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<title>Duncan Prestige Regression Example</title>
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<h1 class="title toc-ignore">Duncan Prestige Regression Example</h1>
<h4 class="author"><em>Jeffrey B. Arnold</em></h4>
<h4 class="date"><em>04/21/2015</em></h4>
</div>
<p>This example will use Duncan’s Occpational Prestige Data, which has data on prestige and other characteristics of 45 U.S. occupations in 1950. See the help page for more info. This data is contained in the <strong>car</strong> package.</p>
<pre class="r"><code>library("car")
data("Duncan")</code></pre>
<pre class="r"><code>?Duncan</code></pre>
<p>We’ll also use a few other packages, which we should load now.</p>
<pre class="r"><code>library("ggplot2")
library("dplyr")
library("broom")</code></pre>
<p>The <code>Duncan</code> data frame contains the names of the profession as row names. This is something that is generally discouraged in modern R, so we will make a new column named <code>occupation</code>.</p>
<p>This uses the <code>$</code> to assign a value to a column in R. We’ve been using <strong>dplyr</strong> so much that we haven’t had to do this, but it can be easier at times. This could have also been done using the dplyr function <code>add_rownames</code>.</p>
<pre class="r"><code>Duncan <- add_rownames(Duncan, var = "occupation")</code></pre>
<p>Without <strong>dplyr</strong> we would have used</p>
<pre class="r"><code># Duncan$occupation <- rownames(Duncan)</code></pre>
<div id="scatterplots" class="section level2">
<h2>Scatterplots</h2>
<p>Before starting let’s create a couple of scatterplots of the data. First, prestige vs. income, labeling the points, and coloring them by type.</p>
<pre class="r"><code>ggplot(Duncan, aes(x = income, y = prestige, colour = type,
label = occupation)) +
geom_text() +
theme_minimal() </code></pre>
<p><img src="Duncan_regression_example_files/figure-html/unnamed-chunk-6-1.png" title="" alt="" width="672" /> Now, prestige vs. income, coloring the points by education level:</p>
<pre class="r"><code>ggplot(Duncan, aes(x = income, y = prestige, colour = education)) +
geom_point() +
theme_minimal() </code></pre>
<p><img src="Duncan_regression_example_files/figure-html/unnamed-chunk-7-1.png" title="" alt="" width="672" /></p>
<p>Run a regression of prestige on income and education</p>
<pre class="r"><code>mod1 <- lm(prestige ~ income + education, data = Duncan)
mod1_summary <- summary(mod1)
mod1_summary</code></pre>
<pre><code>##
## Call:
## lm(formula = prestige ~ income + education, data = Duncan)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29.538 -6.417 0.655 6.605 34.641
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.06466 4.27194 -1.420 0.163
## income 0.59873 0.11967 5.003 1.05e-05 ***
## education 0.54583 0.09825 5.555 1.73e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.37 on 42 degrees of freedom
## Multiple R-squared: 0.8282, Adjusted R-squared: 0.82
## F-statistic: 101.2 on 2 and 42 DF, p-value: < 2.2e-16</code></pre>
<p>This extracts the coefficient estimates <span class="math inline">\(\hat{\beta}\)</span>,</p>
<pre class="r"><code>beta <- coef(mod1)</code></pre>
<p>This extracts the variance covariance matrix of the coefficient estimatates <span class="math inline">\(V(\hat{\beta})\)</span>.</p>
<pre class="r"><code>beta_vc <- vcov(mod1)</code></pre>
<p>Standard errors of the coefficients are <span class="math inline">\(se(\hat{\beta}) = \sqrt{diag(V(\hat{\beta}))}\)</span>,</p>
<pre class="r"><code>beta_se <- sqrt(diag(beta_vc))</code></pre>
<p>The <span class="math inline">\(t\)</span>-statistic of the hypothesis test <span class="math inline">\(\beta = 0\)</span> is <span class="math inline">\(\frac{\hat{\beta}}{se(\hat{\beta})}\)</span>,</p>
<pre class="r"><code>tstat <- beta / beta_se</code></pre>
<p>And the <span class="math inline">\(p\)</span>-value of the two-sided hypothesis test <span class="math inline">\(H_0: \beta = 0\)</span>, <span class="math inline">\(H_a: \beta \neq 0\)</span> is,</p>
<pre class="r"><code>pval <- 2 * (1 - pt(tstat, mod1$df.residual))</code></pre>
<p>This uses the <code>qt</code> function to calculate probabilities from the <span class="math inline">\(t\)</span>-distribution.</p>
<p>The degrees of freedom <span class="math inline">\(n - k - 1\)</span></p>
<pre class="r"><code>mod1$df.residual</code></pre>
<pre><code>## [1] 42</code></pre>
<p>The coefficients, <span class="math inline">\(\hat{\beta}\)</span>:</p>
<pre class="r"><code>coef(mod1)</code></pre>
<pre><code>## (Intercept) income education
## -6.0646629 0.5987328 0.5458339</code></pre>
<p>The coefficients can also be extracted directly from the element in the <code>lm</code> object</p>
<pre class="r"><code>mod1$coefficients</code></pre>
<pre><code>## (Intercept) income education
## -6.0646629 0.5987328 0.5458339</code></pre>
<p>The fitted values, <span class="math inline">\(\hat{y}\)</span> (of the data used for fitting the regression).</p>
<pre class="r"><code>head(fitted(mod1))</code></pre>
<pre><code>## 1 2 3 4 5 6
## 77.99849 78.52748 89.05702 75.99069 79.19595 52.35877</code></pre>
<pre class="r"><code>head(mod1$fitted)</code></pre>
<pre><code>## 1 2 3 4 5 6
## 77.99849 78.52748 89.05702 75.99069 79.19595 52.35877</code></pre>
<p>The residuals, <span class="math inline">\(\hat{epsilon} = y - \hat{y}\)</span>:</p>
<pre class="r"><code>head(residuals(mod1))</code></pre>
<pre><code>## 1 2 3 4 5
## 4.001511779 4.472522658 0.942981643 0.009305892 10.804046136
## 6
## 34.641225280</code></pre>
<pre class="r"><code>head(mod1$residuals)</code></pre>
<pre><code>## 1 2 3 4 5
## 4.001511779 4.472522658 0.942981643 0.009305892 10.804046136
## 6
## 34.641225280</code></pre>
<p>You can check that the residuals are <span class="math inline">\(y - \hat{y}\)</span>,</p>
<pre class="r"><code>head(Duncan$prestige - fitted(mod1))</code></pre>
<pre><code>## 1 2 3 4 5
## 4.001511779 4.472522658 0.942981643 0.009305892 10.804046136
## 6
## 34.641225280</code></pre>
<p>To get the data used in the regression back from the regression,</p>
<pre class="r"><code>head(model.frame(mod1))</code></pre>
<pre><code>## prestige income education
## 1 82 62 86
## 2 83 72 76
## 3 90 75 92
## 4 76 55 90
## 5 90 64 86
## 6 87 21 84</code></pre>
<p>To get the <span class="math inline">\(y\)</span> values (model or design matrix) used in the regression,</p>
<pre class="r"><code>head(model.response(model.frame(mod1)))</code></pre>
<pre><code>## 1 2 3 4 5 6
## 82 83 90 76 90 87</code></pre>
<p>Note that if you have missing values in the regression, R has sophisticated but subtle ways of handling them, especially in <code>predict</code>. See the help for <code>na.omit</code> and follow links about <code>lm</code>.</p>
<p>The <code>anova</code> function returns a table with the total, model, and residual sum of squares.</p>
<pre class="r"><code>anova(mod1)</code></pre>
<pre><code>## Analysis of Variance Table
##
## Response: prestige
## Df Sum Sq Mean Sq F value Pr(>F)
## income 1 30664.8 30664.8 171.570 < 2.2e-16 ***
## education 1 5516.1 5516.1 30.863 1.727e-06 ***
## Residuals 42 7506.7 178.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
</div>
<div id="result-post-processing-with-broom" class="section level2">
<h2>Result post-processing with broom</h2>
<p><a href="https://github.com/dgrtwo/broom">Broom</a> is a relatively new package that works well with the <strong>dplyr</strong> and <code>%>%</code> workflow by converting the results of common models into data frames that can be processed more easily than the default objects R returns.</p>
<p><strong>broom</strong> has three main functions, all of which return data frames (not lists, numeric vectors, or other types of object). <code>glance</code> returns a data frame with a single row summary of the model:</p>
<pre class="r"><code>glance(mod1)</code></pre>
<pre><code>## r.squared adj.r.squared sigma statistic p.value df logLik
## 1 0.8281734 0.8199912 13.36903 101.2162 8.647636e-17 3 -178.9822
## AIC BIC deviance df.residual
## 1 365.9645 373.1911 7506.699 42</code></pre>
<p><code>tidy</code> returns a data frame with a row for each coefficient estimate:</p>
<pre class="r"><code>tidy(mod1)</code></pre>
<pre><code>## term estimate std.error statistic p.value
## 1 (Intercept) -6.0646629 4.27194117 -1.419650 1.630896e-01
## 2 income 0.5987328 0.11966735 5.003310 1.053184e-05
## 3 education 0.5458339 0.09825264 5.555412 1.727192e-06</code></pre>
<p><code>augment</code> returns the original data frame used in the model with additional columns for fitted values, the standard errors of those fitted values, residuals, etc.</p>
<pre class="r"><code>head(augment(mod1))</code></pre>
<pre><code>## prestige income education .fitted .se.fit .resid .hat
## 1 82 62 86 77.99849 3.017029 4.001511779 0.05092832
## 2 83 72 76 78.52748 3.200759 4.472522658 0.05732001
## 3 90 75 92 89.05702 3.527929 0.942981643 0.06963699
## 4 76 55 90 75.99069 3.405677 0.009305892 0.06489441
## 5 90 64 86 79.19595 3.028764 10.804046136 0.05132528
## 6 87 21 84 52.35877 5.561551 34.641225280 0.17305816
## .sigma .cooksd .std.resid
## 1 13.51587 1.688452e-03 0.3072378711
## 2 13.51194 2.406367e-03 0.3445645225
## 3 13.53022 1.334199e-04 0.0731269019
## 4 13.53108 1.198619e-08 0.0007198265
## 5 13.41973 1.241505e-02 0.8297130469
## 6 12.15304 5.663797e-01 2.8494163845</code></pre>
</div>
<div id="coefficient-plots" class="section level2">
<h2>Coefficient Plots</h2>
<p>Also known as airplane plots or ropeladder plots:</p>
<pre class="r"><code>ggplot(tidy(mod1) %>% filter(term != "(Intercept)"),
aes(x = term, y = estimate,
ymin = estimate - 2 * std.error,
ymax = estimate + 2 * std.error)) +
geom_pointrange() +
coord_flip()</code></pre>
<p><img src="Duncan_regression_example_files/figure-html/unnamed-chunk-26-1.png" title="" alt="" width="672" /></p>
<p>This could also be done with the <code>coefplot</code> function from the <strong>coefplot</strong> package:</p>
<pre class="r"><code>library("coefplot")
coefplot(mod1)</code></pre>
<p><img src="Duncan_regression_example_files/figure-html/unnamed-chunk-27-1.png" title="" alt="" width="672" /> or to drop the intercept</p>
<pre class="r"><code>coefplot(mod1, coefficients = c("income", "education"))</code></pre>
<p><img src="Duncan_regression_example_files/figure-html/unnamed-chunk-28-1.png" title="" alt="" width="672" /> Since internally <code>coefplot</code> uses <strong>ggplot2</strong>, you could also edit the object it returns.</p>
</div>
<div id="creating-regression-tables" class="section level2">
<h2>Creating regression tables</h2>
<p>Several packages (<strong>stargazer</strong>, <strong>texreg</strong>, <strong>apsrtable</strong>) are useful for creating publication type regression tables. <strong>stargazer</strong> and <strong>texreg</strong> are the most complete package. Both allow output to either LaTeX or HTML tables for many types of statistical models. We’ll use <strong>stargazer</strong> here:</p>
<pre class="r"><code>library("stargazer")</code></pre>
<pre><code>##
## Please cite as:</code></pre>
<pre><code>## Hlavac, Marek (2015). stargazer: Well-Formatted Regression and Summary Statistics Tables.</code></pre>
<pre><code>## R package version 5.2. http://CRAN.R-project.org/package=stargazer</code></pre>
<pre class="r"><code>stargazer(mod1, type = "text")</code></pre>
<pre><code>##
## ===============================================
## Dependent variable:
## ---------------------------
## prestige
## -----------------------------------------------
## income 0.599***
## (0.120)
##
## education 0.546***
## (0.098)
##
## Constant -6.065
## (4.272)
##
## -----------------------------------------------
## Observations 45
## R2 0.828
## Adjusted R2 0.820
## Residual Std. Error 13.369 (df = 42)
## F Statistic 101.216*** (df = 2; 42)
## ===============================================
## Note: *p<0.1; **p<0.05; ***p<0.01</code></pre>
<p>Now render that as html instead,</p>
<pre class="r"><code>stargazer(mod1, type = "html")</code></pre>
<table style="text-align:center">
<tr>
<td colspan="2" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
<em>Dependent variable:</em>
</td>
</tr>
<tr>
<td>
</td>
<td colspan="1" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
prestige
</td>
</tr>
<tr>
<td colspan="2" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
income
</td>
<td>
0.599<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.120)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
education
</td>
<td>
0.546<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.098)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
Constant
</td>
<td>
-6.065
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(4.272)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
</tr>
<tr>
<td colspan="2" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
Observations
</td>
<td>
45
</td>
</tr>
<tr>
<td style="text-align:left">
R<sup>2</sup>
</td>
<td>
0.828
</td>
</tr>
<tr>
<td style="text-align:left">
Adjusted R<sup>2</sup>
</td>
<td>
0.820
</td>
</tr>
<tr>
<td style="text-align:left">
Residual Std. Error
</td>
<td>
13.369 (df = 42)
</td>
</tr>
<tr>
<td style="text-align:left">
F Statistic
</td>
<td>
101.216<sup>***</sup> (df = 2; 42)
</td>
</tr>
<tr>
<td colspan="2" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
<em>Note:</em>
</td>
<td style="text-align:right">
<sup><em></sup>p<0.1; <sup><strong></sup>p<0.05; <sup></strong></em></sup>p<0.01
</td>
</tr>
</table>
<p>Look at the source <code>.Rmd</code> file for this document; the chunk above used <code>results = "asis"</code> to print and render the HTML rather than R output.</p>
<p>This usefulness of this function is apparent when multiple regressions are plotted:</p>
<pre class="r"><code>mod3 <- lm(prestige ~ income, data = Duncan)
mod4 <- lm(prestige ~ income * type + education * type, data = Duncan)
stargazer(mod1, mod3, mod4, type = "html")</code></pre>
<table style="text-align:center">
<tr>
<td colspan="4" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td colspan="3">
<em>Dependent variable:</em>
</td>
</tr>
<tr>
<td>
</td>
<td colspan="3" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td colspan="3">
prestige
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(1)
</td>
<td>
(2)
</td>
<td>
(3)
</td>
</tr>
<tr>
<td colspan="4" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
income
</td>
<td>
0.599<sup>***</sup>
</td>
<td>
1.080<sup>***</sup>
</td>
<td>
0.783<sup>***</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.120)
</td>
<td>
(0.107)
</td>
<td>
(0.131)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
typeprof
</td>
<td>
</td>
<td>
</td>
<td>
32.008<sup>**</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
(14.109)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
typewc
</td>
<td>
</td>
<td>
</td>
<td>
-7.043
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
(20.638)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
education
</td>
<td>
0.546<sup>***</sup>
</td>
<td>
</td>
<td>
0.320
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(0.098)
</td>
<td>
</td>
<td>
(0.280)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
income:typeprof
</td>
<td>
</td>
<td>
</td>
<td>
-0.369<sup>*</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
(0.204)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
income:typewc
</td>
<td>
</td>
<td>
</td>
<td>
-0.360
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
(0.260)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
typeprof:education
</td>
<td>
</td>
<td>
</td>
<td>
0.019
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
(0.318)
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
typewc:education
</td>
<td>
</td>
<td>
</td>
<td>
0.107
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>