Modeling variability in item parameters in item response models

In some areas of measurement item parameters should not be modeled as fixed but as random. Examples of such areas are: item sampling, computerized item generation, measurement with substantial estimation error in the item parameter estimates, and grouping of items under a common stimulus or in a common context. A hierarchical version of the three-parameter normalogive model is used to model parameter variability in multiple populations of items. Two Bayesian procedures for the estimation of the parameter are given. The first method produces an estimate of the posterior distribution using a Markov Chain Monte Carlo method (Gibbs sampler); the second procedure produces a Bayes modal-estimate. It is shown that the procedure using the Gibbs sampler breaks down if for some of the random item parameters the sampling design yields only one response. However, in this case, marginalization over the item parameters does result in a feasible estimation procedure. Some numerical examples are given.