Hostname: page-component-5cf477f64f-n7lw4 Total loading time: 0 Render date: 2025-04-04T00:38:43.321Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonmetric Multidimensional Scaling: Recovery of Metric Information

Published online by Cambridge University Press:  01 January 2025

Forrest W. Young
Forrest W. Young*
Affiliation:
University of North Carolina
Get access Rights & Permissions [Opens in a new window]

Abstract

The degree of metric determinancy afforded by nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the true dimensionality of the data being scaled, and the amount of error contained in the data being scaled. It was found 1) that if the ratio of the degrees of freedom of the data to that of the coordinates is sufficiently large then metric information is recovered even when random error is present; and 2) when the number of points being scaled increases the stress of the solution increases even though the degree of metric determinacy increases.

Type
Original Paper
Information
Psychometrika , Volume 35 , Issue 4 , December 1970 , pp. 455 - 473
Copyright
Copyright © 1970 The 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.)

Footnotes

*

This report was supported in part by a PHS research grant No. M-10006 from the National Institute of Mental Health, Public Health Service, and in part by a Science Development grant No. GU-2059, from the National Science Foundation. The author is indebted to Charles R. Sherman for his assistance in gathering the data and for his critical re-writing of sections of this report. The assistance of Lyle V. Jones in his critical readings and comments is also deeply appreciated.

References

Bock, R. D. & Jones, L. V. The measurement and prediction of judgment and choice, 1968, San Francisco: Holden DayGoogle Scholar
Coombs, C. H. & Kao, R. C. On a connection between factor analysis and multidimensional unfolding. Psychometrika, 1960, 25, 219–231CrossRefGoogle Scholar
Guttman, L. The development of nonmetric space analysis: A letter to Professor John Ross. Multivariate Behavioral Research, 1967, 2, 71–82CrossRefGoogle ScholarPubMed
Kruskal, J. B. Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–28CrossRefGoogle Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: a numerical method. Psychometrika, 1964, 29, 115–130CrossRefGoogle Scholar
Lingoes, J. C. An IBM-7090 program for Guttman-Lingoes smallest space analysis—RI. Behavioral Science, 1966, 11, 332–332Google Scholar
McGee, V. E. The multidimensional analysis of ‘elastic’ distances. The British Journal of Mathematical and Statistical Psychology, 1966, 19, 181–196CrossRefGoogle Scholar
Ramsay, J. Some statistical considerations in multidimensional scaling. Psychometrika, 1969, 34, 167–182CrossRefGoogle Scholar
Shepard, R. N. The analysis of proximities: multidimensional scaling with an unknown distance function: I. Psychometrika, 1962, 27, 125–140CrossRefGoogle Scholar
Shepard, R. N. The analysis of proximities: multidimensional scaling with an unknown distance function: II. Psychometrika, 1962, 27, 219–246CrossRefGoogle Scholar
Shepard, R. N. Metric structures in ordinal data. Journal of Mathematical Psychology, 1966, 3, 287–315CrossRefGoogle Scholar
Sherman, C. R. and Young, F. W. Nonmetric multidimensional scaling: A monte carlo study. Proceedings of the 76th Annual Convention of the American Psychological Association, 1968, 3, 207–208Google Scholar
Thurstone, L. L. A law of comparative judgment. Psychological Review, 1927, 34, 273–286CrossRefGoogle Scholar
Torgerson, W. S. Theory and methods of scaling, 1958, New York: WileyGoogle Scholar
Young, F. W. TORSCA-9, A FORTRAN IV program for nonmetric multi-dimensional scaling. Behavioral Science, 1968, 13, 343–344CrossRefGoogle Scholar
Young, F. W. & Torgerson, W. S. TORSCA, A FORTRAN IV program for Shepard-Kruskal multidimensional scaling analysis. Behavioral Science, 1967, 12, 498–498Google Scholar