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  "body": "\n## Definition\n\nLet $H_1, \\dots, H_m$ be null hypotheses tested by a procedure that rejects $R$ of them.\nLet $V$ be the number of rejected hypotheses that are true.\nThe *false discovery proportion* is $Q = V/R$ if $R \\ge 1$ and $Q = 0$ if $R = 0$.\nThe *false discovery rate* is $\\mathrm{FDR} = E[Q]$.\n\nLet $p_1, \\dots, p_m$ be [[p-values|0EHYZFZ4PK]] for the $H_i$, ordered as $p_{(1)} \\le \\dots \\le p_{(m)}$ with hypotheses $H_{(1)}, \\dots, H_{(m)}$, and let $q \\in (0, 1)$.\nThe *Benjamini–Hochberg procedure* at level $q$ lets $k$ be the largest $i$ with $p_{(i)} \\le i q/m$.\nIt rejects $H_{(1)}, \\dots, H_{(k)}$, and it rejects nothing if no such $i$ exists.\n\n## Remarks\n\nBenjamini and Hochberg proved the following in 1995.\nIf 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 $\\mathrm{FDR} \\le m_0 q/m \\le q$.\nBenjamini and Yekutieli later extended the bound to a form of positive dependence.\nIf 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|GQCY1N8JZT]].\nIn general $\\mathrm{FDR} \\le \\mathrm{FWER}$, since $Q \\le 1$ and $Q = 0$ when $V = 0$.\nExample: with $m = 4$, $q = 0.05$ and p-values $0.001, 0.02, 0.03, 0.5$, the thresholds are $0.0125, 0.025, 0.0375, 0.05$.\nThe first three p-values are below their thresholds, so $k = 3$ and $H_{(1)}, H_{(2)}, H_{(3)}$ are rejected.\nNon-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 $H_{(1)}$.\n",
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