Sufficient dimension reduction with missing predictors

In high-dimensional data analysis, sufficient dimension reduction (SDR) methods are effective in reducing the predictor dimension, while retaining full regression information and imposing no parametric models. However, it is common in high-dimensional data that a subset of predictors may have missing observations. Existing SDR methods resort to the complete-case analysis by removing all the subjects with missingness in any of the predictors under inquiry. Such an approach does not make effective use of the data and is valid only when missingness is independent of both observed and unobserved quantities. In this article, we propose a new class of SDR estimators under a more general missingness mechanism that allows missingness to depend on the observed data. We focus on a widely used SDR method, sliced inverse regression, and propose an augmented inverse probability weighted sliced inverse regression estimator (AIPW-SIR). We show that AIPW-SIR is doubly robust and asymptotically consistent and demonstrate that AIPW-SIR is more effective than the complete-case analysis through both simulations and real data analysis. We also outline the extension of the AIPW strategy to other SDR methods, including sliced average variance estimation and principal Hessian directions.