Advancing the Bayesian Approach for Multidimensional Polytomous and Nominal IRT Models: Model Formulations and Fit Measures.

It is common to encounter polytomous and nominal responses with latent variables in social or behavior research, and a variety of polytomous and nominal item response theory (IRT) models are available for applied researchers across diverse settings. With its flexibility and scalability, the Bayesian approach using the Markov chain Monte Carlo (MCMC) method demonstrates its great advantages for polytomous and nominal IRT models. However, the potential of the Bayesian approach would not be fully realized without model formulations that can cover various models and effective fit measures for model assessment or criticism. This research first provided formulations for typical models that are representative of different modeling groups. Then, a series of discrepancy measures that can offer diagnostic information for model-data misfit were introduced. Simulation studies showed that the formulation worked as expected, and some of the fit measures were more useful than the others or across different situations.