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| <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> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 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>) |
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