Backward joint model for the joint dynamic prediction of time-to-event and longitudinal data: basic formulation and new developments.

Dynamic prediction of future clinical outcomes based on longitudinally measured predictors plays a crucial role in disease management and patient counseling, particularly when conventional static models are inadequate. Joint modeling of longitudinal and time-to-event data provides a useful framework for addressing this challenge. In this paper, we present a comprehensive development of the recently proposed backward joint model (BJM; Shen and Li, 2021), which factorizes the likelihood into the distribution of time-to-event data and the conditional distribution of longitudinal data given the event time. This structure facilitates computation and is well-suited for multivariate longitudinal data. We introduce several novel developments to the BJM, including the extrapolation and two-part specifications, as well as the incorporation of competing risks. We also address an important yet underexplored problem in the literature: predicting future longitudinal trajectories conditional on predicted event times. Additionally, we explore the connection between BJM and existing joint modeling approaches. All these extensions preserve the computational advantages of the basic BJM formulation, including one-dimensional numerical integration, convex optimization via the EM algorithm, and a quick procedure for consistent estimation using standard software. We evaluate the method's performance through simulation studies and illustrate its utility in a chronic kidney disease application.