In this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models. The parameters of this latent class model will be estimated using an application of the Gibbs sampler. It will be illustrated that the Gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences. Model fit will be investigated using posterior predictive checks. Checks for manifest monotonicity, the agreement between the observed and expected conditional association structure, marginal local homogeneity, and the number of latent classes will be presented.

A lognormal model for response times is used to check response times for aberrances in examinee behavior on computerized adaptive tests. Both classical procedures and Bayesian posterior predictive checks are presented. For a fixed examinee, responses and response times are independent; checks based on response times offer thus information independent of the results of checks on response patterns. Empirical examples of the use of classical and Bayesian checks for detecting two different types of aberrances in response times are presented. The detection rates for the Bayesian checks outperformed those for the classical checks, but at the cost of higher false-alarm rates. A guideline for the choice between the two types of checks is offered.

Two psychometric models are presented for evaluating the difficulty of the distractors in multiple-choice items. They are based on the criterion of rising distractor selection ratios, which facilitates interpretation of the subject and item parameters. Statistical inferential tools are developed in a Bayesian framework: modal a posteriori estimation by application of an EM algorithm and model evaluation by monitoring posterior predictive replications of the data matrix. An educational example with real data is included to exemplify the application of the models and compare them with the nominal categories model.