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Correspondence Analysis and Optimal Structural Representations

Published online by Cambridge University Press:  01 January 2025

Lawrence Hubert  and
Phipps Arabie
Lawrence Hubert*
Affiliation:
University of Illinois, Champaign
Phipps Arabie
Affiliation:
Graduate School of Management, Rutgers University, Newark
*
Requests for reprints should be sent to Lawrence Hubert, Department of Psychology, University of Illinois, 603 East Daniel Street, Champaign, IL 6t820.
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Abstract

Many well-known measures for the comparison of distinct partitions of the same set of n objects are based on the structure of class overlap presented in the form of a contingency table (e.g., Pearson's chi-square statistic, Rand's measure, or Goodman-Kruskal's τb), but they all can be rephrased through the use of a simple cross-product index defined between the corresponding entries from two n ×n proximity matrices that provide particular a priori (numerical) codings of the within- and between-class relationships for each of the partitions. We consider the task of optimally constructing the proximity matrices characterizing the partitions (under suitable restriction) so as to maximize the cross-product measure, or equivalently, the Pearson correlation between their entries. The major result presented states that within the broad classes of matrices that are either symmetric, skew-symmetric, or completely arbitrary, optimal representations are already derivable from what is given by a simple one-dimensional correspondence analysis solution. Besides severely limiting the type of structures that might be of interest to consider for representing the proximity matrices, this result also implies that correspondence analysis beyond one dimension must always be justified from logical bases other than the optimization of a single correlational relationship between the matrices representing the two partitions.

Keywords

partition comparisoncorrespondence analysisoptimal representationproximity matrix comparisonone-dimensional representation
Type
Original Paper
Information
Psychometrika , Volume 57 , Issue 1 , March 1992 , pp. 119 - 140
Copyright
Copyright © 1992 The Psychometric Society

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Footnotes

This research was supported in part by a grant from American Telephone and Telegraph (AT&T) to the Industrial Affiliates Program of the University of Illinois. The acting Editor for this manuscript was Shizuhiko Nishisato.

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