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Hierarchical Classes: Model and Data Analysis

Published online by Cambridge University Press:  01 January 2025

Paul De Boeck  and
Seymour Rosenberg
Paul De Boeck*
Affiliation:
University of Leuven, Belgium
Seymour Rosenberg*
Affiliation:
Rutgers University
*
Requests for reprints should be sent to Paul De Boeck, Department of Psychology, University of Leuven, Tiensestraat 102, 3000 Leuven, BELGIUM; or to Seymour Rosenberg, Department of Psychology, Kilmer Campus—Tillett Hall, Rutgers University, New Brunswick, NJ 08903.
Requests for reprints should be sent to Paul De Boeck, Department of Psychology, University of Leuven, Tiensestraat 102, 3000 Leuven, BELGIUM; or to Seymour Rosenberg, Department of Psychology, Kilmer Campus—Tillett Hall, Rutgers University, New Brunswick, NJ 08903.
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Abstract

A discrete, categorical model and a corresponding data-analysis method are presented for two-way two-mode (objects × attributes) data arrays with 0, 1 entries. The model contains the following two basic components: a set-theoretical formulation of the relations among objects and attributes; a Boolean decomposition of the matrix. The set-theoretical formulation defines a subset of the possible decompositions as consistent with it. A general method for graphically representing the set-theoretical decomposition is described. The data-analysis algorithm, dubbed HICLAS, aims at recovering the underlying structure in a data matrix by minimizing the discrepancies between the data and the recovered structure. HICLAS is evaluated with a simulation study and two empirical applications.

Keywords

binary dataBoolean algebrahierarchical classesset-theoretical modeltwo-way two-mode matrices
Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 3 , September 1988 , pp. 361 - 381
Copyright
Copyright © 1988 The Psychometric Society

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Footnotes

This research was supported in part by a grant from the Belgian NSF (NFWO) to Paul De Boeck and in part by NSF Grant BNS-83-01027 to Seymour Rosenberg. We thank Iven Van Mechelen for clarifying several aspects of the Boolean algebraic formulation of the model and Phipps Arabie for his comments on an earlier draft.

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