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<title>Estimating Sample Sizes for the Negative Binomial.</title>
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<div class="section math_toolkit_dist_stat_tut_weg_neg_binom_eg_neg_binom_size_eg">
<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>
It centres around a routine that prints out a table of minimum sample
sizes (number of trials) for various probability thresholds:
</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>
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>
<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>
Confidence value as % is (1 - alpha) * 100, so alpha 0.05 == 95% confidence
that the desired number of failures will be observed. The values range
from a very low 0.5 or 50% confidence up to an extremely high confidence
of 99.999.
</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>
<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>
find_minimum_number_of_trials returns a double, so <code class="computeroutput"><span class="identifier">ceil</span></code>
rounds this up to ensure we have an integral minimum number of trials.
</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="special">(</span><span class="keyword">int</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="identifier">fixed</span> <span class="special"><<</span> <span class="identifier">setprecision</span><span class="special">(</span><span class="number">1</span><span class="special">)</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"</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>
finally we can produce some tables of minimum trials for the chosen
confidence levels:
</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>
<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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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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