Šidák versus Bonferroni

Question

You run m = 10 independent tests and want the family-wise error rate (the chance of any false positive) held at \alpha = 0.05. What per-test level does the exact Šidák correction give, how does it compare to Bonferroni, and when does the distinction matter?

Accepted Answer

The exact (Šidák) level. If the tests are independent and each runs at level \alpha{	ext{test}}, the probability of no false positive is (1 - \alpha{	ext{test}})^m. To make the family-wise error exactly \alpha, require (1 - \alpha{	ext{test}})^m = 1 - \alpha \;\Longrightarrow\; \alpha{	ext{test}} = 1 - (1 - \alpha)^{1/m}. For m = 10, \alpha = 0.05: \alpha{	ext{test}} = 1 - 0.95^{1/10} = 1 - 0.99488 \approx 0.00512. The Bonferroni level. Bonferroni just divides: \alpha / m = 0.05 / 10 = 0.005. This comes from the union bound and needs no independence assumption. Comparison. The two are al…

Šidák versus Bonferroni

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