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An Item Response Model with Internal Restrictions on Item Difficulty

Published online by Cambridge University Press:  01 January 2025

René Butter ,
Paul De Boeck  and
Norman Verhelst
René Butter
Affiliation:
University of Leuven, Leuven, Belgium
Paul De Boeck*
Affiliation:
University of Leuven, Leuven, Belgium
Norman Verhelst
Affiliation:
CITO, Arnhem, The Netherlands
*
Requests for reprints should be sent to Paul De Boeck, Department of Psychology, Tiensestraat 102, B-3000 Leuven, BELGIUM. E-mail: Paul.DeBoeck@psy.kuleuven.ac.be
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Abstract

An IRT model based on the Rasch model is proposed for composite tasks, that is, tasks that are decomposed into subtasks of different kinds. There is one subtask for each component that is discerned in the composite tasks. A component is a generic kind of subtask of which the subtasks resulting from the decomposition are specific instantiations with respect to the particular composite tasks under study. The proposed model constrains the difficulties of the composite tasks to be linear combinations of the difficulties of the corresponding subtask items, which are estimated together with the weights used in the linear combinations, one weight for each kind of subtask. Although the model does not belong to the exponential family, its parameters can be estimated using conditional maximum likelihood estimation. The approach is demonstrated with an application to spelling tasks.

Keywords

item response theorycomponential modelslinear restrictions
Type
Original Paper
Information
Psychometrika , Volume 63 , Issue 1 , March 1998 , pp. 47 - 63
Copyright
Copyright © 1998 The Psychometric Society

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Footnotes

We thank Eric Maris for his helpful comments.

References

Andersen, E. B.. (1980). Discrete statistical models with social science applications, Amsterdam: North-Holland.Google Scholar
Baker, F. B.. (1992). Item response theory: Parameter estimation techniques, New York: Marcel Dekker.Google Scholar
Butter, R. P.. (1994). Item response models with internal restrictions on item difficulty, Belgium: Catholic University of Leuven.Google Scholar
Carpenter, A. C., Just, M. A., & Shell, P. (1990). What one intelligence test measures: A theoretical account of the processing in the Raven progressive matrices test. Psychological Review, 97, 404–431.CrossRefGoogle Scholar
De Boeck, P.. (1991). Componential IRT models, Belgium: University of Leuven.Google Scholar
Embretson, S. E. (1980). Multicomponent latent trait models for ability tests. Psychometrika, 45, 479–494.Google Scholar
Embretson, S. E. (1984). A general latent trait model for response processes. Psychometrika, 49, 175–186.CrossRefGoogle Scholar
Embretson, S. E. (1985). Introduction to the problem of test design. In Embretson, S. E. (Ed.), Test design: Developments in psychology and psychometrics (pp. 3–17), New York: Academic Press.CrossRefGoogle Scholar
Fischer, G. H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359–374.CrossRefGoogle Scholar
Fischer, G. H.. (1974). Einführung in die Theorie psychologischer Tests [Introduction to mental test theory], Bern: Huber.Google Scholar
Fischer, G. H. (1983). Logistic latent trait models with linear constraints. Psychometrika, 48, 3–26.CrossRefGoogle Scholar
Fischer, G. H., & Formann, A. K. (1982). Some applications of logistic latent trait models with linear constraints on the parameters. Applied Psychological Measurement, 6, 397–416.CrossRefGoogle Scholar
Fischer, G. H., & Kisser, R. (1983). Notes on the exponential latency model and an empirical application. In Wainer, H., & Messick, S. (Eds.), Principals of modern psychological measurement. A festschrift for Frederic M. Lord (pp. 322–336), Hillsdale, NJ: Erlbaum.Google Scholar
Glas, C. A. W., & Verhelst, N. D. (1995). Testing the Rasch model. In Fischer, G. H., & Molenaar, I. W. (Eds.), Rasch models: Their foundations, recent developments and applications (pp. 68–85), New York: Springer-Verlag.Google Scholar
Gustafsson, (1980).. Testing and obtaining fit of data to the Rasch model. British Journal of Mathematical and Statistical Psychology, 33, 205–233.CrossRefGoogle Scholar
Hays, W. L.. (1988). Statistics (4th ed.), New York.: Holt, Rinehart, & Winston.Google Scholar
Hoskens, M., & De Boeck, P. (1995). Componential IRT models for polytomous items. Journal of Educational Measurement, 32, 364–384.CrossRefGoogle Scholar
Janssen, R., & De Boeck, P. (1997). Psychometric modelling of componentially designed synonym tasks. Applied Psychological Measurement, 21, 37–50.CrossRefGoogle Scholar
Lohman, D. F. (1989). Human intelligence: An introduction to advances in theory and research. Review of Educational Research, 59, 333–373.CrossRefGoogle Scholar
Maris, E. (1995). Psychometric latent response models. Psychometrika, 60, 523–547.CrossRefGoogle Scholar
Mislevy, R. J.. (1989). Foundations of a new test theory, Princeton, NJ: Educational Testing Service.Google Scholar
Ramsay, J. O. (1975). Solving implicit equations in psychometric data analysis. Psychometrika, 40, 337–360.CrossRefGoogle Scholar
Scheiblechner, H. (1979). Specifically objective stochastic latency mechanisms. Journal of Mathematical Psychology, 19, 18–38.CrossRefGoogle Scholar
Sternberg, R. J.. (1977). Intelligence, information processing, and analogical reasoning: The componential analysis of human abilities, Hillsdale, NJ: Erlbaum.Google Scholar
Van der Geest, A., Swüste, W., & Raeven, J.. (1978). Spellingwijzer [Spelling indicator], Den Bosch, The Netherlands: Malmberg.Google Scholar
Verhelst, N. D., & Glas, C. A. W. (1995). The one parameter logistic model. In Fischer, G. H., & Molenaar, I. W. (Eds.), Rasch models: Their foundations, recent development, and applications, New York: Springer.Google Scholar
Verhelst, N. D., Glas, C. A. W., & Verstralen, H. H. F. M.. (1995). One-parameter logistic model [Software Manual], Arnhem: Cito.CrossRefGoogle Scholar