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   4 <title>Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation</title>
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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.stat_tut.weg.cs_eg.chi_sq_size"></a><a class="link" href="chi_sq_size.html" title="Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation">Estimating
  28           the Required Sample Sizes for a Chi-Square Test for the Standard Deviation</a>
  29 </h5></div></div></div>
  30 <p>
  31             Suppose we conduct a Chi Squared test for standard deviation and the
  32             result is borderline, a legitimate question to ask is "How large
  33             would the sample size have to be in order to produce a definitive result?"
  34           </p>
  35 <p>
  36             The class template <a class="link" href="../../../dist_ref/dists/chi_squared_dist.html" title="Chi Squared Distribution">chi_squared_distribution</a>
  37             has a static method <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>
  38             that will calculate this value for some acceptable risk of type I failure
  39             <span class="emphasis"><em>alpha</em></span>, type II failure <span class="emphasis"><em>beta</em></span>,
  40             and difference from the standard deviation <span class="emphasis"><em>diff</em></span>.
  41             Please note that the method used works on variance, and not standard
  42             deviation as is usual for the Chi Squared Test.
  43           </p>
  44 <p>
  45             The code for this example is located in <a href="../../../../../../example/chi_square_std_dev_test.cpp" target="_top">chi_square_std_dev_test.cpp</a>.
  46           </p>
  47 <p>
  48             We begin by defining a procedure to print out the sample sizes required
  49             for various risk levels:
  50           </p>
  51 <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">chi_squared_sample_sized</span><span class="special">(</span>
  52      <span class="keyword">double</span> <span class="identifier">diff</span><span class="special">,</span>      <span class="comment">// difference from variance to detect</span>
  53      <span class="keyword">double</span> <span class="identifier">variance</span><span class="special">)</span>  <span class="comment">// true variance</span>
  54 <span class="special">{</span>
  55 </pre>
  56 <p>
  57             The procedure begins by printing out the input data:
  58           </p>
  59 <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span>
  60 <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
  61
  62 <span class="comment">// Print out general info:</span>
  63 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span>
  64    <span class="string">"_____________________________________________________________\n"</span>
  65    <span class="string">"Estimated sample sizes required for various confidence levels\n"</span>
  66    <span class="string">"_____________________________________________________________\n\n"</span><span class="special">;</span>
  67 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
  68 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"True Variance"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">variance</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  69 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">left</span> <span class="special">&lt;&lt;</span> <span class="string">"Difference to detect"</span> <span class="special">&lt;&lt;</span> <span class="string">"=  "</span> <span class="special">&lt;&lt;</span> <span class="identifier">diff</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  70 </pre>
  71 <p>
  72             And defines a table of significance levels for which we'll calculate
  73             sample sizes:
  74           </p>
  75 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">alpha</span><span class="special">[]</span> <span class="special">=</span> <span class="special">{</span> <span class="number">0.5</span><span class="special">,</span> <span class="number">0.25</span><span class="special">,</span> <span class="number">0.1</span><span class="special">,</span> <span class="number">0.05</span><span class="special">,</span> <span class="number">0.01</span><span class="special">,</span> <span class="number">0.001</span><span class="special">,</span> <span class="number">0.0001</span><span class="special">,</span> <span class="number">0.00001</span> <span class="special">};</span>
  76 </pre>
  77 <p>
  78             For each value of alpha we can calculate two sample sizes: one where
  79             the sample variance is less than the true value by <span class="emphasis"><em>diff</em></span>
  80             and one where it is greater than the true value by <span class="emphasis"><em>diff</em></span>.
  81             Thanks to the asymmetric nature of the Chi Squared distribution these
  82             two values will not be the same, the difference in their calculation
  83             differs only in the sign of <span class="emphasis"><em>diff</em></span> that's passed to
  84             <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>.
  85             Finally in this example we'll simply things, and let risk level <span class="emphasis"><em>beta</em></span>
  86             be the same as <span class="emphasis"><em>alpha</em></span>:
  87           </p>
  88 <pre class="programlisting"><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n\n"</span>
  89         <span class="string">"_______________________________________________________________\n"</span>
  90         <span class="string">"Confidence       Estimated          Estimated\n"</span>
  91         <span class="string">" Value (%)      Sample Size        Sample Size\n"</span>
  92         <span class="string">"                (lower one         (upper one\n"</span>
  93         <span class="string">"                 sided test)        sided test)\n"</span>
  94         <span class="string">"_______________________________________________________________\n"</span><span class="special">;</span>
  95 <span class="comment">//</span>
  96 <span class="comment">// Now print out the data for the table rows.</span>
  97 <span class="comment">//</span>
  98 <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">)/</span><span class="keyword">sizeof</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">[</span><span class="number">0</span><span class="special">]);</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
  99 <span class="special">{</span>
 100    <span class="comment">// Confidence value:</span>
 101    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span>
 102    <span class="comment">// calculate df for a lower single sided test:</span>
 103    <span class="keyword">double</span> <span class="identifier">df</span> <span class="special">=</span> <span class="identifier">chi_squared</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
 104       <span class="special">-</span><span class="identifier">diff</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">variance</span><span class="special">);</span>
 105    <span class="comment">// convert to sample size:</span>
 106    <span class="keyword">double</span> <span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
 107    <span class="comment">// Print size:</span>
 108    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span><span class="special">;</span>
 109    <span class="comment">// calculate df for an upper single sided test:</span>
 110    <span class="identifier">df</span> <span class="special">=</span> <span class="identifier">chi_squared</span><span class="special">::</span><span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
 111       <span class="identifier">diff</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">],</span> <span class="identifier">variance</span><span class="special">);</span>
 112    <span class="comment">// convert to sample size:</span>
 113    <span class="identifier">size</span> <span class="special">=</span> <span class="identifier">ceil</span><span class="special">(</span><span class="identifier">df</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">;</span>
 114    <span class="comment">// Print size:</span>
 115    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">fixed</span> <span class="special">&lt;&lt;</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span> <span class="special">&lt;&lt;</span> <span class="identifier">size</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 116 <span class="special">}</span>
 117 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 118 </pre>
 119 <p>
 120             For some example output, consider the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm" target="_top">silicon
 121             wafer data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH
 122             e-Handbook of Statistical Methods.</a>. In this scenario a supplier
 123             of 100 ohm.cm silicon wafers claims that his fabrication process can
 124             produce wafers with sufficient consistency so that the standard deviation
 125             of resistivity for the lot does not exceed 10 ohm.cm. A sample of N =
 126             10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm,
 127             and the question we ask ourselves is "How large would our sample
 128             have to be to reliably detect this difference?".
 129           </p>
 130 <p>
 131             To use our procedure above, we have to convert the standard deviations
 132             to variance (square them), after which the program output looks like
 133             this:
 134           </p>
 135 <pre class="programlisting">_____________________________________________________________
 136 Estimated sample sizes required for various confidence levels
 137 _____________________________________________________________
 138
 139 True Variance                           =  100.00000
 140 Difference to detect                    =  95.16090
 141
 142
 143 _______________________________________________________________
 144 Confidence       Estimated          Estimated
 145  Value (%)      Sample Size        Sample Size
 146                 (lower one         (upper one
 147                  sided test)        sided test)
 148 _______________________________________________________________
 149     50.000               2               2
 150     75.000               2              10
 151     90.000               4              32
 152     95.000               5              51
 153     99.000               7              99
 154     99.900              11             174
 155     99.990              15             251
 156     99.999              20             330
 157 </pre>
 158 <p>
 159             In this case we are interested in a upper single sided test. So for example,
 160             if the maximum acceptable risk of falsely rejecting the null-hypothesis
 161             is 0.05 (Type I error), and the maximum acceptable risk of failing to
 162             reject the null-hypothesis is also 0.05 (Type II error), we estimate
 163             that we would need a sample size of 51.
 164           </p>
 165 </div>
 166 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 167 <td align="left"></td>
 168 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 169       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 170       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 171       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 172       Daryle Walker and Xiaogang Zhang<p>
 173         Distributed under the Boost Software License, Version 1.0. (See accompanying
 174         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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