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Decomposition of a Rasch Partial Credit Item into Independent Binary and Indecomposable Trinary Items

Published online by Cambridge University Press:  01 January 2025

Huynh Huynh
Huynh Huynh*
Affiliation:
University of South Carolina
*
Requests for reprints should be sent to Huynh Huynh, Department of Educational Psychology, College of Education, University of South Carolina, Columbia, SC 29208.
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Abstract

For each Rasch (Masters) partial credit item, there exists a set of independent Rasch binary and indecomposable trinary items for which the sum of the scores and the partial credit score have identical probability density functions. If each indecomposable trinary item is further expressed as the sum of two binary items, then the binary items are positively dependent and cannot be both of the Rasch type.

Keywords

binary itemelementary symmetric functionsfree-response itemindecomposable itemgraded response modelpartial credit modelperformance assessmentopen-ended itemRasch modeltrinary item
Type
Original Paper
Information
Psychometrika , Volume 61 , Issue 1 , March 1996 , pp. 31 - 39
Copyright
© 1996 The Psychometric Society

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Footnotes

This paper was written while the author was working with Steve Ferrara and Hillary Michaels on some technical aspects of the Maryland School Performance Assessment Program. The author had been puzzled by the fact that most MSPAP assessment items have three or less score categories. With a psychometric justification now being apparent, this paper is dedicated to both of them.

References

Andersen, E. B. (1966). Den diskrete malingsmodel af endelig orden med anvendelse pa et socialpsykologisk materiale [A discrete latent trait model for ordered categories with applications to social psychology data], Kobenhavn, Denmark: Statens Trykningskontor.Google Scholar
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573.CrossRefGoogle Scholar
Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more latent categories. Psychometrika, 37, 29–51.CrossRefGoogle Scholar
Burnside, W. S., Panton, A. W. (1892). Theory of equations 3rd ed.,, Dublin: Dublin University Press Series.Google Scholar
Fischer, G. H. (1974). Einführung in die theorie psychologisher tests. Grundlagen und Anwendungen [Introduction to the theory of psychological tests. Foundations and applications], Bern: Huber.Google Scholar
Holland, P. W. (1990). On the sampling theory foundation of item response theory models. Psychometrika, 55, 577–601.CrossRefGoogle Scholar
Huynh, H. (1994). On equivalence between a partial credit item and a set of independent Rasch binary items. Psychometrika, 59, 111–119.CrossRefGoogle Scholar
Huynh, H., Ferrara, S. F. (1994). A comparison of equal percentile and partial credit equatings for performance-based assessments comprised of free-response items. Journal of Educational Measurement, 31, 125–141.CrossRefGoogle Scholar
IMSL (1987). User's manual: Math/library—FORTRAN subroutines for mathematical applications, Houston: Author.Google Scholar
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Muraki, E. (1990). Fitting a polytomous item response model to Likert-type data. Applied Psychological Measurement, 14, 59–71.CrossRefGoogle Scholar
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: Danish Institute for Educational Research.Google Scholar
Rasch, G. (1961). On general laws and the meaning of measurement in psychology. Proceedings of the IV Berkeley Symposium on mathematical statistics and probability (Vol. 4), Berkeley: University of California Press.Google Scholar
Rasch, G. (1967). An informal report on a theory of objectivity in comparisons. In van der Kamp, L. J. Th., Vlek, C. A. J. (Eds.), Measurement theory, Leyden: University of Leyden.Google Scholar
Rasch, G. (1967). An individual-centered approach to item analysis with two categories of answers. In van der Kamp, L. J. Th., Vlek, C. A. J. (Eds.), Measurement theory, Leyden: University of Leyden.Google Scholar
Samejima, F. (1972). A general model for free-response data. Psychometric Monograph, No. 18.Google Scholar
Serret, J. A. (1928). Cours d'algèbre supérieure 7th ed.,, Paris, France: Gauthier-Villars.Google Scholar
Thissen, D., Steinberg, L. (1984). A response model for multiple-choice items. Psychometrika, 49, 501–519.CrossRefGoogle Scholar
Wilson, M. (1988). Detecting and interpreting item local dependence using a family of Rasch models. Applied Psychological Measurement, 12, 353–364.CrossRefGoogle Scholar
Wright, B. D., Masters, G. N. (1982). Rating scale analysis, Chicago: Mesa Press.Google Scholar
Yen, W. M. (1993). Scaling performance assessments: Strategies for managing local item dependence. Journal of Educational Measurement, 30, 187–213.CrossRefGoogle Scholar