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.
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