<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution</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="../binom_eg.html" title="Binomial Distribution Examples"> <link rel="prev" href="binomial_quiz_example.html" title="Binomial Quiz Example"> <link rel="next" href="binom_size_eg.html" title="Estimating Sample Sizes for a Binomial Distribution."> </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="binomial_quiz_example.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../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="binom_size_eg.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.binom_eg.binom_conf"></a><a class="link" href="binom_conf.html" title="Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution"> Calculating Confidence Limits on the Frequency of Occurrence for a Binomial Distribution</a> </h6></div></div></div> <p> Imagine you have a process that follows a binomial distribution: for each trial conducted, an event either occurs or does it does not, referred to as "successes" and "failures". If, by experiment, you want to measure the frequency with which successes occur, the best estimate 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. 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">binomial_distribution</span><span class="special"><>::</span><span class="identifier">find_lower_bound_on_p</span></code> and <code class="computeroutput"><span class="identifier">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 occurrence frequency. </p> <p> The sample program <a href="../../../../../../../../example/binomial_confidence_limits.cpp" target="_top">binomial_confidence_limits.cpp</a> illustrates their use. It begins by defining a procedure that will print a table of confidence limits for various degrees of certainty: </p> <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</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="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">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <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">// </span> <span class="comment">// trials = Total number of trials. </span> <span class="comment">// successes = Total number of observed successes. </span> <span class="comment">// </span> <span class="comment">// Calculate confidence limits for an observed </span> <span class="comment">// frequency of occurrence that follows a binomial </span> <span class="comment">// distribution. </span> <span class="comment">// </span> <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">std</span><span class="special">;</span> <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</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 Ratio\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 Observations"</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">"Sample 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> </pre> <p> The procedure now defines a table of significance levels: these are the probabilities that the true occurrence frequency lies outside the calculated interval: </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> Some pretty printing of the table header follows: </p> <pre class="programlisting"><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 CP Upper CP Lower JP Upper JP\n"</span> <span class="string">" Value (%) Limit Limit Limit Limit\n"</span> <span class="string">"_______________________________________________________________________\n"</span><span class="special">;</span> </pre> <p> And now for the important part - the intervals 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_lower_upper_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. </p> <p> Please note that calculating two separate <span class="emphasis"><em>single sided bounds</em></span>, each with risk level αis not the same thing as calculating a two sided interval. Had we calculate two single-sided intervals each with a risk that the true value is outside the interval of α, then: </p> <div class="itemizedlist"><ul type="disc"><li> The risk that it is less than the lower bound is α. </li></ul></div> <p> and </p> <div class="itemizedlist"><ul type="disc"><li> The risk that it is greater than the upper bound is also α. </li></ul></div> <p> So the risk it is outside <span class="bold"><strong>upper or lower bound</strong></span>, is <span class="bold"><strong>twice</strong></span> alpha, and the probability that it is inside the bounds is therefore not nearly as high as one might have thought. This is why α/2 must be used in the calculations below. </p> <p> In contrast, 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> Finally note that <code class="computeroutput"><span class="identifier">binomial_distribution</span></code> provides a choice of two methods for the calculation, we print out the results from both methods in this example: </p> <pre class="programlisting"> <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 Clopper Pearson bounds: </span> <span class="keyword">double</span> <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">binomial_distribution</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">u</span> <span class="special">=</span> <span class="identifier">binomial_distribution</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 Clopper Pearson 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">l</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">u</span><span class="special">;</span> <span class="comment">// Calculate Jeffreys Prior Bounds: </span> <span class="identifier">l</span> <span class="special">=</span> <span class="identifier">binomial_distribution</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="identifier">binomial_distribution</span><span class="special"><>::</span><span class="identifier">jeffreys_prior_interval</span><span class="special">);</span> <span class="identifier">u</span> <span class="special">=</span> <span class="identifier">binomial_distribution</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="identifier">binomial_distribution</span><span class="special"><>::</span><span class="identifier">jeffreys_prior_interval</span><span class="special">);</span> <span class="comment">// Print Jeffreys Prior 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">l</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">u</span> <span class="special"><<</span> <span class="identifier">std</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> </pre> <p> And that's all there is to it. Let's see some sample output for a 2 in 10 success ratio, first for 20 trials: </p> <pre class="programlisting">___________________________________________ 2-Sided Confidence Limits For Success Ratio ___________________________________________ Number of Observations = 20 Number of successes = 4 Sample frequency of occurrence = 0.2 _______________________________________________________________________ Confidence Lower CP Upper CP Lower JP Upper JP Value (%) Limit Limit Limit Limit _______________________________________________________________________ 50.000 0.12840 0.29588 0.14974 0.26916 75.000 0.09775 0.34633 0.11653 0.31861 90.000 0.07135 0.40103 0.08734 0.37274 95.000 0.05733 0.43661 0.07152 0.40823 99.000 0.03576 0.50661 0.04655 0.47859 99.900 0.01905 0.58632 0.02634 0.55960 99.990 0.01042 0.64997 0.01530 0.62495 99.999 0.00577 0.70216 0.00901 0.67897 </pre> <p> As you can see, even at the 95% confidence level the bounds are really quite wide (this example is chosen to be easily compared to the one in the <a href="http://www.itl.nist.gov/div898/handbook/" target="_top">NIST/SEMATECH e-Handbook of Statistical Methods.</a> <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm" target="_top">here</a>). Note also that the Clopper-Pearson calculation method (CP above) produces quite noticeably more pessimistic estimates than the Jeffreys Prior method (JP above). </p> <p> Compare that with the program output for 2000 trials: </p> <pre class="programlisting">___________________________________________ 2-Sided Confidence Limits For Success Ratio ___________________________________________ Number of Observations = 2000 Number of successes = 400 Sample frequency of occurrence = 0.2000000 _______________________________________________________________________ Confidence Lower CP Upper CP Lower JP Upper JP Value (%) Limit Limit Limit Limit _______________________________________________________________________ 50.000 0.19382 0.20638 0.19406 0.20613 75.000 0.18965 0.21072 0.18990 0.21047 90.000 0.18537 0.21528 0.18561 0.21503 95.000 0.18267 0.21821 0.18291 0.21796 99.000 0.17745 0.22400 0.17769 0.22374 99.900 0.17150 0.23079 0.17173 0.23053 99.990 0.16658 0.23657 0.16681 0.23631 99.999 0.16233 0.24169 0.16256 0.24143 </pre> <p> Now even when the confidence level is very high, the limits are really quite close to the experimentally calculated value of 0.2. Furthermore the difference between the two calculation methods is now really quite small. </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="binomial_quiz_example.html"><img src="../../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../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="binom_size_eg.html"><img src="../../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>