Pseudo-likelihood estimation of multidimensional response models: Polytomous and dichotomous items

Log-multiplicative association (LMA) models have been proposed as uni- and multidimensional item response models for dichotomous and/or polytomous items. A problem that prevents more widespread use of LMA models is that current estimation methods for moderate to large problems are computationally prohibitive. As a special case of a log-linear model, maximum likelihood estimation (MLE) of LMA models requires iteratively computing fitted values for all possible response patterns, the number of which increases exponentially as the number of items and/or response options per item increases. Anderson et al. (J. Stat. Softw. 20, 2007, doi:10.18637/jss.v020.i06) used pseudo-likelihood estimation for linear-by-linear models, which are special cases of LMA models, but in their proposal, the category scores are fixed to specific values. The solution presented here extends pseudo-likelihood estimation to more general LMA models where category scores are estimated. Our simulation studies show that parameter estimates from the new algorithm are nearly identical to parameter estimates from MLE, work for large numbers of items, are insensitive to starting values, and converge in a small number of iterations.