Sufficient dimension reduction via distance covariance for survival data

Sufficient dimension reduction (SDR) plays an essential role in high-dimensional regression analysis, and most existing SDR methods suffer when data are subject to censoring. This paper tries to extend distance covariance (dCov) to SDR for survival data. Unlike most methods requiring strict constraints on predictors and survival time, the method based on dCov only relies on very mild conditions and works well when many predictors are categorical or discrete. The difficulty of SDR for survival data is that we cannot completely observe the real survival time, which may introduce a substantial bias. When the censoring time is independent of survival time, we show that the dCov-SDR method can estimate the SDR directions with a surrogate response. Furthermore, under other censoring assumptions, we extend the dCov-SDR method by proposing a new consistent empirical version of dCov for survival data. We discuss the inherent properties of our method, and under regularity conditions, statistical consistency is established for our estimators. Both simulations and real data analysis demonstrate the promising performance of the proposed method.