Polytomous IRT models and monotone likelihood ratio of the total score

Abstract

IRT models, monotone likelihood ratio, nonparametric IRT models, parametric IRT models, polytomous items,

Monotone Properties of a General Diagnostic Model

“…Simple data summaries are identified to inform about the ordered categories of latent traits. The findings are very much in accordance with the statements made about the GPCM (Hemker, Sijtsma, Molenaar, & Junker, 1996, 1997. On the one hand, by fitting a GDM with equal slopes across items, the observed total score X + = j:q jk =1 x j demonstrates the monotone likelihood ratio (MLR) property in individual coordinate at the cost of losing model fit.…”

Simulation-based Bayesian inference for latent traits of item response models: Introduction to the ltbayes package for R

“…Fortunately, there are several theoretical results that bridge the gap between these two scales. Several researchers have established results concerning the conditional distributions of ζ givenỸ , andỸ given ζ , for a variety of binary (Grayson 1988;Huynh 1994) and polytomous (Hemker et al 1996(Hemker et al , 1997(Hemker et al , 2001; Van der Ark 2005; Van der Ark and Bergsma 2010) item response models (see Hemker 2000, andVan der Ark, 2001, for reviews). Of particular interest here is the property of stochastic ordering of the latent trait by the sum score, meaning that ifỹ 1 <ỹ 2 then P ζ > z|Ỹ =ỹ 1 ≤ P ζ > z|Ỹ =ỹ 2 for any z. Stochastic ordering is implied by the property that a model has a monotone likelihood ratio (MLR), meaning that ifỹ 1 <ỹ 2 then…”

“…Since the sum score is observed but ζ is not, SOL is the useful property for making inferences concerning latent traits using sum scores. All binary item response models with monotonic item response functions exhibit MLR and thus SOL (Grayson 1988;Huynh 1994), but of the polytomous models only the partial credit (Masters 1982) and rating scale (Andersen 1977;Andrich 1978a, b) models have the properties of MLR or SOL (Hemker et al 1996(Hemker et al , 1997(Hemker et al , 2001. Van der Ark (2005), however, showed that polytomous models that do not guarantee SOL may still exhibit it in practice.…”

Examination of Polytomous Items’ Psychometric Properties According to Nonparametric Item Response Theory Models in Different Test Conditions

“…Both binary and polytomous items' psychometric properties are examined through this model. MHM, developed for polytomous items, is defined as nonparametric GRM (Hemker, Sijtsma, Molenaar, & Junker, 1996;Sijtsma & Molenaar, 2002;Sijtsma, Emons, Bouwmeester, Nyklcek, & Roorda, 2008;van Onna, 2004). The main difference is that, even though ICCs are monotone in MHM, they are not as logistic as they are in PIRT.…”