False discovery rate and the Benjamini–Hochberg procedure

Remarks

Benjamini and Hochberg proved the following in 1995. If the p-values of the $m_0$ true nulls are independent, uniform on $[0, 1]$, and independent of the other p-values, then the procedure has $\mathrm{FDR} \le m_0 q/m \le q$. Benjamini and Yekutieli later extended the bound to a form of positive dependence. If all null hypotheses are true, then $Q = 1$ whenever $R \ge 1$, so $\mathrm{FDR} = P(R \ge 1)$ is the family-wise error rate. In general $\mathrm{FDR} \le \mathrm{FWER}$, since $Q \le 1$ and $Q = 0$ when $V = 0$. Example: with $m = 4$, $q = 0.05$ and p-values $0.001, 0.02, 0.03, 0.5$, the thresholds are $0.0125, 0.025, 0.0375, 0.05$. The first three p-values are below their thresholds, so $k = 3$ and $H_{(1)}, H_{(2)}, H_{(3)}$ are rejected. Non-example: the rule "reject $H_{(i)}$ when $p_{(i)} \le q/m$" is the Bonferroni rule, not the Benjamini–Hochberg procedure; here it rejects only $H_{(1)}$.