Bonferroni Adjustment:
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.
If the symbols do not display properly, try
the graphic version of this page