EXAMPLE 7 Bonferroni method of multiple comparisons
Use the Bonferroni method of multiple comparisons to determine which pairs of population mean times differ, for the mice in Groups 0, 1, and 2 in Example 5. Use level of significance .
The Bonferroni method requires that
- the requirements for ANOVA have been met, and
- the null hypothesis that the population means are all equal has been rejected.
In Example 5, we verified both requirements.
Step 1 For each of the c hypothesis tests, state the hypotheses and the rejection rule. There are means, so there will be hypothesis tests. Our hypotheses are
where represents the population mean time spent in the open-ended sections of the maze, for the th group. For each hypothesis test, reject if the Bonferroni-adjusted .
Step 2 Calculate for each hypothesis test. From Figure 11 on page 676, we have the mean square error from the original ANOVA as MSE = 52.9485079 and from Figure 21 we get the sample means and the sample sizes. Thus,
When the requirements are met, follows a distribution with degrees of freedom, where represents the total sample size.
Figure 12.22: FIGURE 22 Unadjusted -values from Excel.
- Step 3 Find the Bonferroni-adjusted -value for each hypothesis test. Figure 22 shows the unadjusted -values for the values of from Step 2, using the function tdist , where and the 2 represents a two-tailed test. Then the Bonferroni-adjusted , for each hypothesis test.
- Test 1: Bonferroni-adjusted .
- Test 2: Bonferroni-adjusted .
- Test 3: Bonferroni-adjusted , but this value exceeds 1, so we set this -value equal to 1.
- Step 4 For each hypothesis test, state the conclusion and the interpretation.
- Test 1: The adjusted , which is ≤0.01; therefore, reject . There is evidence at the 0.01 level of significance that the population mean time spent in the open-ended part of the maze differs between Group 0 and Group 1.
- Test 2: The adjusted , which is ≤0.01; therefore, reject . There is evidence at the 0.01 level of significance that the population mean time differs between Group 0 and Group 2.
- Test 3: The adjusted , which is not ≤0.01; therefore, do not reject . There is insufficient evidence at the 0.01 level of significance that the population mean time differs between Group 1 and Group 2.