Pseudo-likelihood estimation of multidimensional polytomous item response theory models

Log-multiplicative association (LMA) models, special cases of log-linear models, can be used as multidimensional item response theory (MIRT) models for polytomous items (Anderson, Verkuilen and Peyton, 2010; Anderson, 2013). LMA models do not require numerical integration for their estimation nor do they require assumptions regarding the marginal distribution of the latent variables. However, maximum likelihood estimation (MLE) of LMA models requires iteratively computing fitted values for all possible response patterns. Standard estimation methods for large numbers of items fail because the number of possible response patterns increases exponentially as the number of items and response options per item increase. In this study, a new algorithm is proposed to solve this estimation problem. Anderson, Li and Vermunt (2007) proposed using pseudo-likelihood estimation (PLE); however, their solution only applies to models in the Rasch family, which exploits the relationship between log-linear and logistic regression models. Their method is extended to more general models by adding an additional step that estimates slope (item discrimination) parameters for the latent variables. The new algorithm has two basic steps and simplifies for special cases. In Step 1, a (multinomial) logistic regression model is fit by MLE to one item using rest-scores as an explanatory variable to get new estimates of item slopes that are used in the rest-score for the next item. This process is repeated for each item until all item slopes have been up-dated. Step 2 involves fitting a single conditional logistic regression model for a data set formed by stacking the conditional logistic regressions for each item. This yields new estimates of location (item difficulty) parameters and the covariance matrix for the latent variables. Steps 1 and 2 are repeated until all parameter estimates converge. The results of simulation and empirical studies with real data show that the proposed algorithm successfully estimates parameters in more general LMA models with both location and slope parameters as MIRT models