# Hervé Abdi:

# 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-

equality holds:

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 ] .

C times

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

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