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  "body": "\n## Definition\n\nLet $H_1, \\dots, H_m$ be null hypotheses about the law $P$ of the data, tested by a procedure that rejects a random subset of them.\nLet $I_0(P) \\subseteq \\set{1, \\dots, m}$ be the set of indices $i$ such that $H_i$ is true under $P$.\nLet $V$ be the number of rejected $H_i$ with $i \\in I_0(P)$, the number of [[type I errors|0R41PVFA39]].\nThe *family-wise error rate* is $\\mathrm{FWER} = P(V \\ge 1)$.\nThe procedure controls the FWER *strongly* at level $\\alpha$ if $P(V \\ge 1) \\le \\alpha$ for every $P$ in the model.\nIt controls the FWER *weakly* at level $\\alpha$ if this holds for every $P$ under which all $H_i$ are true.\n\n## Remarks\n\nStrong control implies weak control.\nExample: if $m$ tests of exact level $\\alpha$ are independent and all null hypotheses are true, then $\\mathrm{FWER} = 1 - (1 - \\alpha)^m$.\nSo testing each hypothesis at level $\\alpha$ does not control the FWER at level $\\alpha$ for $m \\ge 2$; this is the non-example.\nThe [[Bonferroni correction|8GRD2TMJRR]] tests each hypothesis at level $\\alpha/m$ and gives strong control.\n",
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