The purpose of this note is to focus on the finiteness of the weighted likelihood estimator (WLE) of ability in the context of dichotomous and polytomous item response theory (IRT) models. It is established that the WLE always returns finite ability estimates. This general result is valid for dichotomous (one-, two-, three- and four-parameter logistic) IRT models, the class of polytomous difference models and divide-by-total models, independently of the number of items, the item parameters and the response patterns. Further implications of this result are outlined.

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.

Warm (in Psychometrika, 54, 427–450, 1989) established the equivalence between the so-called Jeffreys modal and the weighted likelihood estimators of proficiency level with some dichotomous item response models. The purpose of this note is to extend this result to polytomous item response models. First, a general condition is derived to ensure the perfect equivalence between these two estimators. Second, it is shown that this condition is fulfilled by two broad classes of polytomous models including, among others, the partial credit, rating scale, graded response, and nominal response models.

Likert-type items and polytomous models are preferred over yes–no items and dichotomous models for the measurement of attitudes, because a broader range of response categories provides superior item and test information functions. Yet, for ability assessment with multiple-choice tests, the dichotomous three-parameter logistic model (3PLM) is often chosen. Because multiple-choice responses are polytomous before they are categorized as correct or incorrect, a polytomous characterization might render more efficient tests. Early studies suggested that the nominal response model (NRM) is advantageous in this respect. We investigate the reasons for those results and the outcomes of a polytomous characterization based on the multiple-choice model (MCM). An empirical data set is used to compare polytomous (NRM and MCM) and dichotomous (3PLM) characterizations of a test. The results revealed superior item and test information functions from polytomous models. Yet, close inspection suggests that these outcomes are artifactual and two simulation studies confirmed this point. These studies revealed a structural inadequacy of the NRM for multiple-choice items and that the MCM characterization outperforms the 3PLM characterization only when distractor endorsement frequencies vary non-monotonically with ability, although this feature is rarely observed in empirical data sets.