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LinearRegression.html
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<h1 class="title toc-ignore">Linear Regression</h1>
</div>
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<p>Determine which explanatory variables have a significant effect on the mean of the quantitative response variable.</p>
<hr />
<div id="simple-linear-regression" class="section level2 tabset tabset-fade tabset-pills">
<h2 class="tabset tabset-fade tabset-pills">Simple Linear Regression</h2>
<div style="float:left;width:125px;" align="center">
<p><img src="Images/QuantYQuantX.png" width=58px;></p>
</div>
<p>Simple linear regression is a good analysis technique when the data consists of a single quantitative response variable <span class="math inline">\(Y\)</span> and a single quantitative explanatory variable <span class="math inline">\(X\)</span>.</p>
<div id="overview" class="section level3 tabset">
<h3 class="tabset">Overview</h3>
<div style="padding-left:125px;">
<p><strong>Mathematical Model</strong></p>
<p>The true regression model assumed by a regression analysis is given by</p>
<div style="float:right;font-size:.8em;background-color:lightgray;padding:5px;border-radius:4px;">
<a style="color:darkgray;" href="javascript:showhide('simplelinearlatexrcode')">Math Code</a>
</div>
<div id="simplelinearlatexrcode" style="display:none;">
<pre><code>$$
\underbrace{Y_i}_\text{Some Label} = \overbrace{\beta_0}^\text{y-int} + \overbrace{\beta_1}^\text{slope} \underbrace{X_i}_\text{Some Label} + \epsilon_i \quad \text{where} \ \epsilon_i \sim N(0, \sigma^2)
$$</code></pre>
</div>
<center>
<span class="tooltipr"> <span class="math inline">\(Y_i\)</span> <span class="tooltiprtext">The response variable. The “i” denotes that this is the y-value for individual “i”, where “i” is 1, 2, 3,… and so on up to <span class="math inline">\(n\)</span>, the sample size.</span> </span><span class="tooltipr"> <span class="math inline">\(=\)</span> <span class="tooltiprtext">This states that we are assuming <span class="math inline">\(Y_i\)</span> was created, or is “equal to” the formula that will follow on the right-hand-side of the equation.</span> </span><span class="tooltipr"> <span class="math inline">\(\underbrace{\overbrace{\beta_0}^\text{y-intercept} + \overbrace{\beta_1}^\text{slope} X_i \ }_\text{true regression relation}\)</span> <span class="tooltiprtext">The true regression relation is a line, a line that is typically unknown in real life. It can be likened to “God’s Law” or “Natural Law”. Something that governs the way the data behaves, but is unkown to us.</span> </span><span class="tooltipr"> <span class="math inline">\(+\)</span> <span class="tooltiprtext">This plus sign emphasizes that the actual data, the <span class="math inline">\(Y_i\)</span>, is created by adding together the value from the true line <span class="math inline">\(\beta_0 + \beta_1 X_i\)</span> and an individual error term <span class="math inline">\(\epsilon_i\)</span>, which allows each dot in the regression to be off of the line by a certain amount called <span class="math inline">\(\epsilon_i\)</span>.</span> </span><span class="tooltipr"> <span class="math inline">\(\overbrace{\epsilon_i}^\text{error term}\)</span> <span class="tooltiprtext">Error term for each individual <span class="math inline">\(i\)</span>. The error terms are “random” and unique for each individual. This provides the statistical relationship of the regression. It is what allows each dot to be different, while still coming from the same line, or underlying law.</span> </span><span class="tooltipr"> <span class="math inline">\(\quad \text{where}\)</span> <span class="tooltiprtext">Some extra comments are needed about <span class="math inline">\(\epsilon_i\)</span>…</span> </span><span class="tooltipr"> <span class="math inline">\(\ \overbrace{\epsilon_i \sim N(0, \sigma^2)}^\text{error term normally distributed}\)</span> <span class="tooltiprtext">The error terms <span class="math inline">\(\epsilon_i\)</span> are assumed to be normally distributed with constant variance. Pay special note that the <span class="math inline">\(\sigma\)</span> does not have an <span class="math inline">\(i\)</span> in it, so it is the same for each individual. In other words, the variance is constant. The mean of the errors is zero, which causes the dots to be spread out symmetrically both above and below the line.</span> </span>
</center>
<p><br/></p>
<p>The estimated regression line obtained from a regression analysis, pronounced “y-hat”, is written as</p>
<div style="float:right;font-size:.8em;background-color:lightgray;padding:5px;border-radius:4px;">
<a style="color:darkgray;" href="javascript:showhide('simplelinearlatexrcodeyhat')">Math Code</a>
</div>
<div id="simplelinearlatexrcodeyhat" style="display:none;">
<pre><code>$$
\underbrace{\hat{Y}_i}_\text{Some Label} = \overbrace{b_0}^\text{est. y-int} + \overbrace{b_1}^\text{est. slope} \underbrace{X_i}_\text{Some Label}
$$</code></pre>
</div>
<center>
<span class="tooltipr"> <span class="math inline">\(\hat{Y}_i\)</span> <span class="tooltiprtext">The estimated average y-value for individual <span class="math inline">\(i\)</span> is denoted by <span class="math inline">\(\hat{Y}_i\)</span>. It is important to recognize that <span class="math inline">\(Y_i\)</span> is the actual value for individual <span class="math inline">\(i\)</span>, and <span class="math inline">\(\hat{Y}_i\)</span> is the average y-value for all individuals with the same <span class="math inline">\(X_i\)</span> value.</span> </span><span class="tooltipr"> <span class="math inline">\(=\)</span> <span class="tooltiprtext">The formula for the average y-value, <span class="math inline">\(\hat{Y}_i\)</span> is equal to what follows…</span> </span><span class="tooltipr"> <span class="math inline">\(\underbrace{\overbrace{\ b_0 \ }^\text{y-intercept} + \overbrace{b_1}^\text{slope} X_i \ }_\text{estimated regression relation}\)</span> <span class="tooltiprtext">Two things are important to notice about this equation. First, it uses <span class="math inline">\(b_0\)</span> and <span class="math inline">\(b_1\)</span> instead of <span class="math inline">\(\beta_0\)</span> and <span class="math inline">\(\beta_1\)</span>. This is because <span class="math inline">\(b_0\)</span> and <span class="math inline">\(b_1\)</span> are the estimated y-intercept and slope, respectively, not the true y-intercept <span class="math inline">\(\beta_0\)</span> and true slope <span class="math inline">\(\beta_1\)</span>. Second, this equation does not include <span class="math inline">\(\epsilon_i\)</span>. In other words, it is the estimated regression line, so it only describes the average y-values, not the actual y-values.</span> </span>
</center>
<p><br/></p>
<div style="font-size:0.8em;">
<p>Note: see the <strong>Explanation</strong> tab <strong>The Mathematical Model</strong> for details about these equations.</p>
</div>
<p><strong>Hypotheses</strong></p>
<div style="float:right;font-size:.8em;background-color:lightgray;padding:5px;border-radius:4px;">
<a style="color:darkgray;" href="javascript:showhide('simplelinearhypecodeslope')">Math Code</a>
</div>
<div id="simplelinearhypecodeslope" style="display:none;">
<pre><code>$$
\left.\begin{array}{ll}
H_0: \beta_1 = 0 \\
H_a: \beta_1 \neq 0
\end{array}
\right\} \ \text{Slope Hypotheses}
$$
$$
\left.\begin{array}{ll}
H_0: \beta_0 = 0 \\
H_a: \beta_0 \neq 0
\end{array}
\right\} \ \text{Intercept Hypotheses}
$$</code></pre>
</div>
<div style="clear:right;">
</div>
<p><span class="math display">\[
\left.\begin{array}{ll}
H_0: \beta_1 = 0 \\
H_a: \beta_1 \neq 0
\end{array}
\right\} \ \text{Slope Hypotheses}^{\quad \text{(most common)}}\quad\quad
\]</span></p>
<p><span class="math display">\[
\left.\begin{array}{ll}
H_0: \beta_0 = 0 \\
H_a: \beta_0 \neq 0
\end{array}
\right\} \ \text{Intercept Hypotheses}^{\quad\text{(sometimes useful)}}
\]</span></p>
<p><br/></p>
<p>If <span class="math inline">\(\beta_1 = 0\)</span>, then the model reduces to <span class="math inline">\(Y_i = \beta_0 + \epsilon_i\)</span>, which is a flat line. This means <span class="math inline">\(X\)</span> does not improve our understanding of the mean of <span class="math inline">\(Y\)</span> if the null hypothesis is true.</p>
<p>If <span class="math inline">\(\beta_0 = 0\)</span>, then the model reduces to <span class="math inline">\(Y_i = \beta_1 X + \epsilon_i\)</span>, a line going through the origin. This means the average <span class="math inline">\(Y\)</span>-value is <span class="math inline">\(0\)</span> when <span class="math inline">\(X=0\)</span> if the null hypothesis is true.</p>
<p><strong>Assumptions</strong></p>
<p>This regression model is appropriate for the data when five assumptions can be made.</p>
<ol style="list-style-type: decimal">
<li><p><strong>Linear Relation</strong>: the true regression relation between <span class="math inline">\(Y\)</span> and <span class="math inline">\(X\)</span> is linear.</p></li>
<li><p><strong>Normal Errors</strong>: the error terms <span class="math inline">\(\epsilon_i\)</span> are normally distributed with a mean of zero.</p></li>
<li><p><strong>Constant Variance</strong>: the variance <span class="math inline">\(\sigma^2\)</span> of the error terms is constant (the same) over all <span class="math inline">\(X_i\)</span> values.</p></li>
<li><p><strong>Fixed X</strong>: the <span class="math inline">\(X_i\)</span> values can be considered fixed and measured without error.</p></li>
<li><p><strong>Independent Errors</strong>: the error terms <span class="math inline">\(\epsilon_i\)</span> are independent.</p></li>
</ol>
<div style="font-size:0.8em;">
<p>Note: see the <strong>Explanation</strong> tab <strong>Residual Plots & Regression Assumptions</strong> for details about checking the regression assumptions.</p>
</div>
<p><strong>Interpretation</strong></p>
<p>The slope is interpreted as, “the change in the average y-value for a one unit change in the x-value.” It <strong>is not</strong> the average change in y. <strong>It is</strong> the change in the average y-value.</p>
<p>The y-intercept is interpreted as, “the average y-value when x is zero.” It is often not meaningful, but is sometimes useful. It just depends if x being zero is meaningful or not within the context of your analysis. For example, knowing the average price of a car with zero miles is useful. However, pretending to know the average height of adult males that weigh zero pounds, is not useful.</p>
<hr />
</div>
</div>
<div id="r-instructions" class="section level3">
<h3>R Instructions</h3>
<div style="padding-left:125px;">
<p><strong>Console</strong> Help Command: <code>?lm()</code></p>
<p><strong>Perform the Regression</strong></p>
<a href="javascript:showhide('simplelinearrcode')">
<div class="hoverchunk">
<p><span class="tooltipr"> mylm <span class="tooltiprtext">This is some name you come up with that will become the R object that stores the results of your linear regression <code>lm(...)</code> command.</span> </span><span class="tooltipr"> <- <span class="tooltiprtext">This is the “left arrow” assignment operator that stores the results of your <code>lm()</code> code into <code>mylm</code> name.</span> </span><span class="tooltipr"> lm( <span class="tooltiprtext">lm(…) is an R function that stands for “Linear Model”. It performs a linear regression analysis for Y ~ X.</span> </span><span class="tooltipr"> Y <span class="tooltiprtext">Y is your quantitative response variable. It is the name of one of the columns in your data set.</span> </span><span class="tooltipr"> ~ <span class="tooltiprtext">The tilde symbol ~ is used to tell R that Y should be treated as the response variable that is being explained by the explanatory variable X.</span> </span><span class="tooltipr"> X, <span class="tooltiprtext">X is the quantitative explanatory variable (at least it is typically quantitative but could be qualitative) that will be used to explain the average Y-value.</span> </span><span class="tooltipr"> data = NameOfYourDataset <span class="tooltiprtext">NameOfYourDataset is the name of the dataset that contains Y and X. In other words, one column of your dataset would be your response variable Y and another column would be your explanatory variable X.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for the lm(…) function.</span> </span><br/><span class="tooltipr"> summary(mylm) <span class="tooltiprtext">The <code>summary</code> command allows you to print the results of your linear regression that were previously saved in <code>mylm</code> name.</span> </span><span class="tooltipr" style="float:right;font-size:.8em;"> Click to Show Output <span class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
<p></a></p>
<div id="simplelinearrcode" style="display:none;">
<p>Example output from a regression. Hover each piece to learn more.</p>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> Call:<br/> lm(formula = dist ~ speed, data = cars) <span class="tooltiprouttext">This is simply a statement of your original lm(…) “call” that you made when performing your regression. It allows you to verify that you ran what you thought you ran in the lm(…).</span> </span>
</td>
</tr>
</table>
<p><br/></p>
<table class="rconsole">
<tr>
<td colspan="2">
<span class="tooltiprout"> Residuals: <span class="tooltiprouttext">Residuals are the vertical difference between each point and the line, <span class="math inline">\(Y_i - \hat{Y}_i\)</span>. The residuals are supposed to be normally distributed, so a quick glance at their five-number summary can give us insight about any skew present in the residuals. </span>
</td>
</tr>
<tr>
<td align="right">
<span class="tooltiprout"> min<br/> -29.069 <span class="tooltiprouttext">“min” gives the value of the residual that is furthest below the regression line. Ideally, the magnitude of this value would be about equal to the magnitude of the largest positive residual (the max) because the hope is that the residuals are normally distributed around the line.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 1Q<br/> -9.525 <span class="tooltiprouttext">“1Q” gives the first quartile of the residuals, which will always be negative, and ideally would be about equal in magnitude to the third quartile.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Median<br/> -2.272 <span class="tooltiprouttext">“Median” gives the median of the residuals, which would ideally would be about equal to zero. Note that because the regression line is the least squares line, the mean of the residuals will ALWAYS be zero, so it is never included in the output summary. This particular median value of -2.272 is a little smaller than zero than we would hope for and suggests a right skew in the data because the mean (0) is greater than the median (-2.272) witnessing the residuals are right skewed. This can also be seen in the maximum being much larger in magnitude than the minimum.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 3Q<br/> 9.215 <span class="tooltiprouttext">“3Q” gives the third quartile of the residuals, which would ideally would be about equal in magnitude to the first quartile. In this case, it is pretty close, which helps us see that the first quartile of residuals on either side of the line is behaving fairly normally.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Max</br> 43.201 <span class="tooltiprouttext">“Max” gives the maximum positive residuals, which would ideally would be about equal in magnitude to the minimum residual. In this case, it is much larger than the minimum, which helps us see that the residuals are likely right skewed.</span> </span>
</td>
</tr>
</table>
<p><br/></p>
<table class="rconsole">
<tr>
<td colspan="2">
<span class="tooltiprout"> Coefficients: <span class="tooltiprouttext">Notice that in your lm(…) you used only <span class="math inline">\(Y\)</span> and <span class="math inline">\(X\)</span>. You did type out any coefficients, i.e., the <span class="math inline">\(\beta_0\)</span> or <span class="math inline">\(\beta_1\)</span> of the regression model. These coefficients are estimated by the lm(…) function and displayed in this part of the output along with standard errors, t-values, and p-values.</span> </span>
</td>
</tr>
<tr>
<td align="left">
</td>
<td align="right">
<span class="tooltiprout"> Estimate <span class="tooltiprouttext">To learn more about the “Estimates” of the “Coefficients” see the “Explanation” tab, “Estimating the Model Parameters” section for details.</span>
</td>
<td align="right">
<span class="tooltiprout"> Std. Error <span class="tooltiprouttext">To learn more about the “Standard Errors” of the “Coefficients” see the “Explanation” tab, “Inference for the Model Parameters” section.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> t value <span class="tooltiprouttext">To learn more about the “t value” of the “Coefficients” see the “Explanation” tab, “Inference for the Model Parameters” section.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Pr(>|t|) <span class="tooltiprouttext">The “Pr” stands for “Probability” and the “(> |t|)” stands for “more extreme than the observed t-value”. Thus, this is the p-value for the hypothesis test of each coefficient being zero.<br/> To learn more about the “p-value” of the “Coefficients” see the “Explanation” tab, “Inference for the Model Parameters” section. </span> </span>
</td>
</tr>
<tr>
<td align="left">
<span class="tooltiprout"> (Intercept) <span class="tooltiprouttext">This always says “Intercept” for any lm(…) you run in R. That is because R always assumes there is a y-intercept for your regression function.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> -17.5791 <span class="tooltiprouttext">This is the estimate of the y-intercept, <span class="math inline">\(\beta_0\)</span>. It is called <span class="math inline">\(b_0\)</span>. It is the average y-value when X is zero.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 6.7584 <span class="tooltiprouttext">This is the standard error of <span class="math inline">\(b_0\)</span>. It tells you how much <span class="math inline">\(b_0\)</span> varies from sample to sample. The closer to zero, the better.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> -2.601 <span class="tooltiprouttext">This is the test statistic t for the test of <span class="math inline">\(\beta_0 = 0\)</span>. It is calculated by dividing the “Estimate” of the intercept (-17.5791) by its standard error (6.7584). It gives the “number of standard errors” away from zero that the “estimate” has landed. In this case, the estimate of -17.5791 is -2.601 standard errors (6.7584) from zero, which is a fairly surprising distance as shown by the p-value.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.0123 <span class="tooltiprouttext">This is the p-value of the test of the hypothesis that <span class="math inline">\(\beta_0 = 0\)</span>. It measures the probability of observing a t-value as extreme as the one observed. To compute it yourself in R, use <code>pt(-abs(your t-value), df of your regression)*2</code>.</span> </span>
</td>
<td align="left">
<span class="tooltiprout"> * <span class="tooltiprouttext">This is called a “star”. One star means significant at the 0.1 level of <span class="math inline">\(\alpha\)</span>.</span> </span>
</td>
</tr>
<tr>
<td align="left">
<span class="tooltiprout"> speed <span class="tooltiprouttext">This is always the name of your X-variable in your lm(Y ~ X, …).</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 3.9324 <span class="tooltiprouttext">This is the estimate of the slope, <span class="math inline">\(\beta_1\)</span>. It is called <span class="math inline">\(b_1\)</span>. It is the change in the average y-value as X is increased by 1 unit.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.4155 <span class="tooltiprouttext">This is the standard error of <span class="math inline">\(b_1\)</span>. It tells you how much <span class="math inline">\(b_1\)</span> varies from sample to sample. The closer to zero, the better.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 9.464 <span class="tooltiprouttext">This is the test statistic t for the test of <span class="math inline">\(\beta_1 = 0\)</span>. It is calculated by dividing the “Estimate” of the slope (3.9324) by its standard error (0.4155). It gives the “number of standard errors” away from zero that the “estimate” has landed. In this case, the estimate of 3.9324 is 9.464 standard errors (0.4155) from zero, which is a really surprising distance as shown by the smallness of the p-value.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 1.49e-12 <span class="tooltiprouttext">This is the p-value of the test of the hypothesis that <span class="math inline">\(\beta_1 = 0\)</span>. To compute it yourself in R, use <code>pt(-abs(your t-value), df of your regression)*2</code></span> </span>
</td>
<td align="left">
<span class="tooltiprout"> *** <span class="tooltiprouttext">This is called a “star”. Three stars means significant at the 0.01 level of <span class="math inline">\(\alpha\)</span>.</span> </span>
</td>
</tr>
</table>
<table class="rconsole">
<tr>
<td>
<span> --- </span>
</td>
</tr>
</table>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 <span class="tooltiprouttext">These “codes” explain what significance level the p-value is smaller than based on how many “stars” * the p-value is labeled with in the Coefficients table above.</span> </span>
</td>
</tr>
</table>
<p><br/></p>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> Residual standard error: <span class="tooltiprouttext">This is the estimate of <span class="math inline">\(\sigma\)</span> in the regression model <span class="math inline">\(Y_i = \beta_0 + \beta_1 X_i + \epsilon_i\)</span> where <span class="math inline">\(\epsilon_i \sim N(0,\sigma^2)\)</span>. It is the square root of the MSE.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 15.38 <span class="tooltiprouttext">For this particular regression, the estimate of <span class="math inline">\(\sigma\)</span> is 15.38. Squaring this number gives you the MSE, which is the estimate of <span class="math inline">\(\sigma^2\)</span>.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> on 48 degrees of freedom <span class="tooltiprouttext">This is <span class="math inline">\(n-p\)</span> where <span class="math inline">\(n\)</span> is the sample size and <span class="math inline">\(p\)</span> is the number of parameters in the regression model. In this case, there is a sample size of 50 and two parameters, <span class="math inline">\(\beta_0\)</span> and <span class="math inline">\(\beta_1\)</span>, so 50-2 = 48.</span> </span>
</td>
</tr>
</table>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> Multiple R-squared: <span class="tooltiprouttext">This is <span class="math inline">\(R^2\)</span>, the percentage of variation in <span class="math inline">\(Y\)</span> that is explained by the regression model. It is equal to the SSR/SSTO or, equivalently, 1 - SSE/SSTO.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.6511, <span class="tooltiprouttext">In this particular regression, 65.11% of the variation in stopping distance <code>dist</code> is explained by the regression model using speed of the car.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Adjusted R-squared: <span class="tooltiprouttext">The adjusted R-squared will always be at least slightly smaller than <span class="math inline">\(R^2\)</span>. The closer to R-squared that it is, the better. When it differs dramatically from <span class="math inline">\(R^2\)</span>, it is a sign that the regression model is over-fitting the data.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.6438 <span class="tooltiprouttext">In this case, the value of 0.6438 is quite close to the original <span class="math inline">\(R^2\)</span> value, so there is no fear of over-fitting with this particular model. That is good.</span> </span>
</td>
</tr>
</table>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> F-statistic: <span class="tooltiprouttext">The F-statistic is found as the ratio of the MSR/MSP where MSR = SSR/(p-1) and MSE = SSE/(n-p) where n is the sample size and p is the number of parameters in the regression model.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 89.57 <span class="tooltiprouttext">This is the value of the F-statistic for the lm(dist ~ speed, data=cars) regression. Note that SSE = sum( cars.lm$res^2 ) = 11353.52 with n - p = 50 - 2 = 48 degrees of freedom for this data. Further, SSR = sum( (cars.lm$fit - mean(cars$dist))^2 ) = 21185.46 with p - 1 = 1 degree of freedom. So MSR = 21185.46 and MSE = 11353.52 / 48 = 236.5317. So MSR / MSE = 21185.46 / 236.5317 = 89.56711.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> on 1 and 48 DF, <span class="tooltiprouttext">The 1 degree of freedom is the SSR degrees of freedom (p-1). The 48 is the SSE degrees of freedom (n-p).</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> p-value: 1.49e-12 <span class="tooltiprouttext">The p-value for an F-statistic is found by the code pf(89.56711, 1, 48, lower.tail=FALSE), which gives the probability of being more extreme than the observed F-statistic in an F distribution with 1 and 48 degrees of freedom.</span> </span>
</td>
</tr>
</table>
</div>
<p><br/></p>
<p><strong>Check Assumptions 1, 2, 3, and 5</strong></p>
<a href="javascript:showhide('assumptionplots')">
<div class="hoverchunk">
<p><span class="tooltipr"> par( <span class="tooltiprtext">The par(…) command stands for “Graphical PARameters”. It allows you to control various aspects of graphics in Base R.</span> </span><span class="tooltipr"> mfrow= <span class="tooltiprtext">This stands for “multiple frames filled by row”, which means, put lots of plots on the same row, starting with the plot on the left, then working towards the right as more plots are created.</span> </span><span class="tooltipr"> c( <span class="tooltiprtext">The combine function c(…) is used to specify how many rows and columns of graphics should be placed together.</span> </span><span class="tooltipr"> 1, <span class="tooltiprtext">This specifies that 1 row of graphics should be produced.</span> </span><span class="tooltipr"> 3 <span class="tooltiprtext">This states that 3 columns of graphics should be produced.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for c(…) function.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for par(…) function.</span> </span><br/><span class="tooltipr"> plot( <span class="tooltiprtext">This version of plot(…) will actually create several regression diagnostic plots by default.</span> </span><span class="tooltipr"> mylm, <span class="tooltiprtext">This is the name of an lm object that you created previously.</span> </span><span class="tooltipr"> which= <span class="tooltiprtext">This allows you to select “which” regression diagnostic plots should be drawn.</span> </span><span class="tooltipr"> 1 <span class="tooltiprtext">Selecting 1, would give the residuals vs. fitted values plot only.</span> </span><span class="tooltipr"> : <span class="tooltiprtext">The colon allows you to select more than just one plot.</span> </span><span class="tooltipr"> 2 <span class="tooltiprtext">Selecting 2 also gives the Q-Q Plot of residuals. If you wanted to instead you could just use which=1 to get the residuals vs fitted values plot, then you could use qqPlot(mylm$residuals) to create a fancier Q-Q Plot of the residuals.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for plot(…) function.</span> </span><br/><span class="tooltipr"> plot( <span class="tooltiprtext">This version of plot(…) will be used to create a time-ordered plot of the residuals. The order of the residuals is the original order of the x-values in the original data set. If the original data set doesn’t have an order, then this plot is not interesting.</span> </span><span class="tooltipr"> mylm <span class="tooltiprtext">The lm object that you created previously.</span> </span><span class="tooltipr"> $ <span class="tooltiprtext">This allows you to access various elements from the regression that was performed.</span> </span><span class="tooltipr"> residuals <span class="tooltiprtext">This grabs the residuals for each observation in the regression.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for plot(…) function.</span> </span><span class="tooltipr" style="float:right;font-size:.8em;"> Click to Show Output <span class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
<p></a></p>
<div id="assumptionplots" style="display:none;">
<p><img src="LinearRegression_files/figure-html/unnamed-chunk-2-1.png" width="672" /></p>
</div>
<p><br/></p>
<p><strong>Plotting the Regression Line</strong></p>
<div class="tab">
<p><button class="tablinks" onclick="openTab(event, 'BaseScatterplot')">Base R</button> <button class="tablinks" onclick="openTab(event, 'ggplotScatterplot')">ggplot2</button></p>
</div>
<div id="BaseScatterplot" class="tabcontent">
<p>
<p>To add the regression line to a scatterplot use the <code>abline(...)</code> command:</p>
<a href="javascript:showhide('regressionline')">
<div class="hoverchunk">
<p><span class="tooltipr"> plot( <span class="tooltiprtext">The plot(…) function is used to create a scatterplot with a y-axis (the vertical axis) and an x-axis (the horizontal axis).</span> </span><span class="tooltipr"> Y <span class="tooltiprtext">This is the “response variable” of your regression. The thing you are interested in predicting. This is the name of a “numeric” column of data from the data set called YourDataSet.</span> </span><span class="tooltipr"> ~ <span class="tooltiprtext">The tilde “~” is used to relate Y to X and can be found on the top-left key of your keyboard.</span> </span><span class="tooltipr"> X, <span class="tooltiprtext">This is the explanatory variable of your regression. It is the name of a “numeric” column of data from YourDataSet. .</span> </span><span class="tooltipr"> data= <span class="tooltiprtext">The data= statement is used to specify the name of the data set where the columns of “X” and “Y” are located.</span> </span><span class="tooltipr"> YourDataSet <span class="tooltiprtext">This is the name of your data set, like KidsFeet or cars or airquality.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for plot(…) function.</span> </span><br/><span class="tooltipr"> abline( <span class="tooltiprtext">This stands for “a” (intercept) “b” (slope) line. It is a function that allows you to add a line to a plot by specifying just the intercept and slope of the line.</span> </span><span class="tooltipr"> mylm <span class="tooltiprtext">This is the name of an lm(…) that you created previoiusly. Since mylm contains the slope and intercept of the estimated line, the abline(…) function will locate these two values from within mylm and use them to add a line to your current plot(…).</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for abline(…) function.</span> </span><span class="tooltipr" style="float:right;font-size:.8em;"> Click to Show Output <span class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
<p></a></p>
<div id="regressionline" style="display:none;">
<pre><code>mylm <- lm(dist ~ speed, data = cars)
plot(dist ~ speed, data = cars)
abline(mylm)</code></pre>
<p><img src="LinearRegression_files/figure-html/unnamed-chunk-3-1.png" width="672" /></p>
</div>
<p>You can customize the look of the regression line with</p>
<a href="javascript:showhide('regressionlinecolor')">
<div class="hoverchunk">
<p><span class="tooltipr"> abline( <span class="tooltiprtext">This stands for “a” (intercept) “b” (slope) line. It is a function that allows you to add a line to a plot by specifying just the intercept and slope of the line.</span> </span><span class="tooltipr"> mylm, <span class="tooltiprtext">This is the name of an lm(…) that you created previoiusly. Since mylm contains the slope and intercept of the estimated line, the abline(…) function will locate these two values from within mylm and use them to add a line to your current plot(…).</span> </span><span class="tooltipr"> lty= <span class="tooltiprtext">The lty= stands for “line type” and allows you to select between 0=blank, 1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash.</span> </span><span class="tooltipr"> 1, <span class="tooltiprtext">This creates a solid line. Remember, other options include: 0=blank, 1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash.</span> </span><span class="tooltipr"> lwd= <span class="tooltiprtext">The lwd= allows you to specify the width of the line. The default width is 1. Using lwd=2 would double the thickness, and so on. Any positive value is allowed.</span> </span><span class="tooltipr"> 1, <span class="tooltiprtext">Default line width. To make a thicker line, us 2 or 3… To make a thinner line, try 0.5, but 1 is already pretty thin.</span> </span><span class="tooltipr"> col= <span class="tooltiprtext">This allows you to specify the color of the line using either a name of a color or rgb(.5,.2,.3,.2) where the format is rgb(percentage red, percentage green, percentage blue, percent opaque).</span> </span><span class="tooltipr"> “someColor” <span class="tooltiprtext">Type colors() in R for options.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for abline(…) function.</span> </span><span class="tooltipr" style="float:right;font-size:.8em;"> Click to Show Output <span class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
<p></a></p>
<div id="regressionlinecolor" style="display:none;">
<pre><code>mylm <- lm(dist ~ speed, data = cars)
plot(dist ~ speed, data = cars)
abline(mylm, lty=1, lwd=1, col="firebrick")</code></pre>
<p><img src="LinearRegression_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
</div>
<p>You can add points to the regression with…</p>
<a href="javascript:showhide('regressionaddpoints')">
<div class="hoverchunk">
<p><span class="tooltipr"> points( <span class="tooltiprtext">This is like plot(…) but adds points to the current plot(…) instead of creating a new plot.</span> </span><span class="tooltipr"> newY <span class="tooltiprtext">newY should be a column of values from some data set. Or, use points(newX, newY) to add a single point to a graph.</span> </span><span class="tooltipr"> ~ <span class="tooltiprtext">This links Y to X in the plot.</span> </span><span class="tooltipr"> newX, <span class="tooltiprtext">newX should be a column of values from some data set. It should be the same length as newY. If just a single value, use points(newX, newY) instead.</span> </span><span class="tooltipr"> data=YourDataSet, <span class="tooltiprtext">If newY and newX come from a dataset, then use data= to tell the points(…) function what data set they come from. If newY and newX are just single values, then data= is not needed.</span> </span><span class="tooltipr"> col=“skyblue”, <span class="tooltiprtext">This allows you to specify the color of the points using either a name of a color or rgb(.5,.2,.3,.2) where the format is rgb(percentage red, percentage green, percentage blue, percent opaque).</span> </span><span class="tooltipr"> pch=16 <span class="tooltiprtext">This allows you to specify the type of plotting symbol to be used for the points. Type ?pch and scroll half way down in the help file that appears to learn about other possible symbols.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for points(…) function.</span> </span><span class="tooltipr" style="float:right;font-size:.8em;"> Click to Show Output <span class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
<p></a></p>
<div id="regressionaddpoints" style="display:none;">
<pre><code>mylm <- lm(dist ~ speed, data = cars)
plot(dist ~ speed, data = cars)
points(7,40, pch=16, col="skyblue", cex=2)
text(7,40, "New Dot", pos=3, cex=0.5)
points(dist ~ speed, data=filter(cars, mylm$res > 2), cex=.8, col="red")
abline(mylm, lty=1, lwd=1, col="firebrick")</code></pre>
<p><img src="LinearRegression_files/figure-html/unnamed-chunk-5-1.png" width="672" /></p>
</div>
</p>
</div>
<div id="ggplotScatterplot" class="tabcontent">
<p>
<p>To add the regression line to a scatterplot using the ggplot2 approach, first ensure:</p>
<p><code>library(ggplot2)</code> or <code>library(tidyverse)</code></p>
<p>is loaded. Then, use the <code>geom_smooth(method = lm)</code> command:</p>
<a href="javascript:showhide('ggplot')">
<div class="hoverchunk">
<p><span class="tooltipr"> ggplot( <span class="tooltiprtext">Every ggplot2 graphic begins with the ggplot() command, which creates a framework, or coordinate system, that you can add layers to. Without adding any layers, ggplot() produces a blank graphic.</span> </span><span class="tooltipr"> YourDataSet, <span class="tooltiprtext">This is simply the name of your data set, like KidsFeet or starwars.</span> </span><span class="tooltipr"> aes( <span class="tooltiprtext">aes stands for aesthetic. Inside of aes(), you place elements that you want to map to the coordinate system, like x and y variables.</span> </span><span class="tooltipr"> x = <span class="tooltiprtext">“x = ” declares which variable will become the x-axis of the graphic, your explanatory variable. Both “x= ” and “y= ” are optional phrasesin the ggplot2 syntax.</span> </span><span class="tooltipr"> X, <span class="tooltiprtext">This is the explanatory variable of the regression: the variable used to <em>explain</em> the mean of y. It is the name of the “numeric” column of YourDataSet.</span> </span><span class="tooltipr"> y = <span class="tooltiprtext">“y= ” declares which variable will become the y-axis of the graphic.</span> </span><span class="tooltipr"> Y <span class="tooltiprtext">This is the response variable of the regression: the variable that you are interested in predicting. It is the name of a “numeric” column of YourDataSet.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for aes(…) function.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for ggplot(…) function.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot(). In this case, you use + to add a geom_point() layer on the next line.</span> </span><br/><span class="tooltipr"> geom_point() <span class="tooltiprtext">geom_point() allows you to add a layer of points, a scatterplot, over the ggplot() framework. The x and y coordinates are received from the previously specified x and y variables declared in the ggplot() aesthetic.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">Here the + is used to add yet another layer to ggplot().</span> </span><br/><span class="tooltipr"> geom_smooth( <span class="tooltiprtext">geom_smooth() is a smoothing function that you can use to add different lines or curves to ggplot(). In this case, you will use it to add the least-squares regression line to the scatterplot.</span> </span><span class="tooltipr"> method = <span class="tooltiprtext">Use “method = ” to tell geom_smooth() that you are going to declare a specific smoothing function, or method, to alter the line or curve..</span> </span><span class="tooltipr"> “lm”, <span class="tooltiprtext">lm stands for linear model. Using method = “lm” tells geom_smooth() to fit a least-squares regression line onto the graphic. The regression line is modeled using y ~ x, which variables were declared in the initial ggplot() aesthetic. There are several other methods that could be used here.</span> </span><span class="tooltipr"> formula = y~x, <span class="tooltiprtext">This tells geom_smooth to place a simple linear regression line on the plot. Other formula statements can be used in the same way as lm(…) to place more complicated models on the plot.</span> </span><span class="tooltipr"> se = FALSE <span class="tooltiprtext">se stands for “standard error”. Specifying FALSE turns this feature off. When TRUE, a gray band showing the “confidence band” for the regression is shown. Unless you know how to interpret this confidence band, leave it turned off.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for the geom_smooth() function.</span> </span><span class="tooltipr" style="float:right;font-size:.8em;"> Click to Show Output <span class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
<p></a></p>
<div id="ggplot" style="display:none;">
<pre><code>ggplot(cars, aes(x = speed, y = dist)) +
geom_point() +
geom_smooth(method = "lm", formula=y~x, se=FALSE)</code></pre>
<p><img src="LinearRegression_files/figure-html/unnamed-chunk-6-1.png" width="672" /></p>
</div>
<p>There are a number of ways to customize the appearance of the regression line:</p>
<a href="javascript:showhide('ggplotline')">
<div class="hoverchunk">
<p><span class="tooltipr"> ggplot( <span class="tooltiprtext">Every ggplot2 graphic begins with the ggplot() command, which creates a framework, or coordinate system, that you can add layers to. Without adding any layers, ggplot() produces a blank graphic.</span> </span><span class="tooltipr"> cars, <span class="tooltiprtext">This is simply the name of your data set, like KidsFeet or starwars.</span> </span><span class="tooltipr"> aes( <span class="tooltiprtext">aes stands for aesthetic. Inside of aes(), you place elements that you want to map to the coordinate system, like x and y variables.</span> </span><span class="tooltipr"> x = <span class="tooltiprtext">“x = ” declares which variable will become the x-axis of the graphic, your explanatory variable. Both “x= ” and “y= ” are optional phrasesin the ggplot2 syntax.</span> </span><span class="tooltipr"> speed, <span class="tooltiprtext">This is the explanatory variable of the regression: the variable used to <em>explain</em> the mean of y. It is the name of the “numeric” column of YourDataSet.</span> </span><span class="tooltipr"> y = <span class="tooltiprtext">“y= ” declares which variable will become the y-axis of the grpahic.</span> </span><span class="tooltipr"> dist <span class="tooltiprtext">This is the response variable of the regression: the variable that you are interested in predicting. It is the name of a “numeric” column of YourDataSet.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for aes(…) function.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for ggplot(…) function.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot(). In this case, you use + to add a geom_point() layer on the next line.</span> </span><br/><span class="tooltipr"> geom_point() <span class="tooltiprtext">geom_point() allows you to add a layer of points, a scatterplot, over the ggplot() framework. The x and y coordinates are received from the previously specified x and y variables declared in the ggplot() aesthetic.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">Here the + is used to add yet another layer to ggplot().</span> </span><br/><span class="tooltipr"> geom_smooth( <span class="tooltiprtext">geom_smooth() is a smoothing function that you can use to add different lines or curves to ggplot(). In this case, you will use it to add the least-squares regression line to the scatterplot.</span> </span><span class="tooltipr"> method = <span class="tooltiprtext">Use “method = ” to tell geom_smooth() that you are going to declare a specific smoothing function, or method, to alter the line or curve..</span> </span><span class="tooltipr"> “lm”, <span class="tooltiprtext">lm stands for linear model. Using method = “lm” tells geom_smooth() to fit a least-squares regression line onto the graphic. The regression line is modeled using y ~ x, which variables were declared in the initial ggplot() aesthetic.</span> </span><span class="tooltipr"> formula = y~x, <span class="tooltiprtext">This tells geom_smooth to place a simple linear regression line on the plot. Other formula statements can be used in the same way as lm(…) to place more complicated models on the plot.</span> </span><span class="tooltipr"> se = FALSE, <span class="tooltiprtext">se stands for “standard error”. Specifying FALSE turns this feature off. When TRUE, a gray band showing the “confidence band” for the regression is shown. Unless you know how to interpret this confidence band, leave it turned off.</span> </span><span class="tooltipr"> size = 2, <span class="tooltiprtext">Use <em>size = 2</em> to adjust the thickness of the line to size 2.</span> </span><span class="tooltipr"> color = “orange”, <span class="tooltiprtext">Use <em>color = “orange”</em> to change the color of the line to orange.</span> </span><br><span class="tooltipr"> linetype = “dashed” <span class="tooltiprtext">Use <em>linetype = “dashed”</em> to change the solid line to a dashed line. Some linetype options include “dashed”, “dotted”, “longdash”, “dotdash”, etc.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for the geom_smooth() function.</span> </span><span class="tooltipr" style="float:right;font-size:.8em;"> Click to Show Output <span class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
<p></a></p>
<div id="ggplotline" style="display:none;">
<p><img src="LinearRegression_files/figure-html/unnamed-chunk-7-1.png" width="672" /></p>
</div>
<p>In addition to customizing the regression line, you can customize the points, add points, add lines, and much more.</p>
<a href="javascript:showhide('ggplotpoints')">
<div class="hoverchunk">
<p><span class="tooltipr"> ggplot( <span class="tooltiprtext">Every ggplot2 graphic begins with the ggplot() command, which creates a framework, or coordinate system, that you can add layers to. Without adding any layers, ggplot() produces a blank graphic.</span> </span><span class="tooltipr"> cars, <span class="tooltiprtext">This is simply the name of your data set, like KidsFeet or starwars.</span> </span><span class="tooltipr"> aes( <span class="tooltiprtext">aes stands for aesthetic. Inside of aes(), you place elements that you want to map to the coordinate system, like x and y variables.</span> </span><span class="tooltipr"> x = <span class="tooltiprtext">“x = ” declares which variable will become the x-axis of the graphic, your explanatory variable. Both “x= ” and “y= ” are optional phrasesin the ggplot2 syntax.</span> </span><span class="tooltipr"> speed, <span class="tooltiprtext">This is the explanatory variable of the regression: the variable used to <em>explain</em> the mean of y. It is the name of the “numeric” column of YourDataSet.</span> </span><span class="tooltipr"> y = <span class="tooltiprtext">“y= ” declares which variable will become the y-axis of the grpahic.</span> </span><span class="tooltipr"> dist <span class="tooltiprtext">This is the response variable of the regression: the variable that you are interested in predicting. It is the name of a “numeric” column of YourDataSet.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for aes(…) function.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for ggplot(…) function.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot(). In this case, you use + to add a geom_point() layer on the next line.</span> </span><br/><span class="tooltipr"> geom_point( <span class="tooltiprtext">geom_point() allows you to add a layer of points, a scatterplot, over the ggplot() framework. The x and y coordinates are received from the previously specified x and y variables declared in the ggplot() aesthetic.</span> </span><span class="tooltipr"> size = 1.5, <span class="tooltiprtext">Use <em>size = 1.5</em> to change the size of the points.</span> </span><span class="tooltipr"> color = “skyblue” <span class="tooltiprtext">Use <em>color = “skyblue”</em> to change the color of the points to Brother Saunders’ favorite color.</span> </span><span class="tooltipr"> alpha = 0.5 <span class="tooltiprtext">Use <em>alpha = 0.5</em> to change the transparency of the points to 0.5.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis of geom_point() function. </span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot().</span> </span><br><span class="tooltipr"> geom_smooth( <span class="tooltiprtext">geom_smooth() is a smoothing function that you can use to add different lines or curves to ggplot(). In this case, you will use it to add the least-squares regression line to the scatterplot.</span> </span><span class="tooltipr"> method = <span class="tooltiprtext">Use “method = ” to tell geom_smooth() that you are going to declare a specific smoothing function, or method, to alter the line or curve..</span> </span><span class="tooltipr"> “lm”, <span class="tooltiprtext">lm stands for linear model. Using method = “lm” tells geom_smooth() to fit a least-squares regression line onto the graphic.</span> </span><span class="tooltipr"> formula = y~x, <span class="tooltiprtext">This tells geom_smooth to place a simple linear regression line on the plot. Other formula statements can be used in ways similar to lm(…) to place more complicated models on the plot.</span> </span><span class="tooltipr"> se = FALSE, <span class="tooltiprtext">se stands for “standard error”. Specifying FALSE turns this feature off. When TRUE, a gray band showing the “confidence band” for the regression is shown. Unless you know how to interpret this confidence band, leave it turned off.</span> </span><span class="tooltipr"> color = “navy”, <span class="tooltiprtext">Use <em>color = “navy”</em> to change the color of the line to navy blue.</span> </span><span class="tooltipr"> size = 1.5 <span class="tooltiprtext">Use <em>size = 1.5</em> to adjust the thickness of the line to 1.5.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis of geom_smooth() function.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot().</span> </span><br><span class="tooltipr"> geom_hline( <span class="tooltiprtext">Use geom_hline() to add a horizontal line at a specified y-intercept. You can also use geom_vline(xintercept = some_number) to add a vertical line to the graph.</span> </span><span class="tooltipr"> yintercept = <span class="tooltiprtext">Use “yintercept =” to tell geom_hline() that you are going to declare a y intercept for the horizontal line.</span> </span><span class="tooltipr"> 75 <span class="tooltiprtext">75 is the value of the y-intercept.</span> </span><span class="tooltipr"> , color = “firebrick” <span class="tooltiprtext">Use <em>color = “firebrick”</em> to change the color of the horizontal line to firebrick red.</span> </span><span class="tooltipr"> , size = 1, <span class="tooltiprtext">Use <em>size = 1</em> to adjust the thickness of the horizontal line to size 1.</span> </span><br><span class="tooltipr"> linetype = “longdash” <span class="tooltiprtext">Use <em>linetype = “longdash”</em> to change the solid line to a dashed line with longer dashes. Some linetype options include “dashed”, “dotted”, “longdash”, “dotdash”, etc.</span> </span><span class="tooltipr"> , alpha = 0.5 <span class="tooltiprtext">Use <em>alpha = 0.5</em> to change the transparency of the horizontal line to 0.5.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis of geom_hline function.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot().</span> </span><br><span class="tooltipr"> geom_segment( <span class="tooltiprtext">geom_segment() allows you to add a line segment to ggplot() by using specified start and end points.</span> </span><span class="tooltipr"> x = <span class="tooltiprtext">“x =” tells geom_segment() that you are going to declare the x-coordinate for the starting point of the line segment.</span> </span><span class="tooltipr"> 14, <span class="tooltiprtext">14 is a number on the x-axis of your graph. It is the x-coordinate of the starting point of the line segment.</span> </span><span class="tooltipr"> y =<br />
<span class="tooltiprtext">“y =” tells geom_segment() that you are going to declare the y-coordinate for the starting point of the line segment.</span> </span><span class="tooltipr"> 75, <span class="tooltiprtext">75 is a number on the y-axis of your graph. It is the y-coordinate of the starting point of the line segment.</span> </span><span class="tooltipr"> xend = <span class="tooltiprtext">“xend =” tells geom_segment() that you are going to declare the x-coordinate for the end point of the line segment.</span> </span><span class="tooltipr"> 14, <span class="tooltiprtext">14 is a number on the x-axis of your graph. It is the x-coordinate of the end point of the line segment.</span> </span><span class="tooltipr"> yend = <span class="tooltiprtext">“yend =” tells geom_segment() that you are going to declare the y-coordinate for the end point of the line segment.</span> </span><span class="tooltipr"> 38, <span class="tooltiprtext">38 is a number on the y-axis of your graph. It is the y-coordinate of the end point of the line segment.</span> </span><br><span class="tooltipr"> size = 1 <span class="tooltiprtext">Use <em>size = 1</em> to adjust the thickness of the line segment.</span> </span><span class="tooltipr"> , color = “lightgray” <span class="tooltiprtext">Use <em>color = “lightgray”</em> to change the color of the line segment to light gray.</span> </span><span class="tooltipr"> , linetype = “longdash” <span class="tooltiprtext">Use *linetype = “longdash* to change the solid line segment to a dashed one. Some linetype options include”dashed“,”dotted“,”longdash“,”dotdash", etc.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for geom_segment() function.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot().</span> </span><br><span class="tooltipr"> geom_point( <span class="tooltiprtext">geom_point() can also be used to add individual points to the graph. Simply declare the x and y coordinates of the point you want to plot.</span> </span><span class="tooltipr"> x = <span class="tooltiprtext">“x =” tells geom_point() that you are going to declare the x-coordinate for the point.</span> </span><span class="tooltipr"> 14, <span class="tooltiprtext">14 is a number on the x-axis of your graph. It is the x-coordinate of the point.</span> </span><span class="tooltipr"> y = <span class="tooltiprtext">“y =” tells geom_point() that you are going to declare the y-coordinate for the point.</span> </span><span class="tooltipr"> 75 <span class="tooltiprtext">75 is a number on the y-axis of your graph. It is the y-coordinate of the point.</span> </span><span class="tooltipr"> , size = 3 <span class="tooltiprtext">Use <em>size = 3</em> to make the point stand out more.</span> </span><span class="tooltipr"> , color = “firebrick” <span class="tooltiprtext">Use <em>color = “firebrick”</em> to change the color of the point to firebrick red.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis of the geom_point() function.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot().</span> </span><br><span class="tooltipr"> geom_text( <span class="tooltiprtext">geom_text() allows you to add customized text anywhere on the graph. It is very similar to the base R equivalent, text(…).</span> </span><span class="tooltipr"> x = <span class="tooltiprtext">“x =” tells geom_text() that you are going to declare the x-coordinate for the text.</span> </span><span class="tooltipr"> 14, <span class="tooltiprtext">14 is a number on the x-axis of your graph. It is the x-coordinate of the text.</span> </span><span class="tooltipr"> y = <span class="tooltiprtext">“y =” tells geom_text() that you are going to declare the y-coordinate for the text.</span> </span><span class="tooltipr"> 84, <span class="tooltiprtext">84 is a number on the y-axis of your graph. It is the y-coordinate of the text.</span> </span><span class="tooltipr"> label = <span class="tooltiprtext">“label =” tells geom_text() that you are going to give it the label.</span> </span><span class="tooltipr"> “My Point (14, 75)”, <span class="tooltiprtext"><em>“My Point (14, 75)”</em> is the text that will appear on the graph.</span> </span><br><span class="tooltipr"> color = “navy” <span class="tooltiprtext">Use <em>color = “navy”</em> to change the color of the text to navy blue.</span> </span><span class="tooltipr"> , size = 3 <span class="tooltiprtext">Use <em>size = 3</em> to change the size of the text.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis of the geom_text() function.</span> </span><span class="tooltipr"> + <span class="tooltiprtext">The + allows you to add more layers to the framework provided by ggplot().</span> </span><br><span class="tooltipr"> theme_minimal() <span class="tooltiprtext">Add a minimalistic theme to the graph. There are many other themes that you can try out.</span> </span><span class="tooltipr" style="float:right;font-size:.8em;"> Click to Show Output <span class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
<p></a></p>
<div id="ggplotpoints" style="display:none;">
<pre><code>## `geom_smooth()` using formula 'y ~ x'</code></pre>
<p><img src="LinearRegression_files/figure-html/unnamed-chunk-8-1.png" width="672" /></p>
</div>
</p>
</div>
<p><br/></p>
<p><strong>Accessing Parts of the Regression</strong></p>
<p>Finally, note that the <code>mylm</code> object contains the <code>names(mylm)</code> of</p>
<a href="javascript:showhide('coeff')">
<div class="hoverchunk">
<p><span class="tooltipr"> mylm$coefficients <span class="tooltiprtext">Contains two values. The first is the estimated <span class="math inline">\(y\)</span>-intercept. The second is the estimated slope.</span> </span></p>
</div>
<p></a></p>
<div id="coeff" style="display:none;">
<pre><code>## (Intercept) speed
## -17.579095 3.932409</code></pre>
</div>
<a href="javascript:showhide('resid')">
<div class="hoverchunk">
<p><span class="tooltipr"> mylm$residuals <span class="tooltiprtext">Contains the residuals from the regression in the same order as the actual dataset.</span> </span></p>
</div>
<p></a></p>
<div id="resid" style="display:none;">
<pre><code>## 1 2 3 4 5 6 7
## 3.849460 11.849460 -5.947766 12.052234 2.119825 -7.812584 -3.744993
## 8 9 10 11 12 13 14
## 4.255007 12.255007 -8.677401 2.322599 -15.609810 -9.609810 -5.609810
## 15 16 17 18 19 20 21
## -1.609810 -7.542219 0.457781 0.457781 12.457781 -11.474628 -1.474628
## 22 23 24 25 26 27 28
## 22.525372 42.525372 -21.407036 -15.407036 12.592964 -13.339445 -5.339445
## 29 30 31 32 33 34 35
## -17.271854 -9.271854 0.728146 -11.204263 2.795737 22.795737 30.795737
## 36 37 38 39 40 41 42
## -21.136672 -11.136672 10.863328 -29.069080 -13.069080 -9.069080 -5.069080
## 43 44 45 46 47 48 49
## 2.930920 -2.933898 -18.866307 -6.798715 15.201285 16.201285 43.201285
## 50
## 4.268876</code></pre>
</div>
<a href="javascript:showhide('fit')">
<div class="hoverchunk">
<p><span class="tooltipr"> mylm$fitted.values <span class="tooltiprtext">The values of <span class="math inline">\(\hat{Y}\)</span> in the same order as the original dataset.</span> </span></p>
</div>
<p></a></p>
<div id="fit" style="display:none;">
<pre><code>## 1 2 3 4 5 6 7 8
## -1.849460 -1.849460 9.947766 9.947766 13.880175 17.812584 21.744993 21.744993
## 9 10 11 12 13 14 15 16
## 21.744993 25.677401 25.677401 29.609810 29.609810 29.609810 29.609810 33.542219
## 17 18 19 20 21 22 23 24
## 33.542219 33.542219 33.542219 37.474628 37.474628 37.474628 37.474628 41.407036
## 25 26 27 28 29 30 31 32
## 41.407036 41.407036 45.339445 45.339445 49.271854 49.271854 49.271854 53.204263
## 33 34 35 36 37 38 39 40
## 53.204263 53.204263 53.204263 57.136672 57.136672 57.136672 61.069080 61.069080
## 41 42 43 44 45 46 47 48
## 61.069080 61.069080 61.069080 68.933898 72.866307 76.798715 76.798715 76.798715
## 49 50
## 76.798715 80.731124</code></pre>
</div>
<div class="hoverchunk">
<p><span class="tooltipr"> mylm$… <span class="tooltiprtext">several other things that will not be explained here.</span> </span></p>
</div>
<p><br/></p>
<p><strong>Making Predictions</strong></p>
<a href="javascript:showhide('prediction')">
<div class="hoverchunk">
<p><span class="tooltipr"> predict( <span class="tooltiprtext">The R function predict(…) allows you to use an lm(…) object to make predictions for specified x-values.</span> </span><span class="tooltipr"> mylm, <span class="tooltiprtext">This is the name of a previously performed lm(…) that was saved into the name <code>mylm <- lm(...)</code>.</span> </span><span class="tooltipr"> data.frame( <span class="tooltiprtext">To specify the values of <span class="math inline">\(x\)</span> that you want to use in the prediction, you have to put those x-values into a data set, or more specifally, a data.frame(…).</span> </span><span class="tooltipr"> X= <span class="tooltiprtext">The value for <code>X=</code> should be whatever x-variable name was used in the original regression. For example, if <code>mylm <- lm(dist ~ speed, data=cars)</code> was the original regression, then this code would read <code>speed =</code> instead of <code>X=</code>… Further, the value of <span class="math inline">\(Xh\)</span> should be some specific number, like <code>speed=12</code> for example.</span> </span><span class="tooltipr"> Xh <span class="tooltiprtext">The value of <span class="math inline">\(Xh\)</span> should be some specific number, like <code>12</code>, as in <code>speed=12</code> for example.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for the data.frame(…) function.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis for the predict(…) function.</span> </span></p>
</div>
<p></a></p>
<div id="prediction" style="display:none;">
<p><code>mylm <- lm(dist ~ speed, data = cars)</code></p>
<p><code>predict(mylm, data.frame(speed = 12))</code></p>
<table class="rconsole">
<tr>
<td align="right">
<span class="tooltiprout"> 1<br/> 29.60981 <span class="tooltiprouttext">The value given is the “fitted-value” or “predicted-value” for the specified x-value. In this case, a car with a speed of 12 is predicted to have a stopping distance of 29.60981 feet.</span> </span>
</td>
</tr>
</table>
</div>
<a href="javascript:showhide('predictionInterval')">
<div class="hoverchunk">
<p><span class="tooltipr"> predict( <span class="tooltiprtext">The R function predict(…) allows you to use an lm(…) object to make predictions for specified x-values.</span> </span><span class="tooltipr"> mylm, <span class="tooltiprtext">This is the name of a previously performed lm(…) that was saved into the name <code>mylm <- lm(...)</code>.</span> </span><span class="tooltipr"> data.frame( <span class="tooltiprtext">To specify the values of <span class="math inline">\(x\)</span> that you want to use in the prediction, you have to put those x-values into a data set, or more specifally, a data.frame(…).</span> </span><span class="tooltipr"> X= <span class="tooltiprtext">The value for <code>X=</code> should be whatever x-variable name was used in the original regression. For example, if <code>mylm <- lm(dist ~ speed, data=cars)</code> was the original regression, then this code would read <code>speed =</code> instead of <code>X=</code>… Further, the value of <span class="math inline">\(Xh\)</span> should be some specific number, like <code>speed=12</code> for example.</span> </span><span class="tooltipr"> Xh <span class="tooltiprtext">The value of <span class="math inline">\(Xh\)</span> should be some specific number, like <code>12</code>, as in <code>speed=12</code> for example.</span> </span><span class="tooltipr"> ), <span class="tooltiprtext">Closing parenthesis for the data.frame(…) function.</span> </span><span class="tooltipr"> interval= <span class="tooltiprtext">This optional command allows you to specify if the predicted value should be accompanied by either a confidence interval or a prediction interval.</span> </span><span class="tooltipr"> “prediction” <span class="tooltiprtext">This specifies that a prediction interval will be included with the predicted value. A prediction interval gives you a 95% confidence interval that captures 95% of the data, or <span class="math inline">\(Y_i\)</span> values for the specific <span class="math inline">\(X\)</span>-value specified in the prediction.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis of the predict(…) function.</span> </span></p>
</div>
<p></a></p>
<div id="predictionInterval" style="display:none;">
<p><code>mylm <- lm(dist ~ speed, data = cars)</code></p>
<p><code>predict(mylm, data.frame(speed = 12), interval = "prediction")</code></p>
<table class="rconsole">
<tr>
<td align="right">
<span class="tooltiprout"> fit <span class="tooltiprouttext">The “fit” is the predicted value.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> lwr <span class="tooltiprouttext">The “lwr” is the lower bound.</span>
</td>
<td align="right">
<span class="tooltiprout"> upr <span class="tooltiprouttext">The “upr” is the upper bound.</span> </span>
</td>
</tr>
<tr>
<td align="left">
<span class="tooltiprout"> 1 29.60981 <span class="tooltiprouttext">In this case, a car with a speed of 12 mph is predicted to have a stopping distance of 29.60981 feet. However, we are wise enough to recognize that the stopping distance for individual cars will vary anywhere from -1.749529 (or 0 because distance can’t go negative) feet to 60.96915 feet.</span> </span>
</td>
<td align="left">
<span class="tooltiprout"> -1.749529 <span class="tooltiprouttext">This is the lower bound of the prediction interval. While we predict a stopping distance of 29.60981 feet, this prediction interval reminds us the stopping distance could be as quick as -1.749529 feet (or 0 because distance can’t go negative).</span> </span>
</td>
<td align="left">
<span class="tooltiprout"> 60.96915 <span class="tooltiprouttext">This is the upper bound of the prediction interval. While we predict a stopping distance of 29.60981 feet, this prediction interval reminds us that the actual stopping distance could be as high as 60.96915 feet.</span> </span>
</td>
</tr>
</table>
</div>
<a href="javascript:showhide('predictionConfidence')">
<div class="hoverchunk">
<p><span class="tooltipr"> predict( <span class="tooltiprtext">The R function predict(…) allows you to use an lm(…) object to make predictions for specified x-values.</span> </span><span class="tooltipr"> mylm, <span class="tooltiprtext">This is the name of a previously performed lm(…) that was saved into the name <code>mylm <- lm(...)</code>.</span> </span><span class="tooltipr"> data.frame( <span class="tooltiprtext">To specify the values of <span class="math inline">\(x\)</span> that you want to use in the prediction, you have to put those x-values into a data set, or more specifally, a data.frame(…).</span> </span><span class="tooltipr"> X= <span class="tooltiprtext">The value for <code>X=</code> should be whatever x-variable name was used in the original regression. For example, if <code>mylm <- lm(dist ~ speed, data=cars)</code> was the original regression, then this code would read <code>speed =</code> instead of <code>X=</code>… Further, the value of <span class="math inline">\(Xh\)</span> should be some specific number, like <code>speed=12</code> for example.</span> </span><span class="tooltipr"> Xh <span class="tooltiprtext">The value of <span class="math inline">\(Xh\)</span> should be some specific number, like <code>12</code>, as in <code>speed=12</code> for example.</span> </span><span class="tooltipr"> ), <span class="tooltiprtext">Closing parenthesis for the data.frame(…) function.</span> </span><span class="tooltipr"> interval= <span class="tooltiprtext">This optional command allows you to specify if the predicted value should be accompanied by either a confidence interval or a prediction interval.</span> </span><span class="tooltipr"> “confidence” <span class="tooltiprtext">This specifies that a confidence interval for the prediction should be provided. This is of use whenever your interest is in just estimating the average y-value, not the actual y-values.</span> </span><span class="tooltipr"> ) <span class="tooltiprtext">Closing parenthesis of the predict(…) function.</span> </span></p>
</div>
<p></a></p>
<div id="predictionConfidence" style="display:none;">
<p><code>mylm <- lm(dist ~ speed, data = cars)</code></p>
<p><code>predict(mylm, data.frame(speed = 12), interval = "confidence")</code></p>
<table class="rconsole">
<tr>
<td align="right">
<span class="tooltiprout"> fit <span class="tooltiprouttext">The “fit” is the predicted value.</span> </span>
</td>