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Geometric Representation of Association between Categories

Published online by Cambridge University Press:  01 January 2025

Willem J. Heiser
Willem J. Heiser*
Affiliation:
Leiden University
*
Correspondence should be addressed to: Willem J. Heiser, Department of Psychology, Leiden University, PO Box 9555, 2300 RB Leiden, The Netherlands (e-mall: Heiser@fsw.leidenuniv.nl).
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Abstract

Categories can be counted, rated, or ranked, but they cannot be measured. Likewise, persons or individuals can be counted, rated, or ranked, but they cannot be measured either. Nevertheless, psychology has realized early on that it can take an indirect road to measurement: What can be measured is the strength of association between categories in samples or populations, and what can be quantitatively compared are counts, ratings, or rankings made under different circumstances, or originating from different persons. The strong demand for quantitative analysis of categorical data has thus created a variety of statistical methods, with substantial contributions from psychometrics and sociometrics. What is the common basis of these methods dealing with categories? The basic element they share is that the sample space has a special geometry, in which categories (or persons) are point masses forming a simplex, while distributions of counts or profiles of ratings are centers of gravity, which are also point masses. Rankings form a discrete subset in the interior of the simplex, known as the permutation polytope, and paired comparisons form another subset on the edges of the simplex. Distances between point masses form the basic tool of analysis. The paper gives some history of major concepts, which naturally leads to a new concept: the shadow point. It is then shown how loglinear models, Luce and Rasch models, unfolding models, correspondence analysis and homogeneity analysis, forced classification and classification trees, as well as other models and methods, fit into this particular geometrical framework.

Keywords

Categorical datasimplextriangular plotpaired comparisonsrank orderspermutation polytopecenter of gravityBTL modelRasch modelinertiaassociation modelvariationmultidimensional unfoldingbiplotmultinomial response modelloglinear modelforced classificationclassification tree
Type
2004 Presidential Address
Information
Psychometrika , Volume 69 , Issue 4 , December 2004 , pp. 513 - 545
Copyright
Copyright © 2004 The Psychometric Society

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Footnotes

This paper is based on my Presidential Address delivered at the 69th Annual Meeting of the Psychometric Society, Pacific Grove, California, June 14–17, 2004. It was completed during a stay as Fellow of the Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS) in Wassenaar, The Netherlands.

I would like to thank Marike Polak, Frank Busing, Elise Dusseldorp, and Angela Jansen for their help in the data analyses and the preparation of the figures, and Laurence Frank for her assistance during the oral presentation. I am also very lucky to have a career-long personal coach, Jacqueline J. Meulman, with whom I share so many interests and perspectives.

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