The Bonferonni and Šidák Corrections
a simpler approximation which is known as the Bonferonni (the most popular name), or Boole, or even Dunn approximation. Tech- nically, it is the first (linear) term of a Taylor expansion of the Šidàk equation. This approximation gives
α[P F ] α[PT ] ≈ C
Šidàk and Bonferonni are linked to each other by the inequality
α[PT ] = 1 − (1 − α[PF ])1/C ≥ α[PF ] . C
They are, in general, very close to each other but the Bonferonni approximation is pessimistic (it always does worse than Šidàk equa- tion). Probably because it is easier to compute, the Bonferonni ap- proximation is more well known (and cited more often) than the exact Šidàk equation.
The Šidàk-Bonferonni equations can be used to find the value of α[PT ] when α[PF ] is fixed. For example, suppose that you want to perform 4 independent tests, and you want to limit the risk of making at least one Type I error to an overall value of α[PF ] = .05, you will consider a test significant if its associated probability is
α[PT ] = 1 − (1 − α[PF ])1/C
= 1 − (1 − .05)1/4
= .0127 .
With the Bonferonni approximation, a test reaches significance if its associated probability is smaller than
α[PT ] = α[PF ] = .05 = .0125 , C4
which is very close to the exact value of .0127.
2.4 Correction for non-independent tests
The Šidàk equation is derived assuming independence of the tests. When they are not independent, it gives a lower bound (cf. Šidàk, 1967; Games, 1977), and then:
α[PF ] ≤ 1 − (1 − α[PT ])C