Analysis of censored survival data with intermittently observed time-dependent binary covariates

In survival analyses that include a time-dependent covariate, the standard partial likelihood approach is often complicated by the fact that the necessary observations of the time-dependent covariate are not always available for the subjects in each risk set. In this article, we consider the situation of a binary time-dependent covariate. We specify a longitudinal model for the binary covariate with missing observations and a proportional hazards model for survival time conditional on fixed covariates and the completed time-dependent covariate history. The joint posterior distribution of the missing covariates and the parameters in the covariate and hazard models is simulated using Gibbs sampling. The joint modeling approach allows the missing covariates to be generated using all of the observed covariates as well as the disease outcome and failure time information. Simulation studies show that this approach often leads to lower bias of relative risk estimates associated with the time-dependent covariate and superior coverage of interval estimates as compared to variations of approaches based on imputing missing covariate values using previous values and then performing partial likelihood analysis. Our research is motivated by a study of survival of lung cancer patients after surgery in which current smoking status is a binary time-dependent covariate. We apply the methods to data from this study and explore the advantages of the joint modeling approach. The application demonstrates that the joint modeling approach can have improved efficiency in estimating the coefficients of both the fixed and the time-dependent covariates. Moreover, it can facilitate analyses using more complicated functions of the time-dependent covariate, such as cumulative smoking, that are difficult or inappropriate to use with the comparison methods.