Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-08T10:14:04.018Z Has data issue: false hasContentIssue false
  • English
  • Français

A Monte Carlo Study of Kruskal’s Variance based Measure on Stress

Published online by Cambridge University Press:  01 January 2025

David M. Levine
David M. Levine*
Affiliation:
Bernard Baruch College (CUNY)
*
Requests for reprints should be sent to David M. Levine, Department of Statistics, Bernard M. Baruch College, 17 Lexington Ave., New York, N. Y. 10010.
Get access Rights & Permissions [Opens in a new window]

Abstract

Researchers in the past ten years have studied various parameters involved in nonmetric multidimensional scaling by utilizing Monte Carlo procedures. This paper develops stress distributions using Kruskal's second stress formula based upon a null hypothesis of equal likelihood in the ranking of a set of proximities. These distributions can serve to determine whether a set of data has other than random structure.

Keywords

nonmetric multidimensional scalingstress distributionstests of randomness
Type
Original Paper
Information
Psychometrika , Volume 43 , Issue 3 , September 1978 , pp. 307 - 315
Copyright
Copyright © 1978 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

The author would like to thank Dr. Joseph Kruskal and three anonymous reviewers for their helpful suggestions on an earlier draft of this paper.

This research was supported in part by a grant from the Baruch College Scholar Assistance program.

References

Reference Notes

Kruskal, J. B. & Carmone, F. J. How to use M-D-SCAL (Version 5M) and other useful information, 1969, Murray Hill, N. J.: Bell Telephone Laboratories.Google Scholar
Levine, D. Estimation of the statistical significance of Kruskal's measure of stress for an asymmetric matrix. Unpublished manuscript, 1976.Google Scholar

References

Cohen, H. S. & Jones, L. E. The effects of random error and subsampling of dimensions on recovery of configurations by nonmetric multidimensional scaling. Psychometrika, 1974, 39, 69–91.CrossRefGoogle Scholar
Green, P. E. On the robustness of multidimensional scaling techniques. Journal of Marketing Research, 1975, 12, 73–81.CrossRefGoogle Scholar
Klahr, D. A. Monte Carlo investigation on the statistical significance of Kruskal's nonmetric scaling procedure. Psychometrika, 1969, 34, 319–333.CrossRefGoogle Scholar
Kruskal, J. B. & Carroll, J. D. Geometric models and badness of fit functions. In Krishnaiah, P. R. (Eds.), Multivariate Analysis (Vol. 2), 1969, New York: Academic Press.Google Scholar
Ling, R. F. A probability theory of cluster analysis. Journal of the American Statistical Association, 1973, 68, 159–164.CrossRefGoogle Scholar
Lingoes, J. C. & Roskam, E. E. A mathematical and empirical analysis of two multidimensional scaling algorithms. Psychometrika Monograph Supplement, 1973, 38 (4, Pt. 2).Google Scholar
Ramsay, J. O. Some statistical considerations in multidimensional scaling. Psychometrika, 1969, 34, 167–182.CrossRefGoogle Scholar
Shepard, R. N. Representation of structure in similarity data: problems and propsects. Psychometrika, 1974, 39, 373–421.CrossRefGoogle Scholar
Spence, I. Multidimensional scaling: an empirical and theoretical investigation. Unpublished Ph.D. Thesis, University of Toronto, 1970.Google Scholar
Spence, I. & Ogilvie, J. C. A table of expected stress values for random rankings in nonmetric multidimensional scaling. Multivariate Behavioral Research, 1973, 8, 511–517.CrossRefGoogle ScholarPubMed
Wagenaar, W. A. & Padmos, P. Quantitative interpretation of stress in Kruskal's multidimensional scaling technique. British Journal of Mathematical and Statistical Psychology, 1971, 24, 101–110.CrossRefGoogle Scholar
Young, F. W. Nonmetric multidimensional scaling: Recovery of metric information. Psychometrika, 1970, 35, 455–473.CrossRefGoogle Scholar