Modeling temporal dependency of longitudinal data: use of multivariate geometric skew-normal copula

Use of copula for the purpose of modeling dependence has been receiving considerable attention in recent times. On the other hand, search for multivariate copulas with desirable dependence properties also is an important area of research. When fitting regression models to non-Gaussian longitudinal data, multivariate Gaussian copula is commonly used to account for temporal dependence of the repeated measurements. But using symmetric multivariate Gaussian copula is not preferable in every situation, since it can not capture non-exchangeable dependence or tail dependence, if present in the data. Hence to ensure reliable inference, it is important to look beyond the Gaussian dependence assumption. In this paper, we construct geometric skew-normal copula from multivariate geometric skew-normal (MGSN) distribution proposed by Kundu (2014) and Kundu (2017) in order to model temporal dependency of non-Gaussian longitudinal data. First we investigate the theoretical properties of the proposed multivariate copula, and then develop regression models for both continuous and discrete longitudinal data. The quantile function of this copula is independent of the correlation matrix of its respective multivariate distribution, which provides computational advantage in terms of likelihood inference compared to the class of copulas derived from skew-elliptical distributions by Azzalini & Valle (1996). Moreover, composite likelihood inference is possible for this multivariate copula, which facilitates to estimate parameters from ordered probit model with same dependence structure as geometric skew-normal distribution. We conduct extensive simulation studies to validate our proposed models and therefore apply them to analyze the longitudinal dependence of two real world data sets. Finally, we report our findings in terms of improvements over multivariate Gaussian copula based regression models.