Benjamini-Hochberg on six p-values

Question

You test m = 6 signals and observe p-values 0.001,\quad 0.008,\quad 0.02,\quad 0.04,\quad 0.15,\quad 0.60. Using Benjamini-Hochberg to control the false discovery rate at q = 0.05, which signals do you declare significant? Compare with Bonferroni.

Accepted Answer

The BH procedure. Controlling the false discovery rate (the expected fraction of your discoveries that are false) at q: sort p{(1)} \le \dots \le p{(m)}, compare each to the sloped threshold 	frac{k}{m}q, take the largest rank k that passes, and reject p{(1)}, \dots, p{(k)}. With m = 6, q = 0.05, the thresholds 	frac{k}{6}(0.05) are: rank k p{(k)} 	frac{k}{6}(0.05) pass? ------------ 1 0.001 0.00833 yes 2 0.008 0.01667 yes 3 0.020 0.02500 yes 4 0.040 0.03333 no 5 0.150 0.04167 no 6 0.600 0.05000 no The largest passing rank is k = 3, so BH rejects the three smallest p-values: 0.001, 0.008, …

Benjamini-Hochberg on six p-values

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