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  "body": "\n## Definition\n\nLet $H_1, \\dots, H_m$ be null hypotheses with [[p-values|0EHYZFZ4PK]] $p_1, \\dots, p_m$, and let $\\alpha \\in (0, 1)$.\nThe *Bonferroni correction* at level $\\alpha$ is the procedure that rejects $H_i$ exactly when $p_i \\le \\alpha/m$.\n\n## Remarks\n\nAssume each p-value is valid: if $H_i$ is true, then $P(p_i \\le u) \\le u$ for all $u \\in [0, 1]$.\nLet $I_0$ be the set of true null hypotheses and $m_0$ its size.\nBy [[Boole's inequality|0EGSEE9J9R]], $P(\\text{some } H_i \\text{ with } i \\in I_0 \\text{ is rejected}) \\le \\sum_{i \\in I_0} P(p_i \\le \\alpha/m) \\le m_0 \\alpha/m \\le \\alpha$.\nSo the correction controls the [[family-wise error rate|GQCY1N8JZT]] strongly at level $\\alpha$, with no assumption on the joint law of the p-values.\nExample: with $m = 10$ and $\\alpha = 0.05$, a hypothesis is rejected when its p-value is at most $0.005$.\nNon-example: rejecting when $p_i \\le \\alpha$ is not the correction, and with $m$ independent true nulls its FWER is $1 - (1 - \\alpha)^m \u003e \\alpha$.\nEquivalently, one rejects $H_i$ when the adjusted p-value $\\min(1, m p_i)$ is at most $\\alpha$.\n",
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