From 448dc91ee314ba0176af8a9975a339a6c25d0bd2 Mon Sep 17 00:00:00 2001 From: Alexander Clark Date: Thu, 18 Apr 2024 22:16:48 -0400 Subject: [PATCH] chi2 exercise --- .DS_Store | Bin 12292 -> 12292 bytes .gitignore | 1 + .../moreAboutSig-checkpoint.ipynb | 51 ++ book/_build/.doctrees/chi2.doctree | Bin 148762 -> 150092 bytes book/_build/.doctrees/environment.pickle | Bin 2027943 -> 2035476 bytes book/_build/.doctrees/moreAboutSig.doctree | Bin 0 -> 5202 bytes book/_build/.doctrees/power.doctree | Bin 0 -> 14926 bytes book/_build/html/_sources/chi2.ipynb | 11 +- book/_build/html/_sources/moreAboutSig.ipynb | 51 ++ book/_build/html/_sources/power.ipynb | 74 ++ book/_build/html/chi2.html | 13 +- book/_build/html/moreAboutSig.html | 630 +++++++++++++++++ book/_build/html/objects.inv | Bin 5150 -> 5222 bytes book/_build/html/power.html | 644 ++++++++++++++++++ book/_build/html/searchindex.js | 2 +- book/_build/jupyter_execute/chi2.ipynb | 17 +- .../_build/jupyter_execute/moreAboutSig.ipynb | 51 ++ book/_build/jupyter_execute/power.ipynb | 74 ++ book/chi2.ipynb | 11 +- book/moreAboutSig.ipynb | 51 ++ book/power.ipynb | 74 ++ 21 files changed, 1745 insertions(+), 10 deletions(-) create mode 100644 book/.ipynb_checkpoints/moreAboutSig-checkpoint.ipynb create mode 100644 book/_build/.doctrees/moreAboutSig.doctree create mode 100644 book/_build/.doctrees/power.doctree create mode 100644 book/_build/html/_sources/moreAboutSig.ipynb create mode 100644 book/_build/html/_sources/power.ipynb create mode 100644 book/_build/html/moreAboutSig.html create mode 100644 book/_build/html/power.html create mode 100644 book/_build/jupyter_execute/moreAboutSig.ipynb create mode 100644 book/_build/jupyter_execute/power.ipynb create mode 100644 book/moreAboutSig.ipynb create mode 100644 book/power.ipynb diff --git a/.DS_Store b/.DS_Store index 65324967877e733a5998554c9b2dac37a62d44b9..2cd6fc1b372991e7850c6272a58f69cfd59824ec 100644 GIT binary patch delta 482 zcmZokXi3 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@@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "68168b94", + "metadata": {}, + "source": [ + "(more)=\n", + "# More About Tests of Significance\n", + "\n", + "```{admonition} Important Readings\n", + ":class: seealso\n", + "- {cite}`freedman2007statistics`, Chapters 29\n", + "```\n", + "\n", + "## \n", + "\n", + "\n", + "## Data Snooping\n", + "\n", + "\n", + "## Was the result important? \n", + "\n", + "\n", + "## Garbage In, Garbage Out\n", + "\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/book/_build/.doctrees/chi2.doctree b/book/_build/.doctrees/chi2.doctree index 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The largest companies cannot power experiments with enough users to detect a \\$10M loss.\n", + "\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/book/_build/html/chi2.html b/book/_build/html/chi2.html index 3885616..0fbe752 100644 --- a/book/_build/html/chi2.html +++ b/book/_build/html/chi2.html @@ -847,11 +847,11 @@

    Using the \(\chi^2\) Distr }); } }(window));
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    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +     + +
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    More About Tests of Significance

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    More About Tests of Significance#

    +
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    Important Readings

    + +
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    #

    +
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    Data Snooping#

    +
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    Was the result important?#

    +
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    Garbage In, Garbage Out#

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    + +
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    + Contents +
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    +By Alexander Clark, Columbia University +

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    + + © Copyright 2022. +
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    Statistical Power

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    Statistical Power#

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    +

    Type I and II Errors#

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    A Type I error is made by rejecting the null hypothesis when the null hypothesis is true. A Type II Error is made when failing to reject the null hypothesis when the null hypothesis is false. While some mistakes, like snooping, are a matter of bad practice and can be avoided, Type I and II errors are unavoidable.

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    Hypothesis testing involves a binary decision. If we compare this to the deliberations of a judge, it is the null hypothesis that is on trial. Rejecting the hypothesis is akin to judging it to be guilty. Failing to reject the null hypothesis is akin to acquitting it, and this might be considered a negative result. This corresponds to the taxonomy in the table below.

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    Continuing the judicial analogy, a Type I error is convicting the true, innocent null hypothesis. A Type II error lets the false, crooked null hypothesis off the hook.

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    Statisticians use \(\alpha\) and \(\beta\) to denote the Type I and II conditional error rates. I call them conditional error rates to emphasize that each is a conditional probability.

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    +\[\alpha = \mathbb{P}(\text{reject }H_0 \mid H_0\text{ true})\]
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    +\[\beta = \mathbb{P}(\text{fail to reject }H_0 \mid H_0\text{ false})\]
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    The value \(\alpha\) is familiar, being directly related to the confidence level. A test has an associated power level, \(1-\beta\).

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    +
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    Power#

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    The power of a test is the probability it will reject the null hypothesis if the null is false.

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    For a given statistical test, the power depends on the significance level \(\alpha\) and the sample size \(n\).

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    First, the significance \(\alpha\) influences the power because you are determining how liberal or conservative to be with rejecting the null. A high \(\alpha\) means you will reject the null more often.

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    TKTK TK

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    Suppose you wanted 95% power in the illustration above (\(\beta = 0.05\)). According to the null hypothesis, a \(z\)-statistic will be drawn from a standard normal distribution (the top panel). We simplify the world to consider a single alternative hypothesis, under which the \(z\)-statistic is actually drawn from a distribution centered at two. To force \(\beta = 0.05\), the vertical line must be 1.645 standard deviations away from the mean of the alternative distribution. Accordingly our critical value is \(z^\star = 2-1.645 = 0.355\). A \(z\)-table helps show this corresponds to \(\alpha \approx 0.361\).

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    Second, \(n\) increases power by lowering the standard errors and thus making the sampling distribution for the sample mean more narrow. This is because the standard error for such a distribution is \(\text{SE} = \frac{\text{SD}}{\sqrt{n}}\). With less overlapping area, greater power is achieved.

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    Is power relevant in the world of big data?#

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    From Ronny Kohavi on LinkedIn:

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    If you think large companies with a massive userbase (Amazon, Google) have an easy time detecting tiny changes in A/B tests, you’re wrong! … The largest companies cannot power experiments with enough users to detect a $10M loss.

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    +By Alexander Clark, Columbia University +

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    + + \ No newline at end of file diff --git a/book/_build/html/searchindex.js b/book/_build/html/searchindex.js index 1a59c9e..467acdb 100644 --- a/book/_build/html/searchindex.js +++ b/book/_build/html/searchindex.js @@ -1 +1 @@ -Search.setIndex({"docnames": ["Homework05lab", "Rbasics", "bayes", "bibliography", "centerspread", "chancevar", "chancevary", "chi2", "confidenceintervals", "correlation", "correlation2", "experiments", "gettingstarted", "ghistogram", "gregression", "gsheets", "histograms", "hypothesistesting", "hypothesistestingB", "intro", "lawofaverages", "natural", "normal", "normal0", "observational", "other", "probability", "probability1", "probtree", "pyHypothesisTesting", "pycalculator", "pylibs", "python", "regression", "sampling", "solutions", "statwrap", "statwrap0", "statwrapregression", "thehistogram", "twosample"], "filenames": ["Homework05lab.ipynb", "Rbasics.ipynb", "bayes.md", "bibliography.md", "centerspread.md", "chancevar.md", "chancevary.ipynb", 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\"},\"shape\":[777],\"dtype\":\"float64\",\"order\":\"little\"}]]}}}],[\"slider\",{\"id\":\"p1046\"}]]},\"code\":\"\\n const data = source.data;\\n const x = data['x'];\\n const df = slider.value;\\n const y_chi2 = data['y_chi2'];\\n for (let i = 0; i < x.length; i++) {\\n y_chi2[i] = Math.pow(x[i], df/2 - 1) * Math.exp(-x[i]/2) / (Math.pow(2, df/2) * gamma(df/2));\\n }\\n source.change.emit();\\n\\n // Gamma function approximation\\n function gamma(z) {\\n const g = 7;\\n const C = [\\n 0.99999999999980993, 676.5203681218851, -1259.1392167224028,\\n 771.32342877765313, -176.61502916214059, 12.507343278686905,\\n -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7\\n ];\\n\\n if (z < 0.5) return Math.PI / (Math.sin(Math.PI * z) * gamma(1 - z));\\n z -= 1;\\n\\n let x = C[0];\\n for (let i = 1; i < g + 2; i++)\\n x += C[i] / (z + i);\\n\\n const t = z + g + 0.5;\\n return Math.sqrt(2 * Math.PI) * Math.pow(t, z + 0.5) * Math.exp(-t) * x;\\n }\\n\"}}]]]},\"title\":\"Degrees of Freedom\",\"start\":1,\"end\":20,\"value\":1}},{\"type\":\"object\",\"name\":\"Figure\",\"id\":\"p1004\",\"attributes\":{\"width\":400,\"height\":400,\"x_range\":{\"type\":\"object\",\"name\":\"DataRange1d\",\"id\":\"p1005\"},\"y_range\":{\"type\":\"object\",\"name\":\"DataRange1d\",\"id\":\"p1006\"},\"x_scale\":{\"type\":\"object\",\"name\":\"LinearScale\",\"id\":\"p1014\"},\"y_scale\":{\"type\":\"object\",\"name\":\"LinearScale\",\"id\":\"p1015\"},\"title\":{\"type\":\"object\",\"name\":\"Title\",\"id\":\"p1007\",\"attributes\":{\"text\":\"Chi-squared Curve\"}},\"renderers\":[{\"type\":\"object\",\"name\":\"GlyphRenderer\",\"id\":\"p1043\",\"attributes\":{\"data_source\":{\"id\":\"p1001\"},\"view\":{\"type\":\"object\",\"name\":\"CDSView\",\"id\":\"p1044\",\"attributes\":{\"filter\":{\"type\":\"object\",\"name\":\"AllIndices\",\"id\":\"p1045\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1040\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y_chi2\"},\"line_color\":\"blue\",\"line_width\":2}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1041\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y_chi2\"},\"line_color\":\"blue\",\"line_alpha\":0.1,\"line_width\":2}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1042\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y_chi2\"},\"line_color\":\"blue\",\"line_alpha\":0.2,\"line_width\":2}}}}],\"toolbar\":{\"type\":\"object\",\"name\":\"Toolbar\",\"id\":\"p1013\",\"attributes\":{\"tools\":[{\"type\":\"object\",\"name\":\"PanTool\",\"id\":\"p1026\"},{\"type\":\"object\",\"name\":\"WheelZoomTool\",\"id\":\"p1027\",\"attributes\":{\"renderers\":\"auto\"}},{\"type\":\"object\",\"name\":\"BoxZoomTool\",\"id\":\"p1028\",\"attributes\":{\"overlay\":{\"type\":\"object\",\"name\":\"BoxAnnotation\",\"id\":\"p1029\",\"attributes\":{\"syncable\":false,\"level\":\"overlay\",\"visible\":false,\"left\":{\"type\":\"number\",\"value\":\"nan\"},\"right\":{\"type\":\"number\",\"value\":\"nan\"},\"top\":{\"type\":\"number\",\"value\":\"nan\"},\"bottom\":{\"type\":\"number\",\"value\":\"nan\"},\"left_units\":\"canvas\",\"right_units\":\"canvas\",\"top_units\":\"canvas\",\"bottom_units\":\"canvas\",\"line_color\":\"black\",\"line_alpha\":1.0,\"line_width\":2,\"line_dash\":[4,4],\"fill_color\":\"lightgrey\",\"fill_alpha\":0.5}}}},{\"type\":\"object\",\"name\":\"SaveTool\",\"id\":\"p1034\"},{\"type\":\"object\",\"name\":\"ResetTool\",\"id\":\"p1035\"},{\"type\":\"object\",\"name\":\"HelpTool\",\"id\":\"p1036\"}]}},\"toolbar_location\":null,\"left\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1021\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1022\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1023\"},\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1024\"}}}],\"below\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1016\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1017\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1018\"},\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1019\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1020\",\"attributes\":{\"axis\":{\"id\":\"p1016\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1025\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p1021\"}}}]}}]}}]}};\n", - " const render_items = [{\"docid\":\"9365b5a3-2f22-41e3-9cd2-c93a4ecfa295\",\"roots\":{\"p1048\":\"ddb3fae3-9d04-4ab5-be5e-c4b864b41822\"},\"root_ids\":[\"p1048\"]}];\n", + " const docs_json = {\"2c3f4bd3-be63-4836-8ef7-426700a4d362\":{\"version\":\"3.3.4\",\"title\":\"Bokeh 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\"},\"shape\":[777],\"dtype\":\"float64\",\"order\":\"little\"}],[\"y_chi2\",{\"type\":\"ndarray\",\"array\":{\"type\":\"bytes\",\"data\":\"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\"},\"shape\":[777],\"dtype\":\"float64\",\"order\":\"little\"}]]}}}],[\"slider\",{\"id\":\"p1046\"}]]},\"code\":\"\\n const data = source.data;\\n const x = data['x'];\\n const df = slider.value;\\n const y_chi2 = data['y_chi2'];\\n for (let i = 0; i < x.length; i++) {\\n y_chi2[i] = Math.pow(x[i], df/2 - 1) * Math.exp(-x[i]/2) / (Math.pow(2, df/2) * gamma(df/2));\\n }\\n source.change.emit();\\n\\n // Gamma function approximation\\n function gamma(z) {\\n const g = 7;\\n const C = [\\n 0.99999999999980993, 676.5203681218851, -1259.1392167224028,\\n 771.32342877765313, -176.61502916214059, 12.507343278686905,\\n -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7\\n ];\\n\\n if (z < 0.5) return Math.PI / (Math.sin(Math.PI * z) * gamma(1 - z));\\n z -= 1;\\n\\n let x = C[0];\\n for (let i = 1; i < g + 2; i++)\\n x += C[i] / (z + i);\\n\\n const t = z + g + 0.5;\\n return Math.sqrt(2 * Math.PI) * Math.pow(t, z + 0.5) * Math.exp(-t) * x;\\n }\\n\"}}]]]},\"title\":\"Degrees of Freedom\",\"start\":1,\"end\":20,\"value\":1}},{\"type\":\"object\",\"name\":\"Figure\",\"id\":\"p1004\",\"attributes\":{\"width\":400,\"height\":400,\"x_range\":{\"type\":\"object\",\"name\":\"DataRange1d\",\"id\":\"p1005\"},\"y_range\":{\"type\":\"object\",\"name\":\"DataRange1d\",\"id\":\"p1006\"},\"x_scale\":{\"type\":\"object\",\"name\":\"LinearScale\",\"id\":\"p1014\"},\"y_scale\":{\"type\":\"object\",\"name\":\"LinearScale\",\"id\":\"p1015\"},\"title\":{\"type\":\"object\",\"name\":\"Title\",\"id\":\"p1007\",\"attributes\":{\"text\":\"Chi-squared Curve\"}},\"renderers\":[{\"type\":\"object\",\"name\":\"GlyphRenderer\",\"id\":\"p1043\",\"attributes\":{\"data_source\":{\"id\":\"p1001\"},\"view\":{\"type\":\"object\",\"name\":\"CDSView\",\"id\":\"p1044\",\"attributes\":{\"filter\":{\"type\":\"object\",\"name\":\"AllIndices\",\"id\":\"p1045\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1040\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y_chi2\"},\"line_color\":\"blue\",\"line_width\":2}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1041\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y_chi2\"},\"line_color\":\"blue\",\"line_alpha\":0.1,\"line_width\":2}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1042\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y_chi2\"},\"line_color\":\"blue\",\"line_alpha\":0.2,\"line_width\":2}}}}],\"toolbar\":{\"type\":\"object\",\"name\":\"Toolbar\",\"id\":\"p1013\",\"attributes\":{\"tools\":[{\"type\":\"object\",\"name\":\"PanTool\",\"id\":\"p1026\"},{\"type\":\"object\",\"name\":\"WheelZoomTool\",\"id\":\"p1027\",\"attributes\":{\"renderers\":\"auto\"}},{\"type\":\"object\",\"name\":\"BoxZoomTool\",\"id\":\"p1028\",\"attributes\":{\"overlay\":{\"type\":\"object\",\"name\":\"BoxAnnotation\",\"id\":\"p1029\",\"attributes\":{\"syncable\":false,\"level\":\"overlay\",\"visible\":false,\"left\":{\"type\":\"number\",\"value\":\"nan\"},\"right\":{\"type\":\"number\",\"value\":\"nan\"},\"top\":{\"type\":\"number\",\"value\":\"nan\"},\"bottom\":{\"type\":\"number\",\"value\":\"nan\"},\"left_units\":\"canvas\",\"right_units\":\"canvas\",\"top_units\":\"canvas\",\"bottom_units\":\"canvas\",\"line_color\":\"black\",\"line_alpha\":1.0,\"line_width\":2,\"line_dash\":[4,4],\"fill_color\":\"lightgrey\",\"fill_alpha\":0.5}}}},{\"type\":\"object\",\"name\":\"SaveTool\",\"id\":\"p1034\"},{\"type\":\"object\",\"name\":\"ResetTool\",\"id\":\"p1035\"},{\"type\":\"object\",\"name\":\"HelpTool\",\"id\":\"p1036\"}]}},\"toolbar_location\":null,\"left\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1021\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1022\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1023\"},\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1024\"}}}],\"below\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1016\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1017\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1018\"},\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1019\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1020\",\"attributes\":{\"axis\":{\"id\":\"p1016\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1025\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p1021\"}}}]}}]}}]}};\n", + " const render_items = [{\"docid\":\"2c3f4bd3-be63-4836-8ef7-426700a4d362\",\"roots\":{\"p1048\":\"cb614269-6119-4e3f-aa2a-c192cc2c3cea\"},\"root_ids\":[\"p1048\"]}];\n", " root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n", " }\n", " if (root.Bokeh !== undefined) {\n", @@ -816,7 +816,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "7b04429d", + "id": "23c2b0fc", "metadata": {}, "outputs": [ { @@ -930,6 +930,15 @@ "5. Is a six-sided die fair? \n", "\n", "```{exercise-end}\n", + "```\n", + "\n", + "\n", + "```{exercise-start}\n", + ":label: chiTest1 \n", + "```\n", + "Travis's playlist contains 4 songs by Taylor Swift, 2 by Pavarotti, and 4 by Little Richard. He listens to 100 songs on shuffle mode, resulting in Swift being played 30 times, Pavarotti being played 25 times, and Little Richard being played 45 times. Use a $\\chi^2$-test to determine if shuffle mode is randomizing over each song with equal probability. \n", + "\n", + "```{exercise-end}\n", "```" ] } diff --git a/book/_build/jupyter_execute/moreAboutSig.ipynb b/book/_build/jupyter_execute/moreAboutSig.ipynb new file mode 100644 index 0000000..3f5429b --- /dev/null +++ b/book/_build/jupyter_execute/moreAboutSig.ipynb @@ -0,0 +1,51 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "68168b94", + "metadata": {}, + "source": [ + "(more)=\n", + "# More About Tests of Significance\n", + "\n", + "```{admonition} Important Readings\n", + ":class: seealso\n", + "- {cite}`freedman2007statistics`, Chapters 29\n", + "```\n", + "\n", + "## \n", + "\n", + "\n", + "## Data Snooping\n", + "\n", + "\n", + "## Was the result important? \n", + "\n", + "\n", + "## Garbage In, Garbage Out\n", + "\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} \ No newline at end of file diff --git a/book/_build/jupyter_execute/power.ipynb b/book/_build/jupyter_execute/power.ipynb new file mode 100644 index 0000000..ffa97bd --- /dev/null +++ b/book/_build/jupyter_execute/power.ipynb @@ -0,0 +1,74 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "fbdfc9d0", + "metadata": {}, + "source": [ + "(power)=\n", + "# Statistical Power\n", + "\n", + "\n", + "## Type I and II Errors\n", + "\n", + "A **Type I error** is made by rejecting the null hypothesis when the null hypothesis is true. A **Type II Error** is made when failing to reject the null hypothesis when the null hypothesis is false. While some mistakes, like snooping, are a matter of bad practice and can be avoided, Type I and II errors are unavoidable.\n", + "\n", + "Hypothesis testing involves a binary decision. If we compare this to the deliberations of a judge, it is the null hypothesis that is on trial. Rejecting the hypothesis is akin to judging it to be guilty. Failing to reject the null hypothesis is akin to acquitting it, and this might be considered a negative result. This corresponds to the taxonomy in the table below. \n", + "\n", + "Continuing the judicial analogy, a Type I error is convicting the true, innocent null hypothesis. A Type II error lets the false, crooked null hypothesis off the hook. \n", + "\n", + "Statisticians use $\\alpha$ and $\\beta$ to denote the Type I and II conditional error rates. I call them *conditional* error rates to emphasize that each is a conditional probability. \n", + "\n", + "$$\\alpha = \\mathbb{P}(\\text{reject }H_0 \\mid H_0\\text{ true})$$\n", + " \n", + "$$\\beta = \\mathbb{P}(\\text{fail to reject }H_0 \\mid H_0\\text{ false})$$\n", + "\n", + "The value $\\alpha$ is familiar, being directly related to the confidence level. A test has an associated **power** level, $1-\\beta$. \n", + "\n", + "## Power\n", + "\n", + "The **power** of a test is the probability it will reject the null hypothesis if the null is false. \n", + "\n", + "For a given statistical test, the power depends on the significance level $\\alpha$ and the sample size $n$. \n", + "\n", + "First, the significance $\\alpha$ influences the power because you are determining how liberal or conservative to be with rejecting the null. A high $\\alpha$ means you will reject the null more often.\n", + "\n", + "\n", + "TKTK TK\n", + "\n", + "Suppose you wanted 95% power in the illustration above ($\\beta = 0.05$). According to the null hypothesis, a $z$-statistic will be drawn from a standard normal distribution (the top panel). We simplify the world to consider a single alternative hypothesis, under which the $z$-statistic is actually drawn from a distribution centered at two. To force $\\beta = 0.05$, the vertical line must be 1.645 standard deviations away from the mean of the alternative distribution. Accordingly our critical value is $z^\\star = 2-1.645 = 0.355$. A $z$-table helps show this corresponds to $\\alpha \\approx 0.361$.\n", + "\n", + "Second, $n$ increases power by lowering the standard errors and thus making the sampling distribution for the sample mean more narrow. This is because the standard error for such a distribution is $\\text{SE} = \\frac{\\text{SD}}{\\sqrt{n}}$. With less overlapping area, greater power is achieved. \n", + "\n", + "\n", + "#### Is power relevant in the world of big data? \n", + "\n", + "From Ronny Kohavi on [LinkedIn](https://www.linkedin.com/posts/ronnyk_abtesting-abtesting-statisticalpower-activity-6950341450593161216-dpTV):\n", + "\n", + "> If you think large companies with a massive userbase (Amazon, Google) have an easy time detecting tiny changes in A/B tests, you’re wrong! ... The largest companies cannot power experiments with enough users to detect a \\$10M loss.\n", + "\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} \ No newline at end of file diff --git a/book/chi2.ipynb b/book/chi2.ipynb index da3c2fe..393ad11 100644 --- a/book/chi2.ipynb +++ b/book/chi2.ipynb @@ -816,7 +816,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "7b04429d", + "id": "23c2b0fc", "metadata": {}, "outputs": [ { @@ -930,6 +930,15 @@ "5. Is a six-sided die fair? \n", "\n", "```{exercise-end}\n", + "```\n", + "\n", + "\n", + "```{exercise-start}\n", + ":label: chiTest1 \n", + "```\n", + "Travis's playlist contains 4 songs by Taylor Swift, 2 by Pavarotti, and 4 by Little Richard. He listens to 100 songs on shuffle mode, resulting in Swift being played 30 times, Pavarotti being played 25 times, and Little Richard being played 45 times. Use a $\\chi^2$-test to determine if shuffle mode is randomizing over each song with equal probability. \n", + "\n", + "```{exercise-end}\n", "```" ] } diff --git a/book/moreAboutSig.ipynb b/book/moreAboutSig.ipynb new file mode 100644 index 0000000..d15e2b9 --- /dev/null +++ b/book/moreAboutSig.ipynb @@ -0,0 +1,51 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "68168b94", + "metadata": {}, + "source": [ + "(more)=\n", + "# More About Tests of Significance\n", + "\n", + "```{admonition} Important Readings\n", + ":class: seealso\n", + "- {cite}`freedman2007statistics`, Chapters 29\n", + "```\n", + "\n", + "## \n", + "\n", + "\n", + "## Data Snooping\n", + "\n", + "\n", + "## Was the result important? \n", + "\n", + "\n", + "## Garbage In, Garbage Out\n", + "\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/book/power.ipynb b/book/power.ipynb new file mode 100644 index 0000000..5c369d4 --- /dev/null +++ b/book/power.ipynb @@ -0,0 +1,74 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "fbdfc9d0", + "metadata": {}, + "source": [ + "(power)=\n", + "# Statistical Power\n", + "\n", + "\n", + "## Type I and II Errors\n", + "\n", + "A **Type I error** is made by rejecting the null hypothesis when the null hypothesis is true. A **Type II Error** is made when failing to reject the null hypothesis when the null hypothesis is false. While some mistakes, like snooping, are a matter of bad practice and can be avoided, Type I and II errors are unavoidable.\n", + "\n", + "Hypothesis testing involves a binary decision. If we compare this to the deliberations of a judge, it is the null hypothesis that is on trial. Rejecting the hypothesis is akin to judging it to be guilty. Failing to reject the null hypothesis is akin to acquitting it, and this might be considered a negative result. This corresponds to the taxonomy in the table below. \n", + "\n", + "Continuing the judicial analogy, a Type I error is convicting the true, innocent null hypothesis. A Type II error lets the false, crooked null hypothesis off the hook. \n", + "\n", + "Statisticians use $\\alpha$ and $\\beta$ to denote the Type I and II conditional error rates. I call them *conditional* error rates to emphasize that each is a conditional probability. \n", + "\n", + "$$\\alpha = \\mathbb{P}(\\text{reject }H_0 \\mid H_0\\text{ true})$$\n", + " \n", + "$$\\beta = \\mathbb{P}(\\text{fail to reject }H_0 \\mid H_0\\text{ false})$$\n", + "\n", + "The value $\\alpha$ is familiar, being directly related to the confidence level. A test has an associated **power** level, $1-\\beta$. \n", + "\n", + "## Power\n", + "\n", + "The **power** of a test is the probability it will reject the null hypothesis if the null is false. \n", + "\n", + "For a given statistical test, the power depends on the significance level $\\alpha$ and the sample size $n$. \n", + "\n", + "First, the significance $\\alpha$ influences the power because you are determining how liberal or conservative to be with rejecting the null. A high $\\alpha$ means you will reject the null more often.\n", + "\n", + "\n", + "TKTK TK\n", + "\n", + "Suppose you wanted 95% power in the illustration above ($\\beta = 0.05$). According to the null hypothesis, a $z$-statistic will be drawn from a standard normal distribution (the top panel). We simplify the world to consider a single alternative hypothesis, under which the $z$-statistic is actually drawn from a distribution centered at two. To force $\\beta = 0.05$, the vertical line must be 1.645 standard deviations away from the mean of the alternative distribution. Accordingly our critical value is $z^\\star = 2-1.645 = 0.355$. A $z$-table helps show this corresponds to $\\alpha \\approx 0.361$.\n", + "\n", + "Second, $n$ increases power by lowering the standard errors and thus making the sampling distribution for the sample mean more narrow. This is because the standard error for such a distribution is $\\text{SE} = \\frac{\\text{SD}}{\\sqrt{n}}$. With less overlapping area, greater power is achieved. \n", + "\n", + "\n", + "#### Is power relevant in the world of big data? \n", + "\n", + "From Ronny Kohavi on [LinkedIn](https://www.linkedin.com/posts/ronnyk_abtesting-abtesting-statisticalpower-activity-6950341450593161216-dpTV):\n", + "\n", + "> If you think large companies with a massive userbase (Amazon, Google) have an easy time detecting tiny changes in A/B tests, you’re wrong! ... The largest companies cannot power experiments with enough users to detect a \\$10M loss.\n", + "\n" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.11.5" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +}