Hierarchical Testing in the High-Dimensional Setting with Correlated Variables

We propose a method for testing whether hierarchically ordered groups of potentially correlated variables are significant for explaining a response in a high-dimensional linear model. In presence of highly correlated variables, as is very common in high-dimensional data, it seems indispensable to go beyond an approach of inferring individual regression coefficients, and we show that detecting smallest groups of variables (MTDs: minimal true detections) is realistic. Thanks to the hierarchy among the groups of variables, powerful multiple testing adjustment is possible which leads to a data-driven choice of the resolution level for the groups. Our procedure, based on repeated sample splitting, is shown to asymptotically control the familywise error rate and we provide empirical results for simulated and real data which complement the theoretical analysis. Supplementary materials for this article are available online.