A Joint Model for Survival and Longitudinal Data Measured with Error

The relationship between a longitudinal covariate and a failure time process can be assessed using the Cox proportional hazards regression model. We consider the problem of estimating the parameters in the Cox model when the longitudinal covariate is measured infrequently and with measurement error. We assume a repeated measures random effects model for the covariate process. Estimates of the parameters are obtained by maximizing the joint likelihood for the covariate process and the failure time process. This approach uses the available information optimally because we use both the covariate and survival data simultaneously. Parameters are estimated using the expectation-maximization algorithm. We argue that such a method is superior to naive methods where one maximizes the partial likelihood of the Cox model using the observed covariate values. It also improves on two-stage methods where, in the first stage, empirical Bayes estimates of the covariate process are computed and then used as time-dependent covariates in a second stage to find the parameters in the Cox model that maximize the partial likelihood.