Abstract
The assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score. Three forms of latent monotonicity can be distinguished: monotonicity of the cumulative probabilities, of the continuation ratios, and of the adjacent-category ratios. Observable consequences of these different forms of latent monotonicity are derived, and Bayes factor methods for testing these consequences are proposed. These methods allow for the quantification of the evidence both in favor and against the tested property. Both item-level and category-level Bayes factors are considered, and their performance is evaluated using a simulation study. The methods are applied to an empirical example consisting of a 10-item Likert scale to investigate whether a polytomous item scoring rule results in item scores that are of ordinal level measurement.
Highlights
The assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score
For item response theory (IRT) models for dichotomous data, the functioning of an item is captured by the item response function πi (θ ) = P(Xi = 1|θ ), which describes the probability of obtaining a positive score for item i as a function of the latent variable θ
This paper proposes Bayes factor (BF) methods for evaluating latent monotonicity in polytomous IRT
Summary
The assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score. It captures the notion that on a wellfunctioning item persons of higher ability should never have a lower probability of providing a correct response than persons of lower ability This makes it a statistical assumption that captures an important qualitative requirement for valid measurement, as a violation suggests that the item does not function adequately. Common parametric IRT models that make use of this building block are the partial credit model and its generalizations (Masters, 1982; Muraki, 1992), and the rating scale model (Andrich, 1978) In each of these models, ψi j (θ ) is assumed to be monotonically nondecreasing in θ for j ∈ [1 : m]. Monotonicity of ψi j (θ ) for j ∈ [1 : m] is a defining assumption of the nonparametric partial credit model (Hemker, Sijtsma, Molenaar, & Junker, 1997)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
