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- [section:f_eg F Distribution Examples]
- Imagine that you want to compare the standard deviations of two
- sample to determine if they differ in any significant way, in this
- situation you use the F distribution and perform an F-test. This
- situation commonly occurs when conducting a process change comparison:
- "is a new process more consistent that the old one?".
- In this example we'll be using the data for ceramic strength from
- [@http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm
- http://www.itl.nist.gov/div898/handbook/eda/section4/eda42a1.htm].
- The data for this case study were collected by Said Jahanmir of the
- NIST Ceramics Division in 1996 in connection with a NIST/industry
- ceramics consortium for strength optimization of ceramic strength.
- The example program is [@../../example/f_test.cpp f_test.cpp],
- program output has been deliberately made as similar as possible
- to the DATAPLOT output in the corresponding
- [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda359.htm
- NIST EngineeringStatistics Handbook example].
- We'll begin by defining the procedure to conduct the test:
- void f_test(
- double sd1, // Sample 1 std deviation
- double sd2, // Sample 2 std deviation
- double N1, // Sample 1 size
- double N2, // Sample 2 size
- double alpha) // Significance level
- {
- The procedure begins by printing out a summary of our input data:
- using namespace std;
- using namespace boost::math;
- // Print header:
- cout <<
- "____________________________________\n"
- "F test for equal standard deviations\n"
- "____________________________________\n\n";
- cout << setprecision(5);
- cout << "Sample 1:\n";
- cout << setw(55) << left << "Number of Observations" << "= " << N1 << "\n";
- cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd1 << "\n\n";
- cout << "Sample 2:\n";
- cout << setw(55) << left << "Number of Observations" << "= " << N2 << "\n";
- cout << setw(55) << left << "Sample Standard Deviation" << "= " << sd2 << "\n\n";
- The test statistic for an F-test is simply the ratio of the square of
- the two standard deviations:
- [expression F = s[sub 1][super 2] / s[sub 2][super 2]]
- where s[sub 1] is the standard deviation of the first sample and s[sub 2]
- is the standard deviation of the second sample. Or in code:
- double F = (sd1 / sd2);
- F *= F;
- cout << setw(55) << left << "Test Statistic" << "= " << F << "\n\n";
- At this point a word of caution: the F distribution is asymmetric,
- so we have to be careful how we compute the tests, the following table
- summarises the options available:
- [table
- [[Hypothesis][Test]]
- [[The null-hypothesis: there is no difference in standard deviations (two sided test)]
- [Reject if F <= F[sub (1-alpha/2; N1-1, N2-1)] or F >= F[sub (alpha/2; N1-1, N2-1)] ]]
- [[The alternative hypothesis: there is a difference in means (two sided test)]
- [Reject if F[sub (1-alpha/2; N1-1, N2-1)] <= F <= F[sub (alpha/2; N1-1, N2-1)] ]]
- [[The alternative hypothesis: Standard deviation of sample 1 is greater
- than that of sample 2]
- [Reject if F < F[sub (alpha; N1-1, N2-1)] ]]
- [[The alternative hypothesis: Standard deviation of sample 1 is less
- than that of sample 2]
- [Reject if F > F[sub (1-alpha; N1-1, N2-1)] ]]
- ]
- Where F[sub (1-alpha; N1-1, N2-1)] is the lower critical value of the F distribution
- with degrees of freedom N1-1 and N2-1, and F[sub (alpha; N1-1, N2-1)] is the upper
- critical value of the F distribution with degrees of freedom N1-1 and N2-1.
- The upper and lower critical values can be computed using the quantile function:
- [expression F[sub (1-alpha; N1-1, N2-1)] = `quantile(fisher_f(N1-1, N2-1), alpha)`]
- [expression F[sub (alpha; N1-1, N2-1)] = `quantile(complement(fisher_f(N1-1, N2-1), alpha))`]
- In our example program we need both upper and lower critical values for alpha
- and for alpha/2:
- double ucv = quantile(complement(dist, alpha));
- double ucv2 = quantile(complement(dist, alpha / 2));
- double lcv = quantile(dist, alpha);
- double lcv2 = quantile(dist, alpha / 2);
- cout << setw(55) << left << "Upper Critical Value at alpha: " << "= "
- << setprecision(3) << scientific << ucv << "\n";
- cout << setw(55) << left << "Upper Critical Value at alpha/2: " << "= "
- << setprecision(3) << scientific << ucv2 << "\n";
- cout << setw(55) << left << "Lower Critical Value at alpha: " << "= "
- << setprecision(3) << scientific << lcv << "\n";
- cout << setw(55) << left << "Lower Critical Value at alpha/2: " << "= "
- << setprecision(3) << scientific << lcv2 << "\n\n";
- The final step is to perform the comparisons given above, and print
- out whether the hypothesis is rejected or not:
- cout << setw(55) << left <<
- "Results for Alternative Hypothesis and alpha" << "= "
- << setprecision(4) << fixed << alpha << "\n\n";
- cout << "Alternative Hypothesis Conclusion\n";
-
- cout << "Standard deviations are unequal (two sided test) ";
- if((ucv2 < F) || (lcv2 > F))
- cout << "ACCEPTED\n";
- else
- cout << "REJECTED\n";
-
- cout << "Standard deviation 1 is less than standard deviation 2 ";
- if(lcv > F)
- cout << "ACCEPTED\n";
- else
- cout << "REJECTED\n";
-
- cout << "Standard deviation 1 is greater than standard deviation 2 ";
- if(ucv < F)
- cout << "ACCEPTED\n";
- else
- cout << "REJECTED\n";
- cout << endl << endl;
- Using the ceramic strength data as an example we get the following
- output:
- [pre
- '''F test for equal standard deviations
- ____________________________________
- Sample 1:
- Number of Observations = 240
- Sample Standard Deviation = 65.549
- Sample 2:
- Number of Observations = 240
- Sample Standard Deviation = 61.854
- Test Statistic = 1.123
- CDF of test statistic: = 8.148e-001
- Upper Critical Value at alpha: = 1.238e+000
- Upper Critical Value at alpha/2: = 1.289e+000
- Lower Critical Value at alpha: = 8.080e-001
- Lower Critical Value at alpha/2: = 7.756e-001
- Results for Alternative Hypothesis and alpha = 0.0500
- Alternative Hypothesis Conclusion
- Standard deviations are unequal (two sided test) REJECTED
- Standard deviation 1 is less than standard deviation 2 REJECTED
- Standard deviation 1 is greater than standard deviation 2 REJECTED'''
- ]
- In this case we are unable to reject the null-hypothesis, and must instead
- reject the alternative hypothesis.
- By contrast let's see what happens when we use some different
- [@http://www.itl.nist.gov/div898/handbook/prc/section3/prc32.htm
- sample data]:, once again from the NIST Engineering Statistics Handbook:
- A new procedure to assemble a device is introduced and tested for
- possible improvement in time of assembly. The question being addressed
- is whether the standard deviation of the new assembly process (sample 2) is
- better (i.e., smaller) than the standard deviation for the old assembly
- process (sample 1).
- [pre
- '''____________________________________
- F test for equal standard deviations
- ____________________________________
- Sample 1:
- Number of Observations = 11.00000
- Sample Standard Deviation = 4.90820
- Sample 2:
- Number of Observations = 9.00000
- Sample Standard Deviation = 2.58740
- Test Statistic = 3.59847
- CDF of test statistic: = 9.589e-001
- Upper Critical Value at alpha: = 3.347e+000
- Upper Critical Value at alpha/2: = 4.295e+000
- Lower Critical Value at alpha: = 3.256e-001
- Lower Critical Value at alpha/2: = 2.594e-001
- Results for Alternative Hypothesis and alpha = 0.0500
- Alternative Hypothesis Conclusion
- Standard deviations are unequal (two sided test) REJECTED
- Standard deviation 1 is less than standard deviation 2 REJECTED
- Standard deviation 1 is greater than standard deviation 2 ACCEPTED'''
- ]
- In this case we take our null hypothesis as "standard deviation 1 is
- less than or equal to standard deviation 2", since this represents the "no change"
- situation. So we want to compare the upper critical value at /alpha/
- (a one sided test) with the test statistic, and since 3.35 < 3.6 this
- hypothesis must be rejected. We therefore conclude that there is a change
- for the better in our standard deviation.
- [endsect][/section:f_eg F Distribution]
- [/
- Copyright 2006 John Maddock and Paul A. Bristow.
- Distributed under 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).
- ]
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