On measurement properties of continuation ratio models

Abstract

acceleration model, adjacent category models, continuation ratio models, cumulative probability models, hierarchical relationships between IRT models, invariant item ordering, monotone likelihood ratio, polytomous IRT models, sequential model, stochastic ordering,

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.…”

Polytomous Latent Scales for the Investigation of the Ordering of Items

“…Polytomous IRT models are commonly divided into the cumulative probability models, continuation ratio models, and adjacent category models (Agresti, 1990;Hemker, Van der Ark, & Sijtsma, 2001;Mellenbergh, 1995;Molenaar, 1983). Each class assumes unidimensionality; that is, the k items in the test share one unidimensional latent variable, and local independence; that is, for a k-dimensional vector of item scores X = x,…”

“…. , m, and M i0 (θ ) = 1 for x < 1, and M ix (θ ) = 0 for x > m. Examples of CRMs are the sequential Rasch model (Tutz, 1990), and the nonparametric sequential model (Hemker et al, 2001). These models assume monotonicity for M ix (θ ) (Equation (7)).…”

“…These models assume monotonicity for M ix (θ ) (Equation (7)). Items typically suited for CRM analysis consist of m subtasks that have to be executed in a fixed order such that failing a subtask implies failing the next subtasks, and the item score reflects that the first x subtasks have been successfully executed (Van Engelenburg, 1997, Chapters 2, 3;Hemker et al, 2001). Adjacent category models (ACMs) define ISRFs as…”