When Benjamini-Hochberg rejects nothing

Question

You test m = 6 hypotheses and control the false discovery rate at q = 0.05. Sorted smallest to largest, the p-values are 0.02,\ 0.03,\ 0.04,\ 0.05,\ 0.06,\ 0.07. How many does the Benjamini-Hochberg procedure reject?

Accepted Answer

Build the ladder. BH compares p{(k)} to \frac{k}{m}q = \frac{k}{6}(0.05) = k 	imes 0.008333, giving approximately 0.0083,\ 0.0167,\ 0.025,\ 0.0333,\ 0.0417,\ 0.05. Compare. k p{(k)} ladder pass? ------------ 1 0.02 0.0083 No 2 0.03 0.0167 No 3 0.04 0.025 No 4 0.05 0.0333 No 5 0.06 0.0417 No 6 0.07 0.05 No No rung clears. Not a single p-value falls at or below its rung, so there is no largest passing k at all. BH rejects 0 hypotheses. The honest reading. Every p-value here is small-ish (0.02 to 0.07), and the smallest, 0.02, would even pass a lone 5% test. But once you account for having run …

When Benjamini-Hochberg rejects nothing

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