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A Nonmetric Variety of Linear Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Joseph B. Kruskal  and
Roger N. Shepard
Joseph B. Kruskal
Affiliation:
Bell Telephone Laboratories
Roger N. Shepard
Affiliation:
Stanford University
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Abstract

The numbers in each column of an n × m matrix of multivariate data are interpreted as giving the measured values of all n of the objects studied on one of m different variables. Except for random error, the rank order of the numbers in such a column is assumed to be determined by a linear rule of combination of latent quantities characterizing each row object with respect to a small number of underlying factors. An approximation to the linear structure assumed to underlie the ordinal properties of the data is obtained by iterative adjustment to minimize an index of over-all departure from monotonicity. The method is “nonmetric” in that the obtained structure in invariant under monotone transformations of the data within each column. Except in certain degenerate cases, the structure is nevertheless determined essentially up to an affine transformation. Tests show (a) that, when the assumed monotone relationships are strictly linear, the recovered structure tends closely to approximate that obtained by standard (metric) factor analysis but (b) that, when these relationships are severely nonlinear, the nonmetric method avoids the inherent tendency of the metric method to yield additional, spurious factors. From the practical standpoint, however, the usefulness of the nonmetric method is limited by its greater computational cost, vulnerability to degeneracy, and sensitivity to error variance.

Keywords

Linear StructureMultivariate DataAffine TransformationDegenerate CaseUnderlying Factor
Type
Original Paper
Information
Psychometrika , Volume 39 , Issue 2 , June 1974 , pp. 123 - 157
Copyright
Copyright © 1974 The Psychometric Society

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Footnotes

1

Although the method described here existed as an operational computer program toward the end of 1962 and although the tests reported here were completed by 1966, this is the first full description of the method and report of the results of the test application. The preparation of this paper was in part supported by NSF grants GS-1302 and GS-2283 to the second author and was completed during the second author's tenure as a John Simon Guggenheim Fellow at the Center for Advanced Study in the Behavioral Sciences, Stanford. The authors are indebted to J. D. Carroll, Mrs. J.-J. Chang, Mrs, C. Brown, and the former Miss M. M. Sheenan, all of the Bell Telephone Laboratories, for their assistance in connection with the test application.

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