The Bonferonni and Šidák Corrections
As previously, we can use a Bonferonni approximation because:
α[PF ] < Cα[PT ] .
Šidàk and Bonferonni are related by the inequality
α[PF ] ≤ 1 − (1 − α[PT ])C < Cα[PT ] .
The Šidàk and Bonferonni inequalities can also be used to find a correction on α[PT ] in order to keep α[PF ] fixed. the Šidàk in- equality gives
α[PT ] ≈ 1 − (1 − α[PF ])1/C .
This is a conservative approximation, because the following in-
α[PT ] ≥ 1 − (1
− α[PF ])1/C
The Bonferonni approximation gives
α[PT ] ≈
α[P F ] C
2.5 Splitting up α[PF ] with unequal slices
With the Bonferonni approximation we can make an unequal allo- cation of α[PF ]. This works because with the Bonferonni approxi- mation, α[PF ] is the sum of the individual α[PT ]:
α[PF ] ≈ Cα[PT ] = α[PT ] + α[PT ] + ··· + α[PT ] .
If some tests are judged more important a priori than some oth- ers, it is possible to allocate unequally α[PF ] (cf. Rosenthal & Ros- now, 1985). For example, suppose we have 3 tests that we want to test with an overall α[PF ] = .05, and we think that the first test is the most important of the set. Then we can decide to test it with α[PT ] = .04, and share the remaining value .01 = .05 − .04 be- tween the last 2 tests, which will be evaluated each with a value of α[PT ] = .005. The overall Type I error for the family is equal to α[PF ] = .04 + .005 + .005 = .05 which was indeed the value we set