A new copula regression model for hierarchical data

Abstract

This article proposes multivariate copula models for hierarchical data. They account for two types of correlation: one is between variables measured on the same unit, and the other is a correlation between units in the same cluster. This model is used to carry out copula regression for hierarchical data that gives cluster-specific prediction curves. In the simple case where a cluster contains two units and where two variables are measured on each one, the new model is constructed with a $D$-vine. The proposed copula density is expressed in terms of three copula families. When the copula families and the marginal distributions are normal, the model is equivalent to a normal linear mixed model with random cluster-specific intercepts. Methods to select the three copula families and to estimate their parameters are proposed. We perform Monte Carlo studies of the sampling properties of these estimators and of out-of-sample predictions. The new model is applied to a dataset on the marks of students in several schools.