  Joint modelling of longitudinal and time-to-event data is usually described
by a joint model which uses shared or correlated latent effects to capture
associations between the two processes. Under this framework, the joint
distribution of the two processes can be derived straightforwardly by assuming
conditional independence given the random effects. Alternative approaches to
induce interdependency into sub-models have also been considered in the
literature and one such approach is using copulas to introduce non-linear
correlation between the marginal distributions of the longitudinal and
time-to-event processes. The multivariate Gaussian copula joint model has been
proposed in the literature to fit joint data by applying a Monte Carlo
expectation-maximisation algorithm. In this paper, we propose an exact
likelihood estimation approach to replace the more computationally expensive
Monte Carlo expectation-maximisation algorithm and we consider results based on
using both the multivariate Gaussian and $t$ copula functions. We also provide
a straightforward way to compute dynamic predictions of survival probabilities,
showing that our proposed model is comparable in prediction performance to the
shared random effects joint model.
