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   4 <title>Estimating Sample Sizes for the Negative Binomial.</title>
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  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.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
  28           Sample Sizes for the Negative Binomial.</a>
  29 </h5></div></div></div>
  30 <p>
  31             Imagine you have an event (let's call it a "failure" - though
  32             we could equally well call it a success if we felt it was a 'good' event)
  33             that you know will occur in 1 in N trials. You may want to know how many
  34             trials you need to conduct to be P% sure of observing at least k such
  35             failures. If the failure events follow a negative binomial distribution
  36             (each trial either succeeds or fails) then the static member function
  37             <code class="computeroutput"><span class="identifier">negative_binomial_distibution</span><span class="special">&lt;&gt;::</span><span class="identifier">find_minimum_number_of_trials</span></code>
  38             can be used to estimate the minimum number of trials required to be P%
  39             sure of observing the desired number of failures.
  40           </p>
  41 <p>
  42             The example program <a href="../../../../../../example/neg_binomial_sample_sizes.cpp" target="_top">neg_binomial_sample_sizes.cpp</a>
  43             demonstrates its usage.
  44           </p>
  45 <p>
  46             It centres around a routine that prints out a table of minimum sample
  47             sizes (number of trials) for various probability thresholds:
  48           </p>
  49 <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>
  50 </pre>
  51 <p>
  52             First define a table of significance levels: these are the maximum acceptable
  53             probability that <span class="emphasis"><em>failure</em></span> or fewer events will be
  54             observed.
  55           </p>
  56 <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>
  57 </pre>
  58 <p>
  59             Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% confidence
  60             that the desired number of failures will be observed. The values range
  61             from a very low 0.5 or 50% confidence up to an extremely high confidence
  62             of 99.999.
  63           </p>
  64 <p>
  65             Much of the rest of the program is pretty-printing, the important part
  66             is in the calculation of minimum number of trials required for each value
  67             of alpha using:
  68           </p>
  69 <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>
  70 </pre>
  71 <p>
  72             find_minimum_number_of_trials returns a double, so <code class="computeroutput"><span class="identifier">ceil</span></code>
  73             rounds this up to ensure we have an integral minimum number of trials.
  74           </p>
  75 <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>
  76 <span class="special">{</span>
  77    <span class="comment">// trials = number of trials</span>
  78    <span class="comment">// failures = number of failures before achieving required success(es).</span>
  79    <span class="comment">// p        = success fraction (0 &lt;= p &lt;= 1.).</span>
  80    <span class="comment">//</span>
  81    <span class="comment">// Calculate how many trials we need to ensure the</span>
  82    <span class="comment">// required number of failures DOES exceed "failures".</span>
  83
  84   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="string">"Target number of failures = "</span> <span class="special">&lt;&lt;</span> <span class="special">(</span><span class="keyword">int</span><span class="special">)</span><span class="identifier">failures</span><span class="special">;</span>
  85   <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">",   Success fraction = "</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">1</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="number">100</span> <span class="special">*</span> <span class="identifier">p</span> <span class="special">&lt;&lt;</span> <span class="string">"%"</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  86    <span class="comment">// Print table header:</span>
  87    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"____________________________\n"</span>
  88            <span class="string">"Confidence        Min Number\n"</span>
  89            <span class="string">" Value (%)        Of Trials \n"</span>
  90            <span class="string">"____________________________\n"</span><span class="special">;</span>
  91    <span class="comment">// Now print out the data for the alpha table values.</span>
  92   <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>
  93    <span class="special">{</span> <span class="comment">// Confidence values %:</span>
  94       <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> <span class="special">&lt;&lt;</span> <span class="string">"      "</span>
  95       <span class="comment">// find_minimum_number_of_trials</span>
  96       <span class="special">&lt;&lt;</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">6</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">right</span>
  97       <span class="special">&lt;&lt;</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>
  98       <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  99    <span class="special">}</span>
 100    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
 101 <span class="special">}</span> <span class="comment">// void find_number_of_trials(double failures, double p)</span>
 102 </pre>
 103 <p>
 104             finally we can produce some tables of minimum trials for the chosen confidence
 105             levels:
 106           </p>
 107 <pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
 108 <span class="special">{</span>
 109     <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>
 110     <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>
 111     <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>
 112     <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>
 113     <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>
 114     <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>
 115
 116     <span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
 117 <span class="special">}</span> <span class="comment">// int main()</span>
 118 </pre>
 119 <div class="note"><table border="0" summary="Note">
 120 <tr>
 121 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../doc/src/images/note.png"></td>
 122 <th align="left">Note</th>
 123 </tr>
 124 <tr><td align="left" valign="top">
 125 <p>
 126               Since we're calculating the <span class="emphasis"><em>minimum</em></span> number of
 127               trials required, we'll err on the safe side and take the ceiling of
 128               the result. Had we been calculating the <span class="emphasis"><em>maximum</em></span>
 129               number of trials permitted to observe less than a certain number of
 130               <span class="emphasis"><em>failures</em></span> then we would have taken the floor instead.
 131               We would also have called <code class="computeroutput"><span class="identifier">find_minimum_number_of_trials</span></code>
 132               like this:
 133             </p>
 134 <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>
 135 </pre>
 136 <p>
 137               which would give us the largest number of trials we could conduct and
 138               still be P% sure of observing <span class="emphasis"><em>failures or less</em></span>
 139               failure events, when the probability of success is <span class="emphasis"><em>p</em></span>.
 140             </p>
 141 </td></tr>
 142 </table></div>
 143 <p>
 144             We'll finish off by looking at some sample output, firstly suppose we
 145             wish to observe at least 5 "failures" with a 50/50 (0.5) chance
 146             of success or failure:
 147           </p>
 148 <pre class="programlisting">Target number of failures = 5,   Success fraction = 50%
 149
 150 ____________________________
 151 Confidence        Min Number
 152  Value (%)        Of Trials
 153 ____________________________
 154     50.000          11
 155     75.000          14
 156     90.000          17
 157     95.000          18
 158     99.000          22
 159     99.900          27
 160     99.990          31
 161     99.999          36
 162
 163 </pre>
 164 <p>
 165             So 18 trials or more would yield a 95% chance that at least our 5 required
 166             failures would be observed.
 167           </p>
 168 <p>
 169             Compare that to what happens if the success ratio is 90%:
 170           </p>
 171 <pre class="programlisting">Target number of failures = 5.000,   Success fraction = 90.000%
 172
 173 ____________________________
 174 Confidence        Min Number
 175  Value (%)        Of Trials
 176 ____________________________
 177     50.000          57
 178     75.000          73
 179     90.000          91
 180     95.000         103
 181     99.000         127
 182     99.900         159
 183     99.990         189
 184     99.999         217
 185 </pre>
 186 <p>
 187             So now 103 trials are required to observe at least 5 failures with 95%
 188             certainty.
 189           </p>
 190 </div>
 191 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 192 <td align="left"></td>
 193 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 194       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 195       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 196       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 197       Daryle Walker and Xiaogang Zhang<p>
 198         Distributed under the Boost Software License, Version 1.0. (See accompanying
 199         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>)
 200       </p>
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