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A Latent Trait Theory Via a Stochastic Learning Theory for a Knowledge Space

Published online by Cambridge University Press:  01 January 2025

Jean-Claude Falmagne
Jean-Claude Falmagne*
Affiliation:
New York University
*
Requests for reprints should be sent to Jean-Claude Falmagne, IRU in Mathematical Behavorial Sciences, School of Social Sciences, University of California, Irvine, Irvine, CA 92717.
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Abstract

To capture the cognitive organization of a set of questions or problems pertaining to a body of information, Doignon and Falmagne have proposed, and analyzed in a number of papers, the concept of a knowledge space, that is, a distinguished collection of subsets of questions, representing the possible knowledge states. This collection of sets is assumed to satisfy a number of conditions. Since this concept is a deterministic one, the problem of empirical testing arises. A stochastic version of a knowledge space is developed in this paper, in which the knowledge states are considered as possible epochs in a subject's learning history. The knowledge space is decomposed as a union of a number of possible learning paths, called gradations. The model specifies how a subject is channelled through and progresses along a gradation. A probabilistic axiom of the “local indepencence” type relates the knowledge states to the observable responses. The predictions of this model are worked out in details in the case of parametric assumptions involving gamma distributions. An application of the model to artificial data is described, based on maximum likelihood methods. The statistical analysis is shown to be capable of revealing the combinatoric core of the model.

Keywords

Guttman scale, knowledge statedevelopmentlocal independencegamma distributionbeta distribution
Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 2 , June 1989 , pp. 283 - 303
Copyright
Copyright © 1989 The Psychometric Society

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Footnotes

This work was supported by NSF grant IST-8418860 and ARI grant DAAG29-84-G-0083 to New York University. I am grateful to Jean-Paul Doignon, Mathieu Koppen, Geoff Iverson and Michael Landy for their reactions to previous drafts of this paper, to Michael Villano for carrying out the computer simulation and the analysis of the simulated data, and especially to one referee for his very useful comments.

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