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Some New and Some Old Results for the Polytomous Rasch Model
Abstract
In the first part, the polytomous Rasch model is discussed as it was presented by Rasch at the Berkeley Symposium in 1960,
and published in the Proceedings from the Symposium in 1961. I shall then discuss what was achieved in the next 10 to 15 years
as regards estimation and check of the model. As Georg Rasch himself never learned to program a computer, this work was carried
out by a handful or so statisticians, who worked on the model. In the next 10 to 15 years much work was done on the polytomous
Rasch model in many directions. But I shall skip this period. In the second part, I shall present a quite recent result for
the estimates in the polytomous Rasch model, which is extremely simple and require only very few elementary combinations of
quantities already computed. In addition it solves the problem of identifying significant points on the graphs suggested by
Rasch.
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This chapter derives polytomous Rasch models from certain sets of assumptions. First, it is shown that the multidimensional polytomous Rasch model follows from the assumption that there exists a vector-valued minimal sufficient statistic T for the vector-valued person parameter θ, where T is independent of the item parameters β i ; the sufficient statistic T is seen to be equivalent to the person marginal vector. Second, it is shown that, within a framework for measuring change between two time points, the partial credit model and the rating scale model follow from the assumption that conditional inference on (or specifically objective assessment of) change is feasible. The essential difference between the partial credit model and the rating scale model is that the former allows for a change of the response distribution in addition to a shift of the person parameter, whereas the rating scale model characterizes change exclusively in terms of the person parameter θ. A family of power series models is derived from the demand for conditional inference which accomodates the partial credit model, the rating scale model, the multiplicative Poisson model, and the dichotomous Rasch model.
The Rasch model is an item analysis model with logistic item characteristic curves of equal slope,i.e. with constant item discriminating powers. The proposed goodness of fit test is based on a comparison between difficulties
estimated from different scoregroups and over-all estimates.
Based on the within scoregroup estimates and the over-all estimates of item difficulties a conditional likelihood ratio is
formed. It is shown that—2 times the logarithm of this ratio isx
2-distributed when the Rasch model is true.
The power of the proposed goodness of fit test is discussed for alternative models with logistic item characteristic curves,
but unequal discriminating items from a scholastic aptitude test.
Residuals for check of model fit in the polytomous Rasch model are examined. Comparisons are made between using counts for all response pattern and using item totals for score groups for the construction of the residuals. Comparisons are also, for the residuals based on score group totals, made between using as basis the item totals, or using the estimated item parameters. The developed methods are illustrated by two examples, one from a psychiatric rating scale, one from a Danish Welfare Study.
The rating scale model Handbook of Modem Item Response Theory
- E B Andersen
- W J Linden
- R K Hambleton
Handbook of Modem Item Response Theory
- E B Andersen
- W J Van Den Linden
- R K Hambleton
- EB Andersen