11-2 Contingency Tables 597 with one or more expected frequencies that are below 5. Fisher’s exact test provides an exact P-value and does not require an approximation technique. Because the calculations are quite complex, it’s a good idea to use technology when using Fisher’s exact test. Statdisk, Minitab, XLSTAT, and StatCrunch all have the ability to perform Fisher’s exact test. Does Yawning Cause Others to Yawn? EXAMPLE 4 The MythBusters show on the Discovery Channel tested the theory that when someone yawns, others are more likely to yawn. The results are summarized in Table 11-8. The methods of Part 1 in this Section should not be used because one of the cells has an expected frequency of 4.480, which violates the requirement that every cell must have an expected frequency E of at least 5. Using Fisher’s exact test results in a P-value of 0.513, so there is not sufficient evidence to support the myth that people exposed to yawning actually yawn more than those not exposed to yawning. (For testing the claim of no difference, the P-value is 1.000, indicating that there is not a significant difference between the two groups.) TABLE 11-8 Yawning Theory Experiment Subject Exposed to Yawning? Yes No Did Subject Yawn? Yes 10 4 No 24 12 McNemar’s Test for Matched Pairs The methods in Part 1 of this section are based on independent data. For 2 * 2 tables consisting of frequency counts that result from matched pairs, the frequency counts within each matched pair are not independent and, for such cases, we can use McNemar’s test of the null hypothesis that the frequencies from the discordant (different) categories occur in the same proportion. Table 11-9 shows a general format for summarizing results from data consisting of frequency counts from matched pairs. Table 11-9 refers to two different treatments (such as two different eyedrop solutions) applied to two different parts of each subject (such as left eye and right eye). We should be careful when reading a table such as Table 11-9. If a = 100, then 100 subjects were cured with both treatments. If b = 50 in Table 11-9, then each of 50 subjects had no cure with treatment X but they were each cured with treatment Y. The total number of subjects is a + b + c + d, and each of those subjects yields results from each of two parts of a matched pair. Remember, the entries in Table 11-9 are frequency counts of subjects, not the total number of individual components in the matched pairs. If 500 people have each eye treated with two different ointments, the value of a + b + c + d is 500 (the number of subjects), not 1000 (the number of treated eyes). TABLE 11-9 2 * 2 Table with Frequency Counts from Matched Pairs Treatment X Cured Not Cured Treatment Y Cured a b Not Cured c d continued
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