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// Copyright John Maddock 2006
// Copyright Paul A. Bristow 2007, 2008, 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)
#ifdef _MSC_VER
# pragma warning(disable: 4512) // assignment operator could not be generated.
# pragma warning(disable: 4510) // default constructor could not be generated.
# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required.
# pragma warning(disable: 4180) // qualifier has no effect (in Fusion).
#endif
#include <iostream>
using std::cout; using std::endl;
using std::left; using std::fixed; using std::right; using std::scientific;
#include <iomanip>
using std::setw;
using std::setprecision;
#include <boost/math/distributions/fisher_f.hpp>
void f_test(
double sd1, // Sample 1 std deviation
double sd2, // Sample 2 std deviation
double N1, // Sample 1 size
double N2, // Sample 2 size
double alpha) // Significance level
{
//
// An F test applied to two sets of data.
// We are testing the null hypothesis that the
// standard deviation of the samples is equal, and
// that any variation is down to chance. We can
// also test the alternative hypothesis that any
// difference is not down to chance.
// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm
//
// Avoid "using namespace boost::math;" because of potential name ambiguity.
using boost::math::fisher_f;
// Print header:
cout <<
"____________________________________\n"
"F test for equal standard deviations\n"
"____________________________________\n\n";
cout << setprecision(5);
cout << "Sample 1:\n";
cout << setw(55) << left << "Number of Observations" << "= " << N1 << "\n";
cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd1 << "\n\n";
cout << "Sample 2:\n";
cout << setw(55) << left << "Number of Observations" << "= " << N2 << "\n";
cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd2 << "\n\n";
//
// Now we can calculate and output some stats:
//
// F-statistic:
double F = (sd1 / sd2);
F *= F;
cout << setw(55) << left << "Test Statistic" << "= " << F << "\n\n";
//
// Finally define our distribution, and get the probability:
//
fisher_f dist(N1 - 1, N2 - 1);
double p = cdf(dist, F);
cout << setw(55) << left << "CDF of test statistic: " << "= "
<< setprecision(3) << scientific << p << "\n";
double ucv = quantile(complement(dist, alpha));
double ucv2 = quantile(complement(dist, alpha / 2));
double lcv = quantile(dist, alpha);
double lcv2 = quantile(dist, alpha / 2);
cout << setw(55) << left << "Upper Critical Value at alpha: " << "= "
<< setprecision(3) << scientific << ucv << "\n";
cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "= "
<< setprecision(3) << scientific << ucv2 << "\n";
cout << setw(55) << left << "Lower Critical Value at alpha: " << "= "
<< setprecision(3) << scientific << lcv << "\n";
cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "= "
<< setprecision(3) << scientific << lcv2 << "\n\n";
//
// Finally print out results of null and alternative hypothesis:
//
cout << setw(55) << left <<
"Results for Alternative Hypothesis and alpha" << "= "
<< setprecision(4) << fixed << alpha << "\n\n";
cout << "Alternative Hypothesis Conclusion\n";
cout << "Standard deviations are unequal (two sided test) ";
if((ucv2 < F) || (lcv2 > F))
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << "Standard deviation 1 is less than standard deviation 2 ";
if(lcv > F)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << "Standard deviation 1 is greater than standard deviation 2 ";
if(ucv < F)
cout << "NOT REJECTED\n";
else
cout << "REJECTED\n";
cout << endl << endl;
}
int main()
{
//
// Run tests for ceramic strength data:
// see http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm
// The data for this case study were collected by Said Jahanmir of the
// NIST Ceramics Division in 1996 in connection with a NIST/industry
// ceramics consortium for strength optimization of ceramic strength.
//
f_test(65.54909, 61.85425, 240, 240, 0.05);
//
// And again for the process change comparison:
// see http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm
// A new procedure to assemble a device is introduced and tested for
// possible improvement in time of assembly. The question being addressed
// is whether the standard deviation of the new assembly process (sample 2) is
// better (i.e., smaller) than the standard deviation for the old assembly
// process (sample 1).
//
f_test(4.9082, 2.5874, 11, 9, 0.05);
return 0;
}
/*
Output:
f_test.cpp
F-test_example1.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Debug\F_test_example1.exe
____________________________________
F test for equal standard deviations
____________________________________
Sample 1:
Number of Observations = 240
Sample Standard Deviation = 65.549
Sample 2:
Number of Observations = 240
Sample Standard Deviation = 61.854
Test Statistic = 1.123
CDF of test statistic: = 8.148e-001
Upper Critical Value at alpha: = 1.238e+000
Upper Critical Value at alpha/2: = 1.289e+000
Lower Critical Value at alpha: = 8.080e-001
Lower Critical Value at alpha/2: = 7.756e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard deviations are unequal (two sided test) REJECTED
Standard deviation 1 is less than standard deviation 2 REJECTED
Standard deviation 1 is greater than standard deviation 2 REJECTED
____________________________________
F test for equal standard deviations
____________________________________
Sample 1:
Number of Observations = 11.00000
Sample Standard Deviation = 4.90820
Sample 2:
Number of Observations = 9.00000
Sample Standard Deviation = 2.58740
Test Statistic = 3.59847
CDF of test statistic: = 9.589e-001
Upper Critical Value at alpha: = 3.347e+000
Upper Critical Value at alpha/2: = 4.295e+000
Lower Critical Value at alpha: = 3.256e-001
Lower Critical Value at alpha/2: = 2.594e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard deviations are unequal (two sided test) REJECTED
Standard deviation 1 is less than standard deviation 2 REJECTED
Standard deviation 1 is greater than standard deviation 2 NOT REJECTED
____________________________________
F test for equal standard deviations
____________________________________
Sample 1:
Number of Observations = 240
Sample Standard Deviation = 65.549
Sample 2:
Number of Observations = 240
Sample Standard Deviation = 61.854
Test Statistic = 1.123
CDF of test statistic: = 8.148e-001
Upper Critical Value at alpha: = 1.238e+000
Upper Critical Value at alpha/2: = 1.289e+000
Lower Critical Value at alpha: = 8.080e-001
Lower Critical Value at alpha/2: = 7.756e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard deviations are unequal (two sided test) REJECTED
Standard deviation 1 is less than standard deviation 2 REJECTED
Standard deviation 1 is greater than standard deviation 2 REJECTED
____________________________________
F test for equal standard deviations
____________________________________
Sample 1:
Number of Observations = 11.00000
Sample Standard Deviation = 4.90820
Sample 2:
Number of Observations = 9.00000
Sample Standard Deviation = 2.58740
Test Statistic = 3.59847
CDF of test statistic: = 9.589e-001
Upper Critical Value at alpha: = 3.347e+000
Upper Critical Value at alpha/2: = 4.295e+000
Lower Critical Value at alpha: = 3.256e-001
Lower Critical Value at alpha/2: = 2.594e-001
Results for Alternative Hypothesis and alpha = 0.0500
Alternative Hypothesis Conclusion
Standard deviations are unequal (two sided test) REJECTED
Standard deviation 1 is less than standard deviation 2 REJECTED
Standard deviation 1 is greater than standard deviation 2 NOT REJECTED
*/
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