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<title>Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution</title>
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<div class="section math_toolkit_dist_stat_tut_weg_neg_binom_eg_neg_binom_conf">
<div class="titlepage"><div><div><h6 class="title">
<a name="math_toolkit.dist.stat_tut.weg.neg_binom_eg.neg_binom_conf"></a><a class="link" href="neg_binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for the Negative Binomial Distribution">Calculating
Confidence Limits on the Frequency of Occurrence for the Negative Binomial
Distribution</a>
</h6></div></div></div>
<p>
Imagine you have a process that follows a negative binomial distribution:
for each trial conducted, an event either occurs or does it does not,
referred to as "successes" and "failures". The
frequency with which successes occur is variously referred to as the
success fraction, success ratio, success percentage, occurrence frequency,
or probability of occurrence.
</p>
<p>
If, by experiment, you want to measure the the best estimate of success
fraction is given simply by <span class="emphasis"><em>k</em></span> / <span class="emphasis"><em>N</em></span>,
for <span class="emphasis"><em>k</em></span> successes out of <span class="emphasis"><em>N</em></span>
trials.
</p>
<p>
However our confidence in that estimate will be shaped by how many
trials were conducted, and how many successes were observed. The static
member functions <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span><span class="special"><>::</span><span class="identifier">find_lower_bound_on_p</span></code>
and <code class="computeroutput"><span class="identifier">negative_binomial_distribution</span><span class="special"><>::</span><span class="identifier">find_upper_bound_on_p</span></code>
allow you to calculate the confidence intervals for your estimate of
the success fraction.
</p>
<p>
The sample program <a href="../../../../../../../../example/neg_binom_confidence_limits.cpp" target="_top">neg_binom_confidence_limits.cpp</a>
illustrates their use.
</p>
<p>
First we need some includes to access the negative binomial distribution
(and some basic std output of course).
</p>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</span><span class="special">;</span>
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setw</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">left</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">fixed</span><span class="special">;</span> <span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">right</span><span class="special">;</span>
</pre>
<p>
</p>
<p>
First define a table of significance levels: these are the probabilities
that the true occurrence frequency lies outside the calculated interval:
</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 true occurence frequency lies <span class="bold"><strong>inside</strong></span>
the calculated interval.
</p>
<p>
We need a function to calculate and print confidence limits for an
observed frequency of occurrence that follows a negative binomial distribution.
</p>
<p>
</p>
<pre class="programlisting"><span class="keyword">void</span> <span class="identifier">confidence_limits_on_frequency</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">successes</span><span class="special">)</span>
<span class="special">{</span>
<span class="comment">// trials = Total number of trials.</span>
<span class="comment">// successes = Total number of observed successes.</span>
<span class="comment">// failures = trials - successes.</span>
<span class="comment">// success_fraction = successes /trials.</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">"2-Sided Confidence Limits For Success Fraction\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">7</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">"Number of trials"</span> <span class="special"><<</span> <span class="string">" = "</span> <span class="special"><<</span> <span class="identifier">trials</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">"Number of successes"</span> <span class="special"><<</span> <span class="string">" = "</span> <span class="special"><<</span> <span class="identifier">successes</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">"Number of failures"</span> <span class="special"><<</span> <span class="string">" = "</span> <span class="special"><<</span> <span class="identifier">trials</span> <span class="special">-</span> <span class="identifier">successes</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">"Observed frequency of occurrence"</span> <span class="special"><<</span> <span class="string">" = "</span> <span class="special"><<</span> <span class="keyword">double</span><span class="special">(</span><span class="identifier">successes</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">trials</span> <span class="special"><<</span> <span class="string">"\n"</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 Lower Upper\n"</span>
<span class="string">" Value (%) Limit Limit\n"</span>
<span class="string">"___________________________________________\n"</span><span class="special">;</span>
</pre>
<p>
</p>
<p>
And now for the important part - the bounds themselves. For each value
of <span class="emphasis"><em>alpha</em></span>, we call <code class="computeroutput"><span class="identifier">find_lower_bound_on_p</span></code>
and <code class="computeroutput"><span class="identifier">find_upper_bound_on_p</span></code>
to obtain lower and upper bounds respectively. Note that since we are
calculating a two-sided interval, we must divide the value of alpha
in two. Had we been calculating a single-sided interval, for example:
<span class="emphasis"><em>"Calculate a lower bound so that we are P% sure that
the true occurrence frequency is greater than some value"</em></span>
then we would <span class="bold"><strong>not</strong></span> have divided by
two.
</p>
<p>
</p>
<pre class="programlisting"> <span class="comment">// Now print out the upper and lower limits 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 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 bounds:</span>
<span class="keyword">double</span> <span class="identifier">lower</span> <span class="special">=</span> <span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_lower_bound_on_p</span><span class="special">(</span><span class="identifier">trials</span><span class="special">,</span> <span class="identifier">successes</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="number">2</span><span class="special">);</span>
<span class="keyword">double</span> <span class="identifier">upper</span> <span class="special">=</span> <span class="identifier">negative_binomial</span><span class="special">::</span><span class="identifier">find_upper_bound_on_p</span><span class="special">(</span><span class="identifier">trials</span><span class="special">,</span> <span class="identifier">successes</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="number">2</span><span class="special">);</span>
<span class="comment">// Print limits:</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">5</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">lower</span><span class="special">;</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">5</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">setw</span><span class="special">(</span><span class="number">15</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">right</span> <span class="special"><<</span> <span class="identifier">upper</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 confidence_limits_on_frequency(unsigned trials, unsigned successes)</span>
</pre>
<p>
</p>
<p>
And then call confidence_limits_on_frequency with increasing numbers
of trials, but always the same success fraction 0.1, or 1 in 10.
</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">confidence_limits_on_frequency</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">2</span><span class="special">);</span> <span class="comment">// 20 trials, 2 successes, 2 in 20, = 1 in 10 = 0.1 success fraction.</span>
<span class="identifier">confidence_limits_on_frequency</span><span class="special">(</span><span class="number">200</span><span class="special">,</span> <span class="number">20</span><span class="special">);</span> <span class="comment">// More trials, but same 0.1 success fraction.</span>
<span class="identifier">confidence_limits_on_frequency</span><span class="special">(</span><span class="number">2000</span><span class="special">,</span> <span class="number">200</span><span class="special">);</span> <span class="comment">// Many more trials, but same 0.1 success fraction.</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>
Let's see some sample output for a 1 in 10 success ratio, first for
a mere 20 trials:
</p>
<pre class="programlisting">______________________________________________
2-Sided Confidence Limits For Success Fraction
______________________________________________
Number of trials = 20
Number of successes = 2
Number of failures = 18
Observed frequency of occurrence = 0.1
___________________________________________
Confidence Lower Upper
Value (%) Limit Limit
___________________________________________
50.000 0.04812 0.13554
75.000 0.03078 0.17727
90.000 0.01807 0.22637
95.000 0.01235 0.26028
99.000 0.00530 0.33111
99.900 0.00164 0.41802
99.990 0.00051 0.49202
99.999 0.00016 0.55574
</pre>
<p>
As you can see, even at the 95% confidence level the bounds (0.012
to 0.26) are really very wide, and very asymmetric about the observed
value 0.1.
</p>
<p>
Compare that with the program output for a mass 2000 trials:
</p>
<pre class="programlisting">______________________________________________
2-Sided Confidence Limits For Success Fraction
______________________________________________
Number of trials = 2000
Number of successes = 200
Number of failures = 1800
Observed frequency of occurrence = 0.1
___________________________________________
Confidence Lower Upper
Value (%) Limit Limit
___________________________________________
50.000 0.09536 0.10445
75.000 0.09228 0.10776
90.000 0.08916 0.11125
95.000 0.08720 0.11352
99.000 0.08344 0.11802
99.900 0.07921 0.12336
99.990 0.07577 0.12795
99.999 0.07282 0.13206
</pre>
<p>
Now even when the confidence level is very high, the limits (at 99.999%,
0.07 to 0.13) are really quite close and nearly symmetric to the observed
value of 0.1.
</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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