Other Considerations
A criticism of the Rasch model is that it is overly restrictive or prescriptive because it does not permit each item to have a different discrimination. A criticism specific to the use of multiple choice items in educational assessment is that there is no provision in the model for guessing because the left asymptote always approaches a zero probability in the Rasch model. These variations are available in models such as the two and three parameter logistic models (Birnbaum, 1968). However, the specification of uniform discrimination and zero left asymptote are necessary properties of the model in order to sustain sufficiency of the simple, unweighted raw score.
In the two-parameter logistic model (2PL-IRT; Lord & Novick, 1968) the weighted raw score is theoretically sufficient for person parameters, where the weights are given by model parameters referred to as discrimination parameters. Lord & Novick's one-parameter logistic model, 1PL, appears similar to the Rasch model in that it does not have discrimination parameters, but 1PL has different motivation and subtly different parameterization. The 1PL is a descriptive model which summarizes the sample as a normal distribution. The dichotomous Rasch model is a measurement model which parameterizes each member of the sample individually. There are other technical differences.
Verhelst & Glas (1995) derive Conditional Maximum Likelihood (CML) equations for a model they refer to as the One Parameter Logistic Model (OPLM). In algebraic form it appears to be identical with the 2PL model, but OPLM contains preset discrimination indexes rather than 2PL's estimated discrimination parameters. As noted by these authors, though, the problem one faces in estimation with estimated discrimination parameters is that the discriminations are unknown, meaning that the weighted raw score "is not a mere statistic, and hence it is impossible to use CML as an estimation method" (Verhelst & Glas, 1995, p. 217). That is, sufficiency of the weighted "score" in the 2PL cannot be used according to the way in which a sufficient statistic is defined. If the weights are imputed instead of being estimated, as in OPLM, conditional estimation is possible and the properties of the Rasch model are retained (Verhelst, Glas & Verstralen, 1995; Verhelst & Glas, 1995). In OPLM, the values of the discrimination index are restricted to between 1 and 15. A limitation of this approach is that in practice, values of discrimination indexes must be preset as a starting point. This means some type of estimation of discrimination is involved when the purpose is to avoid doing so.
The Rasch model for dichotomous data inherently entails a single discrimination parameter which, as noted by Rasch (1960/1980, p. 121), constitutes an arbitrary choice of the unit in terms of which magnitudes of the latent trait are expressed or estimated. However, the Rasch model requires that the discrimination is uniform across interactions between persons and items within a specified frame of reference (i.e. the assessment context given conditions for assessment).
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