ceph/src/boost/libs/math/doc/distributions/f_dist_example.qbk

   1 [section:f_eg F Distribution Examples]
   2
   3 Imagine that you want to compare the standard deviations of two
   4 sample to determine if they differ in any significant way, in this
   5 situation you use the F distribution and perform an F-test.  This
   6 situation commonly occurs when conducting a process change comparison:
   7 "is a new process more consistent that the old one?".
   8
   9 In this example we'll be using the data for ceramic strength from
  10 [@http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm
  11 http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm].
  12 The data for this case study were collected by Said Jahanmir of the
  13 NIST Ceramics Division in 1996 in connection with a NIST/industry
  14 ceramics consortium for strength optimization of ceramic strength.
  15
  16 The example program is [@../../example/f_test.cpp f_test.cpp],
  17 program output has been deliberately made as similar as possible
  18 to the DATAPLOT output in the corresponding
  19 [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm
  20 NIST EngineeringStatistics Handbook example].
  21
  22 We'll begin by defining the procedure to conduct the test:
  23
  24    void f_test(
  25        double sd1,     // Sample 1 std deviation
  26        double sd2,     // Sample 2 std deviation
  27        double N1,      // Sample 1 size
  28        double N2,      // Sample 2 size
  29        double alpha)  // Significance level
  30    {
  31
  32 The procedure begins by printing out a summary of our input data:
  33
  34    using namespace std;
  35    using namespace boost::math;
  36
  37    // Print header:
  38    cout <<
  39       "____________________________________\n"
  40       "F test for equal standard deviations\n"
  41       "____________________________________\n\n";
  42    cout << setprecision(5);
  43    cout << "Sample 1:\n";
  44    cout << setw(55) << left << "Number of Observations" << "=  " << N1 << "\n";
  45    cout << setw(55) << left << "Sample Standard Deviation" << "=  " << sd1 << "\n\n";
  46    cout << "Sample 2:\n";
  47    cout << setw(55) << left << "Number of Observations" << "=  " << N2 << "\n";
  48    cout << setw(55) << left << "Sample Standard Deviation" << "=  " << sd2 << "\n\n";
  49
  50 The test statistic for an F-test is simply the ratio of the square of
  51 the two standard deviations:
  52
  53 F = s[sub 1][super 2] / s[sub 2][super 2]
  54
  55 where s[sub 1] is the standard deviation of the first sample and s[sub 2]
  56 is the standard deviation of the second sample.  Or in code:
  57
  58    double F = (sd1 / sd2);
  59    F *= F;
  60    cout << setw(55) << left << "Test Statistic" << "=  " << F << "\n\n";
  61
  62 At this point a word of caution: the F distribution is asymmetric,
  63 so we have to be careful how we compute the tests, the following table
  64 summarises the options available:
  65
  66 [table
  67 [[Hypothesis][Test]]
  68 [[The null-hypothesis: there is no difference in standard deviations (two sided test)]
  69       [Reject if F <= F[sub (1-alpha/2; N1-1, N2-1)] or F >= F[sub (alpha/2; N1-1, N2-1)] ]]
  70 [[The alternative hypothesis: there is a difference in means (two sided test)]
  71       [Reject if F[sub (1-alpha/2; N1-1, N2-1)] <= F <= F[sub (alpha/2; N1-1, N2-1)] ]]
  72 [[The alternative hypothesis: Standard deviation of sample 1 is greater
  73 than that of sample 2]
  74       [Reject if F < F[sub (alpha; N1-1, N2-1)] ]]
  75 [[The alternative hypothesis: Standard deviation of sample 1 is less
  76 than that of sample 2]
  77       [Reject if F > F[sub (1-alpha; N1-1, N2-1)] ]]
  78 ]
  79
  80 Where F[sub (1-alpha; N1-1, N2-1)] is the lower critical value of the F distribution
  81 with degrees of freedom N1-1 and N2-1, and F[sub (alpha; N1-1, N2-1)] is the upper
  82 critical value of the F distribution with degrees of freedom N1-1 and N2-1.
  83
  84 The upper and lower critical values can be computed using the quantile function:
  85
  86 F[sub (1-alpha; N1-1, N2-1)] = `quantile(fisher_f(N1-1, N2-1), alpha)`
  87
  88 F[sub (alpha; N1-1, N2-1)] = `quantile(complement(fisher_f(N1-1, N2-1), alpha))`
  89
  90 In our example program we need both upper and lower critical values for alpha
  91 and for alpha/2:
  92
  93    double ucv = quantile(complement(dist, alpha));
  94    double ucv2 = quantile(complement(dist, alpha / 2));
  95    double lcv = quantile(dist, alpha);
  96    double lcv2 = quantile(dist, alpha / 2);
  97    cout << setw(55) << left << "Upper Critical Value at alpha: " << "=  "
  98       << setprecision(3) << scientific << ucv << "\n";
  99    cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "=  "
 100       << setprecision(3) << scientific << ucv2 << "\n";
 101    cout << setw(55) << left << "Lower Critical Value at alpha: " << "=  "
 102       << setprecision(3) << scientific << lcv << "\n";
 103    cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "=  "
 104       << setprecision(3) << scientific << lcv2 << "\n\n";
 105
 106 The final step is to perform the comparisons given above, and print
 107 out whether the hypothesis is rejected or not:
 108
 109    cout << setw(55) << left <<
 110       "Results for Alternative Hypothesis and alpha" << "=  "
 111       << setprecision(4) << fixed << alpha << "\n\n";
 112    cout << "Alternative Hypothesis                                    Conclusion\n";
 113
 114    cout << "Standard deviations are unequal (two sided test)          ";
 115    if((ucv2 < F) || (lcv2 > F))
 116       cout << "ACCEPTED\n";
 117    else
 118       cout << "REJECTED\n";
 119
 120    cout << "Standard deviation 1 is less than standard deviation 2    ";
 121    if(lcv > F)
 122       cout << "ACCEPTED\n";
 123    else
 124       cout << "REJECTED\n";
 125
 126    cout << "Standard deviation 1 is greater than standard deviation 2 ";
 127    if(ucv < F)
 128       cout << "ACCEPTED\n";
 129    else
 130       cout << "REJECTED\n";
 131    cout << endl << endl;
 132
 133 Using the ceramic strength data as an example we get the following
 134 output:
 135
 136 [pre
 137 '''F test for equal standard deviations
 138 ____________________________________
 139
 140 Sample 1:
 141 Number of Observations                                 =  240
 142 Sample Standard Deviation                              =  65.549
 143
 144 Sample 2:
 145 Number of Observations                                 =  240
 146 Sample Standard Deviation                              =  61.854
 147
 148 Test Statistic                                         =  1.123
 149
 150 CDF of test statistic:                                 =  8.148e-001
 151 Upper Critical Value at alpha:                         =  1.238e+000
 152 Upper Critical Value at alpha/2:                       =  1.289e+000
 153 Lower Critical Value at alpha:                         =  8.080e-001
 154 Lower Critical Value at alpha/2:                       =  7.756e-001
 155
 156 Results for Alternative Hypothesis and alpha           =  0.0500
 157
 158 Alternative Hypothesis                                    Conclusion
 159 Standard deviations are unequal (two sided test)          REJECTED
 160 Standard deviation 1 is less than standard deviation 2    REJECTED
 161 Standard deviation 1 is greater than standard deviation 2 REJECTED'''
 162 ]
 163
 164 In this case we are unable to reject the null-hypothesis, and must instead
 165 reject the alternative hypothesis.
 166
 167 By contrast let's see what happens when we use some different
 168 [@http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm
 169 sample data]:, once again from the NIST Engineering Statistics Handbook:
 170 A new procedure to assemble a device is introduced and tested for
 171 possible improvement in time of assembly. The question being addressed
 172 is whether the standard deviation of the new assembly process (sample 2) is
 173 better (i.e., smaller) than the standard deviation for the old assembly
 174 process (sample 1).
 175
 176 [pre
 177 '''____________________________________
 178 F test for equal standard deviations
 179 ____________________________________
 180
 181 Sample 1:
 182 Number of Observations                                 =  11.00000
 183 Sample Standard Deviation                              =  4.90820
 184
 185 Sample 2:
 186 Number of Observations                                 =  9.00000
 187 Sample Standard Deviation                              =  2.58740
 188
 189 Test Statistic                                         =  3.59847
 190
 191 CDF of test statistic:                                 =  9.589e-001
 192 Upper Critical Value at alpha:                         =  3.347e+000
 193 Upper Critical Value at alpha/2:                       =  4.295e+000
 194 Lower Critical Value at alpha:                         =  3.256e-001
 195 Lower Critical Value at alpha/2:                       =  2.594e-001
 196
 197 Results for Alternative Hypothesis and alpha           =  0.0500
 198
 199 Alternative Hypothesis                                    Conclusion
 200 Standard deviations are unequal (two sided test)          REJECTED
 201 Standard deviation 1 is less than standard deviation 2    REJECTED
 202 Standard deviation 1 is greater than standard deviation 2 ACCEPTED'''
 203 ]
 204
 205 In this case we take our null hypothesis as "standard deviation 1 is
 206 less than or equal to standard deviation 2", since this represents the "no change"
 207 situation.  So we want to compare the upper critical value at /alpha/
 208 (a one sided test) with the test statistic, and since 3.35 < 3.6 this
 209 hypothesis must be rejected.  We therefore conclude that there is a change
 210 for the better in our standard deviation.
 211
 212 [endsect][/section:f_eg F Distribution]
 213
 214 [/
 215   Copyright 2006 John Maddock and Paul A. Bristow.
 216   Distributed under the Boost Software License, Version 1.0.
 217   (See accompanying file LICENSE_1_0.txt or copy at
 218   http://www.boost.org/LICENSE_1_0.txt).
 219 ]
 220