Dichotomizing Rating Scales and Rasch-Thurstone Thresholds

Findings based on rating scales can be difficult to explain to a non-technical audience. If the rating-scale categories can be bisected by a pass-fail cut-point, such as "agree or not", "competent or not", "impaired or not", "success or failure", then it can simplify communication if the rating scale is dichotomized around the cut-point. Categories above the cut-point are scored "1", and categories below the cut-point are scored "0".

How do measures based on the dichotomized data relate to the measures based on the original ratings? The Figure illustrates the relationship.

The Figure shows a 3-category item, rated 0-1-2. The two cumulative probability ogives (based on the Rasch Partial-Credit Model, PCM) for that rating scale are shown. The intersections of the ogives with 0.5 probability are the Rasch-Thurstone thresholds.

The rating scale can be dichotomized in two ways: 0-1-2 becomes 0-0-1 or 0-1-2 becomes 0-1-1. These two dichotomizations can be analyzed with the dichotomous Rasch model. To make comparison simpler, the person measures are anchored at their rating-scale estimates. The result of the dichotomous analysis is two dichotomous ogives, one for each of the two dichotomizations. The Figures indicates that the two dichotomous ogives approximate the cumulative probability ogives of the rating-scale analysis. Thus the difficulties of the dichotomized items approximate the Rasch-Thurstone cumulative-probability thresholds, not the Rasch-Andrich equal-adjacent-category-probability thresholds (which are generally more central).

This result is reassuring because it indicates that inferences based on the simpler dichotomized data approximate inferences based on the more complex rating-scale data.

This also suggest that, when only the top category of a partial-credit scale is "correct", the Rasch-Thurstone threshold of the top category of the partial-credit (rating-scale) item corresponds to the dichotomous difficulty of the equivalent right-wrong item.

Note: the actual ranges of the categories on the latent variable are given by the 50% Cumulative Probabilities = the Rasch-Thurstone thresholds (not the Rasch-Andrich thresholds). When the interval between two Rasch-Thurstone thresholds is less than about 1.0 logits, the category is no longer modal - the peak probability is lower than the probability of adjacent categories. Todiscover what are the category boundaries on the latent variable, imagine 1000 persons on the latent variable at a category boundary. When they respond, 50% will choose categories below the boundary and 50% above. This happens at the Rasch-Thurstone threshold. It does not happen at the Rasch-Andrich threshold. This is further confirmed by dichotomizing the rating-scale responses - see above. The dichotomized item difficulties are close to the Rasch-Thurstone thresholds, not the usually more central Rasch-Andrich thresholds.

Linacre J.M. (2009) Dichotomizing Rating Scales and Rasch-Thurstone Thresholds, Rasch Measurement Transactions, 2009, 23:3, 1228

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
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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