Bonferroni adjustment is used in multiple comparison procedures to calculate an adjusted probability a of comparison-wise type I error from the desired probability aFW0 of family-wise type I error. The calculation guarantees that the use of the adjusted a in pairwise comparisons keeps the actual probability aFW of family-wise type I errors not higher than the desired level, as specified by the significance level of the test.
Bonferroni, an Italian mathematician, proved the following inequality:
for any value C. Since aFW and a are linked by the formula
(see Family-wise Type I Error), where C is the total number of pairwise comparisons, the Bonferroni inequality gives the following approximate formula
Suppose 4 populations are to be compared, and the maximum allowed family-wise type I error is aFW0 = 0.05. Then, the total number of pairs C=4(4-1)/2=6, and Bonferroni adjustment gives a = 0.05/6 ? 0.0083. Now we may be sure that, if all the 4 populations have the same mean, and we test pairwise differences at significance level a = 0.0083, the probability aFW that you will erroneously conclude that the population means in at least one pair differ is not higher than 0.05.
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