Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-14T07:30:46.161Z Has data issue: false hasContentIssue false
  • English
  • Français

The Conjunctive Model of Hierarchical Classes

Published online by Cambridge University Press:  01 January 2025

Iven Van Mechelen,
Paul De Boeck and
Seymour Rosenberg
Iven Van Mechelen*
Affiliation:
Katholieke Universiteit Leuven
Paul De Boeck
Affiliation:
Katholieke Universiteit Leuven
Seymour Rosenberg
Affiliation:
Rutgers University
*
Please send requests for reprints to Iven Van Mechelen, Department of Psychology, Tiensestraat 102, B-3000 Leuven, BELGIUM. (Email: Iven.VanMechelen@psy.kuleuven.ac.be)
Get accessRights & Permissions [Opens in a new window]

Abstract

This paper describes the conjunctive counterpart of De Boeck and Rosenberg's hierarchical classes model. Both the original model and its conjunctive counterpart represent the set-theoretical structure of a two-way two-mode binary matrix. However, unlike the original model, the new model represents the row-column association as a conjunctive function of a set of hypothetical binary variables. The conjunctive nature of the new model further implies that it may represent some conjunctive higher order dependencies among rows and columns. The substantive significance of the conjunctive model is illustrated with empirical applications. Finally, it is shown how conjunctive and disjunctive hierarchical classes models relate to Galois lattices, and how hierarchical classes analysis can be useful to construct lattice models of empirical data.

Keywords

clusteringpartial orderhierarchical classesconjunctiondisjunctiontwo-way two-mode data

Information

Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 4 , December 1995, pp. 505 - 521
Copyright
Copyright © 1995 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.)

Article purchase

Temporarily unavailable

Footnotes

The research reported in this paper was supported by NATO (Grant CRG.921321 to Iven Van Mechelen and Seymour Rosenberg) and by the Research Fund of Katholieke Universiteit Leuven (Grants PDM92/19 and POR93/3 to Iven Van Mechelen; Grants OT89/9 and F91/56 to Paul De Boeck).

References

Barbut, M., Monjardet, B. (1970). Ordre et classification: Algèbre et combinatoire [Order and classification: Algebra and combinatorius], Paris: Hachette.Google Scholar
Birkhoff, G. (1940). Lattice Theory, Providence: American Mathematical Society.CrossRefGoogle Scholar
De Boeck, P. (1986). HICLAS computer program: Version 1.0, Leuven: Katholieke Universiteit.Google Scholar
De Boeck, P., Rosenberg, S. (1988). Hierarchical classes: Model and data analysis. Psychometrika, 53, 361–381.CrossRefGoogle Scholar
De Boeck, P., Van Damme, G., Van Mechelen, I. (1992). HICLAS computer program: Version 2.0, Leuven: Katholieke Universiteit.Google Scholar
Ford, J. K., Schmitt, N., Schechtman, S. L., Hults, B. M., Doherty, M. L. (1989). Process tracing methods: Contributions, problems, and neglected research questions. Organizational Behavior and Human Decision Processes, 43, 75–117.CrossRefGoogle Scholar
Gara, M. A. (1990). A set-theoretical model of person perception. Multivariate Behavioral Research, 25, 275–293.CrossRefGoogle ScholarPubMed
Gara, M. A., Rosenberg, S. (1979). The identification of persons as supersets and subsets in free-response personality descriptions. Journal of Personality and Social Psychology, 37, 2161–2170.CrossRefGoogle Scholar
Goodman, L. A., Kruskal, W. H. (1954). Measures of association for cross-classifications. Journal of the American Statistical Association, 49, 732–764.Google Scholar
Kim, K. H. (1982). Boolean matrix theory, New York: Marcel Dekker.Google Scholar
Leenen, I., Van Mechelen, I., & De Boeck, P. (1994). A generic disjunctive/conjunctive decomposition model for n-ary relations. Manuscript submitted for publication.Google Scholar
Luce, R. D. (1952). A note on Boolean matrix theory. Proceedings of the American Mathematical Society, 3, 382–388.CrossRefGoogle Scholar
Maris, E. (1992). Psychometric models for psychological processes and structures, Leuven: Katholieke Universiteit.Google Scholar
Maris, E., De Boeck, P., & Van Mechelen, I. (in press). Probability matrix decomposition models. Psychometrika.Google Scholar
Ogilvie, J. R., Schmitt, N. (1979). Situational influences on linear and nonlinear use of information. Organizational Behavior and Human Performance, 4, 337–352.Google Scholar
Rosenberg, S., Jones, R. A. (1972). A method for investigating and representing a person's implicit theory of personality: Theodore Dreiser's view of people. Journal of Personality and Social Psychology, 22, 372–386.CrossRefGoogle Scholar
Rosenberg, S., Sedlak, A. (1972). Structural representations of implicit personality theory. In Berkowitz, L. (Eds.), Advances in experimental social psychology (pp. 235–297). New York: Academic Press.Google Scholar
Rutherford, D. E. (1963). Inverse of Boolean matrices. Proceedings of the Glasgow Mathematical Association, 6, 49–53.CrossRefGoogle Scholar
Van Mechelen, I. (1993). Approximate Galois lattices of formal concepts. In Opitz, O., Lausen, B., Klar, R. (Eds.), Information and classification: Concepts, methods and applications (pp. 108–112). Berlin: Springer.CrossRefGoogle Scholar
Van Mechelen, I., De Boeck, P. (1990). Projection of a binary criterion into a model of hierarchical classes. Psychometrika, 55, 677–694.CrossRefGoogle Scholar
Van Mechelen, I., Van Damme, G. (1994). A latent criteria model for choice data. Acta Psychologica, 87, 85–94.CrossRefGoogle Scholar
Westenberg, M. R. M., Koele, P. (1992). Response modes, decision processes and decision outcomes. Acta Psychologica, 80, 169–184.CrossRefGoogle Scholar
Wille, R. (1982). Restructuring lattice theory: An approach based on hierarchies of concepts. In Rival, O. (Eds.), Ordered sets (pp. 445–470). Boston: Reidel.CrossRefGoogle Scholar