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102 lines
3.8 KiB
C++
102 lines
3.8 KiB
C++
// students_t_example1.cpp
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// Copyright Paul A. Bristow 2006, 2007.
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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 1 of using Student's t
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// http://en.wikipedia.org/wiki/Student's_t-test 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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// WS 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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// 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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// 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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double standard = 38.9;
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const int values = 3;
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double value[values] = {38.9, 37.4, 37.1};
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// Is there any evidence for systematic error?
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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; using std::endl;
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#include <iomanip>
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using std::setprecision;
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#include <cmath>
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using std::sqrt;
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int main()
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{
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cout << "Example 1 using Student's t function. " << endl;
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// Example/test using tabulated value
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// (deliberately coded as naively as possible).
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// Null hypothesis is that there is no difference (greater or less)
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// between measured and standard.
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double degrees_of_freedom = values-1; // 3-1 = 2
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cout << "Measurement 1 = " << value[0] << ", measurement 2 = " << value[1] << ", measurement 3 = " << value[2] << endl;
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double mean = (value[0] + value[1] + value[2]) / static_cast<double>(values);
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cout << "Standard = " << standard << ", mean = " << mean << ", (mean - standard) = " << mean - standard << endl;
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double sd = sqrt(((value[0] - mean) * (value[0] - mean) + (value[1] - mean) * (value[1] - mean) + (value[2] - mean) * (value[2] - mean))/ static_cast<double>(values-1));
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cout << "Standard deviation = " << sd << endl;
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if (sd == 0.)
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{
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cout << "Measured mean is identical to SRM value," << endl;
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cout << "so probability of no difference between measured and standard (the 'null hypothesis') is unity." << endl;
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return 0;
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}
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double t = (mean - standard) * std::sqrt(static_cast<double>(values)) / sd;
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cout << "Student's t = " << t << endl;
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cout.precision(2); // Useful accuracy is only a few decimal digits.
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cout << "Probability of Student's t is " << cdf(students_t(degrees_of_freedom), std::abs(t)) << endl;
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// 0.91, is 1 tailed.
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// So there is insufficient evidence of a difference to meet a 95% (1 in 20) criterion.
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return 0;
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} // int main()
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/*
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Output is:
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Example 1 using Student's t function.
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Measurement 1 = 38.9, measurement 2 = 37.4, measurement 3 = 37.1
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Standard = 38.9, mean = 37.8, (mean - standard) = -1.1
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Standard deviation = 0.964365
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Student's t = -1.97566
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Probability of Student's t is 0.91
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*/
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