Latent and manifest monotonicity in item response models

The monotonicity of item response functions (IRF) is a central feature of most parametric and nonparametric item response models. Monotonicity allows items to be interpreted as measuring a trait, and it allows for a general theory of nonparametric inference for traits. This theory is based on monotone likelihood ratio and stochastic ordering properties. Thus, confirming the monotonicity assumption is essential to applications of nonparametric item response models. The results of two methods of evaluating monotonicity are presented: regressing individual item scores on the total test score and on the "rest " score, which is obtained by omitting the selected item from the total test score. It was found that the item-total regressions of some familia

Cognitive assessment models with few assumptions, and connections with nonparametric item response theory

Some usability and interpretability issues for single-strategy cognitive assessment models are con-sidered. These models posit a stochastic conjunctive relationship between a set of cognitive attributes to be assessed and performance on particular items/tasks in the assessment. The models considered make few assumptions about the relationship between latent attributes and task performance beyond a simple conjunctive structure. An example shows that these models can be sensitive to cognitive attributes, even in data designed to well fit the Rasch model. Several stochastic ordering and monotonicity properties are considered that enhance the interpretability of the models. Simple data summaries are identified that inform about the presence or absence of cognitive attributes when the full computational power needed to estimate the models is not available. Index terms: cognitive diagnosis, conjunctive Bayesian inference networks, multidimensional item response theory, nonparametric item response theory, restricted latent class models, stochastic ordering, transitive reasoning. There has been increasing pressure in educational assessment to make assessments sensitive to specific examinee skills, knowledge, and other cognitive features needed to perform tasks. Fo

Stochastic ordering using the latent trait and the sum score in polytomous IRT models

In a restricted class of item response theory (IRT) models for polytomous items the un-weighted total score has monotone likelihood ratio (MLR) in the latent rait 0. MLR implies two stochastic ordering (SO) properties, denoted SOM and SOL, which are both weaker than MLR, but very useful for measurement with IRT models. Therefore, these SO properties are investigated for a broader class of IRT models for which the MLR property does not hold. In this study, first a taxonomy is given for nonparametric and parametric models for polyto-mous items based on the hierarchical relationship between the models. Next, it is investigated which models have the MLR property and which have the SO properties. It is shown that all models in the taxonomy possess the SOM property. However, counterexamples illustrate that many models do not, in general, possess the even more useful SOL property. Key words: monotone likelihood ratio, nonparametric IRT models, parametric IRT models, poly-tomous IRT models, stochastic ordering

The person response function as a tool in person-fit research

appropriateness measurement, invariant item ordering, nonparametric item response theory, person-fit method, person response function,