A Bayesian approach to person-fit analysis in item response theory models

A Bayesian approach to the evaluation of person fit in item response theory (IRT) models is presented. In a posterior predictive check, the observed value on a discrepancy variable is positioned in its posterior distribution. In a Bayesian framework, a Markov chain Monte Carlo procedure can be used to generate samples of the posterior distribution of the parameters of interest. These draws can also be used to compute the posterior predictive distribution of the discrepancy variable. The procedure is worked out in detail for the three-parameter normal ogive model, but it is also shown that the procedure can be directly generalized to many other IRT models. Type I error rate and the power against some specific model violations are evaluated using a number of simulation studies. Index terms: Bayesian statistics, item response theory, person fit, model fit, 3-parameter normal ogive model, posterior predictive check, power studies, Type I error.