# The Illusion of Measurement: Rasch versus 2-PL

Many researchers who are attempting to measure latent constructs appreciate the special properties of the Rasch model and view it as an ideal model, but, at the same time, they tend to complain about the "inflexibility" of the model when it comes to "explaining" data. The two-parameter logistic model is then seen as a possible resort. However, with discrimination varying from item to item, the very meaning of the construct changes from point to point on the dimension. In other words, measurement in its true sense has not been achieved.

In the dichotomous Rasch model, for each item response only two parameters are relevant, the item location parameter δ and the person location parameter ξ (using Rasch's multiplicative notation). The probability of a correct response then is ξ / ( ξ + δ ) (Rasch 1960/1980, p.107). As Georg Rasch points out, if we knew the exact item parameter and the probability of a correct response, the person location could be computed directly. Vice versa, if we knew the person parameter and the probability, we could compute the item location. Under the two-parameter logistic (2-PL) model, this is not possible without further information because there are infinitely many combination of item difficulty and discrimination which yield the same probability for a given person location.

The plot shows the adjusted log-odds of success on five 2-PL items for persons at five ability levels. The five ability levels are -2, -1, 0, 0.1, and 2 logits. The items have difficulty and (discrimination) of -2 (0.8), -1.0 (1.8), 0.0 (0.4), 0.1 (1.5), and 2 (1.2). For each person-item encounter, the 2-PL probability of success is computed. This is converted into log-odds and adjusted for person ability. The plot thus shows the local Rasch difficulty of each item for each person. If the items were in accord with the Rasch model, this plot would collapse to an identity line. Since the 2-PL item characteristic curves intersect, there is a different "Rasch item difficulty" for each item for each level of person ability. In other words, the meaning of the construct defined by the item difficulty differs for each person location. Thus the apparent advantage of better describing the data set when using the 2-PL, rather than a Rasch model, comes at the expense of a highly fuzzy definition of the latent continuum . "Measurement" becomes an illusion, because there is no precise definition of what is being measured.

Thomas Salzberger
Austria

Rasch, Georg (1960/1980). Probabilistic Models for Some Intelligence and Attainment Tests. Danish Institute for Educational Research & https://www.rasch.org/books.htm

Plot: Local item difficulty of 5 2-PL items for 5 abilities

The illusion of measurement: Rasch versus 2-PL. Salzberger, T. … 16:2 p.882

The illusion of measurement: Rasch versus 2-PL. Salzberger, T. … Rasch Measurement Transactions, 2002, 16:2 p.882

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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