Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-08T10:27:26.338Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on the use of Directional Statistics in Weighted Euclidean Distances Multidimensional Scaling Models

Published online by Cambridge University Press:  01 January 2025

Charles L. Jones
Charles L. Jones*
Affiliation:
McMaster University and University of Toronto
*
Reprint requests should be addressed to Charles L. Jones, Sociology Department, 563 Spadina Avenue, University of Toronto, Toronto, Ontario, M5S 1A1, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

The weighted euclidean distances model in multidimensional scaling (WMDS) represents individual differences as dimension saliences which can be interpreted as the orientations of vectors in a subject space. It has recently been suggested that the statistics of directions would be appropriate for carrying out tests of location with such data. The nature of the directional representation in WMDS is reviewed and it is argued that since dimension saliences are almost always positive, the directional representations will usually be confined to the positive orthant. Conventional statistical techniques are appropriate to angular representations of the individual differences which will yield angles in the interval (0, 90) so long as dimension saliences are nonnegative, a restriction which can be imposed. Ordinary statistical methods are also appropriate with several linear indices which can be derived from WMDS results. Directional statistics may be applied more fruitfully to vector representations of preferences.

Keywords

multidimensional scalingdirectional data
Type
Notes And Comments
Information
Psychometrika , Volume 48 , Issue 3 , September 1983 , pp. 473 - 476
Copyright
Copyright © 1983 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

Material support for work referred to herein was received from SSRC grant HR 1883/1, SSHRC grant 410-77-0860, and from McMaster University.

Thanks are due to A. P. M. Coxon, M. J. Prentice, M. A. Stephens and an anonymous reviewer.

References

Reference Notes

Bloxom, B. Individual differences in multidimensional scaling, Princeton, N.J.: Educational Testing Service, 1968.CrossRefGoogle Scholar
Young, F. W. & Lewyckyj, R. ALSCAL-4 User's Guide 2nd edition, Chapel Hill, NC: Psychometric Laboratory. University of North Carolina, 1979.Google Scholar
Young, F. W. Enhancements in ALSCAL-82. Proceedings of the Seventh Annual SAS Users Group . Cary, NC: The SAS Institute. 1982, 633–642.Google Scholar

References

Batschelet, E. Contribution to the discussion of a paper by K. V. Mardia. Journal of the Royal Statistical Society Series B, 1975, 37, 378–378.Google Scholar
Batschelet, E. Circular statistics in biology, New York: Academic Press, 1981.Google 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, 238–319.CrossRefGoogle Scholar
Coxon, A. P. M. & Jones, C. L. Occupational similarities: subjective aspects of social stratification. Quality and Quantity, 1974, 8, 139–157.CrossRefGoogle Scholar
Coxon, A. P. M. & Jones, C. L. The images of occupational prestige, London: Macmillan, 1978.CrossRefGoogle Scholar
Coxon, A. P. M. & Jones, C. L. Measurement and meanings, London: Macmillan, 1979.CrossRefGoogle Scholar
Coxon, A. P. M. & Jones, C. L. Multidimensional scaling: exploration to confirmation. Quality and Quantity, 1980, 14(1), 31–73.CrossRefGoogle Scholar
Horan, C. B. Multidimensional scaling: combining observations when individuals have different perceptual structures. Psychometrika, 1969, 34, 139–165.CrossRefGoogle Scholar
Hubert, L. J., Golledge, R. G. & Costanzo, C. M. Analysis of variance procedures based on a proximity measure between subjects. Psychological Bulletin, 1982, 91(2), 424–430.CrossRefGoogle Scholar
Jones, C. L. Analysis of preferences as directional data. Quality and Quantity (in press).Google Scholar
Mardia, K. V. Statistics of directional data, New York: Academic Press, 1972.Google Scholar
McCallum, R. C. Effects of conditionality on INDSCAL and ALSCAL weights. Psychometrika, 1977, 42(2), 297–305.CrossRefGoogle Scholar
Schiffman, S. S., Reynolds, M. A. & Young, F. W. Introduction to multidimensional scaling, New York: Academic Press, 1981.Google Scholar
Watson, G. S. & Williams, E. J. On the construction of significance tests on the circle and the sphere. Biometrika, 1956, 43, 344–352.CrossRefGoogle Scholar
Wheeler, S. & Watson, G. S. A distribution-free two sample test on a circle. Biometrika, 1964, 51, 256–257.CrossRefGoogle Scholar