Extended Mixed‐Effects Item Response Models With the MH‐RM Algorithm

Abstract

A mixed‐effects item response theory (IRT) model is presented as a logical extension of the generalized linear mixed‐effects modeling approach to formulating explanatory IRT models. Fixed and random coefficients in the extended model are estimated using a Metropolis‐Hastings Robbins‐Monro (MH‐RM) stochastic imputation algorithm to accommodate for increased dimensionality due to modeling multiple design‐ and trait‐based random effects. As a consequence of using this algorithm, more flexible explanatory IRT models, such as the multidimensional four‐parameter logistic model, are easily organized and efficiently estimated for unidimensional and multidimensional tests. Rasch versions of the linear latent trait and latent regression model, along with their extensions, are presented and discussed, Monte Carlo simulations are conducted to determine the efficiency of parameter recovery of the MH‐RM algorithm, and an empirical example using the extended mixed‐effects IRT model is presented.