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.
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References
Andersen, E. B. (1971). The asymptotic distribution of conditional likelihood ratio tests.Journal of the American Statistical Association, 66, 630–633.
Andersen, E. B. (1973).Conditional inference and models for measuring. Copenhagen: Mentalhygiejnisk Forskningsinstitut.
Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975).Discrete multivariate analysis: Theory and practice. Cambridge, MA: MIT Press.
Birch, M. W. (1964). A new proof of the Pearson-Fisher theorem.Annals of Mathematical Statistics, 35, 817–824.
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: An application of an EM algorithm.Psychometrika, 46, 443–459.
Fischer, G. H. (1974).Einführung in die Theorie psychologischer Tests [Introduction to the theory of psychological tests]. Bern: Verlag Hans Huber.
Fischer, G. H. (1981). On the existence and uniqueness of maximum likelihood estimates in the Rasch model.Psychometrika, 46, 59–77.
Kelderman, H. (1984). Loglinear Rasch model tests.Psychometrika, 49, 223–245.
Martin Löf, P. (1973).Statistika Modeller. Anteckningar fråret 1969–1970 utarbetade av Rolf Sunberg obetydligt ändrat nytryk, oktober 1973 [Statistical models. Proceedings of the Lasåret seminar 1969–1970, edited byRolf Sunberg]. Stockholm: Institutet för Försäkringsmatematik och Matematisk Statistik vid Stockholms Universitet.
Neyman, J., & Scott, E. L. (1948). Consistent estimates based on partially consistent observations.Econometrika, 16, 1–32.
Rao, C. R. (1973).Linear Statistical Inference and its Applications (2nd ed.). New York: Wiley.
Rasch, G. (1960).Probablistic models for some intelligence and attainment tests. Kopenhagen: Danish Institute for Educational Research.
Rasch, G. (1961). On the general laws and the meaning of measurement in psychology. In J. Neyman (Ed.),Proceedings of the Fourth Symposium on Mathematical Statistics and Probability, 4, 321–333.
Rigdon, S. E., & Tsutakawa, R. K. (1983). Parameter estimation in latent trait models.Psychometrika, 48, 567–574.
Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model.Psychometrika, 47, 175–186.
van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model.Psychometrika, 47, 123–140.
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.
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I am indebted to Norman Verhelst and Niels Veldhuijzen for their helpful comments. Requests for reprints should be sent to Cees A. W. Glas, Cito, PO Box 1034, 6801 MG Arnhem, THE NETHERLANDS.
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Glas, C.A.W. The derivation of some tests for the rasch model from the multinomial distribution. Psychometrika 53, 525–546 (1988). https://doi.org/10.1007/BF02294405
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DOI: https://doi.org/10.1007/BF02294405