Bayesian estimation and model selection in ordered latent class models for polytomous items

In a latent class IRT model in which the latent classes are ordered on one dimension, the class specific response probabilities are subject to inequality constraints. The number of these inequality constraints increase dramatically with the number of response categories per item, if assumptions like monotonicity or double monotonicity of the cumulative category response functions are postulated. A Markov chain Monte Carlo method, the Gibbs sampler, can sample from the multivariate posterior distribution of the parameters under the constraints. Bayesian model selection can be done by posterior predictive checks and Bayes factors. A simulation study is done to evaluate results of the application of these methods to ordered latent class models in three realistic situations. Also, an example of the presented methods is given for existing data with polytomous items. It can be concluded that the Bayesian estimation procedure can handle the inequality constraints on the parameters very well. However, the application of Bayesian model selection methods requires more research.