Priors in Bayesian Estimation under the Rasch Model

A review of various priors used in Bayesian estimation under the Rasch model is presented together with clear mathematical definitions of the hierarchical prior distributions. A Bayesian estimation method, Gibbs sampling, was compared with conditional, marginal, and joint maximum likelihood estimation methods using the Knox Cube Test data under the Rasch model. The shrinkage effect of the priors on item and ability parameter estimates was also investigated using the Knox Cube Test data. In addition, item response data for a mathematics test with 14 items by 765 examinees were analyzed with the joint maximum likelihood estimation method and Gibbs sampling under the Rasch model. Both methods yielded nearly identical item parameter estimates. The shrinkage effect was observed in the ability estimates from Gibbs sampling. The computer program OpenBUGS that implemented the rejection sampling method of Gibbs sampling was the main program employed in the study.