# Do Bad Data Refute Good Theory?

Because of its data-descriptive limitation, Item Response Theory (IRT) practice misrepresents Rasch's models, often under the rubric 1-PL, as simple examples of more complicated approaches. IRT implements confused numeration, mislabeled "measurement". In common with all descriptive statistical techniques, IRT tries to fit models to data. Guided by principles of parameter parsimony, the analyst prefers some models to others, but the data always have the last word.

Georg Rasch saw that the road to science was to fit data to models - a data-prescriptive approach. Careful re-analyses of their laboratory diaries show that Mendeleev (with heredity) and Newton (with laws of motion) also employed this approach. Another recent example is the "search for the top quark".

The IRT perspective appeals to superficial fairness. Whatever happened is the truth! "If the child marked the correct box, for whatever reason, then credit must be given!" This deterministic mindset is not new. It also befuddled some ancient Greeks.

Pythagoras vs. IRT?

Pythagoras' Theorem is a model for constructing and recognizing right-triangles. Whenever the sum of the squares of the two sides of a triangle equal the square of the hypotenuse, we recognize a right-triangle, e.g., the 3-4-5 triangle. The Pythagoreans and their opponents, the Sophists, understood the theorem to hold every time a right-triangle was drawn. Then it was discovered that no degree of measurement precision could make the Pythagorean model exactly match a right isosceles triangle, e.g., 1-1-2. This is because the ratio of 1 and 2 cannot be reduced to a ratio of two integers. The Sophists gloated: "The model doesn't work in practice, so it must be rejected!" The Pythagoreans, distressed, tried to hide this failure.

The flaw in the logic of those ancient Greeks is now obvious. Mathematical theorems are not proven or refuted on the basis of whether or not they fit any particular sample of data produced by any particular experimental apparatus. Though this decisive point has been made repeatedly in the history of science, it remains to be integrated into the methods, knowledge, attitudes, and imaginations of social scientists.

Rasch's Separability Theorem is a model for constructing and recognizing mathematically invariant, unidimensional variables. Any time we see that person parameters can be estimated independently of item parameters, we know invariant unidimensionality has been achieved. IRT proponents, imitating the Sophists, understand the Separability Theorem concretely, as holding for every actual item and person. They claim that, since Rasch's models never exactly describe actual data, they are impractical in the psychosocial sciences. On the other hand, some advocates of the Separability Theorem (Streiner & Norman, 1995) follow the Pythagoreans by attempting to conceal its inevitable failure to describe empirical data.

Paradoxically, Pythagoras' Theorem, though never exactly realized in practice, is highly practical when approximated even crudely. Stepping out a 3-4-5 triangle on the ground gives a good-enough 90 angle for most purposes. Similarly with Rasch's Separability Theorem. Administering a simple test of a few items is often good enough to operationalize a construct and to measure respondents along it.

Pythagoras committed suicide over the apparent malfunction of his model. Rasch, however, dismissed so-called failures of his models as a misunderstanding of their purpose: "On the whole we should not overlook that since a model is never true, but only more or less adequate, deficiencies are bound to show, given sufficient data" (Rasch 1980, 1992, p. 92).

William P. Fisher, Jr.

Fisher W.P. Jr. (1998) Do Bad Data Refute Good Theory? Rasch Measurement Transactions 11:4 p. 600.

Streiner DL, Norman GR (1995) Health Measurement Scales, 2nd Ed. New York: Oxford University Press.

Do Bad Data Refute Good Theory? Fisher W.P. Jr. … Rasch Measurement Transactions, 1998, 11:4 p. 600.

Please help with Standard Dataset 4: Andrich Rating Scale Model

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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