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<title>Estimating the Required Sample Sizes for a Chi-Square Test for the Standard Deviation</title>
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<div class="section math_toolkit_dist_stat_tut_weg_cs_eg_chi_sq_size">
<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>
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Lalande, Johan Råde, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<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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