...one of the most highly
regarded and expertly designed C++ library projects in the
world.

— Herb Sutter and Andrei
Alexandrescu, C++
Coding Standards

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 *k* / *N*,
for *k* successes out of *N*
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 `binomial_distribution<>::find_lower_bound_on_p`

and `binomial_distribution<>::find_upper_bound_on_p`

allow you to calculate the confidence intervals for your estimate of
the occurrence frequency.

The sample program binomial_confidence_limits.cpp illustrates their use. It begins by defining a procedure that will print a table of confidence limits for various degrees of certainty:

#include <iostream> #include <iomanip> #include <boost/math/distributions/binomial.hpp> void confidence_limits_on_frequency(unsigned trials, unsigned successes) { // // trials = Total number of trials. // successes = Total number of observed successes. // // Calculate confidence limits for an observed // frequency of occurrence that follows a binomial // distribution. // using namespace std; using namespace boost::math; // Print out general info: cout << "___________________________________________\n" "2-Sided Confidence Limits For Success Ratio\n" "___________________________________________\n\n"; cout << setprecision(7); cout << setw(40) << left << "Number of Observations" << "= " << trials << "\n"; cout << setw(40) << left << "Number of successes" << "= " << successes << "\n"; cout << setw(40) << left << "Sample frequency of occurrence" << "= " << double(successes) / trials << "\n";

The procedure now defines a table of significance levels: these are the probabilities that the true occurrence frequency lies outside the calculated interval:

double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };

Some pretty printing of the table header follows:

cout << "\n\n" "_______________________________________________________________________\n" "Confidence Lower CP Upper CP Lower JP Upper JP\n" " Value (%) Limit Limit Limit Limit\n" "_______________________________________________________________________\n";

And now for the important part - the intervals themselves - for each
value of *alpha*, we call `find_lower_bound_on_p`

and `find_lower_upper_on_p`

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.

Please note that calculating two separate *single sided bounds*,
each with risk level α is 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:

- The risk that it is less than the lower bound is α.

and

- The risk that it is greater than the upper bound is also α.

So the risk it is outside **upper or lower bound**,
is **twice** 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.

In contrast, had we been calculating a single-sided interval, for example:
*"Calculate a lower bound so that we are P% sure that
the true occurrence frequency is greater than some value"*
then we would **not** have divided by
two.

Finally note that `binomial_distribution`

provides a choice of two methods for the calculation, we print out
the results from both methods in this example:

for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i) { // Confidence value: cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]); // Calculate Clopper Pearson bounds: double l = binomial_distribution<>::find_lower_bound_on_p( trials, successes, alpha[i]/2); double u = binomial_distribution<>::find_upper_bound_on_p( trials, successes, alpha[i]/2); // Print Clopper Pearson Limits: cout << fixed << setprecision(5) << setw(15) << right << l; cout << fixed << setprecision(5) << setw(15) << right << u; // Calculate Jeffreys Prior Bounds: l = binomial_distribution<>::find_lower_bound_on_p( trials, successes, alpha[i]/2, binomial_distribution<>::jeffreys_prior_interval); u = binomial_distribution<>::find_upper_bound_on_p( trials, successes, alpha[i]/2, binomial_distribution<>::jeffreys_prior_interval); // Print Jeffreys Prior Limits: cout << fixed << setprecision(5) << setw(15) << right << l; cout << fixed << setprecision(5) << setw(15) << right << u << std::endl; } cout << endl; }

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:

___________________________________________ 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

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 NIST/SEMATECH e-Handbook of Statistical Methods. here). Note also that the Clopper-Pearson calculation method (CP above) produces quite noticeably more pessimistic estimates than the Jeffreys Prior method (JP above).

Compare that with the program output for 2000 trials:

___________________________________________ 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

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.