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We develop a novel multiple hypothesis testing correction with family-wiseerror rate (FWER) control that efficiently exploits positive dependenciesbetween potentially correlated statistical hypothesis tests. Our proposedalgorithm $\texttt{max-rank}$ is conceptually straight-forward, relying on theuse of a $\max$-operator in the rank domain of computed test statistics. Wecompare our approach to the frequently employed Bonferroni correction,theoretically and empirically demonstrating its superiority over Bonferroni inthe case of existing positive dependency, and its equivalence otherwise. Ouradvantage over Bonferroni increases as the number of tests rises, and wemaintain high statistical power whilst ensuring FWER control. We specificallyframe our algorithm in the context of parallel permutation testing, a scenariothat arises in our primary application of conformal prediction, a recentlypopularized approach for quantifying uncertainty in complex predictivesettings.

A powerful rank-based correction to multiple testing under positive dependency