  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
after the References.
