A bivariate functional copula joint model for longitudinal measurements and time-to-event data

A bivariate functional copula joint model, which models the repeatedly measured longitudinal outcome at each time point with the survival data, jointly by both random effects and bivariate functional copulas, is proposed in this paper. A regular joint model normally supposes there are some subject-specific latent random effects or classes shared by the longitudinal and time-to-event processes and they are assumed to be conditionally independent given these latent random variables. Under this assumption, the joint likelihood of the two processes can be easily derived and the association between them, as well as heterogeneity among population are naturally introduced by the unobservable latent random variables. However, because of the unobservable nature of these latent variables, the conditional independence assumption is difficult to verify. Therefore, a bivariate functional copula is introduced into a regular joint model to account for the cases where there could be extra association between the two processes which cannot be captured by the latent random variables. Our proposed model includes a regular joint model as a special case when the correlation function, which is modelled continuously by B-spline basis functions as a function of time $t,$ is constant at 0 under the bivariate Gaussian copula. Simulation studies and dynamic prediction of survival probabilities are conducted to compare the performance of the proposed model with the regular joint model and a real data application on the Primary biliary cirrhosis (PBC) data is performed.