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Relations between Multidimensional Scaling and Three-Mode Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Ledyard R Tucker
Ledyard R Tucker*
Affiliation:
University of Illinois
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Abstract

A combination is achieved of two lines of psychometric interest: a) multidimensional scaling and b) factor analysis. This is accomplished with the use of three-mode factor analysis of scalar product matrices, one for each subject. Two of the modes are the groups of objects scaled and the third mode is the sample of subjects. Results are an object space, a person space, and a system for changing weights given to dimensions and of angles between dimensions in the object space for individuals located at different places in the person space. The development is illustrated with data from an adjective similarity study.

Type
Original Paper
Information
Psychometrika , Volume 37 , Issue 1 , March 1972 , pp. 3 - 27
Copyright
Copyright © 1972 The Psychometric Society

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Footnotes

*

Supported in part by the Personnel and Training Branch of the Office of Naval Research under Contract Number 00014-67-A-0305-0003. A number of very helpful comments were made by an anonymous editorial reviewer for Psychometrika.

†

During 1970–71 at the L. L. Thurstone Psychometric Laboratory, University of North Carolina.

References

Bellman, R. R. Introduction to matrix analysis, 1960, New York: McGraw-HillGoogle Scholar
Carroll, J. D., & Chang, J. J. Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika, 1970, 35, 283–319CrossRefGoogle Scholar
Cliff, N. The “idealized individual” interpretation of individual differences in multidimensional scaling. Psychometrika, 1968, 33, 225–232CrossRefGoogle Scholar
Eckart, C., & Young, G. The approximation of one matrix by another of lower rank. Psychometrika, 1936, 1, 211–218CrossRefGoogle Scholar
Helm, C. E., & Tucker, L. R Individual differences in the structure of color-perception. American Journal of Psychology, 1962, 75, 437–444CrossRefGoogle ScholarPubMed
Horan, C. B. Multidimensional scaling: Combining observations when individuals have different perceptual structures. Psychometrika, 1969, 34, 139–165CrossRefGoogle Scholar
Horst, P. Matrix algebra for social scientists, 1963, New York: Holt, Rinehart, and WinstonGoogle Scholar
Johnson, R. M. On a theorem stated by Eckart and Young. Psychometrika, 1963, 28, 259–263CrossRefGoogle Scholar
Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis. Psychometrika, 1958, 23, 187–200CrossRefGoogle Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method. Psychometrika, 1964, 29, 115–129CrossRefGoogle Scholar
MacDuffee, C. C. The theory of matrices, 1946, New York: ChelseaGoogle Scholar
Ross, J. A remark on Tucker and Messick's “points of view” analysis. Psychometrika, 1966, 31, 27–31CrossRefGoogle Scholar
Torgerson, W. S. Theory and methods of scaling, 1958, New York: WileyGoogle Scholar
Tucker, L. R. Some mathematical notes on three-mode factor analysis. Psychometrika, 1966, 31, 279–311CrossRefGoogle ScholarPubMed
Tucker, L. R. and Messick, S. An individual difference model for multidimensional scaling. Psychometrika, 1963, 28, 333–367CrossRefGoogle Scholar