Current procedures for estimating compensatory multidimensional item response theory (MIRT) models using Markov chain Monte Carlo (MCMC) techniques are inadequate in that they do not directly model the interrelationship between latent traits. This limits the implementation of the model in various applications and further prevents the development of other types of IRT models that offer advantages not realized in existing models. In view of this, an MCMC algorithm is proposed for MIRT models so that the actual latent structure is directly modeled. It is demonstrated that the algorithm performs well in modeling parameters as well as intertrait correlations and that the MIRT model can be used to explore the relative importance of a latent trait in answering each test item.
Sheng, Y., & Headrick, T. C. (2012). A Gibbs Sampler for the Multidimensional Item Response Model. ISRN Applied Mathematics, 2012, 1–14. https://doi.org/10.5402/2012/269385