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<hr />
<p>Regression for a qualitative binary response variable <span
class="math inline">\((Y_i = 0\)</span> or <span
class="math inline">\(1)\)</span>. The explanatory variables can be
either quantitative or qualitative.</p>
<hr />
<div id="simple-logistic-regression-model"
class="section level2 tabset tabset-pills tabset-fade">
<h2 class="tabset tabset-pills tabset-fade">Simple Logistic Regression
Model</h2>
<div style="float:left;width:125px;" align="center">
<p><img src="Images/BinomYQuantX.png" width=58px;></p>
</div>
<p>Regression for a qualitative binary response variable <span
class="math inline">\((Y_i = 0\)</span> or <span
class="math inline">\(1)\)</span> using a single (typically
quantitative) explanatory variable.</p>
<div id="overview" class="section level3">
<h3>Overview</h3>
<div style="padding-left:125px;">
<p>The probability that <span class="math inline">\(Y_i = 1\)</span>
given the observed value of <span class="math inline">\(x_i\)</span> is
called <span class="math inline">\(\pi_i\)</span> and is modeled by the
equation</p>
<div
style="float:right;font-size:.8em;background-color:lightgray;padding:5px;border-radius:4px;">
<a style="color:darkgray;" href="javascript:showhide('simplelogisticlatexrcode')">Math
Code</a>
</div>
<div id="simplelogisticlatexrcode" style="display:none;">
<pre><code>$$
P(Y_i = 1|\, x_i) = \frac{e^{\beta_0 + \beta_1 x_i}}{1+e^{\beta_0 + \beta_1 x_i}} = \pi_i
$$</code></pre>
</div>
<center>
<span class="tooltipr"> <span class="math inline">\(P(\)</span> <span
class="tooltiprtext">The “P” stands for “Probability that…”</span>
</span><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">\(= 1\)</span>
<span class="tooltiprtext">Equals 1… This states that we are assuming
that the probability that the response variable <span
class="math inline">\(Y_i\)</span> is a 1 for the current
individual.</span> </span><span class="tooltipr"> <span
class="math inline">\(| x_i)\)</span> <span class="tooltiprtext">Given
<span class="math inline">\(x_i\)</span>… in other words, the “|” says
“given” and <span class="math inline">\(x_i\)</span> means the x-value
of the current individual.</span> </span><span class="tooltipr"> <span
class="math inline">\(=\)</span> <span class="tooltiprtext">Equals
sign.</span> </span><span class="tooltipr"> <span
class="math inline">\(\displaystyle\frac{e^{\beta_0 + \beta_1 x_i}}{1 +
e^{\beta_0 + \beta_1 x_i}}\)</span> <span class="tooltiprtext">The
logistic regression equation where <span
class="math inline">\(e=2.71828...\)</span> is the “natural constant”
number and <span class="math inline">\(\beta_0\)</span> is the
y-intercept and <span class="math inline">\(\beta_1\)</span> is teh
slope.</span> </span><span class="tooltipr"> <span
class="math inline">\(= \pi_i\)</span> <span class="tooltiprtext">The
<span class="math inline">\(\pi_i\)</span> stands for the probability of
individual <span class="math inline">\(i\)</span> having a y-value equal
to 1 given their <span class="math inline">\(x_i\)</span> value. It is
the short hand notation for <span class="math inline">\(P(Y_i = 1
|x_i)\)</span>. (It is NOT the number 3.14…)</span> </span>
</center>
<p><br/></p>
<p>The coefficents <span class="math inline">\(\beta_0\)</span> and
<span class="math inline">\(\beta_1\)</span> are difficult to interpret
directly. Typicall <span class="math inline">\(e^{\beta_0}\)</span> and
<span class="math inline">\(e^{\beta_1}\)</span> are interpreted
instead. The value of <span class="math inline">\(e^{\beta_0}\)</span>
or <span class="math inline">\(e^{\beta_1}\)</span> denotes the relative
change in the odds that <span class="math inline">\(Y_i=1\)</span>. The
odds that <span class="math inline">\(Y_i=1\)</span> are <span
class="math inline">\(\frac{\pi_i}{1-\pi_i}\)</span>.</p>
<hr />
<p><strong>Examples:</strong> <a
href="./Analyses/Logistic%20Regression/Examples/challengerLogisticReg.html">challenger</a>
| <a
href="./Analyses/Logistic%20Regression/Examples/mouseLogisticReg.html">mouse</a></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>?glm()</code></p>
<div id="perform-a-logistic-regression" class="section level4">
<h4>Perform a Logistic Regression</h4>
<a href="javascript:showhide('logistic1')">
<div class="hoverchunk">
<p><span class="tooltipr"> YourGlmName <span class="tooltiprtext">This
is some name you come up with that will become the R object that stores
the results of your logistic regression <code>glm()</code>
command.</span> </span><span class="tooltipr"> <- <span
class="tooltiprtext">This is the “left arrow” assignment operator that
stores the results of your <code>glm()</code> code into
<code>YourGlmName</code>.</span> </span><span class="tooltipr"> glm(
<span class="tooltiprtext">glm( is an R function that stands for
“General Linear Model”. It works in a similar way that the
<code>lm(</code> function works except that it requires a
<code>family=</code> option to be specified at the end of the
command.</span> </span><span class="tooltipr"> Y <span
class="tooltiprtext">Y is your binary response variable. It must consist
of only 0’s and 1’s. Since TRUE’s = 1’s and FALSE’s = 0’s in R, Y could
be a logical statement like (Price > 100) or (Animal == “Cat”) if
your Y-variable wasn’t currently coded as 0’s and 1’s.</span>
</span><span class="tooltipr"> ~ <span class="tooltiprtext">The tilde
symbol ~ is used to tell R that Y should be treated as a function of the
explanatory variable X.</span> </span><span class="tooltipr"> X, <span
class="tooltiprtext">X is the explanatory variable (typically
quantitative) that will be used to explain the probability that the
response variable Y is a 1.</span> </span><span class="tooltipr"> data
= NameOfYourDataset,<br />
<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 called Y and another column would be called X.</span> </span><span
class="tooltipr"> family=binomial) <span class="tooltiprtext">The
family=binomial command tells the <code>glm(</code> function to perform
a logistic regression. It turns out that <code>glm</code> can perform
many different types of regressions, but we only study it as a tool to
perform a logistic regression in this course.</span> </span><br/><span
class="tooltipr"> summary(YourGlmName) <span class="tooltiprtext">The
<code>summary</code> command allows you to print the results of your
logistic regression that were previously saved in
<code>YourGlmName</code>.</span> </span></p>
</div>
<p></a></p>
<div id="logistic1" 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/> glm(formula = am ~ disp, family =
binomial, data = mtcars) <span class="tooltiprouttext">This is simply a
statement of your original glm(…) “call” that you made when performing
your regression. It allows you to verify that you ran what you thought
you ran in the glm(…).</span> </span>
</td>
</tr>
</table>
<p><br/></p>
<table class="rconsole">
<tr>
<td colspan="2">
<span class="tooltiprout"> Deviance Residuals: <span
class="tooltiprouttext">Deviance residuals are a measure of how far the
fitted probability for <span class="math inline">\(\pi_i\)</span> has
differed from the actual outcome of <span
class="math inline">\(Y_i\)</span> in terms of the log of the fitted
probability space. (This is a fairly complicated idea.) </span>
</td>
</tr>
<tr>
<td align="right">
<span class="tooltiprout"> Min<br/> -1.5651 <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/> -0.6648 <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/> -0.2460 <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 -0.2460 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 (-0.2460) 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/> 0.7276 <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> 2.2691 <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 glm(…) 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 glm(…) 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"> z value <span class="tooltiprouttext">The
test statistic is a regular old z-score. It is most reliable when the
sample size is “large.” It is a measurement of the number of standard
errors the estimate is from 0.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Pr(>|z|) <span
class="tooltiprouttext">This is the p-value, the probability of
observing a test statistic more extreme than Z. </span> </span>
</td>
</tr>
<tr>
<td align="left">
<span class="tooltiprout"> (Intercept) <span
class="tooltiprouttext">This always says “Intercept” for any glm(…) 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"> 2.630849 <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 value of the log of the
odds that <span class="math inline">\(Y_i=1\)</span> when <span
class="math inline">\(x_i\)</span> is zero. Remember to use <span
class="math inline">\(e^{b_0}\)</span> to interpret this values actual
effect on the odds.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 1.050170 <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"> 2.505 <span class="tooltiprouttext">The test
statistic for testing the hypothesis that <span
class="math inline">\(\beta_0 = 0\)</span>.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.01224 <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 z-score as extreme as the one observed. To compute it
yourself in R, use <code>pnorm(-abs(your z-value))*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"> disp <span class="tooltiprouttext">This is
always the name of your X-variable in your glm(Y ~ X, …).</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> -0.014604 <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 log of the
odds that <span class="math inline">\(Y_i = 1\)</span> as X is increased
by 1 unit. Remember to use <span class="math inline">\(e^{b_1}\)</span>
to compute the actual effect on the odds.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.005168 <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"> -2.826 <span class="tooltiprouttext">This is
the test statistic for testing the hypothesis that <span
class="math inline">\(\beta_1 = 0\)</span>.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.00471 <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>pnorm(-abs(your z-value))*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 0.001 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"> (Dispersion parameter for binomial family
taken to be 1) <span class="tooltiprouttext">This is a simplifying
assumption of the logistic regression. Overdispersion is a common
problem with logistic regression data, but is typically ignored. Unless
you become an expert in statistics, this is not something you need to
worry about.</span> </span>
</td>
</tr>
</table>
<p><br/></p>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> Null Deviance: <span
class="tooltiprouttext">The deviance of the null model. This is the
model that excludes any information from the x-variable, i.e., <span
class="math inline">\(\beta_1=0\)</span>.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 43.230 <span class="tooltiprouttext"></span>
</span>
</td>
<td align="right">
<span class="tooltiprout"> on 31 degrees of freedom <span
class="tooltiprouttext">The residual degrees of freedom. The higher this
number, the more reliable the p-values will be from the logistic
regression.</span> </span>
</td>
</tr>
</table>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> Residual deviance: <span
class="tooltiprouttext">The sum of log of the squared residuals.
Essentially the resulting statistic of a goodness of fit test measuring
how well the data works with the logistic regression model. Using
pchisq(residual deviance, df residual deviance, lower.tail=FALSE) gives
the p-value for this goodness of fit test. However, the residual
deviance only follows a chi-squared distribution with df residual
deviance when there are many repeated x-values, and all x-values have at
least a few replicates.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 29.732 <span class="tooltiprouttext">This
can be calculated by <code>sum(log(myglm$res^2))</code>.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> on 30 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 32 and two
parameters, <span class="math inline">\(\beta_0\)</span> and <span
class="math inline">\(\beta_1\)</span>, so 32-2 = 30.</span> </span>
</td>
</tr>
</table>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> AIC: <span class="tooltiprouttext">As stated
in the R Help file for ?glm, “A version of Akaike’s An Information
Criterion…” The AIC is useful for comparing different models for the
same Y-variable. The glm model with the lowest AIC (which can go
negative) is the best model.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 33.732 <span class="tooltiprouttext">The AIC
for this particular model is 33.732. So if a different model (using the
same Y-variable as this model) can get a lower AIC, it is a better
model.</span> </span>
</td>
</tr>
</table>
<p><br/></p>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> Number of Fisher Scoring iterations: <span
class="tooltiprouttext">If you have taken a class in Numerical Analysis,
this tells you how many iterations of the maximization algorithm were
required before converging to the “Estimates” of the parameters <span
class="math inline">\(\beta_0\)</span> and <span
class="math inline">\(\beta_1\)</span> found in the summary.</span>
</span>
</td>
<td align="right">
<span class="tooltiprout"> 5 <span class="tooltiprouttext">This
implementation of glm required 5 Fisher Scoring iterations to converge.
Fewer iterations hints that the model is a better fit than when many
iterations are required.</span> </span>
</td>
</tr>
</table>
</div>
<p><br/></p>
</div>
<div id="diagnose-the-goodness-of-fit" class="section level4">
<h4>Diagnose the Goodness-of-Fit</h4>
<p>There are two ways to check the <strong>goodness of fit</strong> of a
logistic regression model.</p>
<div style="padding-left:25px;">
<p><strong>Option 1</strong>: Hosmer-Lemeshow Goodness-of-Fit Test (Most
Common)</p>
<p>To check the <strong>goodness of fit</strong> of a logistic
regression model where there are <strong>few or no replicated <span
class="math inline">\(x\)</span>-values</strong> use the Hosmer-Lemeshow
Test.</p>
<a href="javascript:showhide('goodnessoffit2')">
<div class="hoverchunk">
<p><span class="tooltipr"> library(ResourceSelection) <span
class="tooltiprtext">This loads the ResourceSelection R package so that
you can access the hoslem.test() function. You may need to run the code:
install.packages(“ResourceSelection”) first.</span> </span><br/><span
class="tooltipr"> hoslem.test( <span class="tooltiprtext">This R
function performs the Hosmer-Lemeshow Goodness of Fit Test. See the
“Explanation” file to learn about this test.</span> </span><span
class="tooltipr"> YourGlmName <span
class="tooltiprtext"><code>YourGlmName</code> is the name of your glm(…)
code that you created previously.</span> </span><span class="tooltipr">
$y, <span class="tooltiprtext">ALWAYS type a “y” here. This gives you
the actual binary (0,1) y-values of your logistic regression. The
goodness of fit test will compare these actual values to your predicted
probabilities for each value in order to see if the model is a “good
fit.”</span> </span><span class="tooltipr"> YourGlmName <span
class="tooltiprtext"><code>YourGlmName</code> is the name you used to
save the results of your glm(…) code.</span> </span><span
class="tooltipr"> $fitted, <span class="tooltiprtext">ALWAYS type
“fitted” here. This gives you the fitted probabilities <span
class="math inline">\(\pi_i\)</span> of your logistic regression.</span>
</span><span class="tooltipr"> g=10) <span class="tooltiprtext">The
“g=10” is the default option for the value of g. The g is the number of
groups to run the goodness of fit test on. Just leave it at 10 unless
you are told to do otherwise. Ask your teacher for more information if
you are interested.</span> </span></p>
</div>
<p></a></p>
<div id="goodnessoffit2" style="display:none;">
<pre><code>##
## Hosmer and Lemeshow goodness of fit (GOF) test
##
## data: myglm$y, myglm$fitted
## X-squared = 5.7327, df = 8, p-value = 0.6771</code></pre>
<p>Note that the null hypothesis of the goodness-of-fit test is that
“the logistic regression is a good fit.” So we actually don’t want to
“reject the null” in this case. So a large p-value here means our
logistic regression fits the data satisfactorily. A small p-value
implies a poor fit and the results of the logistic regression should not
be fully trusted.</p>
</div>
<p><br/></p>
<p><strong>Option 2</strong>: Deviance Goodness-of-fit Test (Less
Common)</p>
<p>In some cases, there are <strong>many replicated <span
class="math inline">\(x\)</span>-values</strong> for
<strong>all</strong> x-values, i.e., each value of x is repeated more
than 50 times. Though this is rare, it is good to use the <em>deviance
goodness-of-fit test</em> whenever this happens.</p>
<a href="javascript:showhide('goodnessoffit1')">
<div class="hoverchunk">
<p><span class="tooltipr"> pchisq( <span class="tooltiprtext">The
<code>pchisq</code> command allows you to compute p-values from the
chi-squared distribution.</span> </span><span class="tooltipr"> residual
deviance, <span class="tooltiprtext">The residual deviance is shown at
the bottom of the output of your <code>summary(YourGlmName)</code> and
should be typed in here as a number like 25.3.</span> </span><span
class="tooltipr"> df for residual deviance, <span
class="tooltiprtext">The df for the residual deviance is also shown at
the bottom of the output of your
<code>summary(YourGlmName)</code>.</span> </span><span class="tooltipr">
lower.tail=FALSE) <span class="tooltiprtext">This command ensures you
find the probability of the chi-squared distribution being as extreme or
more extreme than the observed value of residual deviance.</span>