Abstract: We consider Cox proportional hazards regression when longitudinal measurements are available. In some applications, one major goal is to estimate the effect of the underlying change of the longitudinal measurements on survival. One general approach considers regression analysis when some covariate variables are the underlying regression coefficients of another random effects model. For each subject, the covariate variables to the primary regression model are not observed, but can be estimated from the observed longitudinal measurements. This set-up is often called joint modeling in the literature, but it can be treated as two-stage modeling. In this paper, a corrected score estimator is investigated. Comparisons are made with a naive estimator, a regression calibration estimator, a risk set regression calibration estimator, and a conditional score estimator. Similar to the conditional score estimator, the corrected score estimator does not need the assumption of an underlying distribution of the random effects for each subject. Under some regularity conditions, the proposed corrected score estimator is shown to be consistent and asymptotically normally distributed. Simulation results under various random effects distributions are presented.
Key words and phrases: Empirical process, estimating equation, measurement error, proportional hazards, random effects.