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| <div class="titlepage"><div><div><h6 class="title"> |
| <a name="math_toolkit.dist.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 the Required Sample Sizes for a Chi-Square Test for the Standard |
| Deviation</a> |
| </h6></div></div></div> |
| <p> |
| Suppose we conduct a Chi Squared test for standard deviation and the |
| result is borderline, a legitimate question to ask is "How large |
| would the sample size have to be in order to produce a definitive result?" |
| </p> |
| <p> |
| The class template <a class="link" href="../../../dist_ref/dists/chi_squared_dist.html" title="Chi Squared Distribution">chi_squared_distribution</a> |
| has a static method <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code> |
| that will calculate this value for some acceptable risk of type I failure |
| <span class="emphasis"><em>alpha</em></span>, type II failure <span class="emphasis"><em>beta</em></span>, |
| and difference from the standard deviation <span class="emphasis"><em>diff</em></span>. |
| Please note that the method used works on variance, and not standard |
| deviation as is usual for the Chi Squared Test. |
| </p> |
| <p> |
| 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>. |
| </p> |
| <p> |
| We begin by defining a procedure to print out the sample sizes required |
| for various risk levels: |
| </p> |
| <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">chi_squared_sample_sized</span><span class="special">(</span> |
| <span class="keyword">double</span> <span class="identifier">diff</span><span class="special">,</span> <span class="comment">// difference from variance to detect |
| </span> <span class="keyword">double</span> <span class="identifier">variance</span><span class="special">)</span> <span class="comment">// true variance |
| </span><span class="special">{</span> |
| </pre> |
| <p> |
| The procedure begins by printing out the input data: |
| </p> |
| <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span> |
| <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> |
| |
| <span class="comment">// Print out general info: |
| </span><span class="identifier">cout</span> <span class="special"><<</span> |
| <span class="string">"_____________________________________________________________\n"</span> |
| <span class="string">"Estimated sample sizes required for various confidence levels\n"</span> |
| <span class="string">"_____________________________________________________________\n\n"</span><span class="special">;</span> |
| <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> |
| <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> |
| <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> |
| </pre> |
| <p> |
| And defines a table of significance levels for which we'll calculate |
| sample sizes: |
| </p> |
| <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> |
| </pre> |
| <p> |
| For each value of alpha we can calculate two sample sizes: one where |
| the sample variance is less than the true value by <span class="emphasis"><em>diff</em></span> |
| and one where it is greater than the true value by <span class="emphasis"><em>diff</em></span>. |
| Thanks to the asymmetric nature of the Chi Squared distribution these |
| two values will not be the same, the difference in their calculation |
| differs only in the sign of <span class="emphasis"><em>diff</em></span> that's passed |
| to <code class="computeroutput"><span class="identifier">find_degrees_of_freedom</span></code>. |
| Finally in this example we'll simply things, and let risk level <span class="emphasis"><em>beta</em></span> |
| be the same as <span class="emphasis"><em>alpha</em></span>: |
| </p> |
| <pre class="programlisting"><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n\n"</span> |
| <span class="string">"_______________________________________________________________\n"</span> |
| <span class="string">"Confidence Estimated Estimated\n"</span> |
| <span class="string">" Value (%) Sample Size Sample Size\n"</span> |
| <span class="string">" (lower one (upper one\n"</span> |
| <span class="string">" sided test) sided test)\n"</span> |
| <span class="string">"_______________________________________________________________\n"</span><span class="special">;</span> |
| <span class="comment">// |
| </span><span class="comment">// Now print out the data for the table rows. |
| </span><span class="comment">// |
| </span><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> |
| <span class="special">{</span> |
| <span class="comment">// Confidence value: |
| </span> <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> |
| <span class="comment">// calculate df for a lower single sided test: |
| </span> <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> |
| <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> |
| <span class="comment">// convert to sample size: |
| </span> <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> |
| <span class="comment">// Print size: |
| </span> <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="comment">// calculate df for an upper single sided test: |
| </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> |
| <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> |
| <span class="comment">// convert to sample size: |
| </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> |
| <span class="comment">// Print size: |
| </span> <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> |
| <span class="special">}</span> |
| <span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> |
| </pre> |
| <p> |
| For some example output, consider the <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc23.htm" target="_top">silicon |
| wafer data</a> from the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH |
| e-Handbook of Statistical Methods.</a>. In this scenario a supplier |
| of 100 ohm.cm silicon wafers claims that his fabrication process can |
| produce wafers with sufficient consistency so that the standard deviation |
| of resistivity for the lot does not exceed 10 ohm.cm. A sample of N |
| = 10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm, |
| and the question we ask ourselves is "How large would our sample |
| have to be to reliably detect this difference?". |
| </p> |
| <p> |
| To use our procedure above, we have to convert the standard deviations |
| to variance (square them), after which the program output looks like |
| this: |
| </p> |
| <pre class="programlisting">_____________________________________________________________ |
| Estimated sample sizes required for various confidence levels |
| _____________________________________________________________ |
| |
| True Variance = 100.00000 |
| Difference to detect = 95.16090 |
| |
| |
| _______________________________________________________________ |
| Confidence Estimated Estimated |
| Value (%) Sample Size Sample Size |
| (lower one (upper one |
| sided test) sided test) |
| _______________________________________________________________ |
| 50.000 2 2 |
| 75.000 2 10 |
| 90.000 4 32 |
| 95.000 5 51 |
| 99.000 7 99 |
| 99.900 11 174 |
| 99.990 15 251 |
| 99.999 20 330 |
| </pre> |
| <p> |
| In this case we are interested in a upper single sided test. So for |
| example, if the maximum acceptable risk of falsely rejecting the null-hypothesis |
| is 0.05 (Type I error), and the maximum acceptable risk of failing |
| to reject the null-hypothesis is also 0.05 (Type II error), we estimate |
| that we would need a sample size of 51. |
| </p> |
| </div> |
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| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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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