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<div id="measures-of-fit" class="section level2">
<h2><span class="header-section-number">2.3</span> Measures of Fit</h2>
<p>After fitting a linear regression model, a natural question is how well the model describes the data. Visually, this amounts to assessing whether the observations are tightly clustered around the regression line. Both the <em>coefficient of determination</em> and the <em>standard error of the regression</em> measure how well the OLS Regression line fits the data.</p>
<div id="the-coefficient-of-determination" class="section level3 unnumbered">
<h3>The Coefficient of Determination</h3>
<p><span class="math inline">\(R^2\)</span>, the <em>coefficient of determination</em>, is the fraction of the sample variance of <span class="math inline">\(Y_i\)</span> that is explained by <span class="math inline">\(X_i\)</span>. Mathematically, the <span class="math inline">\(R^2\)</span> can be written as the ratio of the explained sum of squares to the total sum of squares. The <em>explained sum of squares</em> (<span class="math inline">\(ESS\)</span>) is the sum of squared deviations of the predicted values <span class="math inline">\(\hat{Y_i}\)</span>, from the average of the <span class="math inline">\(Y_i\)</span>. The <em>total sum of squares</em> (<span class="math inline">\(TSS\)</span>) is the sum of squared deviations of the <span class="math inline">\(Y_i\)</span> from their average. Thus we have</p>
<p><span class="math display">\[\begin{align}
ESS & = \sum_{i = 1}^n \left( \hat{Y_i} - \overline{Y} \right)^2, \\
TSS & = \sum_{i = 1}^n \left( Y_i - \overline{Y} \right)^2, \\
R^2 & = \frac{ESS}{TSS}.
\end{align}\]</span></p>
<p>Since <span class="math inline">\(TSS = ESS + SSR\)</span> we can also write</p>
<p><span class="math display">\[ R^2 = 1- \frac{SSR}{TSS} \]</span></p>
<p>where <span class="math inline">\(SSR\)</span> is the sum of squared residuals, a measure for the errors made when predicting the <span class="math inline">\(Y\)</span> by <span class="math inline">\(X\)</span>. The <span class="math inline">\(SSR\)</span> is defined as</p>
<p><span class="math display">\[ SSR = \sum_{i=1}^n \hat{u}_i^2. \]</span></p>
<p><span class="math inline">\(R^2\)</span> lies between <span class="math inline">\(0\)</span> and <span class="math inline">\(1\)</span>. It is easy to see that a perfect fit, i.e., no errors made when fitting the regression line, implies <span class="math inline">\(R^2 = 1\)</span> since then we have <span class="math inline">\(SSR=0\)</span>. On the contrary, if our estimated regression line does not explain any variation in the <span class="math inline">\(Y_i\)</span>, we have <span class="math inline">\(ESS=0\)</span> and consequently <span class="math inline">\(R^2=0\)</span>.</p>
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<div id="the-standard-error-of-the-regression" class="section level3 unnumbered">
<h3>The Standard Error of the Regression</h3>
<p>The <em>Standard Error of the Regression</em> (<span class="math inline">\(SER\)</span>) is an estimator of the standard deviation of the residuals <span class="math inline">\(\hat{u}_i\)</span>. As such it measures the magnitude of a typical deviation from the regression line, i.e., the magnitude of a typical residual.</p>
<p><span class="math display">\[ SER = s_{\hat{u}} = \sqrt{s_{\hat{u}}^2} \ \ \ \text{where} \ \ \ s_{\hat{u} }^2 = \frac{1}{n-2} \sum_{i = 1}^n \hat{u}^2_i = \frac{SSR}{n - 2} \]</span></p>
<p>Remember that the <span class="math inline">\(u_i\)</span> are <em>unobserved</em>. This is why we use their estimated counterparts, the residuals <span class="math inline">\(\hat{u}_i\)</span>, instead. See Chapter 4.3 of the book for a more detailed comment on the <span class="math inline">\(SER\)</span>.</p>
</div>
<div id="application-to-the-test-score-data" class="section level3 unnumbered">
<h3>Application to the Test Score Data</h3>
<p>Both measures of fit can be obtained by using the function <tt>summary()</tt> with an <tt>lm</tt> object provided as the only argument. While the function <tt>lm()</tt> only prints out the estimated coefficients to the console, <tt>summary()</tt> provides additional predefined information such as the regression’s <span class="math inline">\(R^2\)</span> and the <span class="math inline">\(SER\)</span>.</p>
<div class="unfolded">
<pre class="sourceCode r"><code class="sourceCode r">mod_summary <-<span class="st"> </span><span class="kw">summary</span>(linear_model)
mod_summary</code></pre>
<pre><code>##
## Call:
## lm(formula = score ~ STR, data = CASchools)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.727 -14.251 0.483 12.822 48.540
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 698.9329 9.4675 73.825 < 2e-16 ***
## STR -2.2798 0.4798 -4.751 2.78e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 18.58 on 418 degrees of freedom
## Multiple R-squared: 0.05124, Adjusted R-squared: 0.04897
## F-statistic: 22.58 on 1 and 418 DF, p-value: 2.783e-06</code></pre>
</div>
<p>The <span class="math inline">\(R^2\)</span> in the output is called <em>Multiple R-squared</em> and has a value of <span class="math inline">\(0.051\)</span>. Hence, <span class="math inline">\(5.1 \%\)</span> of the variance of the dependent variable <span class="math inline">\(score\)</span> is explained by the explanatory variable <span class="math inline">\(STR\)</span>. That is, the regression explains little of the variance in <span class="math inline">\(score\)</span>, and much of the variation in test scores remains unexplained (cf. Figure 4.3 of the book).</p>
<p>The <span class="math inline">\(SER\)</span> is called <em>Residual standard error</em> and equals <span class="math inline">\(18.58\)</span>. The unit of the <span class="math inline">\(SER\)</span> is the same as the unit of the dependent variable. That is, on average the deviation of the actual achieved test score and the regression line is <span class="math inline">\(18.58\)</span> points.</p>
<p>Now, let us check whether <tt>summary()</tt> uses the same definitions for <span class="math inline">\(R^2\)</span> and <span class="math inline">\(SER\)</span> as we do when computing them manually.</p>
<pre class="sourceCode r"><code class="sourceCode r"><span class="co"># compute R^2 manually</span>
SSR <-<span class="st"> </span><span class="kw">sum</span>(mod_summary<span class="op">$</span>residuals<span class="op">^</span><span class="dv">2</span>)
TSS <-<span class="st"> </span><span class="kw">sum</span>((score <span class="op">-</span><span class="st"> </span><span class="kw">mean</span>(score))<span class="op">^</span><span class="dv">2</span>)
R2 <-<span class="st"> </span><span class="dv">1</span> <span class="op">-</span><span class="st"> </span>SSR<span class="op">/</span>TSS
<span class="co"># print the value to the console</span>
R2</code></pre>
<pre><code>## [1] 0.05124009</code></pre>
<pre class="sourceCode r"><code class="sourceCode r"><span class="co"># compute SER manually</span>
n <-<span class="st"> </span><span class="kw">nrow</span>(CASchools)
SER <-<span class="st"> </span><span class="kw">sqrt</span>(SSR <span class="op">/</span><span class="st"> </span>(n<span class="dv">-2</span>))
<span class="co"># print the value to the console</span>
SER</code></pre>
<pre><code>## [1] 18.58097</code></pre>
<p>We find that the results coincide. Note that the values provided by <tt>summary()</tt> are rounded to two decimal places.</p>
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