<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Estimating Sample Sizes for the Negative Binomial.</title> <link rel="stylesheet" href="../../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../../index.html" title="Math Toolkit"> <link rel="up" href="../neg_binom_eg.html" title="Negative Binomial Distribution Examples"> <link rel="prev" href="neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution"> <link rel="next" href="negative_binomial_example1.html" title="Negative Binomial Sales Quota Example."> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="neg_binom_conf.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../neg_binom_eg.html"><img src="../../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../../index.html"><img src="../../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="negative_binomial_example1.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <div class="titlepage"><div><div><h6 class="title"> <a name="math_toolkit.dist.stat_tut.weg.neg_binom_eg.neg_binom_size_eg"></a><a class="link" href="neg_binom_size_eg.html" title="Estimating Sample Sizes for the Negative Binomial."> Estimating Sample Sizes for the Negative Binomial.</a> </h6></div></div></div> <p> Imagine you have an event (let's call it a "failure" - though we could equally well call it a success if we felt it was a 'good' event) that you know will occur in 1 in N trials. You may want to know how many trials you need to conduct to be P% sure of observing at least k such failures. If the failure events follow a negative binomial distribution (each trial either succeeds or fails) then the static member function <code class="computeroutput"><span class="identifier">negative_binomial_distibution</span><span class="special"><>::</span><span class="identifier">find_minimum_number_of_trials</span></code> can be used to estimate the minimum number of trials required to be P% sure of observing the desired number of failures. </p> <p> The example program <a href="../../../../../../../../example/neg_binomial_sample_sizes.cpp" target="_top">neg_binomial_sample_sizes.cpp</a> demonstrates its usage. </p> <p> </p> <p> It centres around a routine that prints out a table of minimum sample sizes for various probability thresholds: </p> <p> </p> <p> </p> <pre class="programlisting"><span class="keyword">void</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">failures</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">p</span><span class="special">);</span></pre> <p> </p> <p> </p> <p> First define a table of significance levels: these are the maximum acceptable probability that <span class="emphasis"><em>failure</em></span> or fewer events will be observed. </p> <p> </p> <p> </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> </p> <p> </p> <p> Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% confidence that the desired number of failures will be observed. </p> <p> </p> <p> Much of the rest of the program is pretty-printing, the important part is in the calculation of minimum number of trials required for each value of alpha using: </p> <p> </p> <pre class="programlisting"><span class="special">(</span><span class="keyword">int</span><span class="special">)</span><span class="identifier">ceil</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="identifier">failures</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]);</span> </pre> <p> </p> <p> find_minimum_number_of_trials returns a double, so ceil rounds this up to ensure we have an integral minimum number of trials. </p> <p> </p> <p> </p> <pre class="programlisting"> <span class="keyword">void</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">failures</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">{</span> <span class="comment">// trials = number of trials </span> <span class="comment">// failures = number of failures before achieving required success(es). </span> <span class="comment">// p = success fraction (0 <= p <= 1.). </span> <span class="comment">// </span> <span class="comment">// Calculate how many trials we need to ensure the </span> <span class="comment">// required number of failures DOES exceed "failures". </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="string">"Target number of failures = "</span> <span class="special"><<</span> <span class="identifier">failures</span><span class="special">;</span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">", Success fraction = "</span> <span class="special"><<</span> <span class="number">100</span> <span class="special">*</span> <span class="identifier">p</span> <span class="special"><<</span> <span class="string">"%"</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span> <span class="comment">// Print table header: </span> <span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n\n"</span> <span class="string">"____________________________\n"</span> <span class="string">"Confidence Min Number\n"</span> <span class="string">" Value (%) Of Trials \n"</span> <span class="string">"____________________________\n"</span><span class="special">;</span> <span class="comment">// Now print out the data for the alpha table values. </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 values %: </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="special"><<</span> <span class="string">" "</span> <span class="comment">// find_minimum_number_of_trials </span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">6</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="special">(</span><span class="keyword">int</span><span class="special">)</span><span class="identifier">ceil</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="identifier">failures</span><span class="special">,</span> <span class="identifier">p</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="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> <span class="special">}</span> <span class="comment">// void find_number_of_trials(double failures, double p)</span></pre> <p> </p> <p> </p> <p> finally we can produce some tables of minimum trials for the chosen confidence levels: </p> <p> </p> <p> </p> <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span> <span class="special">{</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">500</span><span class="special">,</span> <span class="number">0.5</span><span class="special">);</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">50</span><span class="special">,</span> <span class="number">0.1</span><span class="special">);</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">500</span><span class="special">,</span> <span class="number">0.1</span><span class="special">);</span> <span class="identifier">find_number_of_trials</span><span class="special">(</span><span class="number">5</span><span class="special">,</span> <span class="number">0.9</span><span class="special">);</span> <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span> <span class="special">}</span> <span class="comment">// int main() </span> </pre> <p> </p> <p> </p> <div class="note"><table border="0" summary="Note"> <tr> <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../../doc/src/images/note.png"></td> <th align="left">Note</th> </tr> <tr><td align="left" valign="top"> <p> Since we're calculating the <span class="emphasis"><em>minimum</em></span> number of trials required, we'll err on the safe side and take the ceiling of the result. Had we been calculating the <span class="emphasis"><em>maximum</em></span> number of trials permitted to observe less than a certain number of <span class="emphasis"><em>failures</em></span> then we would have taken the floor instead. We would also have called <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code> like this: </p> <pre class="programlisting"><span class="identifier">floor</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_minimum_number_of_trials</span><span class="special">(</span><span class="identifier">failures</span><span class="special">,</span> <span class="identifier">p</span><span class="special">,</span> <span class="identifier">alpha</span><span class="special">[</span><span class="identifier">i</span><span class="special">]))</span> </pre> <p> which would give us the largest number of trials we could conduct and still be P% sure of observing <span class="emphasis"><em>failures or less</em></span> failure events, when the probability of success is <span class="emphasis"><em>p</em></span>. </p> </td></tr> </table></div> <p> We'll finish off by looking at some sample output, firstly suppose we wish to observe at least 5 "failures" with a 50/50 (0.5) chance of success or failure: </p> <pre class="programlisting">Target number of failures = 5, Success fraction = 50% ____________________________ Confidence Min Number Value (%) Of Trials ____________________________ 50.000 11 75.000 14 90.000 17 95.000 18 99.000 22 99.900 27 99.990 31 99.999 36 </pre> <p> So 18 trials or more would yield a 95% chance that at least our 5 required failures would be observed. </p> <p> Compare that to what happens if the success ratio is 90%: </p> <pre class="programlisting">Target number of failures = 5.000, Success fraction = 90.000% ____________________________ Confidence Min Number Value (%) Of Trials ____________________________ 50.000 57 75.000 73 90.000 91 95.000 103 99.000 127 99.900 159 99.990 189 99.999 217 </pre> <p> So now 103 trials are required to observe at least 5 failures with 95% certainty. </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, 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>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="neg_binom_conf.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../neg_binom_eg.html"><img src="../../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../../index.html"><img src="../../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="negative_binomial_example1.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>