Hierarchical Testing of Variable Importance

A frequently encountered challenge in high-dimensional regression is the detection of relevant variables. Variable selection suffers from instability and the power to detect relevant variables is typically low if predictor variables are highly correlated. When taking the multiplicity of the testing problem into account, the power diminishes even further. To gain power and insight, it can be advantageous to look for influence not at the level of individual variables but rather at the level of clusters of highly correlated variables. We propose a hierarchical approach. Variable importance is first tested at the coarsest level, corresponding to the global null hypothesis. If possible, the method tries then to attribute any effect to smaller sub-clusters or even individual variables. The smallest possible clusters which still exhibit a significant influence on the response variable are retained. It is shown that the proposed testing procedure controls the family-wise error rate at a prespecified level, simultaneously over all resolution levels. The method has comparable power to Bonferroni-Holm on the level of individual variables and dramatically larger power for coarser resolution levels. The best resolution level is selected adaptively.