| // students_t_example3.cpp |
| // Copyright Paul A. Bristow 2006, 2007. |
| |
| // 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) |
| |
| // Example 3 of using Student's t. |
| |
| // A general guide to Student's t is at |
| // http://en.wikipedia.org/wiki/Student's_t-test |
| // (and many other elementary and advanced statistics texts). |
| // It says: |
| // The t statistic was invented by William Sealy Gosset |
| // for cheaply monitoring the quality of beer brews. |
| // "Student" was his pen name. |
| // Gosset was statistician for Guinness brewery in Dublin, Ireland, |
| // hired due to Claude Guinness's innovative policy of recruiting the |
| // best graduates from Oxford and Cambridge for applying biochemistry |
| // and statistics to Guinness's industrial processes. |
| // Gosset published the t test in Biometrika in 1908, |
| // but was forced to use a pen name by his employer who regarded the fact |
| // that they were using statistics as a trade secret. |
| // In fact, Gosset's identity was unknown not only to fellow statisticians |
| // but to his employer - the company insisted on the pseudonym |
| // so that it could turn a blind eye to the breach of its rules. |
| |
| // The Students't distribution function is described at |
| // http://en.wikipedia.org/wiki/Student%27s_t_distribution |
| |
| #include <boost/math/distributions/students_t.hpp> |
| using boost::math::students_t; // Probability of students_t(df, t). |
| |
| #include <iostream> |
| using std::cout; using std::endl; |
| #include <iomanip> |
| using std::setprecision; using std::setw; |
| #include <cmath> |
| using std::sqrt; |
| |
| // This example of a two-sided test is from: |
| // B. M. Smith & M. B. Griffiths, Analyst, 1982, 107, 253, |
| // from Statistics for Analytical Chemistry, 3rd ed. (1994), pp 58-59 |
| // J. C. Miller and J. N. Miller, Ellis Horwood ISBN 0 13 0309907 |
| |
| // Concentrations of lead (ug/l) determined by two different methods |
| // for each of four test portions, |
| // the concentration of each portion is significantly different, |
| // the values may NOT be pooled. |
| // (Called a 'paired test' by Miller and Miller |
| // because each portion analysed has a different concentration.) |
| |
| // Portion Wet oxidation Direct Extraction |
| // 1 71 76 |
| // 2 61 68 |
| // 3 50 48 |
| // 4 60 57 |
| |
| const int portions = 4; |
| const int methods = 2; |
| float data [portions][methods] = {{71, 76}, {61,68}, {50, 48}, {60, 57}}; |
| float diffs[portions]; |
| |
| int main() |
| { |
| cout << "Example3 using Student's t function. " << endl; |
| float mean_diff = 0.f; |
| cout << "\n""Portion wet_oxidation Direct_extraction difference" << endl; |
| for (int portion = 0; portion < portions; portion++) |
| { // Echo data and differences. |
| diffs[portion] = data[portion][0] - data[portion][1]; |
| mean_diff += diffs[portion]; |
| cout << setw(4) << portion << ' ' << setw(14) << data[portion][0] << ' ' << setw(18)<< data[portion][1] << ' ' << setw(9) << diffs[portion] << endl; |
| } |
| mean_diff /= portions; |
| cout << "Mean difference = " << mean_diff << endl; // -1.75 |
| |
| float sd_diffs = 0.f; |
| for (int portion = 0; portion < portions; portion++) |
| { // Calculate standard deviation of differences. |
| sd_diffs +=(diffs[portion] - mean_diff) * (diffs[portion] - mean_diff); |
| } |
| int degrees_of_freedom = portions-1; // Use the n-1 formula. |
| sd_diffs /= degrees_of_freedom; |
| sd_diffs = sqrt(sd_diffs); |
| cout << "Standard deviation of differences = " << sd_diffs << endl; // 4.99166 |
| // Standard deviation of differences = 4.99166 |
| double t = mean_diff * sqrt(static_cast<double>(portions))/ sd_diffs; // -0.70117 |
| cout << "Student's t = " << t << ", if " << degrees_of_freedom << " degrees of freedom." << endl; |
| // Student's t = -0.70117, if 3 degrees of freedom. |
| cout << "Probability of the means being different is " |
| << 2.F * cdf(students_t(degrees_of_freedom), t) << "."<< endl; // 0.266846 * 2 = 0.533692 |
| // Double the probability because using a 'two-sided test' because |
| // mean for 'Wet oxidation' could be either |
| // greater OR LESS THAN for 'Direct extraction'. |
| |
| return 0; |
| } // int main() |
| |
| /* |
| |
| Output is: |
| |
| Example3 using Student's t function. |
| Portion wet_oxidation Direct_extraction difference |
| 0 71 76 -5 |
| 1 61 68 -7 |
| 2 50 48 2 |
| 3 60 57 3 |
| Mean difference = -1.75 |
| Standard deviation of differences = 4.99166 |
| Student's t = -0.70117, if 3 degrees of freedom. |
| Probability of the means being different is 0.533692. |
| |
| */ |
| |
| |
| |
| |