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<p>Association testing for two qualitative variables with at least two
levels to each variable.</p>
<hr />
<div id="chi-squared-test-of-independence"
class="section level3 tabset tabset-pills tabset-fade">
<h3 class="tabset tabset-pills tabset-fade">Chi-squared Test of
Independence</h3>
<table>
<tr>
<td style="vertical-align:top;width:20%;">
<div style="float:left;width:125px;" align="center">
<p><img src="Images/QualYQualX.png" width=58px;></p>
</div>
</td>
<td style="vertical-align:top;width:80%;">
<div id="overview" class="section level4">
<h4>Overview</h4>
<p>A method of comparing observed counts to the expected counts in a
contingency table to decide if two qualitative variables are associated
(alternative hypothesis) or are independent, i.e., not associated (null
hypothesis).</p>
<p>Typically, all expected counts are required to be 5 or greater for
the test to be appropriate. However, the test is still appropriate if
all expected counts are at least 1 and the average of the expected
counts is at least 5.</p>
<hr />
</div>
<div id="r-instructions" class="section level4">
<h4>R Instructions</h4>
<p><strong>Console</strong> Help Command: <code>?chisq.test()</code></p>
<div id="to-run-the-test" class="section level5">
<h5>To Run the Test</h5>
<p><code>chisq.test(x)</code></p>
<p><code>x</code> must be a table or a matrix.</p>
<div id="to-make-a-table" class="section level6">
<h6>To Make a Table</h6>
<p>If you have a data set that you want to turn into a table, use the
table command.</p>
<a href="javascript:showhide('table')">
<div class="hoverchunk">
<p><span class="tooltipr"> x <span class="tooltiprtext">The name of your
created table to go in the <code>chisq.test()</code> code.</span>
</span><span class="tooltipr"> <- <span class="tooltiprtext">This
is the “left arrow” assignment operator that stores the results of your
<code>table()</code> code into <code>x</code>.</span> </span><span
class="tooltipr"> table( <span class="tooltiprtext">table is an R
function used to tabulate how many times each value occrus in a given
dataset.</span> </span><span class="tooltipr"> Dataset$FirstVariable,
<span class="tooltiprtext">Use your dataset to call in a row
variable.</span> </span><span class="tooltipr"> Dataset$SecondVariable)
<span class="tooltiprtext">Use your dataset to call in a column
variable.</span> </span></p>
</div>
<p></a></p>
<div id="table" style="display:none;">
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>x <span class="ot"><-</span> <span class="fu">table</span>(mtcars<span class="sc">$</span>am, mtcars<span class="sc">$</span>cyl)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="fu">pander</span>(x)</span></code></pre></div>
<table style="width:31%;">
<colgroup>
<col width="12%" />
<col width="5%" />
<col width="5%" />
<col width="6%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">4</th>
<th align="center">6</th>
<th align="center">8</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>0</strong></td>
<td align="center">3</td>
<td align="center">4</td>
<td align="center">12</td>
</tr>
<tr class="even">
<td align="center"><strong>1</strong></td>
<td align="center">8</td>
<td align="center">3</td>
<td align="center">2</td>
</tr>
</tbody>
</table>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co">#Rename rows and columns of your table:</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="fu">rownames</span>(x) <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"Automatic"</span>, <span class="st">"Manual"</span>)</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="fu">colnames</span>(x) <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"4 cyl"</span>, <span class="st">"6 cyl"</span>, <span class="st">"8 cyl"</span>)</span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="fu">pander</span>(x)</span></code></pre></div>
<table style="width:56%;">
<colgroup>
<col width="22%" />
<col width="11%" />
<col width="11%" />
<col width="11%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">4 cyl</th>
<th align="center">6 cyl</th>
<th align="center">8 cyl</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Automatic</strong></td>
<td align="center">3</td>
<td align="center">4</td>
<td align="center">12</td>
</tr>
<tr class="even">
<td align="center"><strong>Manual</strong></td>
<td align="center">8</td>
<td align="center">3</td>
<td align="center">2</td>
</tr>
</tbody>
</table>
</div>
<p><span style="font-size:.8em;"><em>Click open the example code above
for details on how to rename rows and columns in a
table.</em></span></p>
<p><br/></p>
</div>
<div id="to-make-a-matrix" class="section level6">
<h6>To Make a Matrix</h6>
<p>If you have tally counts of your observations, enter them into
RStudio using a matrix.</p>
<a href="javascript:showhide('matrix1')">
<div class="hoverchunk">
<p><span class="tooltipr"> x <span class="tooltiprtext">The name of your
created matrix to go in the <code>chisq.test()</code> code.</span>
</span><span class="tooltipr"> <- <span class="tooltiprtext">This
is the “left arrow” assignment operator that stores the results of your
<code>cbind()</code> code into <code>x</code>.</span> </span><span
class="tooltipr"> cbind( <span class="tooltiprtext">cbind stands
for”column bind” and is a function that joins together different c()
vectors to make them become columns of a table.</span> </span><span
class="tooltipr"> `Farm 1` = <span class="tooltiprtext">The name of the
first column you want in the matrix. To include spaces in the name, use
the back-ticks ` and `.</span> </span><span class="tooltipr"> c(Pigs =
10, Cats = 5, Dogs = 28, Roosters = 3), <span
class="tooltiprtext">Specify each of the values by name in a c(…)
function to be stored in your first column, in this case,
<code>Farm 1</code>.</span> </span><span class="tooltipr"> `Farm 2` =
<span class="tooltiprtext">The name of the second column you want in the
matrix.</span> </span><span class="tooltipr"> c(Pigs = 15, Cats = 3,
Dogs = 8, Roosters = 1) <span class="tooltiprtext">Each new column must
contain entries for each named element. Technically, you can leave off
the names in all but the first column as only the names from the first
column will be used.</span> </span><br/><span class="tooltipr">
pander(x) <span class="tooltiprtext">Print the matrix to the screen
using pander(…).</span> </span></p>
</div>
<p></a></p>
<div id="matrix1" style="display:none;">
<table style="width:46%;">
<colgroup>
<col width="20%" />
<col width="12%" />
<col width="12%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Farm 1</th>
<th align="center">Farm 2</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Pigs</strong></td>
<td align="center">10</td>
<td align="center">15</td>
</tr>
<tr class="even">
<td align="center"><strong>Cats</strong></td>
<td align="center">5</td>
<td align="center">3</td>
</tr>
<tr class="odd">
<td align="center"><strong>Dogs</strong></td>
<td align="center">28</td>
<td align="center">8</td>
</tr>
<tr class="even">
<td align="center"><strong>Roosters</strong></td>
<td align="center">3</td>
<td align="center">1</td>
</tr>
</tbody>
</table>
</div>
<p>Or, notice how you could rbind instead with slightly different
results…</p>
<a href="javascript:showhide('matrix2')">
<div class="hoverchunk">
<p><span class="tooltipr"> x <span class="tooltiprtext">The name of your
created matrix to go in the <code>chisq.test()</code> code.</span>
</span><span class="tooltipr"> <- <span class="tooltiprtext">This
is the “left arrow” assignment operator that stores the results of your
<code>cbind()</code> code into <code>x</code>.</span> </span><span
class="tooltipr"> rbind( <span class="tooltiprtext">rbind stands for”row
bind” and is a function that joins together different c() vectors to
make them become rows of a table.</span> </span><span class="tooltipr">
`Farm 1` = <span class="tooltiprtext">The name of the first column you
want in the matrix. To include spaces in the name, use the back-ticks `
and `.</span> </span><span class="tooltipr"> c(Pigs = 10, Cats = 5,
Dogs = 28, Roosters = 3), <span class="tooltiprtext">Specify each of the
values by name in a c(…) function to be stored in your first column, in
this case, <code>Farm 1</code>.</span> </span><span class="tooltipr">
`Farm 2` = <span class="tooltiprtext">The name of the second column you
want in the matrix.</span> </span><span class="tooltipr"> c(Pigs = 15,
Cats = 3, Dogs = 8, Roosters = 1) <span class="tooltiprtext">Each new
column must contain entries for each named element. Technically, you can
leave off the names in all but the first column as only the names from
the first column will be used.</span> </span><br/><span
class="tooltipr"> pander(x) <span class="tooltiprtext">Print the matrix
to the screen using pander(…).</span> </span></p>
</div>
<p></a></p>
<div id="matrix2" style="display:none;">
<table style="width:62%;">
<colgroup>
<col width="18%" />
<col width="9%" />
<col width="9%" />
<col width="9%" />
<col width="15%" />
</colgroup>
<thead>
<tr class="header">
<th align="center"> </th>
<th align="center">Pigs</th>
<th align="center">Cats</th>
<th align="center">Dogs</th>
<th align="center">Roosters</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="center"><strong>Farm 1</strong></td>
<td align="center">10</td>
<td align="center">5</td>
<td align="center">28</td>
<td align="center">3</td>
</tr>
<tr class="even">
<td align="center"><strong>Farm 2</strong></td>
<td align="center">15</td>
<td align="center">3</td>
<td align="center">8</td>
<td align="center">1</td>
</tr>
</tbody>
</table>
</div>
<p><br/></p>
</div>
</div>
<div id="diagnostics" class="section level5">
<h5>Diagnostics</h5>
<a href="javascript:showhide('chisq')">
<div class="hoverchunk">
<p><span class="tooltipr"> yourNamedTestResults <span
class="tooltiprtext">This is some name you come up with that will become
the R object that stores the results of your Chi-squated test
<code>chisq.test()</code> command.</span> </span><span class="tooltipr">
<- <span class="tooltiprtext">This is the “left arrow” assignment
operator that stores the results of your <code>chisq.test()</code> code
into <code>yourNamedTestResults</code>.</span> </span><span
class="tooltipr"> chisq.test( <span
class="tooltiprtext"><code>chisq.test()</code> is an R function that
stands for “Chi-Squared Test”. It performs the Chi-squared test for
<code>x</code>.</span> </span><span class="tooltipr"> x) <span
class="tooltiprtext"><code>x</code> is either a matrix or a
table.</span> </span></p>
</div>
<p></a></p>
<div id="chisq" style="display:none;">
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> Pearson’s Chi-squared test <span
class="tooltiprouttext">The name of the test being performed.</span>
</span>
</td>
</tr>
</table>
<p><br/></p>
<table class="rconsole">
<tr>
<td>
<span class="tooltiprout"> data: x <span class="tooltiprouttext">The
data you are using for the Chi-Squared test. <code>x</code> is either a
table or matrix that you created.</span>
</td>
<td>
<span class="tooltiprout"> X-squared = 9.2956, <span
class="tooltiprouttext">The test statistic of the test follows a
chi-squared distribution with the given degrees of freedom. See the end
of the “Making Inference” page of the book to learn more about the
Chi-squared Parametric Distribution.</span>
</td>
<td>
<span class="tooltiprout"> df = 3, <span class="tooltiprouttext">The
degrees of freedom of the test. It is found by (number of rows -
1)x(number of columns - 1)</span>
</td>
<td>
<span class="tooltiprout"> p-value = 0.02561 <span
class="tooltiprouttext">The p-value of the test is found by computing
the probability of being more extreme than given test statistic. In R,
you can compute this yourself using the code pchisq(9.2956, 3,
lower.tail=FALSE).</span>
</td>
</tr>
</table>
</div>
<a href="javascript:showhide('expected')">
<div class="hoverchunk">
<p><span class="tooltipr"> yourNamedTestResults <span
class="tooltiprtext">This is some name you come up with that will become
the R object that stores the results of your Chi-squared test
<code>chisq.test()</code> command.</span> </span><span class="tooltipr">
$ <span class="tooltiprtext">This allows you to access various elements
from the test that was performed.</span> </span><span class="tooltipr">
expected <span class="tooltiprtext">The expected counts of the
chi-squared test. These are computed for each observed count by using
the equation (row tota)*(column total)/(total total).</span> </span></p>
</div>
<p></a></p>
<div id="expected" style="display:none;">
<table class="rconsole">
<tr>
<td align="left">
</td>
<td align="right">
<span class="tooltiprout"> Pigs <span
class="tooltiprouttext">EXPLANATION.</span>
</td>
<td align="right">
<span class="tooltiprout"> Cats <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Dogs <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Roosters <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<tr>
<td align="left">
<span class="tooltiprout"> Farm 1 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 15.753425 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 5.041096 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 22.68493 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 2.520548 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
</td>
</tr>
<tr>
<td align="left">
<span class="tooltiprout"> Farm 2 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 9.246575 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 2.958904 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 13.31507 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 1.479452 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
</td>
</tr>
</table>
</div>
<p><br/></p>
</div>
<div id="interpretation-pearson-residuals" class="section level5">
<h5>Interpretation: Pearson Residuals</h5>
<a href="javascript:showhide('residuals')">
<div class="hoverchunk">
<p><span class="tooltipr"> yourNamedTestResults <span
class="tooltiprtext">This is some name you come up with that will become
the R object that stores the results of your Chi-squated test
<code>chisq.test()</code> command.</span> </span><span class="tooltipr">
$ <span class="tooltiprtext">This allows you to access various elements
from the test that was performed.</span> </span><span class="tooltipr">
residuals <span class="tooltiprtext">This grabs the residuals for each
observation in the Chi-squared test.</span> </span></p>
</div>
<p></a></p>
<div id="residuals" style="display:none;">
<table class="rconsole">
<tr>
<td align="left">
</td>
<td align="right">
<span class="tooltiprout"> Pigs <span
class="tooltiprouttext">EXPLANATION.</span>
</td>
<td align="right">
<span class="tooltiprout"> Cats <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Dogs <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> Roosters <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<tr>
<td align="left">
<span class="tooltiprout"> Farm 1 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> -1.449569 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> -0.01830357 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 1.115938 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.3019936 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
</td>
</tr>
<tr>
<td align="left">
<span class="tooltiprout"> Farm 2 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 1.892065 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> 0.02389092 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> -1.456589 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
<td align="right">
<span class="tooltiprout"> -0.3941801 <span
class="tooltiprouttext">EXPLANATION.</span> </span>
</td>
</td>
</tr>
</table>
</div>
<hr />
</div>
</div>
<div id="explanation" class="section level4">
<h4>Explanation</h4>
<p>To demonstrate the theory behind the Chi-squared Test, consider the
following data about the survival of the Titanic passengers as a
specific example. The rows of the following contingency table (a table
of counts) show the <strong>Class</strong> of the passenger, while the
columns show the <strong>Survival</strong> of the passenger. Note that
each passenger is placed in exactly one <strong>Class</strong> and one
level of <strong>Survival</strong>. Also, the two factors,
<strong>Class</strong> and <strong>Survival</strong>, are qualitative
variables with at least two levels. Thus, the data meets the basic
requirements of the Chi-squared Test.</p>
<div style="padding-left:15%; padding-right:15%;">
<table>
<thead>
<tr class="header">
<th align="left"></th>
<th align="right">No</th>
<th align="right">Yes</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">1st</td>
<td align="right">122</td>
<td align="right">203</td>
</tr>
<tr class="even">
<td align="left">2nd</td>
<td align="right">167</td>
<td align="right">118</td>
</tr>
<tr class="odd">
<td align="left">3rd</td>
<td align="right">528</td>