The Role of Posterior Densities in Latent Variable Models for Ordinal Data

In latent variable models, problems related to the integration of the likelihood function arise since analytical solutions do not exist. Laplace and Adaptive Gauss-Hermite (AGH) approximations have been discussed as good approximating methods. Their performance relies on the assumption of normality of the posterior density of the latent variables, but, in small samples, this is not necessarily assured. Here, we analyze how the shape of the posterior densities varies as function of the model parameters, and we investigate its influence on the performance of AGH and of the Laplace approximation.