# Descriptive IRT vs. Prescriptive Rasch

Item response theory (IRT) generally refers to three probabilistic measurement models: the 1-parameter (identical to the dichotomous Rasch model), the 2-parameter, and the 3-parameter, named by the number of item parameters estimated in each model. All three models can be specified from a single probabilistic function for the occurrence of a right answer by a person to an item:

P(x=1) = ci - (1-ci) exp (ai(t-bi)) / (1 + exp (ai(t-bi)))

where the three item parameters are
ci = low asymptote of ogive (guessing)
bi = median intercept of ogive (difficulty)
ai = slope of ogive at inflection (discrimination)
and the one person parameter is
t = ability of a person on the variable.

Making ci constant produces 2-parameter estimation. Making ai constant produces 1-parameter estimation.

This approach seems to imply that the Rasch model is just a stripped- down version of more complicated models which "must be better" because they account for more of the "presumed reality" of traditional test theory. Quite apart from Occam's razor (that entities are not multiplied beyond necessity), this interpretation is shallow in an essential way. That the Rasch model can be reached by simplifying more complicated models has nothing to do with its genesis or rationale, or with the theory of measurement.

The Rasch model is not intended to fit data or to be evaluated by how well it fits any particular data set. The Rasch model is a definition of measurement derived from the universally accepted measurement requirements that:

1. The measures of objects be free of the particulars of the agents used to estimate these measures and the calibrations of agents be free of the particulars of the objects used to estimate these calibrations.

2. The measures of objects and calibrations of agents function according to the rules of arithmetic on a common scale so that they can be analyzed statistically.

3. Linear combinations of measures and calibrations correspond to plausible concatenations of objects and agents.

Introducing the superfluous parameters, ai and ci, violates these requirements and disqualifies the resulting contrivances as "measurement" models. Were the values of t known prior to analysis, the 2- and 3-parameter formulations could be used to specify two, among many possible, logistic regressions, as they have been in biometric analysis for 60 years. When t is not known and must be estimated from the same data, as is the case for measurement, then it can be demonstrated algebraically that the parameter estimates must diverge unless constrained in an arbitrary manner.

In practice, arbitrary interference, such as limiting the range of ai and ci, does not produce sample-free item calibrations or test-free person measures - unless ai and ci are restricted to constants, when, of course, the formulation becomes a Rasch model and functions accordingly. This is the misunderstanding of those who claim the 2- and 3-parameter computer programs "work". In the few instances where serious 2- and 3-parameter results are reported, their quality is proportional to the extent to which ai and ci were kept from varying - from being influenced by the data. Indeed all important practical IRT applications reported at the 1991 AERA meeting were implemented by the Rasch model.

In theory and in practice, the Rasch model is a unique measurement archetype.

Descriptive IRT vs. Prescriptive Rasch, F Shaw … Rasch Measurement Transactions, 1991, 5:1 p. 131

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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