Inference for the dimension of a regression relationship using pseudo-covariates

In data analysis using dimension reduction methods, the main goal is to summarize how the response is related to the covariates through a few linear combinations. One key issue is to determine the number of independent, relevant covariate combinations, which is the dimension of the sufficient dimension reduction (SDR) subspace. In this work, we propose an easily-applied approach to conduct inference for the dimension of the SDR subspace, based on augmentation of the covariate set with simulated pseudo-covariates. Applying the partitioning principal to the possible dimensions, we use rigorous sequential testing to select the dimensionality, by comparing the strength of the signal arising from the actual covariates to that appearing to arise from the pseudo-covariates. We show that under a "uniform direction" condition, our approach can be used in conjunction with several popular SDR methods, including sliced inverse regression. In these settings, the test statistic asymptotically follows a beta distribution and therefore is easily calibrated. Moreover, the family-wise type I error rate of our sequential testing is rigorously controlled. Simulation studies and an analysis of newborn anthropometric data demonstrate the robustness of the proposed approach, and indicate that the power is comparable to or greater than the alternatives.