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// neg_binomial_sample_sizes.cpp
// Copyright John Maddock 2006
// Copyright Paul A. Bristow 2007, 2010
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt
// or copy at http://www.boost.org/LICENSE_1_0.txt)
#include <boost/math/distributions/negative_binomial.hpp>
using boost::math::negative_binomial;
// Default RealType is double so this permits use of:
double find_minimum_number_of_trials(
double k, // number of failures (events), k >= 0.
double p, // fraction of trails for which event occurs, 0 <= p <= 1.
double probability); // probability threshold, 0 <= probability <= 1.
#include <iostream>
using std::cout;
using std::endl;
using std::fixed;
using std::right;
#include <iomanip>
using std::setprecision;
using std::setw;
//[neg_binomial_sample_sizes
/*`
It centres around a routine that prints out a table of
minimum sample sizes (number of trials) for various probability thresholds:
*/
void find_number_of_trials(double failures, double p);
/*`
First define a table of significance levels: these are the maximum
acceptable probability that /failure/ or fewer events will be observed.
*/
double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
/*`
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.
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:
(int)ceil(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]);
find_minimum_number_of_trials returns a double,
so `ceil` rounds this up to ensure we have an integral minimum number of trials.
*/
void find_number_of_trials(double failures, double p)
{
// trials = number of trials
// failures = number of failures before achieving required success(es).
// p = success fraction (0 <= p <= 1.).
//
// Calculate how many trials we need to ensure the
// required number of failures DOES exceed "failures".
cout << "\n""Target number of failures = " << (int)failures;
cout << ", Success fraction = " << fixed << setprecision(1) << 100 * p << "%" << endl;
// Print table header:
cout << "____________________________\n"
"Confidence Min Number\n"
" Value (%) Of Trials \n"
"____________________________\n";
// Now print out the data for the alpha table values.
for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
{ // Confidence values %:
cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]) << " "
// find_minimum_number_of_trials
<< setw(6) << right
<< (int)ceil(negative_binomial::find_minimum_number_of_trials(failures, p, alpha[i]))
<< endl;
}
cout << endl;
} // void find_number_of_trials(double failures, double p)
/*` finally we can produce some tables of minimum trials for the chosen confidence levels:
*/
int main()
{
find_number_of_trials(5, 0.5);
find_number_of_trials(50, 0.5);
find_number_of_trials(500, 0.5);
find_number_of_trials(50, 0.1);
find_number_of_trials(500, 0.1);
find_number_of_trials(5, 0.9);
return 0;
} // int main()
//] [/neg_binomial_sample_sizes.cpp end of Quickbook in C++ markup]
/*
Output is:
Target number of failures = 5, Success fraction = 50.0%
____________________________
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
Target number of failures = 50, Success fraction = 50.0%
____________________________
Confidence Min Number
Value (%) Of Trials
____________________________
50.000 101
75.000 109
90.000 115
95.000 119
99.000 128
99.900 137
99.990 146
99.999 154
Target number of failures = 500, Success fraction = 50.0%
____________________________
Confidence Min Number
Value (%) Of Trials
____________________________
50.000 1001
75.000 1023
90.000 1043
95.000 1055
99.000 1078
99.900 1104
99.990 1126
99.999 1146
Target number of failures = 50, Success fraction = 10.0%
____________________________
Confidence Min Number
Value (%) Of Trials
____________________________
50.000 56
75.000 58
90.000 60
95.000 61
99.000 63
99.900 66
99.990 68
99.999 71
Target number of failures = 500, Success fraction = 10.0%
____________________________
Confidence Min Number
Value (%) Of Trials
____________________________
50.000 556
75.000 562
90.000 567
95.000 570
99.000 576
99.900 583
99.990 588
99.999 594
Target number of failures = 5, Success fraction = 90.0%
____________________________
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
Target number of failures = 5, Success fraction = 95.0%
____________________________
Confidence Min Number
Value (%) Of Trials
____________________________
50.000 114
75.000 148
90.000 184
95.000 208
99.000 259
99.900 324
99.990 384
99.999 442
*/
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