Approximation for Bayesian Ability Estimation

Abstract

An approximation is proposed for the posterior mean and standard deviation of the ability parameter in an item response model. The procedure assumes that approximations to the posterior mean and covariance matrix of item parameters are available. It is based on the posterior mean of a Taylor series approximation to the posterior mean conditional on the item parameters. The method is illustrated for the two-parameter logistic model using data from an ACT math test with 39 items. A numerical comparison with the empirical Bayes method using n = 400 examinees shows that the point estimates are very similar but the standard deviations under empirical Bayes are about 2% smaller than those under Bayes. Moreover, when the sample size is decreased to n = 100, the standard deviation under Bayes is shown to increase by 14% in some cases.