A new copula regression model for hierarchical data

This paper 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 within a D-vine. Then we focus on situations where two variables are measured on the units of a cluster of arbitrary size. The proposed copula density has an explicit form; it is expressed in terms of three copula families. We study the properties of the model; compare it to the linear mixed model and end with special cases. When the three copula families and the marginal distributions are normal, the model is equivalent to a normal linear mixed model with random, cluster specific, intercepts. The method to select the three copula families and to estimate their parameters are proposed. We perform a Monte Carlo study of the parameter estimators. A data set on the marks of students in several school is used to implement the proposed model and to compare its performance to standard normal mixed linear models.