Copula regression models for discrete and mixed bivariate responses

Estimation of the dependencies between bivariate discrete or mixed responses can be difficult. In this article, we propose a copula-based model with latent variables associated with discrete margins to account for correlations between bivariate discrete responses. Furthermore, we generalize this strategy for jointly modeling the dependencies between mixed responses in regression mixed models. The proposed method allows the adoption of flexible discrete margins and copula functions for various types of data. Maximum likelihood is used for model estimation; particularly, the estimation for bivariate responses in copula-based regression mixed models can be implemented using the SAS PROC NLMIXED procedure via adaptive Gaussian quadrature. In addition, a mixed model with non-Gaussian random effects can also be easily fitted using the same SAS procedure after reformulating the likelihood function by multiplying and dividing by a Gaussian density. Simulation results show good performance for bivariate discrete or mixed outcomes ranging from noncorrelated to highly correlated responses. An analysis of student performance in California schools shows a drastic improvement in estimation precision from the joint model versus two independent fits. © 2017 Grace Scientific Publishing, LLC.