Family-wise Error Rate Control with E-values

The closure principle is a standard tool for achieving strong family-wise error rate (FWER) control in multiple testing problems. We develop an e-value-based closed testing framework that inherits nice properties of e-values, which are common in settings of sequential hypothesis testing or universal inference for irregular parametric models. We prove that e-value-based closed testing strongly controls the post-hoc FWER in the static setting, and has stronger anytime-valid and always-valid FWER-controlling properties in the sequential setting. Furthermore, we extend the celebrated graphical approach for FWER control (Bretz et al. 2009), using the weighted average of e-values for the local test, a strictly more powerful approach than weighted Bonferroni local tests with inverse e-values as p-values. In general, the computational cost for closed testing can be exponential in the number of hypotheses. Although the computational shortcuts for the p-value-based graphical approach are not applicable, we develop an efficient polynomial-time algorithm using dynamic programming for e-value-based graphical approaches with any directed acyclic graph, and tailored algorithms for the e-Holm procedure previously studied by Vovk and Wang (2021) and the e-Fallback procedure.