Bayesian Parameter Estimation Procedures
A more generally applicable approach is to use the MMLEjEM procedures of Chapter 6 to resolve the problem of inconsistent item parameter estimates. Marginal maximum likelihood estimation was distinguished from JMLE by the assumption that examinee abilities have a distribution in a population. When this distribution is known or can be estimated, it permits ability to be integrated out of the likelihood function. This frees item parameters from their dependence on the ability parameters of individual examinees, resulting in maximum likelihood estimates (MLEs) of the item parameters that are consistent for tests of finite length (Bock & Aitkin, 1981). Although MMLEjEM resolved the problem of inconsistent item parameter estimates, the problems of deviant values of the item parameter estimates in some data sets and the lack of a means of estimating an examinee's ability for unusual item response patterns remain. A vehicle for attempting to prevent deviant parameter estimates from occurring is the use of estimation procedures based on a Bayesian approach. One approach to applying these methods to parameter estimation under IRT was due to Swaminathan and Gifford (1982,
1985, 1986) who, in this set of three papers, derived estimation procedures for the one-, two-, and three-parameter logistic ICC models. The Birnbaum paradigm was used to estimate jointly the item parameters of a test and the abilities of the group of examinees. In contrast to the JMLE approach, Swaminathan and Gifford used Bayesian estimation procedures in each of the two stages. While this work employed many of the important concepts of Bayesian estimation within the context of IRT, it will not be included in the present chapter primarily because of the lack of a readily available computer program to analyze test data under the approach.