Joint Models with Multiple Markers and Multiple Time-to-event Outcomes Using Variational Approximations

Joint models are well suited to modelling linked data from laboratories and health registers. However, there are few examples of joint models that allow for (a) multiple markers, (b) multiple survival outcomes (including terminal events, competing events, and recurrent events), (c) delayed entry and (d) scalability. We propose a full likelihood approach for joint models based on a Gaussian variational approximation to satisfy criteria (a)-(d). We provide an open-source implementation for this approach, allowing for flexible sets of models for the longitudinal markers and survival outcomes. Through simulations, we find that the lower bound for the variational approximation is close to the full likelihood. We also find that our approach and implementation are fast and scalable. We provide an application with a joint model for longitudinal measurements of dense and fatty breast tissue and time to first breast cancer diagnosis. The use of variational approximations provides a promising approach for extending current joint models.