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<p>A nonparametric approach to computing the p-value for any test
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<div id="section"
class="section level3 tabset tabset-pills tabset-fade">
<h3 class="tabset tabset-pills tabset-fade"></h3>
<div id="overview" class="section level4">
<h4>Overview</h4>
<div style="padding-left:30px;">
<p>In almost all hypothesis testing scenarios, the null hypothesis can
be interpreted as follows.</p>
<div style="padding-left:15px;padding-right:15px;">
<p><span class="math inline">\(H_0\)</span>: Any pattern that has been
witnessed in the sampled data is simply due to random chance.</p>
</div>
<p>Permutation Tests depend completely on this single idea. If all
patterns in the data really are simply due to random chance, then the
null hypothesis is true. Further, random <strong>re</strong>-samples of
the data should show similar lack of patterns. However, if the pattern
in the data is <em>real</em>, then random <strong>re</strong>-samples of
the data will show very different patterns from the original.</p>
<p>Consider the following image. In that image, the toy blocks on the
left show a clear pattern or structure. They are nicely organized into
colored piles. This suggests a <em>real</em> pattern that is not random.
Someone certainly organized those blocks into that pattern. The blocks
didn’t land that way by random chance. On the other hand, the pile of
toy blocks shown on the right is certainly a random pattern. This is a
pattern that would result if the toy blocks were put into a bag, shaken
up, and dumped out. This is the idea of the permutation test. If there
is structure in the data, then “mixing up the data and dumping it out
again” will show very different patterns from the original. However, if
the data was just random to begin with, then we would see a similar
pattern by “mixing up the data and dumping it out again.”</p>
<center>
<img src="Images/legoPermutationExample.png" width=600px;>
</center>
<p>The process of a permutation test is:</p>
<ol style="list-style-type: decimal">
<li>Compute a test statistic for the original data.</li>
<li>Re-sample the data (“shake it up and dump it out”) thousands of
times, computing a new test statistic each time, to create a sampling
distribution of the test statistic.</li>
<li>Compute the p-value of the permutation test as the percentage of
test statistics that are as extreme or more extreme than the one
originally observed.</li>
</ol>
<p>In review, the sampling distribution is created by permuting
(randomly rearranging) the data thousands of times and calculating a
test statistic on each permuted version of the data. A histogram of the
test statistics then provides the sampling distribution of the test
statistic needed to compute the p-value of the original test
statistic.</p>
</div>
<hr />
</div>
<div id="r-instructions" class="section level4">
<h4>R Instructions</h4>
<div style="padding-left:50px;">
<p>Any permutation test can be performed in R with a <code>for</code>
loop.</p>
<div style="padding-left:80px;">
<a href="javascript:showhide('perm1')">
<div class="hoverchunk">
<p><span class="tooltipr"> #Step 1 <span class="tooltiprtext">Compute a
test statistic for the original data.</span> </span><br/><span
class="tooltipr"> myTest <- …perform the initial test… <span
class="tooltiprtext">This could be a <code>t.test</code>,
<code>wilcox.test</code>, <code>aov</code>, <code>kruskal.test</code>,
<code>lm</code>, <code>glm</code>, <code>chisq.test</code>, or any other
R code that results in a test statistic. It could even simply be the
mean or standard deviation of the data.</span> </span><br/><span
class="tooltipr"> observedTestStat <- …get the test statistic… <span
class="tooltiprtext">Save the test statistic of your test into the
object called <code>observedTestStat</code>. For tests that always
result in a single test statistic like a <code>t.test</code>,
<code>wilcox.test</code>, <code>kruskal.test</code>, and
<code>chisq.test</code> it is <code>myTest$statistic</code>. For an
<code>aov</code>, <code>lm</code>, or <code>glm</code> try printing
<code>summary(myTest)[]</code> to see what values you are interested in
using.</span> </span><br/><span class="tooltipr"> observedTestStat <span
class="tooltiprtext">Print the value of the test statistic of your test
to the screen. This is the value that we now need to use to compute the
P-value from by finding the probability that a randomly computed test
statistic would be “more extreme” than this originally observed
value.</span> </span><br/><br/><span class="tooltipr"> #Step 2 <span
class="tooltiprtext">Re-sample the data (“shake it up and dump it out”)
thousands of times, computing a new test statistic each time, to create
a sampling distribution of the test statistic.</span> </span><br/><span
class="tooltipr"> N <- 2000 <span class="tooltiprtext">N is the
number of times you will reuse the data to create the sampling
distribution of the test statistic. A typical choice is 2000, but
sometimes 10000, or 100000 reuses are needed before useful answers can
be obtained.</span> </span><br/><span class="tooltipr">
permutedTestStats <- <span class="tooltiprtext">This is a storage
container that will be used to store the test statistics from each of
the thousands of reuses of the data.</span> </span><span
class="tooltipr"> rep(NA, N) <span class="tooltiprtext">The rep()
function repeats a given value N times. This particular statement
repeats NA’s or “missing values” N times. This gives us N “empty”
storage spots inside of <code>permutedTestStats</code> that we can use
to store the N test statistics from the N reuses of the data we will
make in our <code>for</code> loop.</span> </span><br/><span
class="tooltipr"> for <span class="tooltiprtext">The <code>for</code>
loop is a programming tool that lets us tell R to run a certain code
over and over again for a certain number of times.</span> </span><span
class="tooltipr"> (i in <span class="tooltiprtext">In R, the
<code>for</code> loop must be followed by a space, then an opening
parenthesis, then a variable name (in this case the variable is called
“i”), then the word “in” then a list of values.</span> </span><span
class="tooltipr"> 1:N <span class="tooltiprtext">The 1:N gives R the
list of values 1, 2, 3, … and so on all the way up to N. These values
are passed into <code>i</code> one at a time and the code inside the
<code>for</code> loop is performed first for <code>i=1</code>, then
again for <code>i=2</code>, then again for <code>i=3</code> and so on
until finally <code>i=N</code>. At that point, the <code>for</code> loop
ends.</span> </span><span class="tooltipr"> ) <span
class="tooltiprtext">Required closing parenthesis on the
<code>for (i in 1:N)</code> statement.</span> </span><span
class="tooltipr"> { <span class="tooltiprtext">This bracket opens the
code section of the <code>for</code> loop. Any code placed between the
opening { and closing } brackets will be performed over and over again
for each value of <code>i=1</code>, <code>i=2</code>, … up through
<code>i=N</code>.</span> </span><br/><span class="tooltipr"> <span
class="tooltiprtext">Two spaces in front of every line inside of the
opening { and closing } brackets helps keep your code organized.</span>
</span><span class="tooltipr"> permutedTest <- …perform test with
permutedData… <span class="tooltiprtext">The same test that was
performed on the original data, should be performed again but with
randomly permuted data. The easiest way to permute data is with
sample(y-variable-name) inside your test. See the Explanation tab for
details.</span> </span><br/><span class="tooltipr"> <span
class="tooltiprtext">Two spaces in front of every line inside of the
opening { and closing } brackets helps keep your code organized.</span>
</span><span class="tooltipr"> permutedTestStats <span
class="tooltiprtext">This is the storage container that was built prior
to the <code>for (i in 1:N)</code> code. Inside the <code>for</code>
loop, this container is filled value by value using the square brackets
<code>[i]</code>.</span> </span><span class="tooltipr"> [i] <span
class="tooltiprtext">The square brackets <code>[i]</code> allows us to
access the “i”th position of <code>permutedTestStats</code>. Remember,
since this code is inside of the <code>for</code> loop, <code>i=1</code>
the first time through the code, then <code>i=2</code> the second time
through the code, <code>i=3</code> the third time through, and so on up
until <code>i=N</code> the <code>N</code>th time through the
code.</span> </span><span class="tooltipr"> <- …get test statistic…
<span class="tooltiprtext">The test statistic from
<code>permutedTest</code> is accessed here and stored into
<code>permutedTestStats[i]</code>.</span> </span><br/><span
class="tooltipr"> } <span class="tooltiprtext">The closing } bracket
ends the code that is repeated over and over again inside the
<code>for</code> loop.</span> </span><br/><span class="tooltipr">
hist(permutedTestStats) <span class="tooltiprtext">Creating a histogram
of the sampling distribution of the test statistics obtained from the
reused and permuted data allows us to visually compare the
<code>observedTestStat</code> to the distribution of test statistics to
visually see the percentage of test statistics that are as extreme or
more extreme than the observed test statistic value. This is the
p-value.</span> </span><br/><span class="tooltipr">
abline(v=observedTestStat) <span class="tooltiprtext">This adds the
<code>observedTestStat</code> to the distribution of test statistics to
visually see the percentage of test statistics that are as extreme or
more extreme than the observed test statistic value. This is the
p-value.</span> </span><br/><br/><span class="tooltipr"> #Step 3 <span
class="tooltiprtext">Compute the p-value of the permutation test as the
percentage of test statistics that are as extreme or more extreme than
the one originally observed.</span> </span><br/><span class="tooltipr">
sum(permutedTestStats >= observedTestStat)/N <span
class="tooltiprtext">This computes a “greater than” p-value. A two-sided
p-value could be obtained by multiplying this value by 2 if the observed
test statistic was on the right hand side of the histogram.</span>
</span><br/><span class="tooltipr"> sum(permutedTestStats <=
observedTestStat)/N <span class="tooltiprtext">This computes a “less
than” p-value. A two-sided p-value could be obtained by multiplying this
value by 2 if the observed test statistic was on the left hand side of
the histogram.</span> </span></p>
</div>
<p></a></p>
<div id="perm1" style="display:none;">
<ul>
<li><p><code>myTest <- ...perform the initial test...</code></p>
<p>This could be a <code>t.test</code>, <code>wilcox.test</code>,
<code>aov</code>, <code>kruskal.test</code>, <code>lm</code>,
<code>glm</code>, <code>chisq.test</code>, or any other R code that
results in a test statistic. It could even simply be the mean or
standard deviation of the data.</p></li>
<li><p><code>observedTestStat <- ...get the test statistic...</code></p>
<p>Save the test statistic of your test into the object called
<code>observedTestStat</code>. For tests that always result in a single
test statistic like a <code>t.test</code>, <code>wilcox.test</code>,
<code>kruskal.test</code>, and <code>chisq.test</code> it is
<code>myTest$statistic</code>. For an <code>aov</code>, <code>lm</code>,
or <code>glm</code> try printing <code>summary(myTest)[]</code> to see
what values you are interested in using.</p></li>
<li><p><code>N <- 2000</code> N is the number of times you will reuse
the data to create the sampling distribution of the test statistic. A
typical choice is 2000, but sometimes 10000, or 100000 reuses are needed
before useful answers can be obtained.</p></li>
<li><p><code>permutedTestStats <- rep(NA, N)</code> This is a storage
container that will be used to store the test statistics from each of
the thousands of reuses of the data. The <code>rep()</code> function
repeats a given value N times. This particular statement repeats NA’s or
“missing values” N times. This gives us N “empty” storage spots inside
of <code>permutedTestStats</code> that we can use to store the N test
statistics from the N reuses of the data we will make in our
<code>for</code> loop.</p></li>
<li><p><code>for (i in 1:N)\{</code> The <code>for</code> loop is a
programming tool that lets us tell R to run a certain code over and over
again for a certain number of times. In R, the <code>for</code> loop
must be followed by a space, then an opening parenthesis, then a
variable name (in this case the variable is called “i”), then the word
“in” then a list of values. The <code>1:N</code> gives R the list of
values 1, 2, 3, … and so on all the way up to N. These values are passed
into <code>i</code> one at a time and the code inside the
<code>for</code> loop is performed first for <code>i=1</code>, then
again for <code>i=2</code>, then again for <code>i=3</code> and so on
until finally <code>i=N</code>. At that point, the <code>for</code> loop
ends. There is a required closing parenthesis on the
<code>for (i in 1:N)</code> statement. Any code placed between the
opening { and closing } brackets will be performed over and over again
for each value of <code>i=1</code>, <code>i=2</code>, … up through
<code>i=N</code>.</p></li>
<li><p>Two spaces in front of every line inside of the opening { and
closing } brackets helps keep your code organized.</p></li>
<li><p><code>permutedData <- ...randomly permute the data...</code>
This is the most important part of the permutation test and takes some
thinking. The data must be randomly reorganized in a way consistent with
the null hypothesis. What that means exactly is specific to each
scenario. Read the Explanation tab for further details on the logic you
should use here.</p></li>
<li><p><code>permutedTest <- ...perform test with permutedData...</code>
The same test that was performed on the original data, should be
performed again on the randomly permuted data.</p></li>
<li><p><code>permutedTestStats[i] <- ...get test statistic...</code>
This is the storage container that was built prior to the
<code>for (i in 1:N)</code> code. Inside the <code>for</code> loop, this
container is filled value by value using the square brackets
<code>[i]</code>. The square brackets <code>[i]</code> allows us to
access the “i”th position of <code>permutedTestStats</code>. Remember,
since this code is inside of the <code>for</code> loop, <code>i=1</code>
the first time through the code, then <code>i=2</code> the second time
through the code, <code>i=3</code> the third time through, and so on up
until <code>i=N</code> the <code>N</code>th time through the code. The
test statistic from <code>permutedTest</code> is accessed here and
stored into <code>permutedTestStats[i]</code>.</p></li>
<li><p><code>}</code> The closing } bracket ends the code that is
repeated over and over again inside the <code>for</code> loop.</p></li>
<li><p><code>hist(permutedTestStats)</code> Creating a histogram of the
sampling distribution of the test statistics obtained from the reused
and permuted data allows us to visually compare the
<code>observedTestStat</code> to the distribution of test statistics to
visually see the percentage of test statistics that are as extreme or
more extreme than the observed test statistic value. This is the
p-value.</p></li>
<li><p><code>abline(v=observedTestStat)</code> This adds the
<code>observedTestStat</code> to the distribution of test statistics to
visually see the percentage of test statistics that are as extreme or
more extreme than the observed test statistic value. This is the
p-value.</p></li>
<li><p><code>sum(permutedTestStats >= observedTestStat)/N</code> This
computes a “greater than” p-value. A two-sided p-value could be obtained
by multiplying this value by 2 if the observed test statistic was on the
right hand side of the histogram.</p></li>
<li><p><code>sum(permutedTestStats <= observedTestStat)/N</code> This
computes a “less than” p-value. A two-sided p-value could be obtained by
multiplying this value by 2 if the observed test statistic was on the
left hand side of the histogram.</p></li>
</ul>
</div>
</div>
</div>
<hr />
</div>
<div id="explanation" class="section level4">
<h4>Explanation</h4>
<div style="padding-left:30px;">
<p>The most difficult part of a permutation test is in the random
permuting of the data. How the permuting is performed depends on the
type of hypothesis test being performed. It is important to remember
that the permutation test only changes the way the p-value is
calculated. Everything else about the original test is unchanged when
switching to a permutation test.</p>
<div id="independent-samples-t-test" class="section level5">
<h5>Independent Samples t Test</h5>
<p>For the independent sample t Test, we will use the data from the <a
href="Analyses/t%20Tests/Examples/SleepIndependentt.html">independent
sleep</a> analysis. In that analysis, we were using the
<code>sleep</code> data to test the hypotheses:</p>
<p><span class="math display">\[
H_0: \mu_\text{Extra Hours of Sleep with Drug 1} - \mu_\text{Extra
Hours of Sleep with Drug 2} = 0
\]</span> <span class="math display">\[
H_a: \mu_\text{Extra Hours of Sleep with Drug 1} - \mu_\text{Extra Hours
of Sleep with Drug 2} \neq 0
\]</span></p>
<p>We used a significance level of <span class="math inline">\(\alpha =
0.05\)</span> and obtained a P-value of <span
class="math inline">\(0.07939\)</span>. Let’s demonstrate how a
permutation test could be used to obtain this same p-value. (Technically
you only need to use a permutation test when the requirements of the
original test were not satisfied. However, it is also reasonable to
perform a permutation test anytime you want. No requirements need to be
checked when performing a permutation test.)</p>
<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><span class="co"># First run the initial test and gain the test statistic:</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>myTest <span class="ot"><-</span> <span class="fu">t.test</span>(extra <span class="sc">~</span> group, <span class="at">data =</span> sleep, <span class="at">mu =</span> <span class="dv">0</span>)</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>observedTestStat <span class="ot"><-</span> myTest<span class="sc">$</span>statistic</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Now we run the permutations to create a distribution of test statistics</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>N <span class="ot"><-</span> <span class="dv">2000</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>permutedTestStats <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, N)</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>N){</span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a> permutedTest <span class="ot"><-</span> <span class="fu">t.test</span>(<span class="fu">sample</span>(extra) <span class="sc">~</span> group, <span class="at">data =</span> sleep, <span class="at">mu =</span> <span class="dv">0</span>)</span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a> permutedTestStats[i] <span class="ot"><-</span> permutedTest<span class="sc">$</span>statistic</span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Now we show a histogram of that distribution</span></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(permutedTestStats, <span class="at">col =</span> <span class="st">"skyblue"</span>)</span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v =</span> observedTestStat, <span class="at">col =</span> <span class="st">"red"</span>, <span class="at">lwd =</span> <span class="dv">3</span>)</span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a><span class="co">#Greater-Than p-value: Not the correct one in this case</span></span>
<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStats <span class="sc">>=</span> observedTestStat)<span class="sc">/</span>N</span>
<span id="cb1-19"><a href="#cb1-19" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-20"><a href="#cb1-20" aria-hidden="true" tabindex="-1"></a><span class="co"># Less-Than p-value: Not the correct one for this data</span></span>
<span id="cb1-21"><a href="#cb1-21" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStats <span class="sc"><=</span> observedTestStat)<span class="sc">/</span>N</span>
<span id="cb1-22"><a href="#cb1-22" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-23"><a href="#cb1-23" aria-hidden="true" tabindex="-1"></a><span class="co"># Two-Sided p-value: This is the one we want based on our alternative hypothesis.</span></span>
<span id="cb1-24"><a href="#cb1-24" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span><span class="sc">*</span><span class="fu">sum</span>(permutedTestStats <span class="sc"><=</span> observedTestStat)<span class="sc">/</span>N</span></code></pre></div>
<p><strong>Note</strong> The Wilcoxon Rank Sum test is run using the
same code except with
<code>myTest <- wilcox.test(y ~ x, data=...)</code> instead of
<code>t.test(...)</code> in both Step’s 1 and 2.</p>
</div>
<div id="other-examples" class="section level5">
<h5>Other Examples</h5>
<p><a href="javascript:showhide('ptTest')">Paired Samples t Test (and
Wilcoxon Signed-Rank) <span style="font-size:8pt;">(click to
show/hide)</span></a></p>
</div>
<div id="ptTest" style="display:none;">
<div id="paired-data-example" class="section level5">
<h5>Paired Data Example</h5>
<p>See the <a
href="./Analyses/t%20Tests/Examples/SleepPairedt.html">Sleep Paired t
Test</a> example for the background and context of the study. Here is
how to perform the test as a permutation test instead of a t test.</p>
<p>The question that this <code>sleep</code> data can answer concerns
which drug is more effective at increasing the amount of extra sleep an
individual receives. The associated hypotheses would be <span
class="math display">\[
H_0: \mu_d = 0
\]</span> <span class="math display">\[
H_a: \mu_d \neq 0
\]</span> where <span class="math inline">\(\mu_d\)</span> denotes the
true mean of the differences between the observations for each drug
obtained from each individual. Differences would be obtained by <span
class="math inline">\(d_i = \text{extra}_{1i} -
\text{extra}_{2i}\)</span>.</p>
<p>To perform a permutation test of the hypothesis that the drugs are
equally effective, we use the following code.</p>
<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"># Perform the initial test:</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>myTest <span class="ot"><-</span> <span class="fu">with</span>(sleep, <span class="fu">t.test</span>(extra[group<span class="sc">==</span><span class="dv">1</span>], extra[group<span class="sc">==</span><span class="dv">2</span>], <span class="at">paired =</span> <span class="cn">TRUE</span>, <span class="at">mu =</span> <span class="dv">0</span>))</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Get the test statistic from the test:</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>observedTestStat <span class="ot"><-</span> myTest<span class="sc">$</span>statistic</span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Obtain the permutation sampling distribution </span></span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a>N <span class="ot"><-</span> <span class="dv">2000</span></span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a>permutedTestStats <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, N)</span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>N){</span>
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a> permuteValues <span class="ot"><-</span> <span class="fu">sample</span>(<span class="fu">c</span>(<span class="sc">-</span><span class="dv">1</span>,<span class="dv">1</span>), <span class="at">size=</span><span class="dv">10</span>, <span class="at">replace=</span><span class="cn">TRUE</span>) </span>
<span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a> permutedTest <span class="ot"><-</span> <span class="fu">with</span>(sleep, <span class="fu">t.test</span>(permuteValues<span class="sc">*</span>(extra[group<span class="sc">==</span><span class="dv">1</span>] <span class="sc">-</span> extra[group<span class="sc">==</span><span class="dv">2</span>]), <span class="at">mu =</span> <span class="dv">0</span>))</span>
<span id="cb2-13"><a href="#cb2-13" aria-hidden="true" tabindex="-1"></a> <span class="co">#Note, t.test(group1 - group2) is the same as t.test(group1, group2, paired=TRUE).</span></span>
<span id="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a> permutedTestStats[i] <span class="ot"><-</span> permutedTest<span class="sc">$</span>statistic</span>
<span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb2-16"><a href="#cb2-16" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(permutedTestStats)</span>
<span id="cb2-17"><a href="#cb2-17" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v=</span>observedTestStat, <span class="at">col=</span><span class="st">'skyblue'</span>, <span class="at">lwd=</span><span class="dv">3</span>)</span></code></pre></div>
<p><img src="PermutationTests_files/figure-html/unnamed-chunk-3-1.png" width="672" /></p>
<div class="sourceCode" id="cb3"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Greater than p-value: (not what we want here)</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStats <span class="sc">>=</span> observedTestStat)<span class="sc">/</span>N</span></code></pre></div>
<pre><code>## [1] 1</code></pre>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Less than p-value:</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStats <span class="sc"><=</span> observedTestStat)<span class="sc">/</span>N</span></code></pre></div>
<pre><code>## [1] 0.004</code></pre>
<div class="sourceCode" id="cb7"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Correct two sided p-value for this study:</span></span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span><span class="sc">*</span><span class="fu">sum</span>(permutedTestStats <span class="sc"><=</span> observedTestStat)<span class="sc">/</span>N</span></code></pre></div>
<pre><code>## [1] 0.008</code></pre>
<p><strong>Note</strong>:</p>
</div>
</div>
<p><a href="javascript:showhide('ANOVA')">ANOVA <span
style="font-size:8pt;">(click to show/hide)</span></a></p>
<div id="ANOVA" style="display:none;">
<div id="one-way-anova" class="section level5">
<h5>One-Way ANOVA</h5>
<p>For this example, we will use the data from the <a
href="Analyses/chickwtsOneWayANOVA.html">chick weights</a> analysis.</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Again, we run the initial test and find the test statistic</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a>myTest <span class="ot"><-</span> <span class="fu">aov</span>(weight <span class="sc">~</span> feed, <span class="at">data =</span> chickwts)</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a>observedTestStat <span class="ot"><-</span> <span class="fu">summary</span>(myTest)[[<span class="dv">1</span>]]<span class="sc">$</span><span class="st">`</span><span class="at">F value</span><span class="st">`</span>[<span class="dv">1</span>]</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a><span class="co"># For this permutation, we need to shake up the groups similar to the Independent Sample example</span></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a>N <span class="ot"><-</span> <span class="dv">2000</span></span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a>permutedTestStats <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, N)</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>N){</span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a> permutedTest <span class="ot"><-</span> <span class="fu">aov</span>(<span class="fu">sample</span>(weight) <span class="sc">~</span> feed, <span class="at">data =</span> chickwts)</span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a> permutedTestStats[i] <span class="ot"><-</span> <span class="fu">summary</span>(permutedTest)[[<span class="dv">1</span>]]<span class="sc">$</span><span class="st">`</span><span class="at">F value</span><span class="st">`</span>[<span class="dv">1</span>]</span>
<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-13"><a href="#cb9-13" aria-hidden="true" tabindex="-1"></a><span class="co"># The histogram of this distribution gives an interesting insight into the results</span></span>
<span id="cb9-14"><a href="#cb9-14" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(permutedTestStats, <span class="at">col =</span> <span class="st">"skyblue"</span>, <span class="at">xlim =</span> <span class="fu">c</span>(<span class="dv">0</span>,<span class="dv">16</span>))</span>
<span id="cb9-15"><a href="#cb9-15" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v =</span> observedTestStat, <span class="at">col =</span> <span class="st">"red"</span>, <span class="at">lwd =</span> <span class="dv">3</span>)</span>
<span id="cb9-16"><a href="#cb9-16" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-17"><a href="#cb9-17" aria-hidden="true" tabindex="-1"></a><span class="co"># Here is the greater-than p-value (since the F-distribution is right skewed</span></span>
<span id="cb9-18"><a href="#cb9-18" aria-hidden="true" tabindex="-1"></a><span class="co"># this is the only p-value of interest.)</span></span>
<span id="cb9-19"><a href="#cb9-19" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStats <span class="sc">>=</span> observedTestStat)<span class="sc">/</span>N</span></code></pre></div>
</div>
<div id="two-way-anova" class="section level5">
<h5>Two-Way ANOVA</h5>
<p>For the two-way ANOVA, I will use the data from the <a
href="Analyses/warpbreaksTwoWayANOVA.html">warpbreaks</a> analysis.</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The initial test is done in the same way as one-way ANOVA but there is a little more to find the test statistic</span></span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a>myTest <span class="ot"><-</span> <span class="fu">aov</span>(breaks <span class="sc">~</span> wool <span class="sc">+</span> tension <span class="sc">+</span> wool<span class="sc">:</span>tension, <span class="at">data=</span>warpbreaks)</span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a><span class="co"># This first test statistic is the comparison between the two types of wool</span></span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a>observedTestStatW <span class="ot"><-</span> <span class="fu">summary</span>(myTest)[[<span class="dv">1</span>]]<span class="sc">$</span><span class="st">`</span><span class="at">F value</span><span class="st">`</span>[<span class="dv">1</span>]</span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a><span class="co"># This second test statistic is the comparison between the three types of tension</span></span>
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a>observedTestStatT <span class="ot"><-</span> <span class="fu">summary</span>(myTest)[[<span class="dv">1</span>]]<span class="sc">$</span><span class="st">`</span><span class="at">F value</span><span class="st">`</span>[<span class="dv">2</span>]</span>
<span id="cb10-9"><a href="#cb10-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-10"><a href="#cb10-10" aria-hidden="true" tabindex="-1"></a><span class="co"># The third test statistic is the comparison of the interaction of wool types and tension</span></span>
<span id="cb10-11"><a href="#cb10-11" aria-hidden="true" tabindex="-1"></a>observedTestStatWT <span class="ot"><-</span> <span class="fu">summary</span>(myTest)[[<span class="dv">1</span>]]<span class="sc">$</span><span class="st">`</span><span class="at">F value</span><span class="st">`</span>[<span class="dv">3</span>]</span>
<span id="cb10-12"><a href="#cb10-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-13"><a href="#cb10-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Now comes three different permutations for the test. First is for wool, second is for tension, and third is the interaction</span></span>
<span id="cb10-14"><a href="#cb10-14" aria-hidden="true" tabindex="-1"></a>N <span class="ot"><-</span> <span class="dv">2000</span></span>
<span id="cb10-15"><a href="#cb10-15" aria-hidden="true" tabindex="-1"></a>permutedTestStatsW <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, N)</span>
<span id="cb10-16"><a href="#cb10-16" aria-hidden="true" tabindex="-1"></a>permutedTestStatsT <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, N)</span>
<span id="cb10-17"><a href="#cb10-17" aria-hidden="true" tabindex="-1"></a>permutedTestStatsWT <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, N)</span>
<span id="cb10-18"><a href="#cb10-18" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>N){</span>
<span id="cb10-19"><a href="#cb10-19" aria-hidden="true" tabindex="-1"></a> permutedTest <span class="ot"><-</span> <span class="fu">aov</span>(<span class="fu">sample</span>(breaks) <span class="sc">~</span> wool <span class="sc">+</span> tension <span class="sc">+</span> wool<span class="sc">:</span>tension, <span class="at">data=</span>warpbreaks)</span>
<span id="cb10-20"><a href="#cb10-20" aria-hidden="true" tabindex="-1"></a> permutedTestStatsW[i] <span class="ot"><-</span> <span class="fu">summary</span>(permutedTest)[[<span class="dv">1</span>]]<span class="sc">$</span><span class="st">`</span><span class="at">F value</span><span class="st">`</span>[<span class="dv">1</span>]</span>
<span id="cb10-21"><a href="#cb10-21" aria-hidden="true" tabindex="-1"></a> permutedTestStatsT[i] <span class="ot"><-</span> <span class="fu">summary</span>(permutedTest)[[<span class="dv">1</span>]]<span class="sc">$</span><span class="st">`</span><span class="at">F value</span><span class="st">`</span>[<span class="dv">2</span>]</span>
<span id="cb10-22"><a href="#cb10-22" aria-hidden="true" tabindex="-1"></a> permutedTestStatsWT[i] <span class="ot"><-</span> <span class="fu">summary</span>(permutedTest)[[<span class="dv">1</span>]]<span class="sc">$</span><span class="st">`</span><span class="at">F value</span><span class="st">`</span>[<span class="dv">3</span>]</span>
<span id="cb10-23"><a href="#cb10-23" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb10-24"><a href="#cb10-24" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-25"><a href="#cb10-25" aria-hidden="true" tabindex="-1"></a><span class="co"># We likewise need three differenct plots to show the distribution. First is wool, second is tension, and third is the interaction</span></span>
<span id="cb10-26"><a href="#cb10-26" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(permutedTestStatsW, <span class="at">col =</span> <span class="st">"skyblue"</span>, <span class="at">xlim =</span> <span class="fu">c</span>(<span class="dv">3</span>,<span class="dv">14</span>))</span>
<span id="cb10-27"><a href="#cb10-27" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v =</span> observedTestStatW, <span class="at">col =</span> <span class="st">"red"</span>, <span class="at">lwd =</span> <span class="dv">3</span>)</span>
<span id="cb10-28"><a href="#cb10-28" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-29"><a href="#cb10-29" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(permutedTestStatsT, <span class="at">col =</span> <span class="st">"skyblue"</span>)</span>
<span id="cb10-30"><a href="#cb10-30" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v =</span> observedTestStatT, <span class="at">col =</span> <span class="st">"red"</span>, <span class="at">lwd =</span> <span class="dv">3</span>)</span>
<span id="cb10-31"><a href="#cb10-31" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-32"><a href="#cb10-32" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(permutedTestStatsWT, <span class="at">col =</span> <span class="st">"skyblue"</span>)</span>
<span id="cb10-33"><a href="#cb10-33" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v =</span> observedTestStatWT, <span class="at">col =</span> <span class="st">"red"</span>, <span class="at">lwd =</span> <span class="dv">3</span>)</span>
<span id="cb10-34"><a href="#cb10-34" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-35"><a href="#cb10-35" aria-hidden="true" tabindex="-1"></a><span class="co"># Greater-than p-value: the three situations are in order</span></span>
<span id="cb10-36"><a href="#cb10-36" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStatsW <span class="sc">>=</span> observedTestStatW)<span class="sc">/</span>N</span>
<span id="cb10-37"><a href="#cb10-37" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-38"><a href="#cb10-38" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStatsT <span class="sc">>=</span> observedTestStatT)<span class="sc">/</span>N</span>
<span id="cb10-39"><a href="#cb10-39" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-40"><a href="#cb10-40" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStatsWT <span class="sc">>=</span> observedTestStatWT)<span class="sc">/</span>N</span>
<span id="cb10-41"><a href="#cb10-41" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-42"><a href="#cb10-42" aria-hidden="true" tabindex="-1"></a><span class="co"># Less-than p-value: again, they are in order</span></span>
<span id="cb10-43"><a href="#cb10-43" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStatsW <span class="sc"><=</span> observedTestStatW)<span class="sc">/</span>N</span>
<span id="cb10-44"><a href="#cb10-44" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-45"><a href="#cb10-45" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStatsT <span class="sc"><=</span> observedTestStatT)<span class="sc">/</span>N</span>
<span id="cb10-46"><a href="#cb10-46" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-47"><a href="#cb10-47" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStatsWT <span class="sc"><=</span> observedTestStatWT)<span class="sc">/</span>N</span>
<span id="cb10-48"><a href="#cb10-48" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-49"><a href="#cb10-49" aria-hidden="true" tabindex="-1"></a><span class="co"># Two-sided p-values:</span></span>
<span id="cb10-50"><a href="#cb10-50" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span><span class="sc">*</span><span class="fu">sum</span>(permutedTestStatsW <span class="sc">>=</span> observedTestStatW)<span class="sc">/</span>N</span>
<span id="cb10-51"><a href="#cb10-51" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-52"><a href="#cb10-52" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span><span class="sc">*</span><span class="fu">sum</span>(permutedTestStatsT <span class="sc">>=</span> observedTestStatT)<span class="sc">/</span>N</span>
<span id="cb10-53"><a href="#cb10-53" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-54"><a href="#cb10-54" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span><span class="sc">*</span><span class="fu">sum</span>(permutedTestStatsWT <span class="sc">>=</span> observedTestStatWT)<span class="sc">/</span>N</span></code></pre></div>
</div>
</div>
<p><a href="javascript:showhide('LinR')">Simple Linear Regression <span
style="font-size:8pt;">(click to show/hide)</span></a></p>
<div id="LinR" style="display:none;">
<p>For this example, I will use the <code>trees</code> dataset to
compare the <code>Girth</code> and <code>Height</code> of black cherry
trees.</p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The test and then the test statistic is found in a similar way to that of an ANOVA (this is the t statistic)</span></span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>myTest <span class="ot"><-</span> <span class="fu">lm</span>(Height <span class="sc">~</span> Girth, <span class="at">data =</span> trees)</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>observedTestStat <span class="ot"><-</span> <span class="fu">summary</span>(myTest)[[<span class="dv">4</span>]][<span class="dv">2</span>,<span class="dv">3</span>]</span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a><span class="co"># The permutation part is set up in this way</span></span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a>N <span class="ot"><-</span> <span class="dv">2000</span></span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a>permutedTestStats <span class="ot"><-</span> <span class="fu">rep</span>(<span class="cn">NA</span>, N)</span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>N){</span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a> permutedTest <span class="ot"><-</span> <span class="fu">lm</span>(<span class="fu">sample</span>(Height) <span class="sc">~</span> Girth, <span class="at">data =</span> trees)</span>
<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a> permutedTestStats[i] <span class="ot"><-</span> <span class="fu">summary</span>(permutedTest)[[<span class="dv">4</span>]][<span class="dv">2</span>,<span class="dv">3</span>]</span>
<span id="cb11-11"><a href="#cb11-11" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb11-12"><a href="#cb11-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-13"><a href="#cb11-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Here, as before, is the histogram of the distribution of the test statistics</span></span>
<span id="cb11-14"><a href="#cb11-14" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(permutedTestStats, <span class="at">col =</span> <span class="st">"skyblue"</span>)</span>
<span id="cb11-15"><a href="#cb11-15" aria-hidden="true" tabindex="-1"></a><span class="fu">abline</span>(<span class="at">v =</span> observedTestStat, <span class="at">col =</span> <span class="st">"red"</span>, <span class="at">lwd =</span> <span class="dv">3</span>)</span>
<span id="cb11-16"><a href="#cb11-16" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-17"><a href="#cb11-17" aria-hidden="true" tabindex="-1"></a><span class="co"># Less-than p-value:</span></span>
<span id="cb11-18"><a href="#cb11-18" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStats <span class="sc"><=</span> observedTestStat)<span class="sc">/</span>N</span>
<span id="cb11-19"><a href="#cb11-19" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-20"><a href="#cb11-20" aria-hidden="true" tabindex="-1"></a><span class="co"># Greater-than p-value:</span></span>
<span id="cb11-21"><a href="#cb11-21" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(permutedTestStats <span class="sc">>=</span> observedTestStat)<span class="sc">/</span>N</span>
<span id="cb11-22"><a href="#cb11-22" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-23"><a href="#cb11-23" aria-hidden="true" tabindex="-1"></a><span class="co"># Two-Sided p-value:</span></span>
<span id="cb11-24"><a href="#cb11-24" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span><span class="sc">*</span><span class="fu">sum</span>(permutedTestStats <span class="sc">>=</span> observedTestStat)<span class="sc">/</span>N</span></code></pre></div>
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