# 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:

loge(Pnix/Pni(x-1)) = Bn - Di - Fk

The {Fk} are the locations along the latent variable, relative to the item difficulty, at which categories k-1 and k are equally likely to be observed. These {Fk} are known as "step measures", "step calibrations", "step difficulties", "tau parameters", and "Rasch-Andrich thresholds". When each item is conceptualized to have its own rating scale, then Di + Fk becomes Di + Fik, or Dik, as in the "Partial Credit" model. When written as Dik, the thresholds are relative to the overall frame of reference, rather than a particular item.

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 Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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