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

TL;DR

This paper introduces a novel four-parameter factor analytic model for binary items, establishing its relationship with four-parameter IRT models and providing Bayesian estimation methods, validated through real data examples.

Contribution

It develops the first four-parameter FA model for binary data and links it to the 4P IRT framework, with Bayesian estimation techniques.

Findings

The 4P FA model accurately captures item and respondent parameters.

The Bayesian estimation method effectively estimates model parameters.

Empirical analysis demonstrates the model's applicability to real datasets.

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.