Kernel Dimension Reduction with Missing Data

Abstract

Kernel dimension reduction (KDR), a form of sufficient dimension reduction (SDR), is a framework for identifying potentially nonlinear multivariate relation- ships between high-dimensional predictors X and outcomes Y , both of which may be multivariate. Here we propose a way to accommodate missing data in either the predictors or the outcomes, enabling KDR to be applied in a much broader range of settings. We cast the problem as that of predicting the missing elements of the kernel matrices using their conditional expected values given all observed data, based on an auxiliary model. We present simulation studies showing that our method is computationally tractable for moderate-sized data sets and has good statistical performance. To aid in interpretation of the nonlinear sufficient predictors, we use Multivariate Adaptive Regression Splines (EARTH/MARS) to estimate the unknown link functions. We illustrate the approach by presenting an analysis of longitudinal data of height for age z-score (HAZ) and systolic blood pressure (SBP) in a sample of people from the Dogon population of Mali.