Bayes Factor Covariance Testing in Item Response Models.

Area of Science:

• Psychometrics
• Statistical modeling
• Item response theory

Background:

• Item response theory (IRT) models are widely used for analyzing test data.
• Traditional IRT models often assume unidimensionality and local independence.
• Evaluating these assumptions, especially with binary data, can be challenging.

Purpose of the Study:

• Introduce two marginal one-parameter IRT models.
• Develop methods for testing covariance structures in binary response data.
• Evaluate unidimensionality and differential item functioning.

Main Methods:

• Integrated out latent variables to derive marginal response models.
• Utilized multivariate probit models with compound symmetry covariance structure.
• Employed fractional Bayes factor tests for hypothesis evaluation.
• Derived closed-form posterior distributions for covariance components.
• Developed a Markov Chain Monte Carlo (MCMC) algorithm for parameter estimation and Bayes factor computation.

Main Results:

• Demonstrated that marginal response models are multivariate probit models.
• Showed that fractional Bayes factor tests have good properties for binary data.
• Successfully estimated model parameters and computed Bayes factors using MCMC.
• Illustrated the method with two real data studies.

Conclusions:

• The proposed marginal IRT models provide a flexible framework for analyzing binary data.
• Fractional Bayes factor tests are effective for assessing covariance structures, unidimensionality, and differential item functioning.
• The MCMC algorithm facilitates practical application of these models.