Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-30T08:15:58.878Z Has data issue: false hasContentIssue false
  • English
  • Français

Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis

Published online by Cambridge University Press:  01 January 2025

J. B. Kruskal
J. B. Kruskal*
Affiliation:
Bell Telephone Laboratories, Murray Hill, N. J.
Get access Rights & Permissions [Opens in a new window]

Abstract

Multidimensional scaling is the problem of representing n objects geometrically by n points, so that the interpoint distances correspond in some sense to experimental dissimilarities between objects. In just what sense distances and dissimilarities should correspond has been left rather vague in most approaches, thus leaving these approaches logically incomplete. Our fundamental hypothesis is that dissimilarities and distances are monotonically related. We define a quantitative, intuitively satisfying measure of goodness of fit to this hypothesis. Our technique of multidimensional scaling is to compute that configuration of points which optimizes the goodness of fit. A practical computer program for doing the calculations is described in a companion paper.

Information

Type
Original Paper
Information
Psychometrika , Volume 29 , Issue 1 , March 1964 , pp. 1 - 27
Copyright
Copyright © 1964 Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abelson, R. P. and Tukey, J. W. Efficient conversion of nonmetric information into metric information. Proc. Amer. statist. Ass. Meetings, Social statist. Section, 1959, 226–230.Google Scholar
Aumann, R. J. and Kruskal, J. B. The coefficients in an allocation problem. Naval Res. Logistics Quart., 1958, 5, 111–123.CrossRefGoogle Scholar
Aumann, R. J. and Kruskal, J. B. Assigning quantitative values to qualitative factors in the Naval electronics problem. Naval Res. Logistics Quart., 1959, 6, 1–16.Google Scholar
Bartholomew, D. J. A test of homogeneity for ordered alternatives. Biometrika, 1959, 46, 36–48.CrossRefGoogle Scholar
Coombs, C. H. An application of a nonmetric model for multidimensional analysis of similarities. Psychol. Rep., 1958, 4, 511–518.CrossRefGoogle Scholar
Coombs, C. H. and Kao, R. C. On a connection between factor analysis and multidimensional unfolding. Psychometrika, 1960, 25, 219–231.CrossRefGoogle Scholar
Ekman, G. Dimensions of color vision. J. Psychol., 1954, 38, 467–474.CrossRefGoogle Scholar
Hardy, G. H., Littlewood, J. E., and Polya, G. Inequalities (2nd ed.), Cambridge, Eng.: Cambridge Univ. Press, 1952.Google Scholar
Indow, T. and Uchizono, T. Multidimensional mapping of Munsell colors varying in hue and chroma. J. exp. Psychol., 1960, 59, 321–329.CrossRefGoogle ScholarPubMed
Indow, T. and Kanazawa, K. Multidimensional mapping of colors varying in hue, chroma and value. J. exp. Psychol., 1960, 59, 330–336.CrossRefGoogle ScholarPubMed
Kolmogorov, A. N. and Fomin, S. V. Elements of the theory of functions and functional analysis. Vol. 1.Metric and normed spaces, Rochester, N. Y.: Graylock Press, 1957.Google Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method. Psychometrika, (accepted for publication, June, 1964).CrossRefGoogle Scholar
Rothkopf, E. Z. A measure of stimulus similarity and errors in some paired-associate learning tasks. J. exp. Psychol., 1957, 53, 94–101.CrossRefGoogle ScholarPubMed
Shepard, R. N. Stimulus and response generalization: A stochastic model relating generalization to distance in psychological space. Psychometrika, 1957, 32, 325–345.CrossRefGoogle Scholar
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function. Psychometrika, 1962, 27, 125–139.CrossRefGoogle Scholar
Shepard, R. N. Analysis of proximities as a technique for the study of information processing in man. Human Factors, 1963, 5, 19–34.CrossRefGoogle Scholar
Torgerson, W. S. Theory and methods of scaling, New York: Wiley, 1958.Google Scholar