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<h1 class="title toc-ignore">Analysis Of Variance (ANOVA)</h1>
</div>
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<hr />
<p>An ANOVA is for testing the equality of several means simultaneously.
A single <em>quantitative</em> response variable is required with one or
more <em>qualitative</em> explanatory variables, i.e., factors.</p>
<div style="font-size:0.8em;">
Note: A factor is defined as a qualitative variable containing at least
two categories. The categories of the factor are referred to as the
“levels” of the factor.
</div>
<hr />
<div id="one-way-anova"
class="section level3 tabset tabset-fade tabset-pills">
<h3 class="tabset tabset-fade tabset-pills">One-way ANOVA</h3>
<div style="float:left;width:125px;" align="center">
<p><img src="Images/QuantYQualXg3plus.png" width=58px;></p>
</div>
<p>Each experimental unit is assigned to exactly one factor-level
combination. Another way to say this is “one measurement per individual”
(no repeated measures) and “equal numbers of individuals per group”.</p>
<div id="overview" class="section level4">
<h4>Overview</h4>
<div style="padding-left:125px;">
<p>An ANOVA is only appropriate when all of the following are
satisfied.</p>
<ol style="list-style-type: decimal">
<li><p>The sample(s) of data can be considered to be representative of
their population(s).</p></li>
<li><p>The data is normally distributed in each group. (This can safely
be assumed to be satisfied when the <em>residuals</em> from the ANOVA
can be assumed to be normally distributed when seen in a Q-Q
Plot.)</p></li>
<li><p>The population variance of each group can be assumed to be the
same. (This can be safely assumed to be satisfied when the
<em>residuals</em> from the ANOVA show constant variance, i.e., are
similarly vertically spread out in a Residuals versus fitted-values
plot.)</p></li>
</ol>
<p><strong>Hypotheses</strong></p>
<p>For a One-way ANOVA</p>
<p><span class="math display">\[
H_0: \mu_1 = \mu_2 = \ldots = \mu
\]</span> <span class="math display">\[
H_a: \mu_i \neq \mu \ \text{for at least one} \ i
\]</span></p>
<p><strong>Mathematical Model</strong></p>
<p>A typical model for a one-way ANOVA is of the form <span
class="math display">\[
Y_{ij} = \mu_i + \epsilon_{ij}
\]</span> where <span class="math inline">\(\mu_i\)</span> is the mean
for level (group) <span class="math inline">\(i\)</span>, and <span
class="math inline">\(\epsilon_{ij} \sim N(0,\sigma^2)\)</span> is the
error term for each point <span class="math inline">\(j\)</span> within
level (group) <span class="math inline">\(i\)</span>.</p>
<hr />
</div>
</div>
<div id="r-instructions" class="section level4">
<h4>R Instructions</h4>
<div style="padding-left:125px;">
<p><strong>Console</strong> Help Command: <code>?aov()</code></p>
<ul>
<li><code>myaov</code> is some name you come up with to store the
results of the <code>aov()</code> test.</li>
<li><code>Y</code> must be a “numeric” vector of the quantitative
response variable.</li>
<li><code>X</code> is a qualitative variable (should have
<code>class(X)</code> equal to <code>factor</code> or
<code>character</code>. If it does not, use <code>as.factor(X)</code>
inside the <code>aov(Y ~ as.factor(X),...)</code> command.</li>
<li><code>YourDataSet</code> is the name of your data set.</li>
</ul>
<p><strong>Perform the ANOVA</strong></p>
<p><code>myaov <- aov(Y ~ X, data=YourDataSet)</code></p>
<p><code>summary(myaov)</code></p>
<p><strong>Diagnose ANOVA Assumptions</strong></p>
<p><code>par(mfrow=c(1,2))</code></p>
<p><code>plot(myaov, which=1:2)</code></p>
<p><br/></p>
<p><strong>Example Code</strong></p>
<p>Hover your mouse over the example codes to learn more.</p>
<p><em>Perform the ANOVA</em></p>
<a href="javascript:showhide('oneWayAnova')">
<div class="hoverchunk">
<p><span class="tooltipr"> chick.aov <- <span class="tooltiprtext">
Saves the results of the ANOVA test as an object named
‘chick.aov’.</span> </span><span class="tooltipr"> aov( <span
class="tooltiprtext">‘aov()’ is a function in R used to perform the
ANOVA.</span> </span><span class="tooltipr"> weight <span
class="tooltiprtext">Y is ‘weight’, which is a numeric variable from the
chickwts dataset.</span> </span><span class="tooltipr"> ~ <span
class="tooltiprtext">‘~’ is the tilde symbol used to separate the Y and
X in a model formula.</span> </span><span class="tooltipr"> feed, <span
class="tooltiprtext">X is ‘feed’, which is a qualitative variable in the
chickwts dataset, or more specifically, a factor with six levels:
“casein”, “horsebean”, and so on… Use str(chickwts) to see this. </span>
</span><span class="tooltipr"> data = chickwts) <span
class="tooltiprtext"> ‘chickwts’ is a dataset in R.</span>
</span><br><span class="tooltipr"> summary( <span class="tooltiprtext">
‘summary()’ shows the results of the ANOVA.</span> </span><span
class="tooltipr"> chick.aov) <span class="tooltiprtext"> ‘chick.aov’ is
the name of the ANOVA.</span> </span><span class="tooltipr"> <br />
<span class="tooltiprtext">Press Enter to run the code if you have typed
it in yourself. You can also click here to view the output.</span>
</span><span class="tooltipr" style="float:right;font-size:.8em;">
Click to View Output <span class="tooltiprtext">Click to View
Output.</span> </span></p>
</div>
</a>
<div id="oneWayAnova" style="display:none;">
<pre><code>## Df Sum Sq Mean Sq F value Pr(>F)
## feed 5 231129 46226 15.37 5.94e-10 ***
## Residuals 65 195556 3009
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
</div>
<p><br/></p>
<p><em>Diagnose the ANOVA</em></p>
<a href="javascript:showhide('oneWayAnovaCheck')">
<div class="hoverchunk">
<p><span class="tooltipr"> par( <span class="tooltiprtext">‘par’ is a R
function that can be used to set or query graphical parameters.</span>
</span><span class="tooltipr"> mfrow = c(1,2)) <span
class="tooltiprtext">The <code>mfrow</code> parameter controls “multiple
frames on a row”. In this case, the c(1,2) specifies 1 row of 2 plots.
This will cause the two diagnostic plots to be placed side-by-side.
</span> </span><br><span class="tooltipr"> plot( <span
class="tooltiprtext">‘plot’ is a R function for the plotting of R
objects.</span> </span><span class="tooltipr"> chick.aov, <span
class="tooltiprtext">‘chick.aov’ is the name of the ANOVA.</span>
</span><span class="tooltipr"> which = 1:2) <span
class="tooltiprtext">The <code>which=1:2</code> selects “which” of 6
available plots we want to have graphed. In this case, 1 shows the
Residuals vs Fitted, and 2 shows the Normal QQ-plot. Both are needed to
check the ANOVA assumptions.</span> </span><span class="tooltipr"
style="float:right;font-size:.8em;"> Click to View Output <span
class="tooltiprtext">Click to View Output.</span> </span></p>
</div>
</a>
<div id="oneWayAnovaCheck" style="display:none;">
<p><img src="ANOVA_files/figure-html/unnamed-chunk-2-1.png" width="672" /></p>
</div>
<hr />
</div>
</div>
<div id="explanation" class="section level4">
<h4>Explanation</h4>
<div style="padding-left:125px;">
<p>Analysis of variance (ANOVA) is often applied to the scenario of
testing for the equality of three or more means from (possibly) separate
normal distributions of data. The normality assumption is required. No
matter the sample size. If the distributions are skewed then a
nonparametric test should be applied instead of ANOVA.</p>
<p><br /></p>
<div id="one-way-anova-1" class="section level5">
<h5>One-Way ANOVA</h5>
<p>One-way ANOVA is when a completely randomized design is used with a
single factor of interest. A typical mathematical model for a one-way
ANOVA is of the form <span class="math display">\[
Y_{ik} = \mu_i + \epsilon_{ik} \quad (\text{sometimes written}\ Y_{ik}
= \mu + \alpha_i + \epsilon_{ik})
\]</span> where <span class="math inline">\(\mu_i\)</span> is the mean
of each group (or level) <span class="math inline">\(i\)</span> of a
factor, and <span class="math inline">\(\epsilon_{ik}\sim
N(0,\sigma^2)\)</span> is the error term. The plot below demonstrates
what these symbols represent. Note that the notation <span
class="math inline">\(\epsilon_{ik}\sim N(0,\sigma^2)\)</span> states
that we are assuming the error term <span
class="math inline">\(\epsilon_{ik}\)</span> is normally distributed
with a mean of 0 and a standard deviation of <span
class="math inline">\(\sigma\)</span>.</p>
<p><img src="ANOVA_files/figure-html/unnamed-chunk-4-1.png" width="672" /></p>
<div id="hypotheses" class="section level6">
<h6>Hypotheses</h6>
<p>The aim of ANOVA is to determine which hypothesis is more plausible,
that the means of the different distributions are all equal (the null),
or that at least one group mean differs (the alternative).
Mathematically, <span class="math display">\[
H_0: \mu_1 = \mu_2 = \ldots = \mu_m = \mu
\]</span> <span class="math display">\[
H_a: \mu_i \neq \mu \quad \text{for at least one}\ i\in\{1,\ldots,m\}.
\]</span> In other words, the goal is to determine if it is more
plausible that each of the <span class="math inline">\(m\)</span>
different samples (where each sample is of size <span
class="math inline">\(n\)</span>) came from the same normal distribution
(this is what the null hypothesis claims) or that at least one of the
samples (and possibly several or all) come from different normal
distributions (this is what the alternative hypothesis claims).</p>
</div>
<div id="visualizing-the-hypotheses" class="section level6">
<h6>Visualizing the Hypotheses</h6>
<p>The first figure below demonstrates what a given scenario might look
like when all <span class="math inline">\(m=3\)</span> samples of data
are from the same normal distribution. In this case, the null hypothesis
<span class="math inline">\(H_0\)</span> is true. Notice that the
variability of the sample means is smaller than the variability of the
points.</p>
<p><img src="ANOVA_files/figure-html/unnamed-chunk-6-1.png" width="672" /></p>
<p>The figure below shows what a given scenario might look like for
<span class="math inline">\(m=3\)</span> samples of data from three
different normal distributions. In this case, the alternative hypothesis
<span class="math inline">\(H_a\)</span> is true. Notice that the
variability of the sample means, i.e., <span
class="math inline">\((\bar{x}_1,\bar{x}_2,\bar{x}_3)\)</span>, is
greater than the variability of the points.</p>
<p><img src="ANOVA_files/figure-html/unnamed-chunk-7-1.png" width="672" /></p>
</div>
<div id="explaining-the-name" class="section level6">
<h6>Explaining the Name</h6>
<p>The above plots are useful in understanding the mathematical details
behind ANOVA and why it is called <em>analysis of variance</em>. Recall
that variance is a measure of the spread of data. When data is very
spread out, the variance is large. When the data is close together, the
variance is small. ANOVA utilizes two important variances, the between
groups variance and the within groups variance.</p>
<ul>
<li><p><strong>Between groups variance</strong>–a measure of the
variability in the sample means, the <span
class="math inline">\(\bar{x}\)</span>’s.</p></li>
<li><p><strong>Within groups variance</strong>–a combined measure of the
variability of the points within each sample.</p></li>
</ul>
<p>The plot below combines the information from the previous plots for
ease of reference. It emphasizes the fact that when the null hypothesis
is true, the points should have a large variance (be really spread out)
while the sample means are relatively close together. On the other hand,
when the points are relative close together <em>within</em> each sample
and the sample means have a large variance (are really spread out) then
the alternative hypothesis is true. This is the theory behind analysis
of variance, or ANOVA.</p>
<p><img src="ANOVA_files/figure-html/unnamed-chunk-8-1.png" width="672" /></p>
</div>
<div id="calculating-the-test-statistic-f" class="section level6">
<h6>Calculating the Test Statistic, <span
class="math inline">\(F\)</span></h6>
<p>The ratio of the “between groups variation” to the “within groups
variation” provides the test statistic for ANOVA. Note that the test
statistic of ANOVA is an <span class="math inline">\(F\)</span>
statistic.</p>
<p><span class="math display">\[
F = \frac{\text{Between groups variation}}{\text{Within groups
variation}}
\]</span></p>
<p>It would be good to take a minute and <a
href="MakingInference.html#fdist">review the <span
class="math inline">\(F\)</span> distribution</a>. The <span
class="math inline">\(p\)</span>-value for ANOVA thus comes from an
<span class="math inline">\(F\)</span> distribution with parameters
<span class="math inline">\(p_1 = m-1\)</span> and <span
class="math inline">\(p_2 = n-m\)</span> where <span
class="math inline">\(m\)</span> is the number of samples and <span
class="math inline">\(n\)</span> is the total number of data points.</p>
<p><br /></p>
</div>
<div id="a-deeper-look-at-variance" class="section level6">
<h6>A Deeper Look at Variance</h6>
<p>It is useful to take a few minutes and explain the word
<em>variance</em> as well as mathematically define the terms “<em>within
group variance</em>” and “<em>between groups variance</em>.”</p>
<p>Variance is a statistical measure of the <em>variability</em> in
data. The square root of the variance is called the <em>standard
deviation</em> and is by far the more typical measure of spread. This is
because standard deviation is easier to interpret. However,
mathematically speaking, the variance is the more important
measurement.</p>
<p>As mentioned previously, the variance turns out to be the key to
determining which hypothesis is the most plausible, <span
class="math inline">\(H_0\)</span> or <span
class="math inline">\(H_a\)</span>, when several means are under
consideration. There are two variances that are important for ANOVA, the
“within groups variance” and the “between groups variance.”</p>
<p>Recall that the formula for computing a <a
href="NumericalSummaries.html#variance">sample variance</a> is given by
<span class="math display">\[
s^2 = \frac{\sum_{i=1}^n(x_i - \bar{x})^2}{n-1} \quad\leftarrow
\frac{\text{sum of squares}}{\text{degrees of freedom}}
\]</span> This formula has a couple of important pieces that are so
important they have been given special names. The <span
class="math inline">\(n-1\)</span> in the denominator of the formula is
called the “degrees of freedom.” The other important part of this
formula is the <span class="math inline">\(\sum_{i=1}^n(x_i -
\bar{x})^2\)</span>, which is called the “sum of squared errors” or
sometimes just the “sum of squares” or “SS” for short. Thus, the sample
variance is calculated by computing a “sum of squares” and dividing this
by the “degrees of freedom.”</p>
<p>It turns out that this general approach works for many different
contexts. Specifically, it allows us to compute the “within groups
variance” and the “between groups variance.” To introduce the
mathematical definitions of these two variances, we need to introduce
some new notation.</p>
<div style="padding-left:15px;">
<p>Let <span class="math inline">\(\bar{y}_{i\bullet}\)</span> represent
the sample mean of group <span class="math inline">\(i\)</span> for
<span class="math inline">\(i=1,\ldots,m\)</span>.</p>
<p>Let <span class="math inline">\(n_i\)</span> denote the sample size
in group <span class="math inline">\(i\)</span>.</p>
<p>Let <span class="math inline">\(\bar{y}_{\bullet\bullet}\)</span>
represent the sample mean of all <span class="math inline">\(n =
n_1+n_2+\cdots+n_m\)</span> data points.</p>
</div>
<p>The mathematical calculations for each of these variances is given as
follows. <span class="math display">\[
\text{Between groups variance} = \frac{\sum_{i=1}^m
(\bar{y}_{i\bullet}-\bar{y}_{\bullet\bullet})^2}{m-1} \leftarrow
\frac{\text{Between groups sum of squares}}{\text{Between groups degrees
of freedom}}
\]</span> <span class="math display">\[
\text{Within groups variance} =
\frac{\sum_{i=1}^m\sum_{k=1}^{n_i}(y_{ik}-\bar{y}_{i\bullet})^2}{n-m}
\leftarrow \frac{\text{Within groups sum of squares}}{\text{Within
groups degrees of freedom}}
\]</span></p>
<p><br /></p>
</div>
<div id="a-fabricated-example" class="section level6">
<h6>A Fabricated Example</h6>
<p>The following table provides three samples of data:
<strong>A</strong>, <strong>B</strong>, and <strong>C</strong>. These
samples were randomly generated from normal distributions using a
computer. The true means <span class="math inline">\(\mu_1,
\mu_2\)</span>, and <span class="math inline">\(\mu_3\)</span> of the
normal distributions are thus known, but withheld from you at this point
of the example.</p>
<table>
<thead>
<tr class="header">
<th align="right">A</th>
<th align="right">B</th>
<th align="right">C</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="right">13.15457</td>
<td align="right">13.17463</td>
<td align="right">16.66831</td>
</tr>
<tr class="even">
<td align="right">12.65225</td>
<td align="right">12.16277</td>
<td align="right">15.54719</td>
</tr>
<tr class="odd">
<td align="right">13.73061</td>
<td align="right">12.76905</td>
<td align="right">16.63074</td>
</tr>
<tr class="even">
<td align="right">14.43471</td>
<td align="right">13.38524</td>
<td align="right">15.06726</td>
</tr>
<tr class="odd">
<td align="right">13.79728</td>
<td align="right">12.02690</td>
<td align="right">15.57534</td>
</tr>
<tr class="even">
<td align="right">13.88599</td>
<td align="right">13.24651</td>
<td align="right">15.99915</td>
</tr>
<tr class="odd">
<td align="right">12.77753</td>
<td align="right">12.58386</td>
<td align="right">15.58995</td>
</tr>
<tr class="even">
<td align="right">13.81536</td>
<td align="right">12.64615</td>
<td align="right">16.99429</td>
</tr>
<tr class="odd">
<td align="right">13.03635</td>
<td align="right">12.52055</td>
<td align="right">15.47153</td>
</tr>
<tr class="even">
<td align="right">14.26062</td>
<td align="right">14.03566</td>
<td align="right">16.13330</td>
</tr>
</tbody>
</table>
<p>An ANOVA will be performed with the sample data to determine which
hypothesis is more plausible: <span class="math display">\[
H_0: \mu_1 = \mu_2 = \mu_3 = \mu
\]</span> <span class="math display">\[
H_a: \mu_i \neq \mu \ \text{for at least one} \ i \in \{1,\ldots,m\}
\]</span></p>
<p>To perform an ANOVA, we must compute the between groups variance and
the within groups variance. This requires the Between groups sums of
squares, within groups sums of squares, between groups degrees of
freedom, and the within groups degrees of freedom. Note that to get the
sums of squares, we first had to calculate <span
class="math inline">\(\bar{y}_{1\bullet}\)</span>, <span
class="math inline">\(\bar{y}_{2\bullet}\)</span>, <span
class="math inline">\(\bar{y}_{3\bullet}\)</span>, and <span
class="math inline">\(\bar{y}_{\bullet\bullet}\)</span> where the 1, 2,
3 corresponds to Samples A, B, and C, respectively. After some work, we
find these values to be <span class="math display">\[
\bar{y}_{1\bullet} = 13.55, \quad \bar{y}_{2\bullet} = 12.86 \quad
\bar{y}_{3\bullet} = 15.97
\]</span> and <span class="math display">\[
\bar{y}_{\bullet\bullet} = \frac{13.55+12.86+15.97}{3} = 14.13
\]</span> Using these values we can then compute the between groups sum
of squares and the within groups sum of squares according to the
formulas stated previously. This process is very tedious and will not be
demonstrated. Only the results are shown in the following table which
summarizes all the important information.</p>
<table>
<colgroup>
<col width="11%" />
<col width="27%" />
<col width="22%" />
<col width="13%" />
<col width="12%" />
<col width="12%" />
</colgroup>
<thead>
<tr class="header">
<th> </th>
<th>Degrees of Freedom</th>
<th>Sum of Squares</th>
<th>Variance</th>
<th>F-value</th>
<th>p-value</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Between groups</td>
<td>2</td>
<td>53.3</td>
<td>26.67</td>
<td>70.2</td>
<td>2e-11</td>
</tr>
<tr class="even">
<td>Within groups</td>
<td>27</td>
<td>10.3</td>
<td>0.38</td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
</div>
<div id="anova-table" class="section level6">
<h6>ANOVA Table</h6>
<p>In general, the ANOVA table is created by</p>
<table>
<colgroup>
<col width="11%" />
<col width="27%" />
<col width="22%" />
<col width="13%" />
<col width="12%" />
<col width="12%" />
</colgroup>
<thead>
<tr class="header">
<th> </th>
<th>Degrees of Freedom</th>
<th>Sum of Squares</th>
<th>Variance</th>
<th>F-value</th>
<th>p-value</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Between groups</td>
<td><span class="math inline">\(m-1\)</span></td>
<td><span class="math inline">\(\sum_{i=1}^m
n_i(\bar{y}_{i\bullet}-\bar{y}_{\bullet\bullet})^2\)</span></td>
<td><span class="math inline">\(\frac{\text{sum of
squares}}{\text{degrees of freedom}}\)</span></td>
<td><span class="math inline">\(\frac{\text{Between groups
variance}}{\text{Within groups variance}}\)</span></td>
<td><span class="math inline">\(F\)</span>-distribution tail
probability</td>
</tr>
<tr class="even">
<td>Within groups</td>
<td><span class="math inline">\(n-m\)</span></td>
<td><span
class="math inline">\(\sum_{i=1}^m\sum_{k=1}^{n_i}(y_{ik}-\bar{y}_{i\bullet})^2\)</span></td>
<td><span class="math inline">\(\frac{\text{sum of
squares}}{\text{degrees of freedom}}\)</span></td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
<p><br /></p>
</div>
</div>
<div id="residuals" class="section level5">
<h5>ANOVA Assumptions</h5>
<p>The requirements for an analysis of variance (the assumptions of the
test) are two-fold and concern only the error terms, the <span
class="math inline">\(\epsilon_{ik}\)</span>.</p>
<ol style="list-style-type: decimal">
<li><p>The errors are normally distributed.</p></li>
<li><p>The variance of the errors is constant.</p></li>
</ol>
<p>Both of these assumptions were stated in the mathematical model where
we assumed that <span class="math inline">\(\epsilon_{ik}\sim
N(0,\sigma^2)\)</span>.</p>
<div id="checking-anova-assumptions" class="section level6">
<h6>Checking ANOVA Assumptions</h6>
<p>To check that the ANOVA assumptions are satisfied, it is required to
check the data in each group for normality using QQ-Plots. Also, the
sample variance of each group must be relatively constant. The fastest
way to check these two assumptions is by analyzing the
<em>residuals</em>.</p>
<ul>
<li>An ANOVA <strong>residual</strong> is defined as the difference
between the observed value of <span
class="math inline">\(y_{ik}\)</span> and the mean <span
class="math inline">\(\bar{y}_{i\bullet}\)</span>. Mathematically, <span
class="math display">\[
r_{ik} = y_{ik} - \bar{y}_{i\bullet}
\]</span> One QQ-Plot of the residuals will provide the necessary
evidence to decide if it is reasonable to assume that the error terms
are normally distributed. Also, the constant variance can be checked
visually by using what is known as a residuals versus fitted values
plot. For the <strong>Fabricated Example</strong> above, the QQ-Plot and
residuals versus fitted values plots show the two assumptions of ANOVA
appear to be satisfied.</li>
</ul>
<p><img src="ANOVA_files/figure-html/unnamed-chunk-10-1.png" width="672" /></p>
<p><br /></p>
<hr />
</div>
</div>
</div>
</div>
</div>
<div id="section" class="section level3">
<h3></h3>
<div style="padding-left:125px;">
<p><strong>Examples:</strong> <a
href="./Analyses/ANOVA/Examples/chickwtsOneWayANOVA.html">chickwts</a>
(One-way)</p>
</div>
<hr />
</div>
<div id="two-way-anova"
class="section level3 tabset tabset-fade tabset-pills">
<h3 class="tabset tabset-fade tabset-pills">Two-way ANOVA</h3>
<div style="float:left;width:125px;" align="center">
<p><img src="Images/QuantYMultQualX.png" width=80px;></p>
</div>
<div id="overview-1" class="section level4">
<h4>Overview</h4>
<div style="padding-left:125px;">
<p>A two-way ANOVA is only appropriate when all of the following are
satisfied.</p>
<ol style="list-style-type: decimal">
<li><p>The sample(s) of data can be considered to be representative of
their population(s).</p></li>
<li><p>The data is normally distributed in each group. (This can safely
be assumed to be satisfied when the <em>residuals</em> from the ANOVA
are normally distributed.)</p></li>
<li><p>The population variance of each group can be assumed to be the
same. (This can be safely assumed to be satisfied when the
<em>residuals</em> from the ANOVA show constant variance, i.e., are
similarly vertically spread out.)</p></li>
</ol>
<p><strong>Hypotheses</strong></p>
<p>With a two-way ANOVA there are three sets of hypotheses. Writing out
the hypotheses can be very involved depending on whether you use the
official “effects model” notation (very mathematically correct) or a
simplified “means model” notation (which isn’t very mathematically
correct, but gets the idea across in an acceptable way).</p>
<div class="tab">
<p><button class="tablinks" onclick="openTab(event, 'meansModel')">Means
Model</button>
<button class="tablinks" onclick="openTab(event, 'effectsModel')">Effects
Model</button></p>
</div>
<div id="meansModel" class="tabcontent" style="display:block;">
<p>
<ol style="list-style-type: decimal">
<li>The first set of hypotheses are a “one-way” set of hypotheses for
the first <em>factor</em> of the ANOVA. Factor: <code>X1</code> with
say, levels <span class="math inline">\(A\)</span> and <span
class="math inline">\(B\)</span>.</li>
</ol>
<p><span class="math display">\[
H_0: \mu_A = \mu_B = \mu
\]</span> <span class="math display">\[
H_a: \mu_A \neq \mu_B
\]</span></p>
<ol start="2" style="list-style-type: decimal">
<li><p>The second set of hypotheses are also a “one-way” set of
hypotheses, but for the second <em>factor</em> of the ANOVA. Factor:
<code>X2</code> with say, levels <span class="math inline">\(C\)</span>,
<span class="math inline">\(D\)</span>, and <span