Groupwise Dimension Reduction via Envelope Method

The family of sufficient dimension reduction (SDR) methods that produce informative combinations of predictors, or indices, are particularly useful for high dimensional regression analysis. In many such analyses, it becomes increasingly common that there is available a priori subject knowledge of the predictors; e.g., they belong to different groups. While many recent SDR proposals have greatly expanded the scope of the methods' applicability, how to effectively incorporate the prior predictor structure information remains a challenge. In this article, we aim at dimension reduction that recovers full regression information while preserving the predictor group structure. Built upon a new concept of the direct sum envelope, we introduce a systematic way to incorporate the group information in most existing SDR estimators. As a result, the reduction outcomes are much easier to interpret. Moreover, the envelope method provides a principled way to build a variety of prior structures into dimension reduction analysis. Both simulations and real data analysis demonstrate the competent numerical performance of the new method.