Weighted estimators for proportional hazards regression with missing covariates

Missing covariate data are common in epidemiologic studies and disease prevention trials. In this article regression parameter estimation in the Cox proportional hazards model is considered when certain covariates are observed for all study subjects and other covariate data are collected only for a subset. The article presents both simple weighted and kernel-assisted fully augmented weighted estimators that use the partially incomplete data nonparametrically. We use nonparametric methods to estimate selection probabilities in the simple weighted estimating functions. We also use nonparametric kernel smoothing techniques to estimate certain conditional expectations in fully augmented weighted estimating functions. The proposed methods are nonparametric in the sense that they require neither a model for the missing-data mechanism nor specification of the conditional distribution of missing covariates given observed covariates. These estimators allow the missing-data mechanism to depend on outcome variables and observed covariates, and they are applicable to various cohort Sampling procedures, including case-cohort and nested case-control designs. We show that the simple and the kernel-assisted fully augmented weighted estimators are typically consistent and asymptotically normal. Moreover, the proposed estimators are more efficient than the simple weighted estimator with the inverse of true selection probability as weight. They also correct the bias of estimates from analysis of the complete data alone when the missing-data mechanism depends on outcome variables. In addition, when covariates are time-independent, certain simple weighted estimators are shown to be asymptotically equivalent to the kernel-assisted fully augmented weighted estimators. Moderate sample size performance of the estimators is examined via simulation and by application to two real datasets.