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On the Determination of Appropriate Dimensionality in Data with Error

Published online by Cambridge University Press:  01 January 2025

Paul D. Isaac  and
David D. S. Poor
Paul D. Isaac
Affiliation:
The Ohio State University
David D. S. Poor
Affiliation:
The Ohio State University
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Abstract

The study deals with the problem of determining true dimensionality of data-with-error scaled by Kruskal's multidimensional scaling technique. Artificial data was constructed for 6, 8, 12, 16, and 30 point configurations of 1, 2, or 3 true dimensions by adding varying amounts of error to the true distances. Results show how stress is affected by error, number of points, and number of dimensions, and indicate that stress and the “elbow” criterion are inadequate for purposes of identifying true dimensionality when there is error in the data. The Wagenaar-Padmos procedure for identifying true dimensionality and error level is discussed. A simplified technique, involving a measure called Constraint, is suggested.

Keywords

Public PolicyStatistical TheoryMultidimensional ScalingError LevelScaling Technique

Information

Type
Original Paper
Information
Psychometrika , Volume 39 , Issue 1 , March 1974 , pp. 91 - 109
Copyright
Copyright © 1974 The Psychometric Society

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Footnotes

*

The authors wish to thank Dr. J. B. Kruskal for his valuable help and suggestions.

†

Computer time for this research was supplied by the Instruction and Research Computer Center, The Ohio State University.

References

Beals, R., Krantz, D. H., & Tversky, A.. Foundations of multidimensional scaling. Psychological Review, 1968, 75, 127–142.CrossRefGoogle ScholarPubMed
Coombs, C. H., & Kao, R. C. On a connection between factor analysis and multidimensional unfolding. Psychometrika, 1960, 25, 219–231.CrossRefGoogle Scholar
Klahr, D. A Monte Carlo investigation of the statistical significance of Kruskal's nonmetric scaling procedure. Psychometrika, 1969, 34, 319–330.CrossRefGoogle Scholar
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–27 (a).CrossRefGoogle Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method. Psychometrika, 1964, 29, 115–129 (b).CrossRefGoogle Scholar
Ramsay, J. O. Some statistical considerations in multidimensional scaling. Psychometrika, 1969, 34, 167–182.CrossRefGoogle Scholar
Shepard, R. N. Metric structures in ordinal data. Journal of Mathematical Psychology, 1966, 3, 287–315.CrossRefGoogle Scholar
Spence, I. Multidimensional scaling: An empirical and theoretical investigation. Ph.D. dissertation, University of Toronto, 1970.Google Scholar
Stenson, H. H., & Knoll, R. L. Goodness of fit for random rankings in Kruskal's nonmetric scaling procedure. Psychological Bulletin, 1969, 71, 122–126.CrossRefGoogle Scholar
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