The Zero-Discrimination Paradox

The dichotomous Rasch model specifies that all items have the same discrimination. But what happens if that discrimination is zero? Some critics perceive here a flaw in the Rasch model (e.g., Bollinger & Hornke, 1978), but, paradoxically, the Rasch model analysis is accurate. It is the 2-PL analysis that is flawed!

Here is a 2-PL IRT model including its usual item discrimination parameter, ai:

If all item discriminations are the same, ai=a and this becomes a Rasch model. Thus data which fit a 2-PL model with uniform item discriminations also fit a Rasch model with Rasch parameters Bn = a.θn and Di = This presents no conceptual difficulties except in the case of a = 0. A 2-PL analysis would, one imagines, report that a = 0, but the Rasch analysis cannot do this, so what would it report?

If all ai=0 in the 2-PL model statement above, then Pni = 0.5 for all n and i. So that, from the Rasch perspective, all Bn = B and all Di = D, and B = D. This is equivalent to coin-tossing. The Rasch analysis would unambiguously report that all person abilities equal all item difficulties, and the data would fit the Rasch model.

In fact, a Rasch analysis can go further. If the items are not discriminating, so that Bn = B and Di = D, but B><D, then this is equivalent to tossing a biased coin. B-D is a measure of the bias in the coin. Rasch would report correctly that all person abilities are equal, and that all item difficulties are equal, but that person ability is unequal to item difficulty. The data would fit the model.

With a biased coin, 2-PL estimation algorithms encounter a paradox. If item discrimination dominates, then ai=a=0 is reported, but the resultant model does not fit the data. This is because a=0 implies Pni=0.5, but in fact Pni><0.5.

If ability and difficulty dominate, then θn = θ and bi = b and ai = a = 1 (or a constant, not equal to 0). The model does fit the data, but 2-PL now misreports the uniform zero discrimination as non-zero! In a situation in which the Rasch measures are straightforward to interpret, it is the 2-PL estimates that are either incorrect or misleading.

John M. Linacre

Bollinger G & Hornke L.F. (1978) The relationship between item discrimination and Rasch scalability. [German]. Archiv für Psychologie, 130, 89-96

The Zero-Discrimination Paradox. Linacre, JM. … Rasch Measurement Transactions, 2003, 16:4 p.904

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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