The multiple-group categorical factor analysis (FA) model and the graded response model (GRM) are commonly used to examine polytomous items for differential item functioning to detect possible measurement bias in educational testing. In this study, the multiple-group categorical factor analysis model (MC-FA) and multiple-group normal-ogive GRM models are unified under the common framework of discretization of a normal variant. We rigorously justify a set of identified parameters and determine possible identifiability constraints necessary to make the parameters just-identified and estimable in the common framework of MC-FA. By doing so, the difference between categorical FA model and normal-ogive GRM is simply the use of two different sets of identifiability constraints, rather than the seeming distinction between categorical FA and GRM. Thus, we compare the performance on DIF assessment between the categorical FA and GRM approaches through simulation studies on the MC-FA models with their corresponding particular sets of identifiability constraints. Our results show that, under the scenarios with varying degrees of DIF for examinees of different ability levels, models with the GRM type of identifiability constraints generally perform better on DIF detection with a higher testing power. General guidelines regarding the choice of just-identified parameterization are also provided for practical use.

A new model, called acceleration model, is proposed in the framework of the heterogenous case of the graded response model, based on processing functions defined for a finite or enumerable number of steps. The model is expected to be useful in cognitive assessment, as well as in more traditional areas of application of latent trait models. Criteria for evaluating models are proposed, and soundness and robustness of the acceleration model are discussed. Graded response models based on individual choice behavior are also discussed, and criticisms on model selection in terms of fitnesses of models to the data are also given.

The ISOP-model or model of twodimensional or bi-isotonicity (Scheiblechner, 1995) postulates that the probabilities of ordered response categories increase isotonically in the order of subject “ability” and item ”easiness”. Adding a conventional cancellation axiom for the factors of subjects and items gives the ADISOP model where the c.d.f.s of response categories are functions of an additive item and subject parameter and an ordinal category parameter. Extending cancellation to the interactions of subjects and categories as well as of items and categories (independence axiom of the category factor from the subject and item factor) gives the CADISOP model (completely additive model) in which the parallel c.d.f.s are functions of the sum of subject, item and category parameters. The CADISOP model is very close to the unidimensional version of the polytomous Rasch model with the logistic item/category characteristic(s) replaced by nonparametric axioms and statistics. The axioms, representation theorems and algorithms for model fitting of the additive models are presented.