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The Holm–Bonferroni method orders p-values and compares the $j$th smallest with $\alpha /(m-j+1)$, stopping at the first failed comparison. It controls the familywise error rate under arbitrary dependence.

Multiple hypothesis testing considers a family of null hypotheses and a procedure for the collection of rejection decisions. Even if every individual test controls its type I error, the collection can produce many false positives. For $m$ independent true nulls, each rejected with probability $\alpha$, the familywise error rate isFor $m=20$ and $\alpha =0.05$, this is about $0.642$. Dependence changes the formula, so independence must not be silently assumed.

Different goals require different error criteria. The familywise error rate is the probability of any false rejection. The false discovery rate is the expected fraction of rejected hypotheses that are true, with fraction zero if nothing is rejected. These do not give interchangeable guarantees.

The Bonferroni correction tests each of $m$ hypotheses at level $\alpha /m$. If $I_0$ is the set of true nulls, the union bound giveswithout independence. Holm–Bonferroni method improves this familywise procedure, whereas the Benjamini-Hochberg procedure targets false discovery rate under its dependence assumptions. Specifying the tested family and intended error criterion is part of the scientific analysis.