Alternative Specifications of Multivariate Multilevel Probit Ordinal Response Models

Abstract

Multivariate multilevel models for ordinal variables are quite complex with respect to both interpretation and estimation. The specification in terms of a multivariate latent distribution and a set of thresholds helps in the interpretation of the variance-covariance parameters. However, most existing estimation algorithms for multilevel models can be used only if the model is reparameterized as a univariate model with an additional dummy bottom level. Moreover, the univariate formulation allows the model to be cast in the framework of Generalized Linear Latent and Mixed Models ( Rabe-Hesketh, Pickles, & Skrondal, 2001a ), a rather general class that includes, as special cases, structural equations and factor models. This article outlines the multivariate latent distribution specification and the corresponding interpretation issues; it then shows the univariate formulation, along with some alternative parameterizations that are useful in the estimation phase. An application to student ratings data illustrates the interpretation of the parameters and the estimation procedures, with a discussion of some computational issues.