Bayesian Estimation of Multidimensional Polytomous Item Response Theory Models with Q-matrices Using Stan.

The Q-matrix is commonly used in diagnostic classification models and has recently been incorporated into the multidimensional item response theory (MIRT) models to add information about the relationship between items and dimensions of the latent trait. The reformulation of the MIRT models with Q-matrix (MIRT-Q) has presented to improve the precision of the parameters of these models and to provide a simple and intuitive method for users to define the item-trait relationship. This paper aims to explore the incorporation of the Q-matrix in the formulation of MIRT models for polytomous item responses. Specifically, we introduce the incorporation of the Q-matrix into two of the polytomous MIRT models most known and used: the multidimensional graded response (MGR) model, hereinafter called MGR-Q, and the multidimensional generalized partial credit (MGPC) model, hereinafter called MGPC-Q. We provide readers the code of the MGR-Q and MGPC-Q models in Stan, a Bayesian estimation software, and we conduct a simulation study in order to evaluate the parameter recovery of the estimation method. To illustrate the use of both models in practice, we fit them to an operational dataset from 2400 individuals on 13 items and demonstrate the estimation of MGR-Q and MGPC-Q using the Stan program.