19. On the Landmark Survival Model for Dynamic Prediction of Event Occurrence Using Longitudinal Data

In longitudinal cohort studies, participants are often monitored through periodic clinical visits until the occurrence of a terminal clinical event. A question of interest to both scientific research and clinical practice is to predict the risk of the terminal event at each visit, using the longitudinal prognostic information collected up to the visit. This problem is called the dynamic prediction: a real-time, personalized prediction of the risk of a future adverse clinical event with longitudinally measured biomarkers and other prognostic information. An important method for dynamic prediction is the landmark Cox model and variants. A fundamental difficulty in the current methodological research of this kind of models is that it is unclear whether there exists a joint distribution of the longitudinal and time-to-event data that satisfies the model assumptions. As a result, this model is often viewed as a working model instead of a probability distribution, and its statistical properties are often studied using data simulated from shared random effect models, where the landmark model works under misspecification. In this paper, we demonstrate that a joint distribution of longitudinal and survival data exists that satisfy the modeling assumptions without additional restrictions, and propose an algorithm to generate data from this joint distribution. We further generalize the results to the more flexible landmark linear transformation models that include the landmark Cox model as a special case. These results facilitate future theoretical and numerical research on landmark survival models for dynamic prediction.