Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-08T04:01:10.811Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

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 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.

References

Birkhoff, G. (1937). Rings of sets. Duke Mathematical Journal, 3, 443–454.CrossRefGoogle Scholar
Chandler, J. P. (1969). STEPIT: Finds local minima of a smooth function of several parameters. Behavioral Science, 14, 81–82.Google Scholar
Degreef, E., Doignon, J.-P., Ducamp, A., Falmagne, J.-C. (1986). Languages for the assessment of knowledge. Journal of Mathematical Psychology, 30(3), 243–256.CrossRefGoogle Scholar
Doignon, J.-P., Falmagne, J.-C. (1985). Spaces for the assessment of knowledge. International Journal of Man-Machine Studies, 23, 175–196.CrossRefGoogle Scholar
Falmagne, J.-C., & Doignon, J.-P. (1988a). A Markovian procedure for assessing the state of a system. Journal of Mathematical Psychology.CrossRefGoogle Scholar
Falmagne, J.-C., & Doignon, J.-P. (1988b). A class of stochastic procedures for the assessment of knowledge. British Journal of Mathematical and Statistical Psychology.CrossRefGoogle Scholar
Johnson, N. L., Kotz, S. (1970). Distributions in statistics. Continuous univariate distributions, 2., New York: Houghton Mifflin.Google Scholar
Lord, F. M., Novick, M. R. (1974). Statistical theories of mental test scores 2nd. ed.,, Reading, MA: Addison Wesley.Google Scholar
Monjardet, B. (1979). Tresses, fuseaux, préordres et topologies. Mathématique et Sciences Humaines, 30, 11–22.Google Scholar
Parzen, E. (1960). Modern probability theory and its applications, New York: John Wiley & Sons.CrossRefGoogle Scholar
Peizer, D. B., Pratt, J. W. (1968). A normal approximation to the binomial, F, beta, and other common, related tail probabilities I. Journal of the American Statistical Association, 63, 1416–1456.Google Scholar
Pratt, J. W. (1968). A normal approximation in the binomiaal, F, beta, and other common, related tail probabilities II. Journal of the American Statistical Association, 63, 1457–1483.Google Scholar
Villano, M., Falmagne, J.-C., Johannesen, L., & Doignon, J.-P. (1987). Stochastic procedures for assessing an individual's state of knowledge. Proceedings of the International Conference on Computer Assisted Learning in Post-Secondary Education (pp. 369–3711). Calgary, Canada.Google Scholar