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221 lines
8.9 KiB
Plaintext
221 lines
8.9 KiB
Plaintext
[section:f_eg F Distribution Examples]
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Imagine that you want to compare the standard deviations of two
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sample to determine if they differ in any significant way, in this
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situation you use the F distribution and perform an F-test. This
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situation commonly occurs when conducting a process change comparison:
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"is a new process more consistent that the old one?".
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In this example we'll be using the data for ceramic strength from
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[@http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm
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http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm].
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The data for this case study were collected by Said Jahanmir of the
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NIST Ceramics Division in 1996 in connection with a NIST/industry
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ceramics consortium for strength optimization of ceramic strength.
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The example program is [@../../example/f_test.cpp f_test.cpp],
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program output has been deliberately made as similar as possible
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to the DATAPLOT output in the corresponding
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[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm
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NIST EngineeringStatistics Handbook example].
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We'll begin by defining the procedure to conduct the test:
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void f_test(
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double sd1, // Sample 1 std deviation
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double sd2, // Sample 2 std deviation
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double N1, // Sample 1 size
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double N2, // Sample 2 size
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double alpha) // Significance level
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{
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The procedure begins by printing out a summary of our input data:
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using namespace std;
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using namespace boost::math;
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// Print header:
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cout <<
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"____________________________________\n"
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"F test for equal standard deviations\n"
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"____________________________________\n\n";
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cout << setprecision(5);
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cout << "Sample 1:\n";
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cout << setw(55) << left << "Number of Observations" << "= " << N1 << "\n";
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cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd1 << "\n\n";
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cout << "Sample 2:\n";
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cout << setw(55) << left << "Number of Observations" << "= " << N2 << "\n";
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cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd2 << "\n\n";
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The test statistic for an F-test is simply the ratio of the square of
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the two standard deviations:
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F = s[sub 1][super 2] / s[sub 2][super 2]
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where s[sub 1] is the standard deviation of the first sample and s[sub 2]
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is the standard deviation of the second sample. Or in code:
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double F = (sd1 / sd2);
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F *= F;
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cout << setw(55) << left << "Test Statistic" << "= " << F << "\n\n";
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At this point a word of caution: the F distribution is asymmetric,
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so we have to be careful how we compute the tests, the following table
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summarises the options available:
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[table
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[[Hypothesis][Test]]
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[[The null-hypothesis: there is no difference in standard deviations (two sided test)]
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[Reject if F <= F[sub (1-alpha/2; N1-1, N2-1)] or F >= F[sub (alpha/2; N1-1, N2-1)] ]]
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[[The alternative hypothesis: there is a difference in means (two sided test)]
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[Reject if F[sub (1-alpha/2; N1-1, N2-1)] <= F <= F[sub (alpha/2; N1-1, N2-1)] ]]
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[[The alternative hypothesis: Standard deviation of sample 1 is greater
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than that of sample 2]
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[Reject if F < F[sub (alpha; N1-1, N2-1)] ]]
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[[The alternative hypothesis: Standard deviation of sample 1 is less
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than that of sample 2]
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[Reject if F > F[sub (1-alpha; N1-1, N2-1)] ]]
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]
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Where F[sub (1-alpha; N1-1, N2-1)] is the lower critical value of the F distribution
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with degrees of freedom N1-1 and N2-1, and F[sub (alpha; N1-1, N2-1)] is the upper
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critical value of the F distribution with degrees of freedom N1-1 and N2-1.
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The upper and lower critical values can be computed using the quantile function:
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F[sub (1-alpha; N1-1, N2-1)] = `quantile(fisher_f(N1-1, N2-1), alpha)`
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F[sub (alpha; N1-1, N2-1)] = `quantile(complement(fisher_f(N1-1, N2-1), alpha))`
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In our example program we need both upper and lower critical values for alpha
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and for alpha/2:
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double ucv = quantile(complement(dist, alpha));
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double ucv2 = quantile(complement(dist, alpha / 2));
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double lcv = quantile(dist, alpha);
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double lcv2 = quantile(dist, alpha / 2);
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cout << setw(55) << left << "Upper Critical Value at alpha: " << "= "
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<< setprecision(3) << scientific << ucv << "\n";
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cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "= "
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<< setprecision(3) << scientific << ucv2 << "\n";
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cout << setw(55) << left << "Lower Critical Value at alpha: " << "= "
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<< setprecision(3) << scientific << lcv << "\n";
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cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "= "
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<< setprecision(3) << scientific << lcv2 << "\n\n";
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The final step is to perform the comparisons given above, and print
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out whether the hypothesis is rejected or not:
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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 << "Standard deviations are unequal (two sided test) ";
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if((ucv2 < F) || (lcv2 > F))
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cout << "ACCEPTED\n";
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else
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cout << "REJECTED\n";
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cout << "Standard deviation 1 is less than standard deviation 2 ";
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if(lcv > F)
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cout << "ACCEPTED\n";
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else
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cout << "REJECTED\n";
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cout << "Standard deviation 1 is greater than standard deviation 2 ";
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if(ucv < F)
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cout << "ACCEPTED\n";
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else
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cout << "REJECTED\n";
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cout << endl << endl;
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Using the ceramic strength data as an example we get the following
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output:
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[pre
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'''F test for equal standard deviations
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____________________________________
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Sample 1:
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Number of Observations = 240
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Sample Standard Deviation = 65.549
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Sample 2:
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Number of Observations = 240
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Sample Standard Deviation = 61.854
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Test Statistic = 1.123
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CDF of test statistic: = 8.148e-001
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Upper Critical Value at alpha: = 1.238e+000
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Upper Critical Value at alpha/2: = 1.289e+000
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Lower Critical Value at alpha: = 8.080e-001
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Lower Critical Value at alpha/2: = 7.756e-001
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Results for Alternative Hypothesis and alpha = 0.0500
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Alternative Hypothesis Conclusion
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Standard deviations are unequal (two sided test) REJECTED
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Standard deviation 1 is less than standard deviation 2 REJECTED
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Standard deviation 1 is greater than standard deviation 2 REJECTED'''
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]
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In this case we are unable to reject the null-hypothesis, and must instead
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reject the alternative hypothesis.
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By contrast let's see what happens when we use some different
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[@http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm
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sample data]:, once again from the NIST Engineering Statistics Handbook:
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A new procedure to assemble a device is introduced and tested for
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possible improvement in time of assembly. The question being addressed
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is whether the standard deviation of the new assembly process (sample 2) is
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better (i.e., smaller) than the standard deviation for the old assembly
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process (sample 1).
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[pre
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'''____________________________________
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F test for equal standard deviations
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____________________________________
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Sample 1:
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Number of Observations = 11.00000
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Sample Standard Deviation = 4.90820
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Sample 2:
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Number of Observations = 9.00000
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Sample Standard Deviation = 2.58740
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Test Statistic = 3.59847
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CDF of test statistic: = 9.589e-001
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Upper Critical Value at alpha: = 3.347e+000
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Upper Critical Value at alpha/2: = 4.295e+000
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Lower Critical Value at alpha: = 3.256e-001
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Lower Critical Value at alpha/2: = 2.594e-001
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Results for Alternative Hypothesis and alpha = 0.0500
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Alternative Hypothesis Conclusion
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Standard deviations are unequal (two sided test) REJECTED
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Standard deviation 1 is less than standard deviation 2 REJECTED
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Standard deviation 1 is greater than standard deviation 2 ACCEPTED'''
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]
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In this case we take our null hypothesis as "standard deviation 1 is
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less than or equal to standard deviation 2", since this represents the "no change"
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situation. So we want to compare the upper critical value at /alpha/
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(a one sided test) with the test statistic, and since 3.35 < 3.6 this
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hypothesis must be rejected. We therefore conclude that there is a change
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for the better in our standard deviation.
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[endsect][/section:f_eg F Distribution]
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[/
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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