Simulated maximum likelihood estimation in joint models for multiple longitudinal markers and recurrent events of multiple types, in the presence of a terminal event

In medical studies we are often confronted with complex longitudinal data. During the follow-up period, which can be ended prematurely by a terminal event (e.g. death), a subject can experience recurrent events of multiple types. In addition, we collect repeated measurements from multiple markers. An adverse health status, represented by ‘bad’ marker values and an abnormal number of recurrent events, is often associated with the risk of experiencing the terminal event. In this situation, the missingness of the data is not at random and, to avoid bias, it is necessary to model all data simultaneously using a joint model. The correlations between the repeated observations of a marker or an event type within an individual are captured by normally distributed random effects. Because the joint likelihood contains an analytically intractable integral, Bayesian approaches or quadrature approximation techniques are necessary to evaluate the likelihood. However, when the number of recurrent event types and markers is large, the dimensionality of the integral is high and these methods are too computationally expensive. As an alternative, we propose a simulated maximum-likelihood approach based on quasi-Monte Carlo integration to evaluate the likelihood of joint models with multiple recurrent event types and markers.