# Measurement Scale in the Rasch Model

The issue of the type of measurement scale in the Rasch model, as well as in any other IRT model, is usually settled by the criterion of invariance of model fit under transformation of scale. This seems to be a sound criterion. If a model still fits when its unit of scale is changed, then apparently the unit is arbitrary and does not tell us anything about reality. But if the model fit degrades, then the unit is absolute. Reality punishes us if we don't adhere to the measurement scale dictated by the data.

Absolute Unit?

Given this empirical criterion, it seems straightforward to answer the question of scale type in the Rasch model. Suppose we have the well-known exponential representation of the Rasch model and know that the model fits a given test perfectly for a population of examinees. Then, the only transformation of the ability scale that leaves the fit of the model intact seems to be a linear transformation adding the same constant to each parameter value. Apparently, the model allows for ability measurement on a quantitative scale with an absolute unit but an arbitrary zero.

Arbitrary Unit?

The linear transformation of the ability scale that changes the zero is not the only algebraic transformation that has been applied to define the ability scale in the Rasch model. Another is a composite transformation in which first a logarithm of the parameters is taken and all parameter values are subsequently multiplied by a constant. This transformation results in a zero that now seems to be absolute and a unit that is arbitrary. But how is this possible? How can we ever have a class of transformations that leave the fit of the model intact, with one member showing that the scale unit is absolute and another that it is not the unit but the zero that is absolute. Is this not an outright paradox?

Ordinal Scale

Assume that we have a graphical representation of the response functions of the items under the same model and that we start distorting the scale, stretching at some points and squeezing it at others. However we stretch and squeeze, the examinees are just moved up and down along the scale but keep their own probabilities of success on each of the items in the test. We are not able to undo the perfect fit of the model! Surprisingly, model fit now appears to be invariant under monotonic transformations and, therefore, the ability scale is only an ordinal scale.

Our reaction to the paradox should now be clear: To determine the type of scale in a measurement model, we must take the largest possible class of transformations under which the model retains its fit. For IRT models this is the class of monotonic transformations. Hence, both the unit and the zero are arbitrary. The fact that we do not have an algebraic representation of this class of transformations is not a requirement included in the criterion of invariance.

Wim J. van der Linden

Measurement scale in the Rasch model. van der Linden WJ. … 1993, 7:2 p.287

Measurement scale in the Rasch model. van der Linden WJ. … Rasch Measurement Transactions, 1993, 1993, 7:2 p.287

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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