The relationship between a time-dependent covariate and a failure time process can be assessed using the Cox model. Time-dependent covariates are generally available as longitudinal data collected regularly during the course of the study. A frequently encountered problem in practice is the occurence of missing covariate data. Several imputation methods have been proposed for handling missing covariates in the Cox model. Dupuy and Mesbah (2002) have proposed an alternative approach to the problem of Cox regression with a time-dependent covariate when its value at failure time is not observed. In this talk, we describe this approach which jointly models the time-dependent covariate and the time-to-failure process. Identifiability of the proposed joint model is proved. Estimation of the parameters in the joint model is conducted by maximization of a likelihood obtained from both the covariate process and the time-to-failure data. Direct maximization of this likelihood is not possible because the baseline hasard is not specified. Hence we modify the likelihood by constraining the cumulative baseline hazard to be a step function with jumps at the observed distinct failure times. We prove existence of maximum likelihood estimators of the parameters of the joint model, based on this modified likelihood. Then we show that these estimators are consistent and asymptotically normally distributed
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