Estimation of Optimal Sufficient Dimension Reduction Score and Its Extension to Censored Data

Rank-based measures, which is used to access the concordance between the univariate response variable y and a linear composite score of its p-dimensional explanatory variable z, has been studied because of its applicability to a wide variety of survival data. In this article, a new rank-based measure is developed for extending a linear score to a multivariate polynomial one, which captures the most information of z with respect to y. This new measure explores the simplicity of the graphical view of regression; that is, we can regress y against this multivariate polynomial score based on dimension reduction framework. To introduce this concept, a general semiparametric model, which characterizes the dependence of a response on covariates through a multivariate polynomial transformation of the central subspace (CS) directions with unknown structural degree and dimension, is proposed. In light of the monotonic model structure and defined concordance index (C-index) function, such a composite score, which is referred to as the optimal sufficient dimension reduction (SDR) score, is shown to enjoy the existence, optimality, and uniqueness up to scale and location. By means of these properties and the generalized single-index (SI) representation of any multivariate polynomial function, the concordance-based generalized Bayesian information criterion (BIC) is proposed to estimate the optimal SDR score and its corresponding C-index, say Cmax. Meanwhile, effective computational algorithms are offered to carry out the presented estimation procedure. With estimated structural degree and dimension from this BIC, an alternative approach is further developed to estimate the optimal SDR score and Cmax. In addition, we establish the consistency of structural degree and dimension estimators and the asymptotic normality of optimal SDR score and Cmax estimators. As for significant covariates, a variable selection is proposed to retain important confounding variables when p is large. In general, survival data is partially observed due to right-censoring. In this case, a partial rank-based approach allows us to follow the similar estimation procedure with the completed data. Further, we adopt an imputation method to recover unobserved counting process for estimating Cmax. The performance and practicality of our proposal are also investigated by a series of simulations and illustrated examples.