A Copula-based GLMM Model For Multivariate Longitudinal Data With Mixed-types Of Outcomes

Modelling multivariate longitudinal data is much more challenging than for the univariate case.This is due to the correlation between different outcomes must be taken into account besides the correlation over time.In this paper,we propose a copula-based generalized linear mixed model(GLMM)to jointly analyze multivariate longitudinal data with mixed types,including continuous,count and binary outcomes.The association of repeated measurements is modelled through the GLMM model,meanwhile a pair-copula construction(D-vine)is adopted to measure the dependency structure between different outcomes.By combining mixed models and D-vine copulas,our proposed approach could not only deal with unbalanced data with arbitrary margins but also handle high-dimensional problems due to the efficiency and flexibility of D-vines.Based on D-vine copulas,algorithms for sampling mixed data and for computing likelihood are also developed.Leaving the random effects distribution unspecified,we use nonparametric maximum likelihood for model fitting.Then an EM algorithm is used to obtain the maximum likelihood estimates of parameters.Both simulations and real data analysis show that the nonparametric models are more efficient and flexible than the parametric models.