Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-07T18:30:10.228Z Has data issue: false hasContentIssue false
  • English
  • Français

The Dutch Identity: A New Tool for the Study of Item Response Models

Published online by Cambridge University Press:  01 January 2025

Paul W. Holland
Paul W. Holland*
Affiliation:
Educational Testing Service
*
Requests for reprints should be sent to Paul W. Holland, Educational Testing Service, 21-T, Princeton, NJ 08541-0001
Get access Rights & Permissions [Opens in a new window]

Abstract

The Dutch Identity is a useful way to reexpress the basic equations of item response models that relate the manifest probabilities to the item response functions (IRFs) and the latent trait distribution. The identity may be exploited in several ways. For example: (a) to suggest how item response models behave for large numbers of items—they are approximate submodels of second-order loglinear models for 2J tables; (b) to suggest new ways to assess the dimensionality of the latent trait—principle components analysis of matrices composed of second-order interactions from loglinear models; (c) to give insight into the structure of latent class models; and (d) to illuminate the problem of identifying the IRFs and the latent trait distribution from sample data.

Keywords

IRTmanifest probabilitiesloglinear modelssecond-order exponential modelsmoment problemslatent class modelsRasch modeldimensionality
Type
Original Paper
Information
Psychometrika , Volume 55 , Issue 1 , March 1990 , pp. 5 - 18
Copyright
Copyright © 1990 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research was supported in part by contract number N00014-87-K-0730 from the Cognitive Science Program of the Office of Naval Research. I realized the usefulness of the identity in Theorem 1 while lecturing in the Netherlands during October, 1986. Because this was in no small part due to the stimulating psychometric atmosphere there, I call the result the Dutch Identity.

References

Basawa, I. V., Rao, B. L. S. P. (1980). Statistical inference for stochastic processes, New York: Academic Press.Google Scholar
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F., Novick, M. (Eds.), Statistical theories of mental test scores (pp. 395–479). Reading, MA: Addison-Wesley.Google Scholar
Bishop, Y., Fienberg, S., Holland, P. (1975). Discrete multivariate analysis: Theory and practice, Cambridge, MA: MIT Press.Google Scholar
Bock, R. D., Aitken, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of the EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R. D., Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.CrossRefGoogle Scholar
Bock, R. D., Mislevy, R. (1982). Adaptive EAP estimation of ability in a micro-computer environment. Applied Psychological Measurement, 6, 431–444.CrossRefGoogle Scholar
Cressie, N. (1982). A useful empirical Bayes identity. Annals of Statistics, 10, 625–629.CrossRefGoogle Scholar
Cressie, N., Holland, P. W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika, 48, 129–141.CrossRefGoogle Scholar
Dawid, A. P. (1970). On the limiting normality of posterior distributions. Proceedings of the Cambridge Philosophical Society, 67, 625–633.CrossRefGoogle Scholar
de Leeuw, J., Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics, 11, 183–196.CrossRefGoogle Scholar
Holland, P. (1981). When are item response theory models consistent with observed data?. Psychometrika, 46, 79–92.CrossRefGoogle Scholar
Holland, P., Rosenbaum, P. (1986). Conditional association and unidimensionality in monotone latent variable models. Annals of Statistics, 14, 1523–1543.CrossRefGoogle Scholar
Kemperman, J. H. B. (1968). The general moment problem, a geometric approach. Annals of Mathematical Statistics, 39, 93–122.CrossRefGoogle Scholar
Lord, F. (1962). Review of: Studies in Item Analysis and Prediction. Psychometrika, 27, 207–213.CrossRefGoogle Scholar
Mislevy, R., Bock, R. D. (1982). BILOG: Item analysis and test scoring with binary logistic models [computer program], Mooresville, IN: Scientific Software.Google Scholar
Thissen, D., Steinberg, L., Wainer, H. (1988). Use of item response theory in the study of group differences in trace lines. In Wainer, H., Braun, H. (Eds.), Test validity (pp. 147–170). Hillsdale, NJ: Erlbaum.Google Scholar
Tsao, Rhett. (1967). A second order exponential model for multidimensional dichotomous contingency tables with applications in medical diagnosis. Unpublished doctoral thesis, Harvard University, Department of Statistics.Google Scholar
Wingersky, M. (1983). LOGIST: A program for computing maximum likelihood procedures for logistic test models. In Hambleton, R. K. (Eds.), Applications of item response theory (pp. 45–56). Vancouver, BC: Educational Research Institute of British Columbia.Google Scholar