Rasch and Pseudo-Rasch Models: Suitableness for Practical Test Applications1

Abstract The Rasch model has been suggested for psychological test data (subjects × items) for various scales of measurement. It is defined to be specifically objective. If the data are dichotomous, the use of the dichotomous model of Rasch for psychological test construction is almost inevitable. The two- and three-parameter logistic models of Birnbaum and further models with additional parameters are not always identifiable. The linear logistic model is useful for the construction of item pools. For polytomous graded response data, there are useful models (Samejima, 1969; Tutz, 1990; and again by Rasch, cf. Fischer, 1974, or Kubinger, 1989) which, however, are not specifically objective. The partial credit model (Masters, 1982) is not meaningful in a measurement theory sense. For polytomous nominal data, the multicategorical Rasch model is much too rarely applied. There are limited possibilities for locally dependent data. The mixed Rasch model is not a true Rasch model, but useful for model controls and heuristic purposes. The models for frequency data and continuous data are not discussed here. The nonparametric ISOP-models are sample (ordinally specifically objective) models for (up to 3 dependent) graded responses providing ordinal scales or interval scales for subject-, item- and response-scale-parameters. The true achievement of sample-independent Rasch models is an extraordinary generalizeability of psychological assessment procedures. Key words: specific objectivity; measurement structures; graded responses; local dependence; generalized assessment procedures 1. Basic concepts Any kind of measurement requires comparison. There are no directly available observable outcomes of the contact of the 'intelligences' of two subjects (or of the 'difficulties' of two items). We can only observe the reactions of subjects to items which do depend on both simultaneously. We need instruments to compare latent dimensions. The items are the instruments used for the comparison of subjects and vice versa. The difficulty in (psychological) measurement is to achieve comparisons on a set of subjects (or a set of items) in regard to a specific latent dimension which do not depend on the choice of instruments used for the comparison. Definition: Let be a 'frame of reference' (Rasch, 1961) or a '(probabilistic) instrumental paired comparison system' (Irtel, 1987) where A = a,b,c,...} is a set of subjects and Q = x, y,z,. . .} is a set of items and P[t;a,x) = PIT Definition: An instrumental paired comparison system is a Rasch model if it is specifically objective, i.e. if there exist comparison functions on the sets A×A and Q×Q which do not depend on the parameters of the respective other set, i.e. v(a,b) independent of jc,v,z,...} and w(x,y) independent of {a,b, c,... .} Pseudo-Rasch models are models which call themselves Rasch models without being specifically objective. The present section defines the basic problem. The subsections of Section 2 are organized along the lines of Fischer's (1974) proof of the uniqueness of the specific objectivity of the dichotomous Rasch model among dichotomous models. Each assumption will be examined with respect to usefulness and necessity. Fischer's list of necessary and sufficient conditions ensures that no important aspect is overlooked. Section 3 illustrates the true advantages of Rasch models for theory and practice. Section 4 presents nonparametric Rasch models together with an empirical application. The discussion in Section 5 summarizes the results. 2. Dichotomous indicators If & is a third item i ≠ j≠k then the probability that two of the discriminations are equal or that a parameter is equal to the sum of the two others is zero; and so on for finite numbers k of items. …