Testing normality of latent variables in the polychoric correlation

Abstract

This paper explores the feasibility of simultaneously facing three sources of complexity in Bayesian testing, namely (i) testing a parametric against a non-parametric alternative (ii) adjusting for partial observability (iii) developing a test under a Bayesian encompassing principle. Testing the normality of latent variables in the polychoric correlation model is taken as a case study. This paper starts from the specification of the model defining the polychoric correlation in the framework of manifest ordinal variables viewed as discretizations of underlying latent variables. Taking advantage of the fact that in this model, the marginal distributions of the latent variables are not identified, we use the approach of copula. Some identification issues are analysed. Next, we develop a Bayesian encompassing specification test for testing the Gaussianity of the underlying copula and consider the discretization model as a case of partial observability. The computational feasibility, the numerical stability and the discriminating power of the procedure are checked through a simulation experiment. An application completes the paper by illustrating the working of the procedure on a meta-analysis of clinical trials on acute migraine. The final section proposes, in the form of conclusions, an evaluation of the actual achievements of the paper.