Category, Step and Threshold: Definitions & Disordering

There is ambiguity in the Rasch literature over the use of the terms "Category", "Step" and "Threshold". This leads to enigmatic statements such as "Since the thresholds are disordered, the intended category order has been refuted by the data."

What is a "category"? Agreement is almost universal that it is "a class or division formed for the purposes of discussion or classification" (Webster's New Collegiate). An "ordered category" implies that the categories have been numbered so that a higher numbered category is thought to imply more of the latent variable under investigation. Numerically ordering of categories as qualitative advances along the variable is a prerequisite to Rasch measurement. But, in the course of Rasch analysis, one may discover that the imagined category ordering is not supported by the data. Remedies include renumbering the categories, collapsing adjacent categories or dropping items.

What is a "step"? Here the ambiguity becomes more greater. In some contexts, a step is the transition from one ordered category to the next. This transition can be conceptualized in various ways, often termed "thresholds". In other contexts, steps are the categories renumbered sequentially up from 0.

What are "thresholds"? They are the boundaries between categories. Again they can be conceptualized in various ways. The "Thurstone threshold" for a category corresponds to a point on the variable at which the probability of being observed in that category or above equals that of being observed in the categories below.

The "Rasch-Andrich threshold" is a parameter of a Rasch rating scale model. Here is such a model:

What about disordering? When the analyst-assigned category order does not accord with the latent variable, then the empirical average measures for each category are out of sequence and there is misfit (RMT 13:1 p. 675). When the thresholds are "Rasch-Thurstone thresholds", then threshold disorder can never be observed. When the thresholds are "Rasch-Andrich thresholds" or "step calibrations", then disordering occurs when some categories never become modal, i.e., they are not observed frequently enough. This implies that they correspond to intervals on the latent variable narrower than about 1 logit in terms of Rasch-Thurstone thresholds.

John Michael Linacre

Category, Step and Threshold: Definitions & Disordering. Linacre J.M. … Rasch Measurement Transactions, 2001, 15:1 p.794

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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