Bridging Item Response Theory and Factor Analysis: A Four-Parameter Mixture-Dichotomized Model with Bayesian Estimation

Abstract: Item Response Theory (IRT) and Factor Analysis (FA) are two major frameworks used to model multi-item measurements of latent traits. While the relationship between two-parameter IRT models and dichotomized FA models is well established, IRT models with additional parameters have lacked corresponding FA formulations. This work introduces a four-parameter factor analytic (4P FA) model for multi-item measurements composed of binary items, building on the traditional dichotomized single-factor FA model. We derive the relationship between the proposed 4P FA model and its counterpart in the IRT framework, the 4P IRT model. A Bayesian estimation method is developed to estimate the four item parameters, the respondents' latent scores, and the scores adjusted for guessing and inattention effects. The proposed algorithm is implemented in R and Python, and the relationship between the 4P FA and 4P IRT models is empirically examined using two real datasets: a standardized admission test and a psychological anxiety inventory.