Alternating imputation posterior estimation of models with crossed random effects

Generalized linear mixed models or latent variable models for categorical data are difficult to estimate if the random effects or latent variables vary at non-nested levels, such as persons and test items. Clayton and Rasbash (1999) suggested an Alternating Imputation Posterior (AIP) algorithm for approximate maximum likelihood estimation. For item response models with random item effects, the algorithm iterates between an item wing in which the item mean and variance are estimated for given person effects and a person wing in which the person mean and variance are estimated for given item effects. The person effects used for the item wing are sampled from the conditional posterior distribution estimated in the person wing and vice versa. Clayton and Rasbash (1999) used marginal quasi-likelihood (MQL) and penalized quasi-likelihood (PQL) estimation within the AIP algorithm, but this method has been shown to produce biased estimates in many situations, so we use maximum likelihood estimation with adaptive quadrature. We apply the proposed algorithm to the famous salamander mating data, comparing the estimates with many other methods, and to an educational testing dataset. We also present a simulation study to assess performance of the AIP algorithm and the Laplace approximation with different numbers of items and persons and a range of item and person variances.