A Joint Model for Longitudinal Data with Terminal Event: Integrating Time-Varying Coefficients, Residual Life Effects, and Censored Data

Abstract

In longitudinal studies, a terminal event such as death typically halts data collection, and patients may exhibit marked changes as they approach the event. Existing methods face three related challenges: interpreting the effect of residual life on the longitudinal response, accommodating time-varying covariates and coefficients, and retaining information from censored individuals. We propose a joint model that addresses these challenges in a unified likelihood framework. The longitudinal submodel includes an explicit residual-life effect g(Ti−t,ξ), time-varying covariates Xi(t) with time-varying coefficients β(t), and shared random effects. Exponential-decay and Gaussian-kernel specifications are considered for g(·). The survival submodel depends on the latent state mi(t)=Xi(t)⊤β(t)+Zi(t)⊤bi. The coefficient functions are approximated by B-splines, and the full observed-data likelihood is maximized numerically with random-effects integrals evaluated by Gauss–Hermite quadrature. Under the stated regularity conditions, we establish consistency, root-n asymptotic normality of the finite-dimensional parameters, and the sieve convergence rate of the time-varying coefficient functions. Simulation studies demonstrate accurate recovery of the coefficient and residual-life functions. In the MADIT application, the fitted hazard ratio for ICD implantation is 0.433, and the residual-life effect indicates an increase in medical costs approximately three weeks before death.