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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