Efficient Multiple Testing Adjustment for Hierarchical Inference

TL;DR

This paper introduces a new hierarchical multiple testing adjustment method that guarantees strong control of familywise error rate and improves power in high-dimensional regression inference, especially for sparse signals.

Contribution

It proposes a novel, adaptive hierarchical adjustment procedure compatible with any significance test, enhancing power over existing methods in high-dimensional settings.

Findings

Guarantees strong control of familywise error rate.

At least as powerful as depth-wise hierarchical Bonferroni.

Substantial power gains for sparse signals along few branches.

Abstract

Hierarchical inference in (generalized) regression problems is powerful for finding significant groups or even single covariates, especially in high-dimensional settings where identifiability of the entire regression parameter vector may be ill-posed. The general method proceeds in a fully data-driven and adaptive way from large to small groups or singletons of covariates, depending on the signal strength and the correlation structure of the design matrix. We propose a novel hierarchical multiple testing adjustment that can be used in combination with any significance test for a group of covariates to perform hierarchical inference. Our adjustment passes on the significance level of certain hypotheses that could not be rejected and is shown to guarantee strong control of the familywise error rate. Our method is at least as powerful as a so-called depth-wise hierarchical Bonferroni adjustment. It provides a substantial gain in power over other previously proposed inheritance hierarchical procedures if the underlying alternative hypotheses occur sparsely along a few branches in the tree-structured hierarchy.