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On Compensation in Multidimensional Response Modeling

Published online by Cambridge University Press:  01 January 2025

Wim J. van der Linden
Wim J. van der Linden*
Affiliation:
CTB/McGraw-Hill
*
Requests for reprints should be sent to Wim J. van der Linden, CTB/McGraw-Hill, 20 Ryan Ranch Road, Monterey, CA 93940, USA. E-mail: wim_vanderlinden@ctb.com
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Abstract

The issue of compensation in multidimensional response modeling is addressed. We show that multidimensional response models are compensatory in their ability parameters if and only if they are monotone. In addition, a minimal set of assumptions is presented under which the MLEs of the ability parameters are also compensatory. In a recent series of articles, beginning with Hooker, Finkelman, and Schwartzman (2009) in this journal, the second type of compensation was presented as a paradoxical result for certain multidimensional response models, leading to occasional unfairness in maximum-likelihood test scoring. First, it is indicated that the compensation is not unique and holds generally for any multiparameter likelihood with monotone score functions. Second, we analyze why, in spite of its generality, the compensation may give the impression of a paradox or unfairness.

Keywords

compensatory modelmaximum-likelihood estimationmonotone score functionmultidimensional response model

Information

Type
Original Paper
Information
Psychometrika , Volume 77 , Issue 1 , January 2012 , pp. 21 - 30
Copyright
Copyright © 2011 The Psychometric Society

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References

Andrich, D., Luo, G. (1993). A hyperbolic cosine latent trait model for unfolding dichotomous single-stimulus responses. Applied Psychological Measurement, 17, 253–276.CrossRefGoogle Scholar
Finkelman, M., Hooker, G., Wang, Z. (2010). Prevalence and magnitude of paradoxical results in multidimensional item response theory. Journal of Educational and Behavioral Statistics, 35, 744–761.CrossRefGoogle Scholar
Hoijtink, H. (1990). A latent trait model for dichotomous choice data. Psychometrika, 55, 641–656.CrossRefGoogle Scholar
Hooker, G. (2010). On separable test, correlated priors, and paradoxical results in multidimensional item response theory. Psychometrika, 75, 694–707.CrossRefGoogle Scholar
Hooker, G., Finkelman, M. (2010). Paradoxical results and item bundles. Psychometrika, 75, 249–271.CrossRefGoogle Scholar
Hooker, G., Finkelman, M., Schwartzman, A. (2009). Paradoxical results in multidimensional item response theory. Psychometrika, 74, 419–442.CrossRefGoogle Scholar
McDonald, R.P. (1997). Normal-ogive multidimensional model. In van der Linden, W.J., Hambleton, R.K. (Eds.), Handbook of modern item response theory (pp. 258–286). New York: Springer.Google Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Chicago: University of Chicago Press.Google Scholar
Reckase, M.D. (2009). Multidimensional item response theory, New York: Springer.CrossRefGoogle Scholar
Roberts, J.S., Laughlin, J.E. (1996). A unidimensional item response model for unfolding responses from a graded disagree-agree response scale. Applied Psychological Measurement, 20, 231–255.CrossRefGoogle Scholar
Sympson, J.B. (1978). A model for testing with multiple choice items. In Weiss, D.J. (Eds.), Proceedings of the 1977 computerized adaptive testing conference (pp. 82–98). Minneapolis: Department of Psychology, University of Minnesota.Google Scholar
van der Linden, W.J. (1998). Stochastic order in dichotomous item response models for fixed, adaptive, and multidimensional tests. Psychometrika, 63, 211–226.CrossRefGoogle Scholar
van der Linden, W.J. (2005). Linear models for optimal test design, New York: Springer.CrossRefGoogle Scholar
van der Linden, W.J., Vos, J.H. (1996). A compensatory approach to optimal selection with mastery scores. Psychometrika, 61, 155–172.CrossRefGoogle Scholar
Verhelst, N.D., Verstralen, H.H.F.M. (1993). A stochastic unfolding model derived from the partial credit model. Kwantitatieve Methoden, 42, 73–92.Google Scholar
Whitely, S. (1980). Multicomponent latent trait models for ability tests. Psychometrika, 45, 479–494.CrossRefGoogle Scholar