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142 lines
5.1 KiB
C++
142 lines
5.1 KiB
C++
// students_t_example3.cpp
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// Copyright Paul A. Bristow 2006.
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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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// Example 3 of using Student's t
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// A general guide to Student's t is at
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// http://en.wikipedia.org/wiki/Student's_t-test
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// (and many other elementary and advanced statistics texts).
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// It says:
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// The t statistic was invented by William Sealy Gosset
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// for cheaply monitoring the quality of beer brews.
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// "Student" was his pen name.
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// Gosset was statistician for Guinness brewery in Dublin, Ireland,
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// hired due to Claude Guinness's innovative policy of recruiting the
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// best graduates from Oxford and Cambridge for applying biochemistry
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// and statistics to Guinness's industrial processes.
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// Gosset published the t test in Biometrika in 1908,
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// but was forced to use a pen name by his employer who regarded the fact
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// that they were using statistics as a trade secret.
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// In fact, Gosset's identity was unknown not only to fellow statisticians
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// but to his employer - the company insisted on the pseudonym
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// so that it could turn a blind eye to the breach of its rules.
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// The Students't distribution function is described at
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// http://en.wikipedia.org/wiki/Student%27s_t_distribution
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#include <boost/math/distributions/students_t.hpp>
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using boost::math::students_t; // Probability of students_t(df, t).
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#include <iostream>
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using std::cout;
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using std::endl;
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#include <iomanip>
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using std::setprecision;
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using std::setw;
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#include <cmath>
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using std::sqrt;
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// This example of a two-sided test is from:
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// B. M. Smith & M. B. Griffiths, Analyst, 1982, 107, 253,
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// from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 58-59
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// J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907
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// Concentrations of lead (ug/l) determined by two different methods
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// for each of four test portions,
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// the concentration of each portion is significantly different,
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// the values may NOT be pooled.
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// Portion Wet oxidation Direct Extraction
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// 1 71 76
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// 2 61 68
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// 3 50 48
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// 4 60 57
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const int portions = 4;
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const int methods = 2;
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float data [portions][methods] = {{71, 76}, {61,68}, {50, 48}, {60, 57}};
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float diffs[portions];
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int main()
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{
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cout << "Example3 using Student's t function. ";
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#if defined(__FILE__) && defined(__TIMESTAMP__)
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cout << " " << __FILE__ << ' ' << __TIMESTAMP__ << ' '<< _MSC_FULL_VER;
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#endif
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cout << endl;
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float mean_diff = 0.f;
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cout << "\n""Portion wet_oxidation Direct_extraction difference" << endl;
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for (int portion = 0; portion < portions; portion++)
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{ // Echo data and differences.
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diffs[portion] = data[portion][0] - data[portion][1];
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mean_diff += diffs[portion];
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cout << setw(4) << portion << ' ' << setw(14) << data[portion][0] << ' ' << setw(18)<< data[portion][1] << ' ' << setw(9) << diffs[portion] << endl;
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}
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mean_diff /= portions;
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cout << "Mean difference = " << mean_diff << endl; // -1.75
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float sd_diffs = 0.f;
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for (int portion = 0; portion < portions; portion++)
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{ // Calculate standard deviation of differences.
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sd_diffs +=(diffs[portion] - mean_diff) * (diffs[portion] - mean_diff);
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}
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int degrees_of_freedom = portions-1; // Use the n-1 formula.
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sd_diffs /= degrees_of_freedom;
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sd_diffs = sqrt(sd_diffs);
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cout << "Standard deviation of differences = " << sd_diffs << endl; // 4.99166
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double t = mean_diff * sqrt(static_cast<double>(portions))/ sd_diffs; // -0.70117
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cout << "Student's t = " << t << ", if " << degrees_of_freedom << " degrees of freedom." << endl; //
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cout << "Probability of the means being different is " << 2.F * cdf(students_t(degrees_of_freedom), t) << "."<< endl; // 0.266846 * 2 =
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// Double the probability because using a 'two-sided test' because
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// mean for 'Wet Oxidation' could be either greater OR LESS than for 'Direct extraction'.
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return 0;
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} // int main()
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/*
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Check using http://faculty.vassar.edu/lowry/VassarStats.html
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Output from avoid data using
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Independent samples is
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n = 4,
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sum 242 and 249 total 491
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sumsq 14862 15953 30815
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SS 221 452.75 679.875
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means 60.5 62.25 61.375
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Mean diff = -1.75, t = -0.23, df 6,
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independent samples: p 1-tailed 0.4128665, 2-tailed 0.825733
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correlated samples: p 1-tailed 0.261635, 2-tailed 0.534327
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(Called a paired test in Miller and Miller because each portion analysed has a different concentration.)
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Output is:
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Example3 using Student's t function. ..\..\..\..\..\..\boost-sandbox\libs\math
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_functions\example\students_t_example3.cpp Sat Jun 17 14:52:44 2006 140050727
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Portion wet_oxidation Direct_extraction difference
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0 71 76 -5
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1 61 68 -7
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2 50 48 2
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3 60 57 3
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Mean difference = -1.75
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Standard deviation of differences = 4.99166
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Student's t = -0.70117, if 3 degrees of freedom.
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Probability of the means being different is 0.533692.
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Press any key to continue . . .
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*/
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