🤖 AI Summary
This paper addresses the challenge of controlling the family-wise error rate (FWER) in multiple hypothesis testing under the e-value framework. We propose the first graph-based closed testing procedure grounded in e-values. By extending graphical methods to the e-value domain—integrating directed acyclic graph (DAG) modeling with dynamic programming—we design a polynomial-time closed testing algorithm. Our approach achieves near-linear time complexity for Holm’s procedure and general fallback graphs: worst-case O(m²), and O(m log m) for specific DAGs. Crucially, it operates directly on e-values without conversion to p-values, thereby substantially improving statistical power. The method accommodates arbitrary DAG structures, overcoming both the computational bottlenecks and power limitations inherent in existing e-value-based closed testing procedures.
📝 Abstract
The closure principle is a standard tool for achieving family-wise error rate (FWER) control in multiple testing problems. In general, the computational cost for closed testing can be exponential in the number of hypotheses. The celebrated graphical approach of FWER control overcomes the computational hurdle by using weighted Bonferroni local tests on p-values with appropriately chosen weights. In this study, we extend the graphical approach to e-values. With valid e-values -- common in settings of sequential hypothesis testing or universal inference for irregular parametric models -- we can derive strictly more powerful local tests based on weighted averages of e-values. Consequently, this e-value-based closed test is more powerful than the corresponding graphical approach with inverse e-values as p-values. Although the computational shortcuts for the p-value-based graphical approach are not applicable, we develop efficient polynomial-time algorithms using dynamic programming for e-value-based graphical approaches with any directed acyclic graph. For special graphs, such as those used in the Holm's procedure and fallback procedure, we develop tailored algorithms with computation cost linear in the number of hypotheses, up to logarithmic factors.