Bayesian Item Response model: a generalised approach for the abilities' distribution using mixtures

TL;DR

This paper introduces a Bayesian mixture model for item response theory that allows for flexible ability distributions beyond Gaussian assumptions, improving modeling of diverse population traits.

Contribution

It develops a generalized Bayesian IRT model using mixtures of normals for abilities, with a novel MCMC algorithm ensuring good convergence and interpretability.

Findings

Effective in modeling skewed and multimodal ability distributions

Demonstrates improved fit over traditional Gaussian-based models

Applicable to both simulated and real data examples

Abstract

Traditional Item Response Theory models assume the distribution of the abilities of the population in study to be Gaussian. However, this may not always be a reasonable assumption, which motivates the development of more general models. This paper presents a generalised approach for the distribution of the abilities in dichotomous 3-parameter Item Response models. A mixture of normal distributions is considered, allowing for features like skewness, multimodality and heavy tails. A solution is proposed to deal with model identifiability issues without compromising the flexibility and practical interpretation of the model. Inference is carried out under the Bayesian Paradigm through a novel MCMC algorithm. The algorithm is designed in a way to favour good mixing and convergence properties and is also suitable for inference in traditional IRT models. The efficiency and applicability of our methodology is illustrated in simulated and real examples.