Backward Joint Model for the Dynamic Prediction of Both Competing Risk and Longitudinal Outcomes

Abstract: Joint modeling is a useful approach to dynamic prediction of clinical outcomes using longitudinally measured predictors. When the outcomes are competing risk events, fitting the conventional shared random effects joint model often involves intensive computation, especially when multiple longitudinal biomarkers are be used as predictors, as is often desired in prediction problems. This paper proposes a new joint model for the dynamic prediction of competing risk outcomes. The model factorizes the likelihood into the distribution of the competing risks data and the distribution of longitudinal data given the competing risks data. It extends the basic idea of the recently published backward joint model (BJM) to the competing risk setting, and we call this model crBJM. This model also enables the prediction of future longitudinal data trajectories conditional on being at risk at a future time, a practically important problem that has not been studied in the statistical literature. The model fitting with the EM algorithm is efficient, stable and computationally fast, with a one-dimensional integral in the E-step and convex optimization for most parameters in the M-step, regardless of the number of longitudinal predictors. The model also comes with a consistent albeit less efficient estimation method that can be quickly implemented with standard software, ideal for model building and diagnostics. We study the numerical properties of the proposed method using simulations and illustrate its use in a chronic kidney disease study.