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On Metric Multidimensional Unfolding

Published online by Cambridge University Press:  01 January 2025

Peter H. Schönemann
Peter H. Schönemann*
Affiliation:
The Ohio State University
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Abstract

The problem of locating two sets of points in a joint space, given the Euclidean distances between elements from distinct sets, is solved algebraically. For error free data the solution is exact, for fallible data it has least squares properties.

Information

Type
Original Paper
Information
Psychometrika , Volume 35 , Issue 3 , September 1970 , pp. 349 - 366
Copyright
Copyright © 1970 The Psychometric Society

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Footnotes

†

This remedy did not work as well in practice as had been hoped when these lines were written. If M has nonpositive roots the program should be terminated. One of my students, Miss Wang, is presently working on a more robust least squares solution to handle the fallible case.

References

Bennett, J. F. Determination of the number of independent parameters of a score matrix from the examination of rank orders. Psychometrika, 1956, 21, 383–393.CrossRefGoogle Scholar
Bennett, J. F., & Hays, W. L. Multidimensional unfolding: Determining the dimensionality of ranked preference data. Psychometrika, 1960, 25, 27–43.CrossRefGoogle Scholar
Bloxom, B. Individual differences in multidimensional scaling, 1968, Princeton: Educational Testing Service.CrossRefGoogle Scholar
Carroll, J. D., & Chang, Jih-Jie Relating preference data to multidimensional scaling solutions via a generalization of Coombs' unfolding model. Paper read at the Spring Meeting of the Psychometric Society, Madison, Wisconsin, 1967.Google Scholar
Carroll, J. D., & Chang, Jih-Jie Analysis of individual differences in multidimensional scaling via an N-way generalization of “Eckart-Young” decomposition. Psychometrika, 1970, 35, 283–319.CrossRefGoogle Scholar
Coombs, C. H. Inconsistencies of preferences in psychological measurement. Journal of Experimental Psychology, 1958, 55, 1–7.CrossRefGoogle ScholarPubMed
Coombs, C. H. A theory of data, 1964, New York: Wiley.Google Scholar
Coombs, C. H., & Kao, R. C. Nonmetric factor analysis, 1955, Ann Arbor: University of Michigan.Google Scholar
Eckart, C., & Young, G.. The approximation of one matrix by another of lower rank. Psychometrika, 1936, 1, 211–218.CrossRefGoogle Scholar
Goode, F. M. Interval scale representation of ordered metric scales. Dittoed manuscript, University of Michigan, 1957.Google Scholar
Hays, W. L., &Bennett, J. F. Multidimensional unfolding: Determining configuration from complete rank order preference data. Psychometrika, 1961, 26, 221–238.CrossRefGoogle Scholar
Krantz, D. H. Rational distances functions for multidimensional scaling. Journal of Mathematical Psychology, 1967, 4, 226–244.CrossRefGoogle Scholar
Kruskal, J. B. How to use MDSCAL, a program to do multidimensional scaling and multidimensional unfolding. Unpublished report, Bell Telephone Laboratories, 1968.Google Scholar
Lingoes, J. C. A general nonparametric model for representing objects and attributes in a joint space. In Gardin, J. C. (Ed.), Les Comptes-rendus de Colloque International sur L'emploi des Calculateurs en Archeologie: Problèmes Sémiologiques et Mathématiques. Marseilles: Centre Nationale de la Recherche Scientifiques, in press.Google Scholar
Ross, J., & Cliff, N. A generalization of the interpoint distance model. Psychometrika, 1964, 29, 167–176.CrossRefGoogle Scholar
Schönemann, P. H. Fitting a simplex symmetrically. Psychometrika, 1970, 35, 1–21.CrossRefGoogle Scholar