A General Approach for Estimating Projective IRT Models

Projective Item Response Theory (PIRT) modeling is a measurement framework in which multidimensional item response models are expressed in terms of lower-dimensional (e.g., unidimensional) proxy IRT models. Existing PIRT methods and their associated estimators currently rely on logistic function approximations that are limited to a narrow class of ordered, monotonic response functions. Though unexplored to date, these methods also require computationally intensive procedures to obtain sampling variability estimates for their resulting PIRT estimates. These and other limitations restrict the practical utility and broad application of PIRT models, particularly in empirical settings where only moderate sample sizes are available. To address these challenges, we introduce a general maximum marginal likelihood PIRT (MML-PIRT) approach that leverages components of the expectation–maximization (EM) algorithm commonly used in MML estimation. The proposed method uses expected count information generated during the EM-MML to fit proxy response functions for any focal trait of interest, and for a much broader class of multidimensional IRT models. In addition, MML-PIRT provides accurate and efficient large-sample variability estimates of the resulting PIRT model, thereby enhancing both the flexibility and statistical efficacy of PIRT model applications.