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425 lines
16 KiB
C++
425 lines
16 KiB
C++
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// Copyright John Maddock 2006
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// Copyright Paul A. Bristow 2007, 2010
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifdef _MSC_VER
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# pragma warning(disable: 4512) // assignment operator could not be generated.
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# pragma warning(disable: 4510) // default constructor could not be generated.
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# pragma warning(disable: 4610) // can never be instantiated - user defined constructor required.
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#endif
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#include <boost/math/distributions/students_t.hpp>
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// avoid "using namespace std;" and "using namespace boost::math;"
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// to avoid potential ambiguity with names in std random.
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#include <iostream>
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using std::cout; using std::endl;
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using std::left; using std::fixed; using std::right; using std::scientific;
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#include <iomanip>
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using std::setw;
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using std::setprecision;
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void confidence_limits_on_mean(double Sm, double Sd, unsigned Sn)
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{
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//
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// Sm = Sample Mean.
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// Sd = Sample Standard Deviation.
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// Sn = Sample Size.
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//
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// Calculate confidence intervals for the mean.
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// For example if we set the confidence limit to
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// 0.95, we know that if we repeat the sampling
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// 100 times, then we expect that the true mean
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// will be between out limits on 95 occations.
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// Note: this is not the same as saying a 95%
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// confidence interval means that there is a 95%
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// probability that the interval contains the true mean.
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// The interval computed from a given sample either
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// contains the true mean or it does not.
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// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
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using boost::math::students_t;
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// Print out general info:
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cout <<
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"__________________________________\n"
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"2-Sided Confidence Limits For Mean\n"
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"__________________________________\n\n";
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cout << setprecision(7);
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cout << setw(40) << left << "Number of Observations" << "= " << Sn << "\n";
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cout << setw(40) << left << "Mean" << "= " << Sm << "\n";
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cout << setw(40) << left << "Standard Deviation" << "= " << Sd << "\n";
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//
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// Define a table of significance/risk levels:
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//
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double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
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//
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// Start by declaring the distribution we'll need:
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//
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students_t dist(Sn - 1);
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//
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// Print table header:
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//
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cout << "\n\n"
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"_______________________________________________________________\n"
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"Confidence T Interval Lower Upper\n"
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" Value (%) Value Width Limit Limit\n"
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"_______________________________________________________________\n";
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//
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// Now print out the data for the table rows.
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//
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for(unsigned i = 0; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
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{
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// Confidence value:
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cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
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// calculate T:
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double T = quantile(complement(dist, alpha[i] / 2));
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// Print T:
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cout << fixed << setprecision(3) << setw(10) << right << T;
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// Calculate width of interval (one sided):
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double w = T * Sd / sqrt(double(Sn));
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// Print width:
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if(w < 0.01)
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cout << scientific << setprecision(3) << setw(17) << right << w;
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else
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cout << fixed << setprecision(3) << setw(17) << right << w;
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// Print Limits:
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cout << fixed << setprecision(5) << setw(15) << right << Sm - w;
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cout << fixed << setprecision(5) << setw(15) << right << Sm + w << endl;
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}
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cout << endl;
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} // void confidence_limits_on_mean
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void single_sample_t_test(double M, double Sm, double Sd, unsigned Sn, double alpha)
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{
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//
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// M = true mean.
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// Sm = Sample Mean.
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// Sd = Sample Standard Deviation.
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// Sn = Sample Size.
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// alpha = Significance Level.
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//
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// A Students t test applied to a single set of data.
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// We are testing the null hypothesis that the true
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// mean of the sample is M, and that any variation is down
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// to chance. We can also test the alternative hypothesis
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// that any difference is not down to chance.
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// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda352.htm
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using boost::math::students_t;
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// Print header:
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cout <<
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"__________________________________\n"
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"Student t test for a single sample\n"
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"__________________________________\n\n";
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cout << setprecision(5);
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cout << setw(55) << left << "Number of Observations" << "= " << Sn << "\n";
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cout << setw(55) << left << "Sample Mean" << "= " << Sm << "\n";
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cout << setw(55) << left << "Sample Standard Deviation" << "= " << Sd << "\n";
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cout << setw(55) << left << "Expected True Mean" << "= " << M << "\n\n";
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//
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// Now we can calculate and output some stats:
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//
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// Difference in means:
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double diff = Sm - M;
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cout << setw(55) << left << "Sample Mean - Expected Test Mean" << "= " << diff << "\n";
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// Degrees of freedom:
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unsigned v = Sn - 1;
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cout << setw(55) << left << "Degrees of Freedom" << "= " << v << "\n";
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// t-statistic:
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double t_stat = diff * sqrt(double(Sn)) / Sd;
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cout << setw(55) << left << "T Statistic" << "= " << t_stat << "\n";
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//
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// Finally define our distribution, and get the probability:
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//
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students_t dist(v);
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double q = cdf(complement(dist, fabs(t_stat)));
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cout << setw(55) << left << "Probability that difference is due to chance" << "= "
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<< setprecision(3) << scientific << 2 * q << "\n\n";
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//
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// Finally print out results of alternative hypothesis:
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//
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cout << setw(55) << left <<
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"Results for Alternative Hypothesis and alpha" << "= "
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<< setprecision(4) << fixed << alpha << "\n\n";
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cout << "Alternative Hypothesis Conclusion\n";
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cout << "Mean != " << setprecision(3) << fixed << M << " ";
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if(q < alpha / 2)
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cout << "NOT REJECTED\n";
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else
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cout << "REJECTED\n";
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cout << "Mean < " << setprecision(3) << fixed << M << " ";
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if(cdf(complement(dist, t_stat)) > alpha)
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cout << "NOT REJECTED\n";
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else
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cout << "REJECTED\n";
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cout << "Mean > " << setprecision(3) << fixed << M << " ";
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if(cdf(dist, t_stat) > alpha)
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cout << "NOT REJECTED\n";
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else
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cout << "REJECTED\n";
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cout << endl << endl;
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} // void single_sample_t_test(
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void single_sample_find_df(double M, double Sm, double Sd)
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{
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//
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// M = true mean.
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// Sm = Sample Mean.
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// Sd = Sample Standard Deviation.
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//
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using boost::math::students_t;
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// Print out general info:
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cout <<
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"_____________________________________________________________\n"
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"Estimated sample sizes required for various confidence levels\n"
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"_____________________________________________________________\n\n";
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cout << setprecision(5);
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cout << setw(40) << left << "True Mean" << "= " << M << "\n";
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cout << setw(40) << left << "Sample Mean" << "= " << Sm << "\n";
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cout << setw(40) << left << "Sample Standard Deviation" << "= " << Sd << "\n";
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//
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// Define a table of significance intervals:
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//
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double alpha[] = { 0.5, 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.00001 };
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//
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// Print table header:
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//
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cout << "\n\n"
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"_______________________________________________________________\n"
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"Confidence Estimated Estimated\n"
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" Value (%) Sample Size Sample Size\n"
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" (one sided test) (two sided test)\n"
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"_______________________________________________________________\n";
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//
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// Now print out the data for the table rows.
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//
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for(unsigned i = 1; i < sizeof(alpha)/sizeof(alpha[0]); ++i)
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{
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// Confidence value:
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cout << fixed << setprecision(3) << setw(10) << right << 100 * (1-alpha[i]);
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// calculate df for single sided test:
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double df = students_t::find_degrees_of_freedom(
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fabs(M - Sm), alpha[i], alpha[i], Sd);
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// convert to sample size, always one more than the degrees of freedom:
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double size = ceil(df) + 1;
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// Print size:
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cout << fixed << setprecision(0) << setw(16) << right << size;
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// calculate df for two sided test:
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df = students_t::find_degrees_of_freedom(
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fabs(M - Sm), alpha[i]/2, alpha[i], Sd);
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// convert to sample size:
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size = ceil(df) + 1;
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// Print size:
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cout << fixed << setprecision(0) << setw(16) << right << size << endl;
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}
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cout << endl;
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} // void single_sample_find_df
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int main()
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{
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//
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// Run tests for Heat Flow Meter data
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// see http://www.itl.nist.gov/div898/handbook/eda/section4/eda428.htm
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// The data was collected while calibrating a heat flow meter
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// against a known value.
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//
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confidence_limits_on_mean(9.261460, 0.2278881e-01, 195);
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single_sample_t_test(5, 9.261460, 0.2278881e-01, 195, 0.05);
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single_sample_find_df(5, 9.261460, 0.2278881e-01);
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//
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// Data for this example from:
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// P.K.Hou, O. W. Lau & M.C. Wong, Analyst (1983) vol. 108, p 64.
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// from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 54-55
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// J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907
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//
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// Determination of mercury by cold-vapour atomic absorption,
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// the following values were obtained fusing a trusted
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// Standard Reference Material containing 38.9% mercury,
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// which we assume is correct or 'true'.
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//
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confidence_limits_on_mean(37.8, 0.964365, 3);
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// 95% test:
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single_sample_t_test(38.9, 37.8, 0.964365, 3, 0.05);
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// 90% test:
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single_sample_t_test(38.9, 37.8, 0.964365, 3, 0.1);
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// parameter estimate:
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single_sample_find_df(38.9, 37.8, 0.964365);
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return 0;
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} // int main()
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/*
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Output:
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------ Rebuild All started: Project: students_t_single_sample, Configuration: Release Win32 ------
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students_t_single_sample.cpp
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Generating code
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Finished generating code
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students_t_single_sample.vcxproj -> J:\Cpp\MathToolkit\test\Math_test\Release\students_t_single_sample.exe
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__________________________________
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2-Sided Confidence Limits For Mean
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__________________________________
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Number of Observations = 195
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Mean = 9.26146
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Standard Deviation = 0.02278881
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_______________________________________________________________
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Confidence T Interval Lower Upper
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Value (%) Value Width Limit Limit
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_______________________________________________________________
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50.000 0.676 1.103e-003 9.26036 9.26256
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75.000 1.154 1.883e-003 9.25958 9.26334
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90.000 1.653 2.697e-003 9.25876 9.26416
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95.000 1.972 3.219e-003 9.25824 9.26468
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99.000 2.601 4.245e-003 9.25721 9.26571
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99.900 3.341 5.453e-003 9.25601 9.26691
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99.990 3.973 6.484e-003 9.25498 9.26794
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99.999 4.537 7.404e-003 9.25406 9.26886
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__________________________________
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Student t test for a single sample
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__________________________________
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Number of Observations = 195
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Sample Mean = 9.26146
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Sample Standard Deviation = 0.02279
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Expected True Mean = 5.00000
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Sample Mean - Expected Test Mean = 4.26146
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Degrees of Freedom = 194
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T Statistic = 2611.28380
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Probability that difference is due to chance = 0.000e+000
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Results for Alternative Hypothesis and alpha = 0.0500
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Alternative Hypothesis Conclusion
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Mean != 5.000 NOT REJECTED
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Mean < 5.000 REJECTED
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Mean > 5.000 NOT REJECTED
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_____________________________________________________________
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Estimated sample sizes required for various confidence levels
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_____________________________________________________________
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True Mean = 5.00000
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Sample Mean = 9.26146
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Sample Standard Deviation = 0.02279
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_______________________________________________________________
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Confidence Estimated Estimated
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Value (%) Sample Size Sample Size
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(one sided test) (two sided test)
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_______________________________________________________________
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75.000 2 2
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90.000 2 2
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95.000 2 2
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99.000 2 2
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99.900 3 3
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99.990 3 3
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99.999 4 4
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__________________________________
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2-Sided Confidence Limits For Mean
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__________________________________
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Number of Observations = 3
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Mean = 37.8000000
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Standard Deviation = 0.9643650
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_______________________________________________________________
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Confidence T Interval Lower Upper
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Value (%) Value Width Limit Limit
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_______________________________________________________________
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50.000 0.816 0.455 37.34539 38.25461
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75.000 1.604 0.893 36.90717 38.69283
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90.000 2.920 1.626 36.17422 39.42578
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95.000 4.303 2.396 35.40438 40.19562
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|
99.000 9.925 5.526 32.27408 43.32592
|
||
|
|
99.900 31.599 17.594 20.20639 55.39361
|
||
|
|
99.990 99.992 55.673 -17.87346 93.47346
|
||
|
|
99.999 316.225 176.067 -138.26683 213.86683
|
||
|
|
|
||
|
|
__________________________________
|
||
|
|
Student t test for a single sample
|
||
|
|
__________________________________
|
||
|
|
|
||
|
|
Number of Observations = 3
|
||
|
|
Sample Mean = 37.80000
|
||
|
|
Sample Standard Deviation = 0.96437
|
||
|
|
Expected True Mean = 38.90000
|
||
|
|
|
||
|
|
Sample Mean - Expected Test Mean = -1.10000
|
||
|
|
Degrees of Freedom = 2
|
||
|
|
T Statistic = -1.97566
|
||
|
|
Probability that difference is due to chance = 1.869e-001
|
||
|
|
|
||
|
|
Results for Alternative Hypothesis and alpha = 0.0500
|
||
|
|
|
||
|
|
Alternative Hypothesis Conclusion
|
||
|
|
Mean != 38.900 REJECTED
|
||
|
|
Mean < 38.900 NOT REJECTED
|
||
|
|
Mean > 38.900 NOT REJECTED
|
||
|
|
|
||
|
|
|
||
|
|
__________________________________
|
||
|
|
Student t test for a single sample
|
||
|
|
__________________________________
|
||
|
|
|
||
|
|
Number of Observations = 3
|
||
|
|
Sample Mean = 37.80000
|
||
|
|
Sample Standard Deviation = 0.96437
|
||
|
|
Expected True Mean = 38.90000
|
||
|
|
|
||
|
|
Sample Mean - Expected Test Mean = -1.10000
|
||
|
|
Degrees of Freedom = 2
|
||
|
|
T Statistic = -1.97566
|
||
|
|
Probability that difference is due to chance = 1.869e-001
|
||
|
|
|
||
|
|
Results for Alternative Hypothesis and alpha = 0.1000
|
||
|
|
|
||
|
|
Alternative Hypothesis Conclusion
|
||
|
|
Mean != 38.900 REJECTED
|
||
|
|
Mean < 38.900 NOT REJECTED
|
||
|
|
Mean > 38.900 REJECTED
|
||
|
|
|
||
|
|
|
||
|
|
_____________________________________________________________
|
||
|
|
Estimated sample sizes required for various confidence levels
|
||
|
|
_____________________________________________________________
|
||
|
|
|
||
|
|
True Mean = 38.90000
|
||
|
|
Sample Mean = 37.80000
|
||
|
|
Sample Standard Deviation = 0.96437
|
||
|
|
|
||
|
|
|
||
|
|
_______________________________________________________________
|
||
|
|
Confidence Estimated Estimated
|
||
|
|
Value (%) Sample Size Sample Size
|
||
|
|
(one sided test) (two sided test)
|
||
|
|
_______________________________________________________________
|
||
|
|
75.000 3 4
|
||
|
|
90.000 7 9
|
||
|
|
95.000 11 13
|
||
|
|
99.000 20 22
|
||
|
|
99.900 35 37
|
||
|
|
99.990 50 53
|
||
|
|
99.999 66 68
|
||
|
|
|
||
|
|
*/
|
||
|
|
|