Holm's step-down beats plain Bonferroni

Question

You test m = 4 hypotheses and want the family-wise error rate held at \alpha = 0.05. Sorted smallest to largest, the p-values are 0.004,\ 0.013,\ 0.030,\ 0.045. How many does the Holm step-down procedure reject, and how does that compare to plain Bonferroni?

Accepted Answer

The Holm idea. Plain Bonferroni uses the same tiny cutoff \alpha/m for every test. Holm starts just as strict for the smallest p-value but relaxes the cutoff as it moves down the sorted list, because once you have "used up" some hypotheses, fewer remain to protect against. Concretely, the k-th smallest p-value is compared to \frac{\alpha}{m - k + 1}. Walk down the list. With \alpha = 0.05 and m = 4: k p{(k)} cutoff \frac{\alpha}{m-k+1} decision ------------ 1 0.004 0.05/4 = 0.0125 reject, continue 2 0.013 0.05/3 = 0.0167 reject, continue 3 0.030 0.05/2 = 0.025 fail, stop Stop at the first fai…

Holm's step-down beats plain Bonferroni

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