
-<li class="chapter" data-level="26.3" data-path="reporting-statistics.html"><a href="reporting-statistics.html#additional-resources-21"><i class="fa fa-check"></i><b>26.3</b> Additional resources</a>
+<li class="chapter" data-level="26.3" data-path="reporting-statistics.html"><a href="reporting-statistics.html#additional-resources-22"><i class="fa fa-check"></i><b>26.3</b> Additional resources</a>
 <ul>
 <li class="chapter" data-level="26.3.1" data-path="reporting-statistics.html"><a href="reporting-statistics.html#misc-12"><i class="fa fa-check"></i><b>26.3.1</b> Misc</a></li>
 </ul></li>
@@ -1762,10 +1769,10 @@ <h2><span class="header-section-number">18.6</span> Session info<a href="linear-
 
 loaded via a namespace (and not attached):
  [1] sjlabelled_1.2.0   tidyselect_1.2.0   viridisLite_0.4.2  farver_2.1.1      
- [5] fastmap_1.1.1      bayestestR_0.13.1  digest_0.6.33      estimability_1.4.1
+ [5] fastmap_1.1.1      bayestestR_0.13.1  digest_0.6.34      estimability_1.4.1
  [9] timechange_0.2.0   lifecycle_1.0.4    magrittr_2.0.3     compiler_4.3.2    
-[13] rlang_1.1.2        sass_0.4.8         tools_4.3.2        utf8_1.2.4        
-[17] yaml_2.3.8         labeling_0.4.3     xml2_1.3.6         withr_2.5.2       
+[13] rlang_1.1.3        sass_0.4.8         tools_4.3.2        utf8_1.2.4        
+[17] yaml_2.3.8         labeling_0.4.3     xml2_1.3.6         withr_3.0.0       
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 [29] MASS_7.3-60        insight_0.19.7     cli_3.6.2          mvtnorm_1.2-4     
@@ -1774,13 +1781,13 @@ <h2><span class="header-section-number">18.6</span> Session info<a href="linear-
 [41] splines_4.3.2      rvest_1.0.3        parallel_4.3.2     vctrs_0.6.5       
 [45] boot_1.3-28.1      webshot_0.5.5      jsonlite_1.8.8     bookdown_0.37     
 [49] patchwork_1.2.0    hms_1.1.3          ggrepel_0.9.4      pbkrtest_0.5.2    
-[53] listenv_0.9.0      systemfonts_1.0.5  jquerylib_0.1.4    glue_1.6.2        
-[57] parallelly_1.36.0  nloptr_2.0.3       codetools_0.2-19   stringi_1.8.3     
+[53] listenv_0.9.1      systemfonts_1.0.5  jquerylib_0.1.4    glue_1.7.0        
+[57] parallelly_1.37.0  nloptr_2.0.3       codetools_0.2-19   stringi_1.8.3     
 [61] gtable_0.3.4       munsell_0.5.0      furrr_0.3.1        pillar_1.9.0      
 [65] htmltools_0.5.7    R6_2.5.1           evaluate_0.23      lattice_0.22-5    
 [69] haven_2.5.4        highr_0.10         backports_1.4.1    broom_1.0.5       
-[73] snakecase_0.11.1   bslib_0.6.1        Rcpp_1.0.11        svglite_2.1.3     
-[77] coda_0.19-4        nlme_3.1-164       mgcv_1.9-1         xfun_0.41         
+[73] snakecase_0.11.1   bslib_0.6.1        Rcpp_1.0.12        svglite_2.1.3     
+[77] coda_0.19-4.1      nlme_3.1-164       mgcv_1.9-1         xfun_0.41         
 [81] pkgconfig_2.0.3   </code></pre>
 
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-[["generalized-linear-model.html", "Chapter 21 Generalized linear model 21.1 Learning goals 21.2 Load packages and set plotting theme 21.3 Load data set 21.4 Logistic regression 21.5 Simulate a logistic regression 21.6 Testing hypotheses 21.7 Logistic mixed effects model 21.8 Additional information 21.9 Session info", " Chapter 21 Generalized linear model 21.1 Learning goals Logistic regression. Logit transform. Fitting a logistic regression in R. Visualizing and interpreting model predictions. Simulating data from a logistic regression. Assessing model fit. Testing hypotheses. Reporting results. Mixed effects logistic regression. 21.2 Load packages and set plotting theme library(&quot;knitr&quot;) # for knitting RMarkdown library(&quot;titanic&quot;) # titanic dataset library(&quot;kableExtra&quot;) # for making nice tables library(&quot;janitor&quot;) # for cleaning column names library(&quot;broom&quot;) # for tidying up linear models library(&quot;lme4&quot;) # for linear mixed effects models library(&quot;boot&quot;) # for bootstrapping (also has an inverse logit function) library(&quot;ggeffects&quot;) # for computing marginal effects library(&quot;afex&quot;) # for significance testing of mixed effects models library(&quot;emmeans&quot;) # for the joint_tests() function library(&quot;tidyverse&quot;) # for wrangling, plotting, etc. theme_set(theme_classic() + #set the theme theme(text = element_text(size = 20))) #set the default text size opts_chunk$set(comment = &quot;&quot;, fig.show = &quot;hold&quot;) 21.3 Load data set df.titanic = titanic_train %&gt;% clean_names() %&gt;% mutate(sex = as.factor(sex)) Let’s take a quick look at the data: df.titanic %&gt;% glimpse() Rows: 891 Columns: 12 $ passenger_id &lt;int&gt; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17… $ survived &lt;int&gt; 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, … $ pclass &lt;int&gt; 3, 1, 3, 1, 3, 3, 1, 3, 3, 2, 3, 1, 3, 3, 3, 2, 3, 2, 3, … $ name &lt;chr&gt; &quot;Braund, Mr. Owen Harris&quot;, &quot;Cumings, Mrs. John Bradley (F… $ sex &lt;fct&gt; male, female, female, female, male, male, male, male, fem… $ age &lt;dbl&gt; 22, 38, 26, 35, 35, NA, 54, 2, 27, 14, 4, 58, 20, 39, 14,… $ sib_sp &lt;int&gt; 1, 1, 0, 1, 0, 0, 0, 3, 0, 1, 1, 0, 0, 1, 0, 0, 4, 0, 1, … $ parch &lt;int&gt; 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 0, 0, 5, 0, 0, 1, 0, 0, … $ ticket &lt;chr&gt; &quot;A/5 21171&quot;, &quot;PC 17599&quot;, &quot;STON/O2. 3101282&quot;, &quot;113803&quot;, &quot;3… $ fare &lt;dbl&gt; 7.2500, 71.2833, 7.9250, 53.1000, 8.0500, 8.4583, 51.8625… $ cabin &lt;chr&gt; &quot;&quot;, &quot;C85&quot;, &quot;&quot;, &quot;C123&quot;, &quot;&quot;, &quot;&quot;, &quot;E46&quot;, &quot;&quot;, &quot;&quot;, &quot;&quot;, &quot;G6&quot;, &quot;… $ embarked &lt;chr&gt; &quot;S&quot;, &quot;C&quot;, &quot;S&quot;, &quot;S&quot;, &quot;S&quot;, &quot;Q&quot;, &quot;S&quot;, &quot;S&quot;, &quot;S&quot;, &quot;C&quot;, &quot;S&quot;, &quot;S… # Table of the first 10 entries df.titanic %&gt;% head(10) %&gt;% kable(digits = 2) %&gt;% kable_styling(bootstrap_options = &quot;striped&quot;, full_width = F) passenger_id survived pclass name sex age sib_sp parch ticket fare cabin embarked 1 0 3 Braund, Mr. Owen Harris male 22 1 0 A/5 21171 7.25 S 2 1 1 Cumings, Mrs. John Bradley (Florence Briggs Thayer) female 38 1 0 PC 17599 71.28 C85 C 3 1 3 Heikkinen, Miss. Laina female 26 0 0 STON/O2. 3101282 7.92 S 4 1 1 Futrelle, Mrs. Jacques Heath (Lily May Peel) female 35 1 0 113803 53.10 C123 S 5 0 3 Allen, Mr. William Henry male 35 0 0 373450 8.05 S 6 0 3 Moran, Mr. James male NA 0 0 330877 8.46 Q 7 0 1 McCarthy, Mr. Timothy J male 54 0 0 17463 51.86 E46 S 8 0 3 Palsson, Master. Gosta Leonard male 2 3 1 349909 21.08 S 9 1 3 Johnson, Mrs. Oscar W (Elisabeth Vilhelmina Berg) female 27 0 2 347742 11.13 S 10 1 2 Nasser, Mrs. Nicholas (Adele Achem) female 14 1 0 237736 30.07 C 21.4 Logistic regression Let’s see if we can predict whether or not a passenger survived based on the price of their ticket. Let’s run a simple regression first: # fit a linear model fit.lm = lm(formula = survived ~ 1 + fare, data = df.titanic) # summarize the results fit.lm %&gt;% summary() Call: lm(formula = survived ~ 1 + fare, data = df.titanic) Residuals: Min 1Q Median 3Q Max -0.9653 -0.3391 -0.3222 0.6044 0.6973 Coefficients: Estimate Std. Error t value Pr(&gt;|t|) (Intercept) 0.3026994 0.0187849 16.114 &lt; 2e-16 *** fare 0.0025195 0.0003174 7.939 6.12e-15 *** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 Residual standard error: 0.4705 on 889 degrees of freedom Multiple R-squared: 0.06621, Adjusted R-squared: 0.06516 F-statistic: 63.03 on 1 and 889 DF, p-value: 6.12e-15 Look’s like fare is a significant predictor of whether or not a person survived. Let’s visualize the model’s predictions: ggplot(data = df.titanic, mapping = aes(x = fare, y = survived)) + geom_smooth(method = &quot;lm&quot;) + geom_point() + labs(y = &quot;survived&quot;) This doesn’t look good! The model predicts intermediate values of survived (which doesn’t make sense given that a person either survived or didn’t survive). Furthermore, the model predicts values greater than 1 for fares greather than ~ 300. Let’s run a logistic regression instead. # fit a logistic regression fit.glm = glm(formula = survived ~ 1 + fare, family = &quot;binomial&quot;, data = df.titanic) fit.glm %&gt;% summary() Call: glm(formula = survived ~ 1 + fare, family = &quot;binomial&quot;, data = df.titanic) Coefficients: Estimate Std. Error z value Pr(&gt;|z|) (Intercept) -0.941330 0.095129 -9.895 &lt; 2e-16 *** fare 0.015197 0.002232 6.810 9.79e-12 *** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 1186.7 on 890 degrees of freedom Residual deviance: 1117.6 on 889 degrees of freedom AIC: 1121.6 Number of Fisher Scoring iterations: 4 And let’s visualize the predictions of the logistic regression: ggplot(data = df.titanic, mapping = aes(x = fare, y = survived)) + geom_smooth(method = &quot;glm&quot;, method.args = list(family = &quot;binomial&quot;)) + geom_point() + labs(y = &quot;p(survived)&quot;) Much better! Note that we’ve changed the interpretation of our dependent variable. We are now predicting the probability that a person survived based on their fare. The model now only predicts values between 0 and 1. To achieve this, we apply a logit transform to the outcome variable like so: \\[ \\ln(\\frac{\\pi_i}{1-\\pi_i}) = b_0 + b_1 \\cdot X_i + e_i \\] where \\(\\pi_i\\) is the probability of passenger \\(i\\) having survived. Importantly, this affects our interpretation of the model parameters. They are now defined in log-odds, and can apply an inverse logit transformation to turn this back into a probability: With \\[ \\pi = P(Y = 1) \\] and the logit transformation \\[ \\ln(\\frac{\\pi}{1-\\pi}) = V, \\] where \\(V\\) is just a placeholder for our linear model formula, we can go back to \\(\\pi\\) through the inverse logit transformation like so: \\[ \\pi = \\frac{e^V}{1 + e^V} \\] In R, we can use log(x) to calculate the natural logarithm \\(\\ln(x)\\), and exp(x) to calculate e^x. 21.4.1 Interpreting the parameters fit.glm %&gt;% summary() Call: glm(formula = survived ~ 1 + fare, family = &quot;binomial&quot;, data = df.titanic) Coefficients: Estimate Std. Error z value Pr(&gt;|z|) (Intercept) -0.941330 0.095129 -9.895 &lt; 2e-16 *** fare 0.015197 0.002232 6.810 9.79e-12 *** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 1186.7 on 890 degrees of freedom Residual deviance: 1117.6 on 889 degrees of freedom AIC: 1121.6 Number of Fisher Scoring iterations: 4 The estimate for the intercept and fare are in log-odds. Let’s take a look at the linear model’s predictions in log-odds space. df.plot = fit.glm %&gt;% augment() %&gt;% clean_names() ggplot(data = df.plot, mapping = aes(x = fare, y = fitted)) + geom_line() Nice, looks like a good old linear model. But what’s the y-axis here? It’s in log-odds (buh!). This is difficult to interpret. Let’s transform the y-axis back to probabilities to make it easier to see what’s going on. ggplot(data = df.plot, mapping = aes(x = fare, y = inv.logit(fitted))) + geom_line() Great! Now the y-axis is back in probability space. We used the inverse logit function inv.logit() to transfer to log-odds back into probabilities. Let’s check what the intercept of our model is in probability space: fit.glm %&gt;% pluck(coefficients, 1) %&gt;% inv.logit() [1] 0.2806318 Here, we see that the intercept is \\(p = 0.28\\). That is, the predicted chance of survival for someone who didn’t pay any fare at all is 28% according to the model. Interpreting the slope is a little more tricky. Let’s look at a situation first where we have a binary predictor. 21.4.1.1 Logit transform Here is a visualization of what the odds and log odds transformation look like. # going from probabilities to odds (ranges from 0 to +infinity) ggplot(data = tibble(x = seq(0, 1, 0.1)), mapping = aes(x = x)) + stat_function(fun = ~ ./(1 - .), size = 1) Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0. ℹ Please use `linewidth` instead. This warning is displayed once every 8 hours. Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated. # going from probabilities to log odds (ranges from -infinity to +infinity) ggplot(data = tibble(x = seq(0, 1, 0.1)), mapping = aes(x = x)) + stat_function(fun = ~ log(./(1 - .)), size = 1) 21.4.2 Binary predictor Let’s see whether the probability of survival differed between male and female passengers. fit.glm2 = glm(formula = survived ~ 1 + sex, family = &quot;binomial&quot;, data = df.titanic) fit.glm2 %&gt;% summary() Call: glm(formula = survived ~ 1 + sex, family = &quot;binomial&quot;, data = df.titanic) Coefficients: Estimate Std. Error z value Pr(&gt;|z|) (Intercept) 1.0566 0.1290 8.191 2.58e-16 *** sexmale -2.5137 0.1672 -15.036 &lt; 2e-16 *** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 1186.7 on 890 degrees of freedom Residual deviance: 917.8 on 889 degrees of freedom AIC: 921.8 Number of Fisher Scoring iterations: 4 It looks like it did! Let’s visualize: df.titanic %&gt;% mutate(survived = factor(survived, labels = c(&quot;died&quot;, &quot;survived&quot;))) %&gt;% ggplot(data = ., mapping = aes(x = sex, fill = survived)) + geom_bar(position = &quot;fill&quot;, color = &quot;black&quot;) + scale_fill_brewer(palette = &quot;Set1&quot;) + labs(x = &quot;&quot;, fill = &quot;&quot;, y = &quot;probability&quot;) And let’s interpret the parameters by applying the inverse logit transform. To get the prediction for female passengers we get \\[ \\widehat{\\ln(\\frac{\\pi_i}{1-\\pi_i})} = b_0 + b_1 \\cdot \\text{sex}_i = b_0 + b_1 \\cdot 0 = b_0 \\] since we dummy coded the predictor and female is our reference category. To get the predicted probability of survival for women we do the logit transform: \\[ \\pi = \\frac{e^{b_0}}{1 + e^{b_0}} \\] The predicted probability is: fit.glm2 %&gt;% pluck(coefficients, 1) %&gt;% inv.logit() [1] 0.7420382 To get the prediction for male passengers we have: \\[ \\widehat{\\ln(\\frac{\\pi_i}{1-\\pi_i})} = b_0 + b_1 \\cdot \\text{sex}_i = b_0 + b_1 \\cdot 1 = b_0 + b_1 \\] Applying the logit transform like so \\[ \\pi = \\frac{e^{b_0 + b_1}}{1 + e^{b_0 + b_1}} \\] The predicted probability of male passengers surviving is: fit.glm2 %&gt;% pluck(coefficients) %&gt;% sum() %&gt;% inv.logit() [1] 0.1889081 Here is the same information in a table: df.titanic %&gt;% count(sex, survived) %&gt;% mutate(p = n / sum(n)) %&gt;% group_by(sex) %&gt;% mutate(`p(survived|sex)` = p / sum(p)) # A tibble: 4 × 5 # Groups: sex [2] sex survived n p `p(survived|sex)` &lt;fct&gt; &lt;int&gt; &lt;int&gt; &lt;dbl&gt; &lt;dbl&gt; 1 female 0 81 0.0909 0.258 2 female 1 233 0.262 0.742 3 male 0 468 0.525 0.811 4 male 1 109 0.122 0.189 21.4.3 Continuous predictor To interpret the predictions when a continuous predictor is involved, it’s easiest to consider a few concrete cases. Here, I use the augment() function from the “broom” package to get the model’s predictions for some values of interest: fit.glm %&gt;% augment(newdata = tibble(fare = c(0, 10, 50, 100, 500))) %&gt;% clean_names() %&gt;% select(fare, prediction = fitted) %&gt;% mutate(`p(survival)` = inv.logit(prediction)) # A tibble: 5 × 3 fare prediction `p(survival)` &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; 1 0 -0.941 0.281 2 10 -0.789 0.312 3 50 -0.181 0.455 4 100 0.578 0.641 5 500 6.66 0.999 21.4.4 Several predictors Let’s fit a logistic regression that predicts the probability of survival based both on the passenger’s sex and what fare they paid (allowing for an interaction of the two predictors): fit.glm3 = glm(formula = survived ~ 1 + sex * fare, family = &quot;binomial&quot;, data = df.titanic) fit.glm3 %&gt;% summary() Call: glm(formula = survived ~ 1 + sex * fare, family = &quot;binomial&quot;, data = df.titanic) Coefficients: Estimate Std. Error z value Pr(&gt;|z|) (Intercept) 0.408428 0.189999 2.150 0.031584 * sexmale -2.099345 0.230291 -9.116 &lt; 2e-16 *** fare 0.019878 0.005372 3.701 0.000215 *** sexmale:fare -0.011617 0.005934 -1.958 0.050252 . --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 1186.66 on 890 degrees of freedom Residual deviance: 879.85 on 887 degrees of freedom AIC: 887.85 Number of Fisher Scoring iterations: 5 Make sure not to interpret the significance test on the coefficients here as main effects. Based on this summary table, you cannot say whether there is a significant difference between male vs. female passenger in their probability of survival. What coefficient for sexmale captures is whether there is a significant difference between male and female passengers who paid a fare of 0. That is, it’s the predicted difference between the reference category (female) and the other category (male) when all other predictors are 0. Let’s visualize the model predictions: ggplot(data = df.titanic, mapping = aes(x = fare, y = survived, color = sex)) + geom_point(alpha = 0.1, size = 2) + geom_smooth(method = &quot;glm&quot;, method.args = list(family = &quot;binomial&quot;), alpha = 0.2, aes(fill = sex)) + scale_color_brewer(palette = &quot;Set1&quot;) Just for kicks, to get a better sense for what the interaction looks like, here is the visualization in log-odds space: fit.glm3 %&gt;% augment() %&gt;% clean_names() %&gt;% ggplot(data = ., mapping = aes(x = fare, color = sex, y = fitted)) + geom_line() + scale_color_brewer(palette = &quot;Set1&quot;) Let’s see how large the difference between genders is once we take into account how much each person paid for the fair: ggpredict(fit.glm3, terms = c(&quot;sex&quot;)) # Predicted probabilities of survived sex | Predicted | 95% CI --------------------------------- female | 0.74 | [0.69, 0.79] male | 0.19 | [0.16, 0.23] Adjusted for: * fare = 32.20 We notice that there is one outlier who was male and paid a $500 fare (or maybe this is a mistake in the data entry?!). Let’s remove this outlier and see what happens: fit.glm3_no_outlier = glm(formula = survived ~ 1 + sex * fare, family = &quot;binomial&quot;, data = df.titanic %&gt;% filter(fare &lt; 500)) fit.glm3_no_outlier %&gt;% summary() Call: glm(formula = survived ~ 1 + sex * fare, family = &quot;binomial&quot;, data = df.titanic %&gt;% filter(fare &lt; 500)) Coefficients: Estimate Std. Error z value Pr(&gt;|z|) (Intercept) 0.408436 0.190019 2.149 0.031598 * sexmale -2.085344 0.232260 -8.979 &lt; 2e-16 *** fare 0.019878 0.005373 3.699 0.000216 *** sexmale:fare -0.012178 0.006066 -2.008 0.044688 * --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 1180.89 on 887 degrees of freedom Residual deviance: 879.51 on 884 degrees of freedom AIC: 887.51 Number of Fisher Scoring iterations: 5 df.titanic %&gt;% filter(fare &lt; 500) %&gt;% mutate(sex = as.factor(sex)) %&gt;% ggplot(data = ., mapping = aes(x = fare, y = survived, color = sex)) + geom_point(alpha = 0.1, size = 2) + stat_smooth(method = &quot;glm&quot;, method.args = list(family = &quot;binomial&quot;), alpha = 0.2, fullrange = T, aes(fill = sex)) + scale_color_brewer(palette = &quot;Set1&quot;) + scale_x_continuous(limits = c(0, 500)) There is still a clear difference between female and male passengers, but the prediction for male passengers has changed a bit. Let’s look at a concrete example: # with the outlier: # predicted probability of survival for a male passenger who paid $200 for their fare inv.logit(fit.glm3$coefficients[1] + fit.glm3$coefficients[2] + fit.glm3$coefficients[3] * 200 + fit.glm3$coefficients[4] * 200) (Intercept) 0.4903402 # without the outlier: # predicted probability of survival for a male passenger who paid $200 for their fare inv.logit(fit.glm3_no_outlier$coefficients[1] + fit.glm3_no_outlier$coefficients[2] + fit.glm3_no_outlier$coefficients[3] * 200 + fit.glm3_no_outlier$coefficients[4] * 200) (Intercept) 0.4658284 With the outlier removed, the predicted probability of survival for a male passenger who paid $200 decreases from 49% to 47%. 21.4.5 Using the “ggeffects” package The “ggeffects” package helps with the interpretation of the results. It applies the inverse logit transform for us, and shows the predictions for a range of cases. # show effects ggeffect(model = fit.glm, terms = &quot;fare [1, 100, 200, 300, 400, 500]&quot;) # Predicted probabilities of survived fare | Predicted | 95% CI ------------------------------- 1 | 0.28 | [0.25, 0.32] 100 | 0.64 | [0.56, 0.72] 200 | 0.89 | [0.79, 0.95] 300 | 0.97 | [0.92, 0.99] 400 | 0.99 | [0.97, 1.00] 500 | 1.00 | [0.99, 1.00] I’ve used the [] construction to specify for what values of the predictor fare, I’d like get the predicted values. Here, the prediction is marginalized across both women and men. We can also get a plot of the model predictions like so: ggeffect(model = fit.glm, terms = &quot;fare [1, 100, 200, 300, 400, 500]&quot;) %&gt;% plot() And, we can also get the predicted probability of survival for sex marginalized over the fare, using the model which included both sex and fare, as well as its interaction as predictors. ggeffect(model = fit.glm3, terms = &quot;sex&quot;) # Predicted probabilities of survived sex | Predicted | 95% CI --------------------------------- female | 0.74 | [0.69, 0.79] male | 0.19 | [0.16, 0.23] Finally, we can ask for predictions for specific combinations of our predictor variables, by using the ggpredict() function. ggpredict(model = fit.glm3, terms = c(&quot;sex&quot;, &quot;fare [200]&quot;)) # Predicted probabilities of survived sex | Predicted | 95% CI --------------------------------- female | 0.99 | [0.93, 1.00] male | 0.49 | [0.29, 0.70] The example above, shows the predicted probability of survival for male vs. female passengers, assuming that they paid 200 for the fare. 21.5 Simulate a logistic regression As always, to better understand a statistical modeling procedure, it’s helpful to simulate data from the assumed data-generating process, fit the model, and see whether we can reconstruct the parameters. # make example reproducible set.seed(1) # set parameters sample_size = 1000 b0 = 0 b1 = 1 # b1 = 8 # generate data df.data = tibble(x = rnorm(n = sample_size), y = b0 + b1 * x, p = inv.logit(y)) %&gt;% mutate(response = rbinom(n(), size = 1, p = p)) # fit model fit = glm(formula = response ~ 1 + x, family = &quot;binomial&quot;, data = df.data) # model summary fit %&gt;% summary() Call: glm(formula = response ~ 1 + x, family = &quot;binomial&quot;, data = df.data) Coefficients: Estimate Std. Error z value Pr(&gt;|z|) (Intercept) -0.06214 0.06918 -0.898 0.369 x 0.92905 0.07937 11.705 &lt;2e-16 *** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 1385.4 on 999 degrees of freedom Residual deviance: 1209.6 on 998 degrees of freedom AIC: 1213.6 Number of Fisher Scoring iterations: 3 df.data %&gt;% head(10) %&gt;% kable(digits = 2) %&gt;% kable_styling(bootstrap_options = &quot;striped&quot;, full_width = F) x y p response -0.63 -0.63 0.35 1 0.18 0.18 0.55 0 -0.84 -0.84 0.30 1 1.60 1.60 0.83 1 0.33 0.33 0.58 1 -0.82 -0.82 0.31 0 0.49 0.49 0.62 1 0.74 0.74 0.68 1 0.58 0.58 0.64 1 -0.31 -0.31 0.42 0 Nice! The inferred estimates are very close to the parameter values we used to simulate the data. Let’s visualize the result: ggplot(data = df.data, mapping = aes(x = x, y = response)) + geom_smooth(method = &quot;glm&quot;, method.args = list(family = &quot;binomial&quot;)) + geom_point(alpha = 0.1) + labs(y = &quot;p(response)&quot;) 21.5.1 Calculate the model’s likelihood To calculate the likelihood of the data for a given logistic model, we look at the actual response, and the probability of the predicted response, and then determine the likelihood of the observation assuming a Bernoulli process. To get the overall likelihood of the data, we then multiply the likelihood of each data point (or take the logs first and then the sum to get the log-likelihood). This table illustrate the steps involved: fit %&gt;% augment() %&gt;% clean_names() %&gt;% mutate(p = inv.logit(fitted)) %&gt;% select(response, p) %&gt;% mutate(p_response = ifelse(response == 1, p, 1-p), log_p = log(p_response)) %&gt;% rename(`p(Y = 1)` = p, `p(Y = response)` = p_response, `log(p(Y = response))` = log_p) # A tibble: 1,000 × 4 response `p(Y = 1)` `p(Y = response)` `log(p(Y = response))` &lt;int&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; 1 1 0.344 0.344 -1.07 2 0 0.527 0.473 -0.749 3 1 0.302 0.302 -1.20 4 1 0.805 0.805 -0.217 5 1 0.561 0.561 -0.579 6 0 0.305 0.695 -0.364 7 1 0.596 0.596 -0.517 8 1 0.651 0.651 -0.429 9 1 0.616 0.616 -0.484 10 0 0.414 0.586 -0.535 # ℹ 990 more rows Let’s calculate the log-likelihood by hand: fit %&gt;% augment() %&gt;% clean_names() %&gt;% mutate(p = inv.logit(fitted), log_likelihood = response * log(p) + (1 - response) * log(1 - p)) %&gt;% summarize(log_likelihood = sum(log_likelihood)) # A tibble: 1 × 1 log_likelihood &lt;dbl&gt; 1 -605. And compare it with the model summary fit %&gt;% glance() %&gt;% select(logLik, AIC, BIC) # A tibble: 1 × 3 logLik AIC BIC &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; 1 -605. 1214. 1223. We’re getting the same result – neat! Now we know how the likelihood of the data is calculated for a logistic regression model. 21.6 Testing hypotheses To test hypotheses, we can use our good old model comparison approach: # fit compact model fit.compact = glm(formula = survived ~ 1 + fare, family = &quot;binomial&quot;, data = df.titanic) # fit augmented model fit.augmented = glm(formula = survived ~ 1 + sex + fare, family = &quot;binomial&quot;, data = df.titanic) # likelihood ratio test anova(fit.compact, fit.augmented, test = &quot;LRT&quot;) Analysis of Deviance Table Model 1: survived ~ 1 + fare Model 2: survived ~ 1 + sex + fare Resid. Df Resid. Dev Df Deviance Pr(&gt;Chi) 1 889 1117.57 2 888 884.31 1 233.26 &lt; 2.2e-16 *** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 Note that in order to get a p-value out of this, we need to specify what statistical test we’d like to run. In this case, we use the likelihood ratio test (“LRT”). We can also test for both effects of survived and fare in one go using the joint_tests() function from the “emmeans” package like so: glm(formula = survived ~ 1 + sex + fare, family = &quot;binomial&quot;, data = df.titanic) %&gt;% joint_tests() model term df1 df2 F.ratio Chisq p.value sex 1 Inf 201.881 201.881 &lt;.0001 fare 1 Inf 23.869 23.869 &lt;.0001 Notice that the F.ratio reported using joint_tests() (201.881) is not quite the same as the deviance value that we get through the likelihood ratio test (233.26). You can read more about why these two can come apart here. 21.7 Logistic mixed effects model Just like we can build linear mixed effects models using lmer() instead of lm(), we can also build a logistic mixed effects regression using glmer() instead of glm(). Let’s read in some data: # load bdf data set from nlme package data(bdf, package = &quot;nlme&quot;) df.language = bdf %&gt;% clean_names() %&gt;% filter(repeatgr != 2) %&gt;% mutate(repeatgr = repeatgr %&gt;% as.character() %&gt;% as.numeric()) rm(bdf) Fit the model, and print out the results: fit = glmer(repeatgr ~ 1 + ses + minority + (1 | school_nr), data = df.language, family = &quot;binomial&quot;) fit %&gt;% summary() Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) [glmerMod] Family: binomial ( logit ) Formula: repeatgr ~ 1 + ses + minority + (1 | school_nr) Data: df.language AIC BIC logLik deviance df.resid 1659.1 1682.1 -825.6 1651.1 2279 Scaled residuals: Min 1Q Median 3Q Max -0.9235 -0.4045 -0.3150 -0.2249 5.8372 Random effects: Groups Name Variance Std.Dev. school_nr (Intercept) 0.2489 0.4989 Number of obs: 2283, groups: school_nr, 131 Fixed effects: Estimate Std. Error z value Pr(&gt;|z|) (Intercept) -0.506291 0.197568 -2.563 0.01039 * ses -0.060086 0.007524 -7.986 1.39e-15 *** minorityY 0.673612 0.238659 2.822 0.00477 ** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 Correlation of Fixed Effects: (Intr) ses ses -0.898 minorityY -0.308 0.208 To visualize the results, we can use the ggeffects package. ggpredict(model = fit, terms = c(&quot;ses [all]&quot;, &quot;minority&quot;)) %&gt;% plot() And for significance testing, we can use the the joint_tests() function from the “emmeans” package glmer(formula = repeatgr ~ 1 + ses + minority + (1 | school_nr), data = df.language, family = &quot;binomial&quot;) %&gt;% joint_tests() model term df1 df2 F.ratio Chisq p.value ses 1 Inf 63.783 63.783 &lt;.0001 minority 1 Inf 7.966 7.966 0.0048 The results show that there was both a significant effect of ses and of minority. Note: This post here says a little more about the relationship of the F.ratio in the joint_tests() function, and what a likelihood ratio test yields. In short, it’s roughly the same thing. If you’d like to compute the likelihood ratio test, a convenient way of doing so is by using the mixed() function from the “afex” package. mixed(formula = repeatgr ~ 1 + ses + minority + (1 | school_nr), family = &quot;binomial&quot;, data = df.language, method = &quot;LRT&quot;) Contrasts set to contr.sum for the following variables: minority, school_nr Numerical variables NOT centered on 0: ses If in interactions, interpretation of lower order (e.g., main) effects difficult. Mixed Model Anova Table (Type 3 tests, LRT-method) Model: repeatgr ~ 1 + ses + minority + (1 | school_nr) Data: df.language Df full model: 4 Effect df Chisq p.value 1 ses 1 75.39 *** &lt;.001 2 minority 1 7.53 ** .006 --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;+&#39; 0.1 &#39; &#39; 1 And we can compare that the model comparison approach gives us the same result: fit_a = glmer(repeatgr ~ 1 + ses + minority + (1 | school_nr), data = df.language, family = &quot;binomial&quot;) # dropping ses as a predictor fit_c = glmer(repeatgr ~ 1 + minority + (1 | school_nr), data = df.language, family = &quot;binomial&quot;) anova(fit_a, fit_c, test = &quot;LRT&quot;) Data: df.language Models: fit_c: repeatgr ~ 1 + minority + (1 | school_nr) fit_a: repeatgr ~ 1 + ses + minority + (1 | school_nr) npar AIC BIC logLik deviance Chisq Df Pr(&gt;Chisq) fit_c 3 1732.5 1749.7 -863.27 1726.5 fit_a 4 1659.1 1682.1 -825.57 1651.1 75.395 1 &lt; 2.2e-16 *** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 21.8 Additional information 21.8.1 Misc Nice logistic regression explainer StatQuest: Logistic regression 21.8.2 Datacamp Multiple and logistic regression Generalized linear models in R Categorical data in the tidyverse 21.9 Session info Information about this R session including which version of R was used, and what packages were loaded. sessionInfo() R version 4.3.2 (2023-10-31) Platform: aarch64-apple-darwin20 (64-bit) Running under: macOS Sonoma 14.1.2 Matrix products: default BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0 locale: [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8 time zone: America/Los_Angeles tzcode source: internal attached base packages: [1] stats graphics grDevices utils datasets methods base other attached packages: [1] lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1 dplyr_1.1.4 [5] purrr_1.0.2 readr_2.1.4 tidyr_1.3.0 tibble_3.2.1 [9] ggplot2_3.4.4 tidyverse_2.0.0 emmeans_1.9.0 afex_1.3-0 [13] ggeffects_1.3.4 boot_1.3-28.1 lme4_1.1-35.1 Matrix_1.6-4 [17] broom_1.0.5 janitor_2.2.0 kableExtra_1.3.4 titanic_0.1.0 [21] knitr_1.45 loaded via a namespace (and not attached): [1] sjlabelled_1.2.0 tidyselect_1.2.0 viridisLite_0.4.2 [4] farver_2.1.1 fastmap_1.1.1 digest_0.6.33 [7] timechange_0.2.0 estimability_1.4.1 lifecycle_1.0.4 [10] survival_3.5-7 magrittr_2.0.3 compiler_4.3.2 [13] rlang_1.1.2 sass_0.4.8 tools_4.3.2 [16] utf8_1.2.4 yaml_2.3.8 labeling_0.4.3 [19] RColorBrewer_1.1-3 plyr_1.8.9 xml2_1.3.6 [22] abind_1.4-5 withr_2.5.2 numDeriv_2016.8-1.1 [25] effects_4.2-2 nnet_7.3-19 datawizard_0.9.1 [28] grid_4.3.2 fansi_1.0.6 xtable_1.8-4 [31] colorspace_2.1-0 scales_1.3.0 MASS_7.3-60 [34] insight_0.19.7 survey_4.2-1 cli_3.6.2 [37] mvtnorm_1.2-4 crayon_1.5.2 rmarkdown_2.25 [40] generics_0.1.3 rstudioapi_0.15.0 httr_1.4.7 [43] reshape2_1.4.4 tzdb_0.4.0 DBI_1.2.0 [46] minqa_1.2.6 cachem_1.0.8 splines_4.3.2 [49] rvest_1.0.3 parallel_4.3.2 mitools_2.4 [52] vctrs_0.6.5 webshot_0.5.5 jsonlite_1.8.8 [55] carData_3.0-5 bookdown_0.37 car_3.1-2 [58] hms_1.1.3 systemfonts_1.0.5 jquerylib_0.1.4 [61] glue_1.6.2 nloptr_2.0.3 stringi_1.8.3 [64] gtable_0.3.4 lmerTest_3.1-3 munsell_0.5.0 [67] pillar_1.9.0 htmltools_0.5.7 R6_2.5.1 [70] evaluate_0.23 lattice_0.22-5 haven_2.5.4 [73] highr_0.10 backports_1.4.1 snakecase_0.11.1 [76] bslib_0.6.1 Rcpp_1.0.11 svglite_2.1.3 [79] coda_0.19-4 nlme_3.1-164 mgcv_1.9-1 [82] xfun_0.41 pkgconfig_2.0.3 "],["404.html", "Page not found", " Page not found The page you requested cannot be found (perhaps it was moved or renamed). 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+[["bayesian-data-analysis-2.html", "Chapter 23 Bayesian data analysis 2 23.1 Learning goals 23.2 Load packages and set plotting theme 23.3 Load data sets 23.4 Poker 23.5 Sleep study 23.6 Titanic study 23.7 Politeness data 23.8 Additional resources 23.9 Session info", " Chapter 23 Bayesian data analysis 2 23.1 Learning goals Building Bayesian models with brms. Model evaluation: Visualizing and interpreting results. Testing hypotheses. Inference evaluation: Did things work out? 23.2 Load packages and set plotting theme library(&quot;knitr&quot;) # for knitting RMarkdown library(&quot;kableExtra&quot;) # for making nice tables library(&quot;janitor&quot;) # for cleaning column names library(&quot;tidybayes&quot;) # tidying up results from Bayesian models library(&quot;brms&quot;) # Bayesian regression models with Stan library(&quot;patchwork&quot;) # for making figure panels library(&quot;GGally&quot;) # for pairs plot library(&quot;broom.mixed&quot;) # for tidy lmer results library(&quot;bayesplot&quot;) # for visualization of Bayesian model fits library(&quot;modelr&quot;) # for modeling functions library(&quot;lme4&quot;) # for linear mixed effects models library(&quot;afex&quot;) # for ANOVAs library(&quot;car&quot;) # for ANOVAs library(&quot;emmeans&quot;) # for linear contrasts library(&quot;ggeffects&quot;) # for help with logistic regressions library(&quot;titanic&quot;) # titanic dataset library(&quot;gganimate&quot;) # for animations library(&quot;parameters&quot;) # for getting parameters library(&quot;transformr&quot;) # for gganimate # install via: devtools::install_github(&quot;thomasp85/transformr&quot;) library(&quot;tidyverse&quot;) # for wrangling, plotting, etc. theme_set(theme_classic() + # set the theme theme(text = element_text(size = 20))) # set the default text size opts_chunk$set(comment = &quot;&quot;, fig.show = &quot;hold&quot;) options(dplyr.summarise.inform = F) # set default color scheme in ggplot options(ggplot2.discrete.color = RColorBrewer::brewer.pal(9,&quot;Set1&quot;)) 23.3 Load data sets # poker df.poker = read_csv(&quot;data/poker.csv&quot;) %&gt;% mutate(skill = factor(skill, levels = 1:2, labels = c(&quot;expert&quot;, &quot;average&quot;)), skill = fct_relevel(skill, &quot;average&quot;, &quot;expert&quot;), hand = factor(hand, levels = 1:3, labels = c(&quot;bad&quot;, &quot;neutral&quot;, &quot;good&quot;)), limit = factor(limit, levels = 1:2, labels = c(&quot;fixed&quot;, &quot;none&quot;)), participant = 1:n()) %&gt;% select(participant, everything()) # sleep df.sleep = sleepstudy %&gt;% as_tibble() %&gt;% clean_names() %&gt;% mutate(subject = as.character(subject)) %&gt;% select(subject, days, reaction) %&gt;% bind_rows(tibble(subject = &quot;374&quot;, days = 0:1, reaction = c(286, 288)), tibble(subject = &quot;373&quot;, days = 0, reaction = 245)) # titanic df.titanic = titanic_train %&gt;% clean_names() %&gt;% mutate(sex = as.factor(sex)) # politeness df.politeness = read_csv(&quot;data/politeness_data.csv&quot;) %&gt;% rename(pitch = frequency) 23.4 Poker 23.4.1 1. Visualize the data Let’s visualize the data first. set.seed(1) df.poker %&gt;% ggplot(mapping = aes(x = hand, y = balance, fill = hand, group = skill, shape = skill)) + geom_point(alpha = 0.2, position = position_jitterdodge(dodge.width = 0.5, jitter.height = 0, jitter.width = 0.2)) + stat_summary(fun.data = &quot;mean_cl_boot&quot;, position = position_dodge(width = 0.5), size = 1) + labs(y = &quot;final balance (in Euros)&quot;) + scale_shape_manual(values = c(21, 22)) + guides(fill = guide_legend(override.aes = list(shape = 21, fill = RColorBrewer::brewer.pal(3, &quot;Set1&quot;))), shape = guide_legend(override.aes = list(alpha = 1, fill = &quot;black&quot;))) 23.4.2 2. Specify and fit the model 23.4.2.1 Frequentist model And let’s now fit a simple (frequentist) ANOVA model. You have multiple options to do so: # Option 1: Using the &quot;afex&quot; package aov_ez(id = &quot;participant&quot;, dv = &quot;balance&quot;, between = c(&quot;hand&quot;, &quot;skill&quot;), data = df.poker) Contrasts set to contr.sum for the following variables: hand, skill Anova Table (Type 3 tests) Response: balance Effect df MSE F ges p.value 1 hand 2, 294 16.16 79.17 *** .350 &lt;.001 2 skill 1, 294 16.16 2.43 .008 .120 3 hand:skill 2, 294 16.16 7.08 *** .046 &lt;.001 --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;+&#39; 0.1 &#39; &#39; 1 # Option 2: Using the car package (here we have to remember to set the contrasts to sum # contrasts!) lm(balance ~ hand * skill, contrasts = list(hand = &quot;contr.sum&quot;, skill = &quot;contr.sum&quot;), data = df.poker) %&gt;% car::Anova(type = 3) Anova Table (Type III tests) Response: balance Sum Sq Df F value Pr(&gt;F) (Intercept) 28644.7 1 1772.1137 &lt; 2.2e-16 *** hand 2559.4 2 79.1692 &lt; 2.2e-16 *** skill 39.3 1 2.4344 0.1197776 hand:skill 229.0 2 7.0830 0.0009901 *** Residuals 4752.3 294 --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 # Option 3: Using the emmeans package (I like this one the best! It let&#39;s us use the # general lm() syntax and we don&#39;t have to remember to set the contrast) fit.lm_poker = lm(balance ~ hand * skill, data = df.poker) fit.lm_poker %&gt;% joint_tests() model term df1 df2 F.ratio p.value hand 2 294 79.169 &lt;.0001 skill 1 294 2.434 0.1198 hand:skill 2 294 7.083 0.0010 All three options give the same result. Personally, I like Option 3 the best. 23.4.2.2 Bayesian model Now, let’s fit a Bayesian regression model using the brm() function (starting with a simple model that only considers hand as a predictor): fit.brm_poker = brm(formula = balance ~ 1 + hand, data = df.poker, seed = 1, file = &quot;cache/brm_poker&quot;) # we&#39;ll use this model here later fit.brm_poker2 = brm(formula = balance ~ 1 + hand * skill, data = df.poker, seed = 1, file = &quot;cache/brm_poker2&quot;) fit.brm_poker %&gt;% summary() Family: gaussian Links: mu = identity; sigma = identity Formula: balance ~ 1 + hand Data: df.poker (Number of observations: 300) Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; total post-warmup draws = 4000 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS Intercept 5.94 0.42 5.11 6.76 1.00 3434 2747 handneutral 4.41 0.60 3.22 5.60 1.00 3669 2860 handgood 7.09 0.61 5.91 8.27 1.00 3267 2841 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 4.12 0.17 3.82 4.48 1.00 3378 3010 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). I use the file = argument to save the model’s results so that when I run this code chunk again, the model doesn’t need to be fit again (fitting Bayesian models takes a while …). And I used the seed = argument to make this example reproducible. 23.4.2.2.1 Full specification So far, we have used the defaults that brm() comes with and not bothered about specifiying the priors, etc. Notice that we didn’t specify any priors in the model. By default, “brms” assigns weakly informative priors to the parameters in the model. We can see what these are by running the following command: fit.brm_poker %&gt;% prior_summary() prior class coef group resp dpar nlpar lb ub (flat) b (flat) b handgood (flat) b handneutral student_t(3, 9.5, 5.6) Intercept student_t(3, 0, 5.6) sigma 0 source default (vectorized) (vectorized) default default We can also get information about which priors need to be specified before fitting a model: get_prior(formula = balance ~ 1 + hand, family = &quot;gaussian&quot;, data = df.poker) prior class coef group resp dpar nlpar lb ub (flat) b (flat) b handgood (flat) b handneutral student_t(3, 9.5, 5.6) Intercept student_t(3, 0, 5.6) sigma 0 source default (vectorized) (vectorized) default default Here is an example for what a more complete model specification could look like: fit.brm_poker_full = brm(formula = balance ~ 1 + hand, family = &quot;gaussian&quot;, data = df.poker, prior = c(prior(normal(0, 10), class = &quot;b&quot;, coef = &quot;handgood&quot;), prior(normal(0, 10), class = &quot;b&quot;, coef = &quot;handneutral&quot;), prior(student_t(3, 3, 10), class = &quot;Intercept&quot;), prior(student_t(3, 0, 10), class = &quot;sigma&quot;)), inits = list(list(Intercept = 0, sigma = 1, handgood = 5, handneutral = 5), list(Intercept = -5, sigma = 3, handgood = 2, handneutral = 2), list(Intercept = 2, sigma = 1, handgood = -1, handneutral = 1), list(Intercept = 1, sigma = 2, handgood = 2, handneutral = -2)), iter = 4000, warmup = 1000, chains = 4, file = &quot;cache/brm_poker_full&quot;, seed = 1) fit.brm_poker_full %&gt;% summary() Family: gaussian Links: mu = identity; sigma = identity Formula: balance ~ 1 + hand Data: df.poker (Number of observations: 300) Draws: 4 chains, each with iter = 4000; warmup = 1000; thin = 1; total post-warmup draws = 12000 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS Intercept 5.96 0.41 5.15 6.76 1.00 10533 8949 handneutral 4.39 0.58 3.26 5.52 1.00 11458 9173 handgood 7.06 0.58 5.91 8.20 1.00 10754 8753 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 4.13 0.17 3.81 4.48 1.00 11877 8882 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). We can also take a look at the Stan code that the brm() function creates: fit.brm_poker_full %&gt;% stancode() // generated with brms 2.20.4 functions { } data { int&lt;lower=1&gt; N; // total number of observations vector[N] Y; // response variable int&lt;lower=1&gt; K; // number of population-level effects matrix[N, K] X; // population-level design matrix int&lt;lower=1&gt; Kc; // number of population-level effects after centering int prior_only; // should the likelihood be ignored? } transformed data { matrix[N, Kc] Xc; // centered version of X without an intercept vector[Kc] means_X; // column means of X before centering for (i in 2:K) { means_X[i - 1] = mean(X[, i]); Xc[, i - 1] = X[, i] - means_X[i - 1]; } } parameters { vector[Kc] b; // regression coefficients real Intercept; // temporary intercept for centered predictors real&lt;lower=0&gt; sigma; // dispersion parameter } transformed parameters { real lprior = 0; // prior contributions to the log posterior lprior += normal_lpdf(b[1] | 0, 10); lprior += normal_lpdf(b[2] | 0, 10); lprior += student_t_lpdf(Intercept | 3, 3, 10); lprior += student_t_lpdf(sigma | 3, 0, 10) - 1 * student_t_lccdf(0 | 3, 0, 10); } model { // likelihood including constants if (!prior_only) { target += normal_id_glm_lpdf(Y | Xc, Intercept, b, sigma); } // priors including constants target += lprior; } generated quantities { // actual population-level intercept real b_Intercept = Intercept - dot_product(means_X, b); } One thing worth noticing: by default, “brms” centers the predictors which makes it easier to assign a default prior over the intercept. 23.4.3 3. Model evaluation 23.4.3.1 a) Did the inference work? So far, we’ve assumed that the inference has worked out. We can check this by running plot() on our brm object: plot(fit.brm_poker, N = 7, ask = F) The posterior distributions (left hand side), and the trace plots of the samples from the posterior (right hand side) look good. Let’s make our own version of a trace plot for one parameter in the model: fit.brm_poker %&gt;% spread_draws(b_Intercept) %&gt;% clean_names() %&gt;% mutate(chain = as.factor(chain)) %&gt;% ggplot(aes(x = iteration, y = b_intercept, group = chain, color = chain)) + geom_line() + scale_color_brewer(type = &quot;seq&quot;, direction = -1) We can also take a look at the auto-correlation plot. Ideally, we want to generate independent samples from the posterior. So we don’t want subsequent samples to be strongly correlated with each other. Let’s take a look: variables = fit.brm_poker %&gt;% get_variables() %&gt;% .[1:4] fit.brm_poker %&gt;% as_draws() %&gt;% mcmc_acf(pars = variables, lags = 4) Warning: The `facets` argument of `facet_grid()` is deprecated as of ggplot2 2.2.0. ℹ Please use the `rows` argument instead. ℹ The deprecated feature was likely used in the bayesplot package. Please report the issue at &lt;https://github.com/stan-dev/bayesplot/issues/&gt;. This warning is displayed once every 8 hours. Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated. Looking good! The autocorrelation should become very small as the lag increases (indicating that we are getting independent samples from the posterior). 23.4.3.1.0.1 When things go wrong Let’s try to fit a model to very little data (just two observations) with extremely uninformative priors: df.data = tibble(y = c(-1, 1)) fit.brm_wrong = brm(data = df.data, family = gaussian, formula = y ~ 1, prior = c(prior(uniform(-1e10, 1e10), class = Intercept), prior(uniform(0, 1e10), class = sigma)), inits = list(list(Intercept = 0, sigma = 1), list(Intercept = 0, sigma = 1)), iter = 4000, warmup = 1000, chains = 2, file = &quot;cache/brm_wrong&quot;) Let’s take a look at the posterior distributions of the model parameters: summary(fit.brm_wrong) Warning: There were 1396 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup Family: gaussian Links: mu = identity; sigma = identity Formula: y ~ 1 Data: df.data (Number of observations: 2) Draws: 2 chains, each with iter = 4000; warmup = 1000; thin = 1; total post-warmup draws = 6000 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Intercept 228357056.15 984976734.30 -616012010.65 4388257793.68 1.04 45 Tail_ESS Intercept 59 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 812525872.95 1745959463.85 282591.91 6739518200.93 1.04 44 63 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). Not looking good – The estimates and credible intervals are off the charts. And the effective samples sizes in the chains are very small. Let’s visualize the trace plots: plot(fit.brm_wrong, N = 2, ask = F) fit.brm_wrong %&gt;% spread_draws(b_Intercept) %&gt;% clean_names() %&gt;% mutate(chain = as.factor(chain)) %&gt;% ggplot(aes(x = iteration, y = b_intercept, group = chain, color = chain)) + geom_line() + scale_color_brewer(direction = -1) Given that we have so little data in this case, we need to help the model a little bit by providing some slighlty more specific priors. fit.brm_right = brm(data = df.data, family = gaussian, formula = y ~ 1, prior = c(prior(normal(0, 10), class = Intercept), # more reasonable priors prior(cauchy(0, 1), class = sigma)), iter = 4000, warmup = 1000, chains = 2, seed = 1, file = &quot;cache/brm_right&quot;) Let’s take a look at the posterior distributions of the model parameters: summary(fit.brm_right) Family: gaussian Links: mu = identity; sigma = identity Formula: y ~ 1 Data: df.data (Number of observations: 2) Draws: 2 chains, each with iter = 4000; warmup = 1000; thin = 1; total post-warmup draws = 6000 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS Intercept -0.05 1.55 -3.35 3.11 1.00 1730 1300 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 2.01 1.84 0.62 6.81 1.00 1206 1568 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). This looks much better. There is still quite a bit of uncertainty in our paremeter estimates, but it has reduced dramatically. Let’s visualize the trace plots: plot(fit.brm_right, N = 2, ask = F) fit.brm_right %&gt;% spread_draws(b_Intercept, sigma) %&gt;% clean_names() %&gt;% mutate(chain = as.factor(chain)) %&gt;% pivot_longer(cols = c(b_intercept, sigma)) %&gt;% ggplot(aes(x = iteration, y = value, group = chain, color = chain)) + geom_line() + facet_wrap(vars(name), ncol = 1) + scale_color_brewer(direction = -1) Looking mostly good! 23.4.3.2 b) Visualize model predictions 23.4.3.2.1 Posterior predictive check To check whether the model did a good job capturing the data, we can simulate what future data the Bayesian model predicts, now that it has learned from the data we feed into it. pp_check(fit.brm_poker, ndraws = 100) This looks good! The predicted shaped of the data based on samples from the posterior distribution looks very similar to the shape of the actual data. Let’s make a hypothetical outcome plot that shows what concrete data sets the model would predict. The add_predicted_draws() function from the “tidybayes” package is helpful for generating predictions from the posterior. df.predictive_samples = df.poker %&gt;% add_predicted_draws(newdata = ., object = fit.brm_poker2, ndraws = 10) p = ggplot(data = df.predictive_samples, mapping = aes(x = hand, y = .prediction, fill = hand, group = skill, shape = skill)) + geom_point(alpha = 0.2, position = position_jitterdodge(dodge.width = 0.5, jitter.height = 0, jitter.width = 0.2)) + stat_summary(fun.data = &quot;mean_cl_boot&quot;, position = position_dodge(width = 0.5), size = 1) + labs(y = &quot;final balance (in Euros)&quot;) + scale_shape_manual(values = c(21, 22)) + guides(fill = guide_legend(override.aes = list(shape = 21)), shape = guide_legend(override.aes = list(alpha = 1, fill = &quot;black&quot;))) + transition_manual(.draw) animate(p, nframes = 120, width = 800, height = 600, res = 96, type = &quot;cairo&quot;) nframes and fps adjusted to match transition 23.4.3.2.2 Prior predictive check fit.brm_poker_prior = brm(formula = balance ~ 0 + Intercept + hand * skill, family = &quot;gaussian&quot;, data = df.poker, prior = c(prior(normal(0, 10), class = &quot;b&quot;), prior(student_t(3, 0, 10), class = &quot;sigma&quot;)), iter = 4000, warmup = 1000, chains = 4, file = &quot;cache/brm_poker_prior&quot;, sample_prior = &quot;only&quot;, seed = 1) # generate prior samples df.prior_samples = df.poker %&gt;% add_predicted_draws(newdata = ., object = fit.brm_poker_prior, ndraws = 10) # plot the results as an animation p = ggplot(data = df.prior_samples, mapping = aes(x = hand, y = .prediction, fill = hand, group = skill, shape = skill)) + geom_point(alpha = 0.2, position = position_jitterdodge(dodge.width = 0.5, jitter.height = 0, jitter.width = 0.2)) + stat_summary(fun.data = &quot;mean_cl_boot&quot;, position = position_dodge(width = 0.5), size = 1) + labs(y = &quot;final balance (in Euros)&quot;) + scale_shape_manual(values = c(21, 22)) + guides(fill = guide_legend(override.aes = list(shape = 21, fill = RColorBrewer::brewer.pal(3, &quot;Set1&quot;))), shape = guide_legend(override.aes = list(alpha = 1, fill = &quot;black&quot;))) + transition_manual(.draw) animate(p, nframes = 120, width = 800, height = 600, res = 96, type = &quot;cairo&quot;) nframes and fps adjusted to match transition # anim_save(&quot;poker_prior_predictive.gif&quot;) 23.4.4 4. Interpret the model parameters 23.4.4.1 Visualize the posteriors Let’s visualize what the posterior for the different parameters looks like. We use the stat_halfeye() function from the “tidybayes” package to do so: fit.brm_poker %&gt;% as_draws_df() %&gt;% select(starts_with(&quot;b_&quot;), sigma) %&gt;% pivot_longer(cols = everything(), names_to = &quot;variable&quot;, values_to = &quot;value&quot;) %&gt;% ggplot(data = ., mapping = aes(y = fct_rev(variable), x = value)) + stat_halfeye(fill = &quot;lightblue&quot;) + theme(axis.title.y = element_blank()) 23.4.4.2 Compute highest density intervals To compute the MAP (maximum a posteriori probability) estimate and highest density interval, we use the mean_hdi() function that comes with the “tidybayes” package. fit.brm_poker %&gt;% as_draws_df() %&gt;% select(starts_with(&quot;b_&quot;), sigma) %&gt;% mean_hdi() %&gt;% pivot_longer(cols = -c(.width:.interval), names_to = &quot;index&quot;, values_to = &quot;value&quot;) %&gt;% select(index, value) %&gt;% mutate(index = ifelse(str_detect(index, fixed(&quot;.&quot;)), index, str_c(index, &quot;.mean&quot;))) %&gt;% separate(index, into = c(&quot;parameter&quot;, &quot;type&quot;), sep = &quot;\\\\.&quot;) %&gt;% pivot_wider(names_from = type, values_from = value) # A tibble: 4 × 4 parameter mean lower upper &lt;chr&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; 1 b_Intercept 5.94 5.10 6.75 2 b_handneutral 4.41 3.21 5.58 3 b_handgood 7.09 5.92 8.27 4 sigma 4.12 3.80 4.46 23.4.5 5. Test specific hypotheses 23.4.5.1 with hypothesis() One key advantage of Bayesian over frequentist analysis is that we can test hypothesis in a very flexible manner by directly probing our posterior samples in different ways. We may ask, for example, what the probability is that the parameter for the difference between a bad hand and a neutral hand (b_handneutral) is greater than 0. Let’s plot the posterior distribution together with the criterion: fit.brm_poker %&gt;% as_draws_df() %&gt;% select(b_handneutral) %&gt;% pivot_longer(cols = everything(), names_to = &quot;variable&quot;, values_to = &quot;value&quot;) %&gt;% ggplot(data = ., mapping = aes(y = variable, x = value)) + stat_halfeye(fill = &quot;lightblue&quot;) + geom_vline(xintercept = 0, color = &quot;red&quot;) We see that the posterior is definitely greater than 0. We can ask many different kinds of questions about the data by doing basic arithmetic on our posterior samples. The hypothesis() function makes this even easier. Here are some examples: # the probability that the posterior for handneutral is less than 0 hypothesis(fit.brm_poker, hypothesis = &quot;handneutral &lt; 0&quot;) Hypothesis Tests for class b: Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob 1 (handneutral) &lt; 0 4.41 0.6 3.4 5.37 0 0 Star 1 --- &#39;CI&#39;: 90%-CI for one-sided and 95%-CI for two-sided hypotheses. &#39;*&#39;: For one-sided hypotheses, the posterior probability exceeds 95%; for two-sided hypotheses, the value tested against lies outside the 95%-CI. Posterior probabilities of point hypotheses assume equal prior probabilities. # the probability that the posterior for handneutral is greater than 4 hypothesis(fit.brm_poker, hypothesis = &quot;handneutral &gt; 4&quot;) %&gt;% plot() # the probability that good hands make twice as much as bad hands hypothesis(fit.brm_poker, hypothesis = &quot;Intercept + handgood &gt; 2 * Intercept&quot;) Hypothesis Tests for class b: Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio 1 (Intercept+handgo... &gt; 0 1.15 0.96 -0.4 2.73 7.93 Post.Prob Star 1 0.89 --- &#39;CI&#39;: 90%-CI for one-sided and 95%-CI for two-sided hypotheses. &#39;*&#39;: For one-sided hypotheses, the posterior probability exceeds 95%; for two-sided hypotheses, the value tested against lies outside the 95%-CI. Posterior probabilities of point hypotheses assume equal prior probabilities. We can also make a plot of what the posterior distribution of the hypothesis looks like: hypothesis(fit.brm_poker, hypothesis = &quot;Intercept + handgood &gt; 2 * Intercept&quot;) %&gt;% plot() # the probability that neutral hands make less than the average of bad and good hands hypothesis(fit.brm_poker, hypothesis = &quot;Intercept + handneutral &lt; (Intercept + Intercept + handgood) / 2&quot;) Hypothesis Tests for class b: Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio 1 (Intercept+handne... &lt; 0 0.86 0.52 0.01 1.72 0.05 Post.Prob Star 1 0.05 --- &#39;CI&#39;: 90%-CI for one-sided and 95%-CI for two-sided hypotheses. &#39;*&#39;: For one-sided hypotheses, the posterior probability exceeds 95%; for two-sided hypotheses, the value tested against lies outside the 95%-CI. Posterior probabilities of point hypotheses assume equal prior probabilities. Let’s double check one example, and calculate the result directly based on the posterior samples: df.hypothesis = fit.brm_poker %&gt;% as_draws_df() %&gt;% clean_names() %&gt;% select(starts_with(&quot;b_&quot;)) %&gt;% mutate(neutral = b_intercept + b_handneutral, bad_good_average = (b_intercept + b_intercept + b_handgood)/2, hypothesis = neutral &lt; bad_good_average) Warning: Dropping &#39;draws_df&#39; class as required metadata was removed. df.hypothesis %&gt;% summarize(p = sum(hypothesis)/n()) # A tibble: 1 × 1 p &lt;dbl&gt; 1 0.0485 23.4.5.2 with emmeans() We can also use the emmeans() function to compute contrasts. fit.brm_poker %&gt;% emmeans(specs = consec ~ hand) Loading required namespace: rstanarm $emmeans hand emmean lower.HPD upper.HPD bad 5.94 5.10 6.74 neutral 10.34 9.58 11.24 good 13.03 12.19 13.87 Point estimate displayed: median HPD interval probability: 0.95 $contrasts contrast estimate lower.HPD upper.HPD neutral - bad 4.41 3.21 5.58 good - neutral 2.67 1.56 3.92 Point estimate displayed: median HPD interval probability: 0.95 Here, it computed the estimated means for each group for us, as well as the consecutive contrasts between each group. Let’s visualize the contrasts. First, let’s just use the plot() function as it’s been adapted by the emmeans package: fit.brm_poker %&gt;% emmeans(specs = consec ~ hand) %&gt;% pluck(&quot;contrasts&quot;) %&gt;% plot() To get full posterior distributions instead of summaries, we can use the “tidybayes” package like so: fit.brm_poker %&gt;% emmeans(specs = consec ~ hand) %&gt;% pluck(&quot;contrasts&quot;) %&gt;% gather_emmeans_draws() %&gt;% ggplot(mapping = aes(y = contrast, x = .value)) + stat_halfeye(fill = &quot;lightblue&quot;, point_interval = mean_hdi, .width = c(0.5, 0.75, 0.95)) To see whether neutral hands did differently from bad and good hands (combined), we can define the following contrast. contrasts = list(neutral_vs_rest = c(-1, 2, -1)) fit.brm_poker %&gt;% emmeans(specs = &quot;hand&quot;, contr = contrasts) %&gt;% pluck(&quot;contrasts&quot;) %&gt;% gather_emmeans_draws() %&gt;% mean_hdi() # A tibble: 1 × 7 contrast .value .lower .upper .width .point .interval &lt;chr&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;chr&gt; &lt;chr&gt; 1 neutral_vs_rest 1.72 -0.247 3.82 0.95 mean hdi Here, the HDP does not exclude 0. Let’s double check that we get the same result using the hypothesis() function, or by directly computing from the posterior samples. # using hypothesis() fit.brm_poker %&gt;% hypothesis(&quot;(Intercept + handneutral)*2 &lt; (Intercept + Intercept + handgood)&quot;) Hypothesis Tests for class b: Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio 1 ((Intercept+handn... &lt; 0 1.72 1.03 0.01 3.43 0.05 Post.Prob Star 1 0.05 --- &#39;CI&#39;: 90%-CI for one-sided and 95%-CI for two-sided hypotheses. &#39;*&#39;: For one-sided hypotheses, the posterior probability exceeds 95%; for two-sided hypotheses, the value tested against lies outside the 95%-CI. Posterior probabilities of point hypotheses assume equal prior probabilities. # directly computing from the posterior fit.brm_poker %&gt;% as_draws_df() %&gt;% clean_names() %&gt;% mutate(contrast = (b_intercept + b_handneutral) * 2 - (b_intercept + b_intercept + b_handgood)) %&gt;% summarize(contrast = mean(contrast)) # A tibble: 1 × 1 contrast &lt;dbl&gt; 1 1.72 The emmeans() function becomes particularly useful when our model has several categorical predictors, and we are interested in comparing differences along one predictor while marginalizing over the values of the other predictor. Let’s take a look for a model that considers both skill and hand as predictors (as well as the interaction). fit.brm_poker2 = brm(formula = balance ~ hand * skill, data = df.poker, seed = 1, file = &quot;cache/brm_poker2&quot;) fit.brm_poker2 %&gt;% summary() Family: gaussian Links: mu = identity; sigma = identity Formula: balance ~ 1 + hand * skill Data: df.poker (Number of observations: 300) Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; total post-warmup draws = 4000 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Intercept 4.58 0.56 3.48 5.69 1.00 2402 handneutral 5.26 0.79 3.72 6.86 1.00 2617 handgood 9.23 0.80 7.65 10.77 1.00 2251 skillexpert 2.73 0.78 1.17 4.25 1.00 2125 handneutral:skillexpert -1.71 1.11 -3.84 0.46 1.00 2238 handgood:skillexpert -4.28 1.12 -6.43 -2.05 1.00 2024 Tail_ESS Intercept 2615 handneutral 2925 handgood 2504 skillexpert 2476 handneutral:skillexpert 3040 handgood:skillexpert 2295 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 4.03 0.17 3.72 4.38 1.00 3369 2728 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). In the summary table above, skillexpert captures the difference between an expert and an average player when they have a bad hand. To see whether there was a difference in expertise overall (i.e. across all three kinds of hands), we can calculate a linear contrast. fit.brm_poker2 %&gt;% emmeans(pairwise ~ skill) NOTE: Results may be misleading due to involvement in interactions $emmeans skill emmean lower.HPD upper.HPD average 9.41 8.77 10.0 expert 10.14 9.48 10.8 Results are averaged over the levels of: hand Point estimate displayed: median HPD interval probability: 0.95 $contrasts contrast estimate lower.HPD upper.HPD average - expert -0.722 -1.68 0.0885 Results are averaged over the levels of: hand Point estimate displayed: median HPD interval probability: 0.95 It looks like overall, skilled players weren’t doing much better than average players. We can even do something like an equivalent of an ANOVA using emmeans(), like so: joint_tests(fit.brm_poker2) model term df1 df2 F.ratio Chisq p.value hand 2 Inf 82.446 164.892 &lt;.0001 skill 1 Inf 2.549 2.549 0.1103 hand:skill 2 Inf 7.347 14.694 0.0006 The values we get here are very similar to what we would get from a frequentist ANOVA: aov_ez(id = &quot;participant&quot;, dv = &quot;balance&quot;, between = c(&quot;hand&quot;, &quot;skill&quot;), data = df.poker) Contrasts set to contr.sum for the following variables: hand, skill Anova Table (Type 3 tests) Response: balance Effect df MSE F ges p.value 1 hand 2, 294 16.16 79.17 *** .350 &lt;.001 2 skill 1, 294 16.16 2.43 .008 .120 3 hand:skill 2, 294 16.16 7.08 *** .046 &lt;.001 --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;+&#39; 0.1 &#39; &#39; 1 23.4.5.3 Bayes factor Another way of testing hypothesis is via the Bayes factor. Let’s fit the two models we are interested in comparing with each other: fit.brm_poker_bf1 = brm(formula = balance ~ 1 + hand, data = df.poker, save_pars = save_pars(all = T), file = &quot;cache/brm_poker_bf1&quot;) fit.brm_poker_bf2 = brm(formula = balance ~ 1 + hand + skill, data = df.poker, save_pars = save_pars(all = T), file = &quot;cache/brm_poker_bf2&quot;) And then compare the models using the bayes_factor() function: bayes_factor(fit.brm_poker_bf2, fit.brm_poker_bf1) Iteration: 1 Iteration: 2 Iteration: 3 Iteration: 4 Iteration: 1 Iteration: 2 Iteration: 3 Iteration: 4 Iteration: 5 Estimated Bayes factor in favor of fit.brm_poker_bf2 over fit.brm_poker_bf1: 3.83098 Bayes factors don’t have a very good reputation (see here and here). Instead, the way to go these days appears to be via approximate leave one out cross-validation. 23.4.5.4 Approximate leave one out cross-validation fit.brm_poker_bf1 = add_criterion(fit.brm_poker_bf1, criterion = &quot;loo&quot;, reloo = T, file = &quot;cache/brm_poker_bf1&quot;) fit.brm_poker_bf2 = add_criterion(fit.brm_poker_bf2, criterion = &quot;loo&quot;, reloo = T, file = &quot;cache/brm_poker_bf2&quot;) loo_compare(fit.brm_poker_bf1, fit.brm_poker_bf2) elpd_diff se_diff fit.brm_poker_bf2 0.0 0.0 fit.brm_poker_bf1 -0.3 1.5 23.5 Sleep study 23.5.1 1. Visualize the data set.seed(1) ggplot(data = df.sleep %&gt;% mutate(days = as.factor(days)), mapping = aes(x = days, y = reaction)) + geom_point(alpha = 0.2, position = position_jitter(width = 0.1)) + stat_summary(fun.data = &quot;mean_cl_boot&quot;) 23.5.2 2. Specify and fit the model 23.5.2.1 Frequentist analysis fit.lmer_sleep = lmer(formula = reaction ~ 1 + days + (1 + days | subject), data = df.sleep) fit.lmer_sleep %&gt;% summary() Linear mixed model fit by REML. t-tests use Satterthwaite&#39;s method [ lmerModLmerTest] Formula: reaction ~ 1 + days + (1 + days | subject) Data: df.sleep REML criterion at convergence: 1771.4 Scaled residuals: Min 1Q Median 3Q Max -3.9707 -0.4703 0.0276 0.4594 5.2009 Random effects: Groups Name Variance Std.Dev. Corr subject (Intercept) 582.72 24.140 days 35.03 5.919 0.07 Residual 649.36 25.483 Number of obs: 183, groups: subject, 20 Fixed effects: Estimate Std. Error df t value Pr(&gt;|t|) (Intercept) 252.543 6.433 19.295 39.257 &lt; 2e-16 *** days 10.452 1.542 17.163 6.778 3.06e-06 *** --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 Correlation of Fixed Effects: (Intr) days -0.137 23.5.2.2 Bayesian analysis fit.brm_sleep = brm(formula = reaction ~ 1 + days + (1 + days | subject), data = df.sleep, seed = 1, file = &quot;cache/brm_sleep&quot;) 23.5.3 3. Model evaluation 23.5.3.1 a) Did the inference work? fit.brm_sleep %&gt;% summary() Family: gaussian Links: mu = identity; sigma = identity Formula: reaction ~ 1 + days + (1 + days | subject) Data: df.sleep (Number of observations: 183) Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; total post-warmup draws = 4000 Group-Level Effects: ~subject (Number of levels: 20) Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sd(Intercept) 26.15 6.51 15.13 40.55 1.00 1511 2248 sd(days) 6.57 1.52 4.12 9.90 1.00 1419 2223 cor(Intercept,days) 0.09 0.29 -0.48 0.65 1.00 888 1670 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS Intercept 252.18 6.95 238.65 265.96 1.00 1755 2235 days 10.47 1.75 7.07 14.00 1.00 1259 1608 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS sigma 25.82 1.54 22.96 29.05 1.00 3254 2958 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). fit.brm_sleep %&gt;% plot(N = 6) 23.5.3.2 b) Visualize model predictions pp_check(fit.brm_sleep, ndraws = 100) 23.5.4 4. Interpret the parameters fit.brm_sleep %&gt;% tidy(conf.method = &quot;HPDinterval&quot;) # A tibble: 6 × 8 effect component group term estimate std.error conf.low conf.high &lt;chr&gt; &lt;chr&gt; &lt;chr&gt; &lt;chr&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; 1 fixed cond &lt;NA&gt; (Intercept) 252. 6.95 238. 266. 2 fixed cond &lt;NA&gt; days 10.5 1.75 6.82 13.6 3 ran_pars cond subject sd__(Interc… 26.2 6.51 14.3 39.4 4 ran_pars cond subject sd__days 6.57 1.52 4.00 9.67 5 ran_pars cond subject cor__(Inter… 0.0909 0.293 -0.491 0.635 6 ran_pars cond Residual sd__Observa… 25.8 1.54 22.7 28.7 23.5.4.1 Summary of posterior distributions # all parameters fit.brm_sleep %&gt;% as_draws_df() %&gt;% select(-c(lp__, contains(&quot;[&quot;))) %&gt;% pivot_longer(cols = everything(), names_to = &quot;variable&quot;, values_to = &quot;value&quot;) %&gt;% ggplot(data = ., mapping = aes(x = value)) + stat_halfeye(point_interval = mode_hdi, fill = &quot;lightblue&quot;) + facet_wrap(~ variable, ncol = 2, scales = &quot;free&quot;) + theme(text = element_text(size = 12)) # just the parameter of interest fit.brm_sleep %&gt;% as_draws_df() %&gt;% select(b_days) %&gt;% ggplot(data = ., mapping = aes(x = b_days)) + stat_halfeye(point_interval = mode_hdi, fill = &quot;lightblue&quot;) + theme(text = element_text(size = 12)) 23.5.5 5. Test specific hypotheses Here, we were just interested in how the number of days of sleep deprivation affected reaction time (and we can see that by inspecting the posterior for the days predictor in the model). 23.5.6 6. Report results 23.5.6.1 Model prediction with posterior draws (aggregate) df.model = tibble(days = 0:9) %&gt;% add_linpred_draws(newdata = ., object = fit.brm_sleep, ndraws = 10, seed = 1, re_formula = NA) ggplot(data = df.sleep, mapping = aes(x = days, y = reaction)) + geom_point(alpha = 0.2, position = position_jitter(width = 0.1)) + geom_line(data = df.model, mapping = aes(y = .linpred, group = .draw), color = &quot;lightblue&quot;) + stat_summary(fun.data = &quot;mean_cl_boot&quot;) + scale_x_continuous(breaks = 0:9) 23.5.6.2 Model prediction with credible intervals (aggregate) df.model = fit.brm_sleep %&gt;% fitted(re_formula = NA, newdata = tibble(days = 0:9)) %&gt;% as_tibble() %&gt;% mutate(days = 0:9) %&gt;% clean_names() ggplot(data = df.sleep, mapping = aes(x = days, y = reaction)) + geom_point(alpha = 0.2, position = position_jitter(width = 0.1)) + geom_ribbon(data = df.model, mapping = aes(y = estimate, ymin = q2_5, ymax = q97_5), fill = &quot;lightblue&quot;, alpha = 0.5) + geom_line(data = df.model, mapping = aes(y = estimate), color = &quot;lightblue&quot;, size = 1) + stat_summary(fun.data = &quot;mean_cl_boot&quot;) + scale_x_continuous(breaks = 0:9) Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0. ℹ Please use `linewidth` instead. This warning is displayed once every 8 hours. Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated. 23.5.6.3 Model prediction with credible intervals (individual participants) fit.brm_sleep %&gt;% fitted() %&gt;% as_tibble() %&gt;% clean_names() %&gt;% bind_cols(df.sleep) %&gt;% ggplot(data = ., mapping = aes(x = days, y = reaction)) + geom_ribbon(aes(ymin = q2_5, ymax = q97_5), fill = &quot;lightblue&quot;) + geom_line(aes(y = estimate), color = &quot;blue&quot;) + geom_point() + facet_wrap(~subject, ncol = 5) + labs(x = &quot;Days of sleep deprivation&quot;, y = &quot;Average reaction time (ms)&quot;) + scale_x_continuous(breaks = 0:4 * 2) + theme(strip.text = element_text(size = 12), axis.text.y = element_text(size = 12)) 23.5.6.4 Model prediction for random samples df.model = df.sleep %&gt;% complete(subject, days) %&gt;% add_linpred_draws(newdata = ., object = fit.brm_sleep, ndraws = 10, seed = 1) df.sleep %&gt;% ggplot(data = ., mapping = aes(x = days, y = reaction)) + geom_line(data = df.model, aes(y = .linpred, group = .draw), color = &quot;lightblue&quot;, alpha = 0.5) + geom_point() + facet_wrap(~subject, ncol = 5) + labs(x = &quot;Days of sleep deprivation&quot;, y = &quot;Average reaction time (ms)&quot;) + scale_x_continuous(breaks = 0:4 * 2) + theme(strip.text = element_text(size = 12), axis.text.y = element_text(size = 12)) 23.5.6.5 Animated model prediction for random samples df.model = df.sleep %&gt;% complete(subject, days) %&gt;% add_linpred_draws(newdata = ., object = fit.brm_sleep, ndraws = 10, seed = 1) p = df.sleep %&gt;% ggplot(data = ., mapping = aes(x = days, y = reaction)) + geom_line(data = df.model, aes(y = .linpred, group = .draw), color = &quot;black&quot;) + geom_point() + facet_wrap(~subject, ncol = 5) + labs(x = &quot;Days of sleep deprivation&quot;, y = &quot;Average reaction time (ms)&quot;) + scale_x_continuous(breaks = 0:4 * 2) + theme(strip.text = element_text(size = 12), axis.text.y = element_text(size = 12)) + transition_states(.draw, 0, 1) + shadow_mark(past = TRUE, alpha = 1/5, color = &quot;gray50&quot;) animate(p, nframes = 10, fps = 1, width = 800, height = 600, res = 96, type = &quot;cairo&quot;) # anim_save(&quot;sleep_posterior_predictive.gif&quot;) 23.6 Titanic study 23.6.1 1. Visualize the data df.titanic %&gt;% mutate(sex = as.factor(sex)) %&gt;% ggplot(data = ., mapping = aes(x = fare, y = survived, color = sex)) + geom_point(alpha = 0.1, size = 2) + geom_smooth(method = &quot;glm&quot;, method.args = list(family = &quot;binomial&quot;), alpha = 0.2, aes(fill = sex)) + scale_color_brewer(palette = &quot;Set1&quot;) 23.6.2 2. Specify and fit the model 23.6.2.1 Frequentist analysis fit.glm_titanic = glm(formula = survived ~ 1 + fare * sex, family = &quot;binomial&quot;, data = df.titanic) fit.glm_titanic %&gt;% summary() Call: glm(formula = survived ~ 1 + fare * sex, family = &quot;binomial&quot;, data = df.titanic) Coefficients: Estimate Std. Error z value Pr(&gt;|z|) (Intercept) 0.408428 0.189999 2.150 0.031584 * fare 0.019878 0.005372 3.701 0.000215 *** sexmale -2.099345 0.230291 -9.116 &lt; 2e-16 *** fare:sexmale -0.011617 0.005934 -1.958 0.050252 . --- Signif. codes: 0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 1186.66 on 890 degrees of freedom Residual deviance: 879.85 on 887 degrees of freedom AIC: 887.85 Number of Fisher Scoring iterations: 5 23.6.2.2 Bayesian analysis fit.brm_titanic = brm(formula = survived ~ 1 + fare * sex, family = &quot;bernoulli&quot;, data = df.titanic, file = &quot;cache/brm_titanic&quot;, seed = 1) 23.6.3 3. Model evaluation 23.6.3.1 a) Did the inference work? fit.brm_titanic %&gt;% summary() Family: bernoulli Links: mu = logit Formula: survived ~ 1 + fare * sex Data: df.titanic (Number of observations: 891) Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1; total post-warmup draws = 4000 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS Intercept 0.40 0.19 0.03 0.76 1.00 1900 2394 fare 0.02 0.01 0.01 0.03 1.00 1414 1684 sexmale -2.10 0.23 -2.54 -1.64 1.00 1562 1841 fare:sexmale -0.01 0.01 -0.02 -0.00 1.00 1399 1585 Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size measures, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). fit.brm_titanic %&gt;% plot() 23.6.3.2 b) Visualize model predictions pp_check(fit.brm_titanic, ndraws = 100) Let’s visualize what the posterior predictive would have looked like for a linear model (instead of a logistic model). fit.brm_titanic_linear = brm(formula = survived ~ 1 + fare * sex, data = df.titanic, file = &quot;cache/brm_titanic_linear&quot;, seed = 1) pp_check(fit.brm_titanic_linear, ndraws = 100) 23.6.4 4. Interpret the parameters fit.brm_titanic %&gt;% as_draws_df() %&gt;% select(-lp__) %&gt;% pivot_longer(cols = everything(), names_to = &quot;variable&quot;, values_to = &quot;value&quot;) %&gt;% ggplot(data = ., mapping = aes(y = variable, x = value)) + stat_intervalh() + scale_color_brewer() fit.brm_titanic %&gt;% parameters(centrality = &quot;mean&quot;, ci = 0.95) fit.brm_titanic %&gt;% ggpredict(terms = c(&quot;fare [0:500]&quot;, &quot;sex&quot;)) %&gt;% plot() 23.6.5 5. Test specific hypotheses Difference between men and women in survival? fit.brm_titanic %&gt;% emmeans(specs = pairwise ~ sex, type = &quot;response&quot;) NOTE: Results may be misleading due to involvement in interactions $emmeans sex response lower.HPD upper.HPD female 0.744 0.691 0.794 male 0.194 0.164 0.229 Point estimate displayed: median Results are back-transformed from the logit scale HPD interval probability: 0.95 $contrasts contrast odds.ratio lower.HPD upper.HPD female / male 12.1 8.43 16.9 Point estimate displayed: median Results are back-transformed from the log odds ratio scale HPD interval probability: 0.95 Difference in how fare affected the chances of survival for men and women? fit.brm_titanic %&gt;% emtrends(specs = pairwise ~ sex, var = &quot;fare&quot;) $emtrends sex fare.trend lower.HPD upper.HPD female 0.0206 0.01065 0.0311 male 0.0085 0.00346 0.0135 Point estimate displayed: median HPD interval probability: 0.95 $contrasts contrast estimate lower.HPD upper.HPD female - male 0.0121 0.0011 0.0242 Point estimate displayed: median HPD interval probability: 0.95 23.6.6 6. Report results df.model = add_linpred_draws(newdata = expand_grid(sex = c(&quot;female&quot;, &quot;male&quot;), fare = 0:500) %&gt;% mutate(sex = factor(sex, levels = c(&quot;female&quot;, &quot;male&quot;))), object = fit.brm_titanic, ndraws = 10) ggplot(data = df.titanic, mapping = aes(x = fare, y = survived, color = sex)) + geom_point(alpha = 0.1, size = 2) + geom_line(data = df.model %&gt;% filter(sex == &quot;male&quot;), aes(y = .linpred, group = .draw, color = sex)) + geom_line(data = df.model %&gt;% filter(sex == &quot;female&quot;), aes(y = .linpred, group = .draw, color = sex)) + scale_color_brewer(palette = &quot;Set1&quot;) 23.7 Politeness data The data is drawn from Winter and Grawunder (2012), and this section follows the excellent tutorial by Franke and Roettger (2019). (I’m skipping some of the steps of our recipe for Bayesian data analysis here.) 23.7.1 1. Visualize the data ggplot(data = df.politeness, mapping = aes(x = attitude, y = pitch, fill = gender, color = gender)) + geom_point(alpha = 0.2, position = position_jitter(width = 0.1, height = 0)) + stat_summary(fun.data = &quot;mean_cl_boot&quot;, shape = 21, size = 1, color = &quot;black&quot;) Warning: Removed 1 rows containing non-finite values (`stat_summary()`). Warning: Removed 1 rows containing missing values (`geom_point()`). 23.7.2 2. Specify and fit the model 23.7.2.1 Frequentist analysis fit.lm_polite = lm(formula = pitch ~ gender * attitude, data = df.politeness) 23.7.2.2 Bayesian analysis fit.brm_polite = brm(formula = pitch ~ gender * attitude, data = df.politeness, file = &quot;cache/brm_polite&quot;, seed = 1) 23.7.3 5. Test specific hypotheses 23.7.3.1 Frequentist fit.lm_polite %&gt;% joint_tests() model term df1 df2 F.ratio p.value gender 1 79 191.028 &lt;.0001 attitude 1 79 6.170 0.0151 gender:attitude 1 79 1.029 0.3135 It looks like there are significant main effects of gender and attitude, but no interaction effect. Let’s check whether there is a difference in attitude separately for each gender: fit.lm_polite %&gt;% emmeans(specs = pairwise ~ attitude | gender) %&gt;% pluck(&quot;contrasts&quot;) gender = F: contrast estimate SE df t.ratio p.value inf - pol 27.4 11.0 79 2.489 0.0149 gender = M: contrast estimate SE df t.ratio p.value inf - pol 11.5 11.1 79 1.033 0.3048 There was a significant difference of attitude for female participants but not for male participants. 23.7.3.2 Bayesian Let’s whether there was a main effect of gender. # main effect of gender fit.brm_polite %&gt;% emmeans(specs = pairwise ~ gender) %&gt;% pluck(&quot;contrasts&quot;) NOTE: Results may be misleading due to involvement in interactions contrast estimate lower.HPD upper.HPD F - M 108 91.7 124 Results are averaged over the levels of: attitude Point estimate displayed: median HPD interval probability: 0.95 Let’s take a look what the full posterior distribution over this contrast looks like: fit.brm_polite %&gt;% emmeans(specs = pairwise ~ gender) %&gt;% pluck(&quot;contrasts&quot;) %&gt;% gather_emmeans_draws() %&gt;% ggplot(mapping = aes(x = .value)) + stat_halfeye() NOTE: Results may be misleading due to involvement in interactions Looks neat! And let’s confirm that we really estimated the main effect here. Let’s fit a model that only has gender as a predictor, and then compare: fit.brm_polite_gender = brm(formula = pitch ~ 1 + gender, data = df.politeness, file = &quot;cache/brm_polite_gender&quot;, seed = 1) # using the gather_emmeans_draws to get means rather than medians fit.brm_polite %&gt;% emmeans(spec = pairwise ~ gender) %&gt;% pluck(&quot;contrasts&quot;) %&gt;% gather_emmeans_draws() %&gt;% mean_hdi() NOTE: Results may be misleading due to involvement in interactions # A tibble: 1 × 7 contrast .value .lower .upper .width .point .interval &lt;chr&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;chr&gt; &lt;chr&gt; 1 F - M 108. 91.9 124. 0.95 mean hdi fit.brm_polite_gender %&gt;% fixef() %&gt;% as_tibble(rownames = &quot;term&quot;) # A tibble: 2 × 5 term Estimate Est.Error Q2.5 Q97.5 &lt;chr&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; 1 Intercept 247. 5.72 236. 258. 2 genderM -108. 8.16 -124. -92.3 Yip, both of these methods give us the same result (the sign is flipped but that’s just because emmeans computed F-M, whereas the other method computed M-F)! Again, the emmeans() route is more convenient because we can more easily check for several main effects (and take a look at specific contrast, too). # main effect attitude fit.brm_polite %&gt;% emmeans(specs = pairwise ~ attitude) %&gt;% pluck(&quot;contrasts&quot;) NOTE: Results may be misleading due to involvement in interactions contrast estimate lower.HPD upper.HPD inf - pol 19.8 5.16 35.9 Results are averaged over the levels of: gender Point estimate displayed: median HPD interval probability: 0.95 # effect of attitude separately for each gender fit.brm_polite %&gt;% emmeans(specs = pairwise ~ attitude | gender) %&gt;% pluck(&quot;contrasts&quot;) gender = F: contrast estimate lower.HPD upper.HPD inf - pol 27.9 5.96 49.5 gender = M: contrast estimate lower.HPD upper.HPD inf - pol 12.0 -9.29 34.2 Point estimate displayed: median HPD interval probability: 0.95 # in case you want the means instead of medians fit.brm_polite %&gt;% emmeans(specs = pairwise ~ attitude | gender) %&gt;% pluck(&quot;contrasts&quot;) %&gt;% gather_emmeans_draws() %&gt;% mean_hdi() # A tibble: 2 × 8 contrast gender .value .lower .upper .width .point .interval &lt;fct&gt; &lt;fct&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;dbl&gt; &lt;chr&gt; &lt;chr&gt; 1 inf - pol F 27.8 5.96 49.5 0.95 mean hdi 2 inf - pol M 11.9 -9.29 34.2 0.95 mean hdi Here is a way to visualize the contrasts: fit.brm_polite %&gt;% emmeans(specs = pairwise ~ attitude | gender) %&gt;% pluck(&quot;contrasts&quot;) %&gt;% gather_emmeans_draws() %&gt;% ggplot(aes(x = .value, y = gender, fill = stat(x &gt; 0))) + facet_wrap(~ contrast) + stat_halfeye(show.legend = F) + geom_vline(xintercept = 0, linetype = 2) + scale_fill_manual(values = c(&quot;gray80&quot;, &quot;skyblue&quot;)) Warning: `stat(x &gt; 0)` was deprecated in ggplot2 3.4.0. ℹ Please use `after_stat(x &gt; 0)` instead. This warning is displayed once every 8 hours. Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated. Here is one way to check whether there was an interaction between attitude and gender (see this vignette for more info). fit.brm_polite %&gt;% emmeans(pairwise ~ attitude | gender) %&gt;% pluck(&quot;emmeans&quot;) %&gt;% contrast(interaction = c(&quot;consec&quot;), by = NULL) attitude_consec gender_consec estimate lower.HPD upper.HPD pol - inf M - F 16.1 -15.1 47.2 Point estimate displayed: median HPD interval probability: 0.95 23.8 Additional resources Bayesian regression: Theory &amp; Practice Tutorial on visualizing brms posteriors with tidybayes Hypothetical outcome plots Visual MCMC diagnostics Visualization of different MCMC algorithms Article describing the different inference algorithms 23.9 Session info Information about this R session including which version of R was used, and what packages were loaded. sessionInfo() R version 4.3.2 (2023-10-31) Platform: aarch64-apple-darwin20 (64-bit) Running under: macOS Sonoma 14.1.2 Matrix products: default BLAS: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0 locale: [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8 time zone: America/Los_Angeles tzcode source: internal attached base packages: [1] stats graphics grDevices utils datasets methods base other attached packages: [1] lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1 [4] dplyr_1.1.4 purrr_1.0.2 readr_2.1.4 [7] tidyr_1.3.0 tibble_3.2.1 tidyverse_2.0.0 [10] transformr_0.1.4.9000 parameters_0.21.3 gganimate_1.0.8 [13] titanic_0.1.0 ggeffects_1.3.4 emmeans_1.9.0 [16] car_3.1-2 carData_3.0-5 afex_1.3-0 [19] lme4_1.1-35.1 Matrix_1.6-4 modelr_0.1.11 [22] bayesplot_1.10.0 broom.mixed_0.2.9.4 GGally_2.2.0 [25] ggplot2_3.4.4 patchwork_1.2.0 brms_2.20.4 [28] Rcpp_1.0.12 tidybayes_3.0.6 janitor_2.2.0 [31] kableExtra_1.3.4 knitr_1.45 loaded via a namespace (and not attached): [1] svUnit_1.0.6 shinythemes_1.2.0 splines_4.3.2 [4] later_1.3.2 datawizard_0.9.1 rpart_4.1.23 [7] xts_0.13.1 lifecycle_1.0.4 sf_1.0-15 [10] StanHeaders_2.26.28 vroom_1.6.5 globals_0.16.2 [13] lattice_0.22-5 MASS_7.3-60 insight_0.19.7 [16] crosstalk_1.2.1 ggdist_3.3.1 backports_1.4.1 [19] magrittr_2.0.3 Hmisc_5.1-1 sass_0.4.8 [22] rmarkdown_2.25 jquerylib_0.1.4 yaml_2.3.8 [25] httpuv_1.6.13 gifski_1.12.0-2 pkgbuild_1.4.3 [28] DBI_1.2.0 minqa_1.2.6 RColorBrewer_1.1-3 [31] abind_1.4-5 rvest_1.0.3 nnet_7.3-19 [34] tensorA_0.36.2.1 tweenr_2.0.2 inline_0.3.19 [37] listenv_0.9.1 units_0.8-5 bridgesampling_1.1-2 [40] parallelly_1.37.0 svglite_2.1.3 codetools_0.2-19 [43] DT_0.31 xml2_1.3.6 tidyselect_1.2.0 [46] rstanarm_2.26.1 farver_2.1.1 matrixStats_1.2.0 [49] stats4_4.3.2 base64enc_0.1-3 webshot_0.5.5 [52] jsonlite_1.8.8 e1071_1.7-14 Formula_1.2-5 [55] ellipsis_0.3.2 survival_3.5-7 systemfonts_1.0.5 [58] tools_4.3.2 progress_1.2.3 glue_1.7.0 [61] gridExtra_2.3 mgcv_1.9-1 xfun_0.41 [64] distributional_0.3.2 loo_2.6.0 withr_3.0.0 [67] numDeriv_2016.8-1.1 fastmap_1.1.1 boot_1.3-28.1 [70] fansi_1.0.6 shinyjs_2.1.0 digest_0.6.34 [73] timechange_0.2.0 R6_2.5.1 mime_0.12 [76] estimability_1.4.1 colorspace_2.1-0 lpSolve_5.6.20 [79] gtools_3.9.5 markdown_1.12 threejs_0.3.3 [82] utf8_1.2.4 generics_0.1.3 data.table_1.14.10 [85] class_7.3-22 prettyunits_1.2.0 httr_1.4.7 [88] htmlwidgets_1.6.4 ggstats_0.5.1 pkgconfig_2.0.3 [91] dygraphs_1.1.1.6 gtable_0.3.4 furrr_0.3.1 [94] htmltools_0.5.7 bookdown_0.37 scales_1.3.0 [97] posterior_1.5.0 snakecase_0.11.1 rstudioapi_0.15.0 [100] tzdb_0.4.0 reshape2_1.4.4 coda_0.19-4.1 [103] checkmate_2.3.1 nlme_3.1-164 curl_5.2.0 [106] nloptr_2.0.3 proxy_0.4-27 cachem_1.0.8 [109] zoo_1.8-12 sjlabelled_1.2.0 KernSmooth_2.23-22 [112] parallel_4.3.2 miniUI_0.1.1.1 foreign_0.8-86 [115] pillar_1.9.0 grid_4.3.2 vctrs_0.6.5 [118] shinystan_2.6.0 promises_1.2.1 arrayhelpers_1.1-0 [121] cluster_2.1.6 xtable_1.8-4 htmlTable_2.4.2 [124] evaluate_0.23 mvtnorm_1.2-4 cli_3.6.2 [127] compiler_4.3.2 rlang_1.1.3 crayon_1.5.2 [130] rstantools_2.3.1.1 labeling_0.4.3 classInt_0.4-10 [133] plyr_1.8.9 stringi_1.8.3 rstan_2.32.3 [136] viridisLite_0.4.2 QuickJSR_1.0.9 lmerTest_3.1-3 [139] munsell_0.5.0 colourpicker_1.3.0 Brobdingnag_1.2-9 [142] bayestestR_0.13.1 V8_4.4.1 hms_1.1.3 [145] bit64_4.0.5 future_1.33.1 shiny_1.8.0 [148] haven_2.5.4 highr_0.10 igraph_1.6.0 [151] broom_1.0.5 RcppParallel_5.1.7 bslib_0.6.1 [154] bit_4.0.5 References "],["404.html", "Page not found", " Page not found The page you requested cannot be found (perhaps it was moved or renamed). 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