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The Derivation of Some Tests for the Rasch Model from the Multinomial Distribution

Published online by Cambridge University Press:  01 January 2025

Cees A. W. Glas
Cees A. W. Glas*
Affiliation:
National Institute for Educational Measurement (CITO), Arnhem, The Netherlands
*
Requests for reprints should be sent to Cees A. W. Glas, Cito, PO Box 1034, 6801 MG Arnhem, THE NETHERLANDS.
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Abstract

The present paper is concerned with testing the fit of the Rasch model. It is shown that this can be achieved by constructing functions of the data, on which model tests can be based that have power against specific model violations. It is shown that the asymptotic distribution of these tests can be derived by using the theoretical framework of testing model fit in general multinomial and product-multinomial models. The model tests are presented in two versions: one that can be used in the context of marginal maximum likelihood estimation and one that can be applied in the context of conditional maximum likelihood estimation.

Keywords

Item response theoryRasch modelmodel test

Information

Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 4 , December 1988, pp. 525 - 546
Copyright
Copyright © 1988 The Psychometric Society

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Footnotes

I am indebted to Norman Verhelst and Niels Veldhuijzen for their helpful comments.

References

Andersen, E. B. (1971). The asymptotic distribution of conditional likelihood ratio tests. Journal of the American Statistical Association, 66, 630–633.CrossRefGoogle Scholar
Andersen, E. B. (1973). Conditional inference and models for measuring, Copenhagen: Mentalhygiejnisk Forskningsinstitut.Google Scholar
Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975). Discrete multivariate analysis: Theory and practice, Cambridge, MA: MIT Press.Google Scholar
Birch, M. W. (1964). A new proof of the Pearson-Fisher theorem. Annals of Mathematical Statistics, 35, 817–824.CrossRefGoogle Scholar
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: An application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Fischer, G. H. (1974). Einführung in die Theorie psychologischer Tests [Introduction to the theory of psychological tests], Bern: Verlag Hans Huber.Google Scholar
Fischer, G. H. (1981). On the existence and uniqueness of maximum likelihood estimates in the Rasch model. Psychometrika, 46, 59–77.CrossRefGoogle Scholar
Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika, 49, 223–245.CrossRefGoogle Scholar
Martin Löf, P. (1973). Statistika Modeller. Anteckningar fråret 1969–1970 utarbetade av Rolf Sunberg obetydligt ändrat nytryk, oktober 1973. In Sunberg, Rolf (Eds.), Statistical models. Proceedings of the Lasåret seminar 1969–1970, Stockholm: Institutet för Försäkringsmatematik och Matematisk Statistik vid Stockholms Universitet.Google Scholar
Neyman, J., & Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrika, 16, 1–32.CrossRefGoogle Scholar
Rao, C. R. (1973). Linear Statistical Inference and its Applications 2nd ed.,, New York: Wiley.CrossRefGoogle Scholar
Rasch, G. (1960). Probablistic models for some intelligence and attainment tests, Kopenhagen: Danish Institute for Educational Research.Google Scholar
Rasch, G. (1961). On the general laws and the meaning of measurement in psychology. Proceedings of the Fourth Symposium on Mathematical Statistics and Probability, 4, 321–333.Google Scholar
Rigdon, S. E., & Tsutakawa, R. K. (1983). Parameter estimation in latent trait models. Psychometrika, 48, 567–574.CrossRefGoogle Scholar
Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175–186.CrossRefGoogle Scholar
van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.CrossRefGoogle Scholar
Verhelst, N. D., Glas, C. A. W., & van der Sluis, A. (1984). Estimation problems in the Rasch model: The basic symmetric functions. Computational Statistics Quarterly, 1, 245–262.Google Scholar