A comparison among different solutions for assessing the goodness of fit of a generalized linear latent variable model for ordinal data

Generalized linear latent variable models (GLLVM) aim to explain the interrelationships among a set of continuous and/or categorical observed variables through a smaller set of latent variables. In this paper we analyze different solutions to evaluate the goodness of fit of a GLLVM with ordinal observed variables in presence of sparse data. In this situation the theoretical distribution of classical goodness of fit statistics is badly approximated by the chi square distribution and alternative strategies have to be used. Through a Monte Carlo simulation study the performance of the classical statistics are compared with the ones based on the residuals associated to the bivariate marginal distributions of the observed variables. In particular we present an extension of a test proposed by Reiser (1996) for a model with binary data.