Šidák cutoff for just 2 independent tests

Question

You run 2 independent tests and want the family-wise error rate held at 5%. Bonferroni would use $0.05/2 = 0.025$ per test.

Compute the exact Šidák per-test cutoff, and note the clean form it takes for two tests.

Accepted Answer

Exact formula. For m independent tests with family-wise target \alpha, the Šidák cutoff solves 1 - (1-\alpha')^m = \alpha, giving \alpha' = 1 - (1-\alpha)^{1/m}. Two tests give a square root. With m = 2, the exponent 1/m = 1/2, so the formula becomes a square root: \alpha' = 1 - (0.95)^{1/2} = 1 - \sqrt{0.95} \approx 1 - 0.97468 = 0.0253. Compare. Bonferroni gives 0.05/2 = 0.025. Šidák's 0.0253 is slightly larger, the widest relative gap you will see, since the over-counting in the union bound is largest when there are few tests each carrying a non-tiny alarm chance. Sanity check. Plug \alpha…

Šidák cutoff for just 2 independent tests

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