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On Holland's Dutch Identity Conjecture

Published online by Cambridge University Press:  01 January 2025

Jinming Zhang  and
William Stout
Jinming Zhang*
Affiliation:
Educational Testing Service
William Stout
Affiliation:
Department of Statistics, University of Illinois at Urbana-Champaign
*
Requests for reprints should be sent to Jinming Zhang, Educational Testing Service, MS 02-T, Rosedale Road, Princeton, NJ 08541. E-mail: [email protected]
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Abstract

The manifest probabilities of observed examinee response patterns resulting from marginalization with respect to the latent ability distribution produce the marginal likelihood function in item response theory. Under the conditions that the posterior distribution of examinee ability given some test response pattern is normal and the item logit functions are linear, Holland (1990a) gives a quadratic form for the log-manifest probabilities by using the Dutch Identity. Further, Holland conjectures that this special quadratic form is a limiting one for all “smooth” unidimensional item response models as test length tends to infinity. The purpose of this paper is to give three counterexamples to demonstrate that Holland's Dutch Identity conjecture does not hold in general. The counterexamples suggest that only under strong assumptions can it be true that the limits of log-manifest probabilities are quadratic. Three propositions giving sets of such strong conditions are given.

Keywords

item response theoryIRTmanifest probabilitiesDutch Identityposterior distribution
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 3 , September 1997 , pp. 375 - 392
Copyright
Copyright © 1997 The Psychometric Society

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Footnotes

The authors would like to thank Xuming He for his comments and suggestions. The research of the first author was partially supported by an ETS/GREB Psychometric Fellowship. The research of the second author was partially supported by NSF grand DMS 94-04327.

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