Bayesian Estimation of Item Response Curves

We will consider dichotomous responses to a set of test items which are designed to measure the abilities of individuals. We assume that each item is characterized by an item response curve, a function of ability which is indexed by unknown parameters called item parameters. We consider a Bayesian method for estimating these parameters when the abilities of the individuals are assumed to have a normal prior distribution. A standard method for estimating abilities and item parameters in the absence of prior information is maximum likelihood (see Lord, 1980). Under the assumption that abilities are normally distributed, the maximum likelihood (ML) estimates of item parameters have been studied through various applications of the EM algorithm (Dempster, Laird, & Rubin, 1977) by Bock and Aitkin (1981), and Rigdon and Tsutakawa (1983), among others. The Bayesian hierarchical approach developed for linear models by Lindley and Smith (1972) has been adapted to estimating item parameters by Swaminathan (1981) Swaminathan & Gifford (1982). The procedure is dependent on obtaining modal estimates as a solution to a simultaneous system of a large number of equations. As suggested by the illustration in Novick, Jackson; Thayer, and Cole (1972), one of the issues concerning the hierarchical approach is the problem of specifying.the prior for the hyperparameter, about which information is usually limited. In this paper we show how the computational simplicity of the EM algorithm for ML estimation of item parameters continues to hold in finding the posterior mode, provided the item parameters have independent prior distributions. This simplicity consists of being able to work iterative!y one item at a time rather than with all items simultaneously. We also introduce a new family of priors for the item parameters, which we believe can be more readily specified in practice. It is based on the user’s prior belief about the probability of correct response to each item for subjects at given percentiles of the ability distribution.