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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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| 25 | <div class="section"> | |
| 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"><<</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"><<</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"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"True Variance"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">variance</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span> | |
| 69 | <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">40</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">left</span> <span class="special"><<</span> <span class="string">"Difference to detect"</span> <span class="special"><<</span> <span class="string">"= "</span> <span class="special"><<</span> <span class="identifier">diff</span> <span class="special"><<</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"><<</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"><</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"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">3</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">10</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</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"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</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"><<</span> <span class="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">16</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">size</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> | |
| 116 | <span class="special">}</span> | |
| 117 | <span class="identifier">cout</span> <span class="special"><<</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> | |
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| 168 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 169 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 170 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 171 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 172 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
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