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<title>Anderson-Darling k-Sample Test</title>

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<h1 class="title toc-ignore">Anderson-Darling k-Sample Test</h1>
<h4 class="author">Stefan Kloppenborg</h4>
<h4 class="date">20-Jan-2019</h4>



<p>This vignette explores the Anderson–Darling k-Sample test. CMH-17-1G <span class="citation">[1]</span> provides a formulation for this test that appears different than the formulation given by Scholz and Stephens in their 1987 paper <span class="citation">[2]</span>.</p>
<p>Both references use different nomenclature, which is summarized as follows:</p>
<table>
<thead>
<tr class="header">
<th>Term</th>
<th>CMH-17-1G</th>
<th>Scholz and Stephens</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>A sample</td>
<td><span class="math inline">\(i\)</span></td>
<td><span class="math inline">\(i\)</span></td>
</tr>
<tr class="even">
<td>The number of samples</td>
<td><span class="math inline">\(k\)</span></td>
<td><span class="math inline">\(k\)</span></td>
</tr>
<tr class="odd">
<td>An observation within a sample</td>
<td><span class="math inline">\(j\)</span></td>
<td><span class="math inline">\(j\)</span></td>
</tr>
<tr class="even">
<td>The number of observations within the sample <span class="math inline">\(i\)</span></td>
<td><span class="math inline">\(n_i\)</span></td>
<td><span class="math inline">\(n_i\)</span></td>
</tr>
<tr class="odd">
<td>The total number of observations within all samples</td>
<td><span class="math inline">\(n\)</span></td>
<td><span class="math inline">\(N\)</span></td>
</tr>
<tr class="even">
<td>Distinct values in combined data, ordered</td>
<td><span class="math inline">\(z_{(1)}\)</span>…<span class="math inline">\(z_{(L)}\)</span></td>
<td><span class="math inline">\(Z_1^*\)</span>…<span class="math inline">\(Z_L^*\)</span></td>
</tr>
<tr class="odd">
<td>The number of distinct values in the combined data</td>
<td><span class="math inline">\(L\)</span></td>
<td><span class="math inline">\(L\)</span></td>
</tr>
</tbody>
</table>
<p>Given the possibility of ties in the data, the discrete version of the test must be used Scholz and Stephens (1987) give the test statistic as:</p>
<p><span class="math display">\[
A_{a k N}^2 = \frac{N - 1}{N}\sum_{i=1}^k \frac{1}{n_i}\sum_{j=1}^{L}\frac{l_j}{N}\frac{\left(N M_{a i j} - n_i B_{a j}\right)^2}{B_{a j}\left(N - B_{a j}\right) - N l_j / 4}
\]</span></p>
<p>CMH-17-1G gives the test statistic as:</p>
<p><span class="math display">\[
ADK = \frac{n - 1}{n^2\left(k - 1\right)}\sum_{i=1}^k\frac{1}{n_i}\sum_{j=1}^L h_j \frac{\left(n F_{i j} - n_i H_j\right)^2}{H_j \left(n - H_j\right) - n h_j / 4}
\]</span></p>
<p>By inspection, the CMH-17-1G version of this test statistic contains an extra factor of <span class="math inline">\(\frac{1}{\left(k - 1\right)}\)</span>.</p>
<p>Scholz and Stephens indicate that one rejects <span class="math inline">\(H_0\)</span> at a significance level of <span class="math inline">\(\alpha\)</span> when:</p>
<p><span class="math display">\[
\frac{A_{a k N}^2 - \left(k - 1\right)}{\sigma_N} \ge t_{k - 1}\left(\alpha\right)
\]</span></p>
<p>This can be rearranged to give a critical value:</p>
<p><span class="math display">\[
A_{c r i t}^2 = \left(k - 1\right) + \sigma_N t_{k - 1}\left(\alpha\right)
\]</span></p>
<p>CHM-17-1G gives the critical value for <span class="math inline">\(ADK\)</span> for <span class="math inline">\(\alpha=0.025\)</span> as:</p>
<p><span class="math display">\[
ADC = 1 + \sigma_n \left(1.96 + \frac{1.149}{\sqrt{k - 1}} - \frac{0.391}{k - 1}\right)
\]</span></p>
<p>The definition of <span class="math inline">\(\sigma_n\)</span> from the two sources differs by a factor of <span class="math inline">\(\left(k - 1\right)\)</span>.</p>
<p>The value in parentheses in the CMH-17-1G critical value corresponds to the interpolation formula for <span class="math inline">\(t_m\left(\alpha\right)\)</span> given in Scholz and Stephen’s paper. It should be noted that this is <em>not</em> the student’s t-distribution, but rather a distribution referred to as the <span class="math inline">\(T_m\)</span> distribution.</p>
<p>The <code>cmstatr</code> package use the package <code>kSamples</code> to perform the k-sample Anderson–Darling tests. This package uses the original formulation from Scholz and Stephens, so the test statistic will differ from that given software based on the CMH-17-1G formulation by a factor of <span class="math inline">\(\left(k-1\right)\)</span>. The conclusions about the null hypothesis drawn, however, will be the same.</p>
<div id="references" class="section level1 unnumbered">
<h1 class="unnumbered">References</h1>
<div id="refs" class="references csl-bib-body">
<div id="ref-CMH-17-1G" class="csl-entry">
<div class="csl-left-margin">[1] </div><div class="csl-right-inline"><span>“Composite Materials Handbook, Volume 1. Polymer Matrix Composites Guideline for Characterization of Structural Materials,”</span> SAE International, CMH-17-1G, Mar. 2012.</div>
</div>
<div id="ref-Stephens1987" class="csl-entry">
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">F. W. Scholz and M. A. Stephens, <span>“K-Sample Anderson--Darling Tests,”</span> <em>Journal of the American Statistical Association</em>, vol. 82, no. 399. pp. 918–924, Sep-1987.</div>
</div>
</div>
</div>



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