BH step-up: a later rung rescues an earlier miss

Question

You test m = 4 hypotheses and control the false discovery rate at q = 0.05. Sorted smallest to largest, the p-values are 0.011,\ 0.030,\ 0.031,\ 0.045. How many does the Benjamini-Hochberg procedure reject?

Accepted Answer

Build the ladder. BH compares p{(k)} to \frac{k}{m}q = \frac{k}{4}(0.05) = k 	imes 0.0125, giving 0.0125,\ 0.025,\ 0.0375,\ 0.05. Compare. k p{(k)} ladder pass? ------------ 1 0.011 0.0125 Yes 2 0.030 0.025 No 3 0.031 0.0375 Yes 4 0.045 0.05 Yes The step-up rescue. Notice k = 2 fails its own rung (0.030 0.025). A naive "stop at the first failure" reading would reject only one. But BH is a step-up rule: it looks for the largest k that passes, then rejects everything at or below it. The largest passing rung is k = 4 (0.045 \le 0.05), so all four are rejected, including the number 2 that indivi…

BH step-up: a later rung rescues an earlier miss

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