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Nonmetric Multidimensional Scaling: A Monte Carlo Study of the Basic Parameters

Published online by Cambridge University Press:  01 January 2025

Charles R. Sherman
Charles R. Sherman*
Affiliation:
University of North Carolina
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Abstract

Metric determinacy of nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the amount of error in the data being scaled, and the accuracy of estimation of the Minkowski distance function parameters, dimensionality and the r-constant. It was found that nonmetric scaling may provide better models if (1) the true structure is of low dimensionality, (2) the dimensionality of recovered structure is not less than the dimensionality of the true structure, (3) degree of error is low, and (4) the degrees of freedom ratio is greater than about 2.5. It was also found that (5) accurate estimation of the Minkowski constant leads to a better model only if the dimensionality has been properly estimated.

Keywords

Public PolicyDistance FunctionAccurate EstimationStatistical TheoryFunction Parameter
Type
Original Paper
Information
Psychometrika , Volume 37 , Issue 3 , September 1972 , pp. 323 - 355
Copyright
Copyright © 1972 The Psychometric Society

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Footnotes

*

This report is based on a thesis submitted in partial fulfillment of the degree of Master of Arts at the University of North Carolina, April, 1970. The thesis is an outgrowth of earlier work done with Forrest W. Young. The author is indebted to Forrest W. Young, Norman Cliff, and Lyle V. Jones for their assistance in the preparation of this report. This report was supported in part by PHS research grant No. M-10006 from the National Institute of Mental Health, Public Health Service.

References

Armstrong, J. S. Derivation of theory by means of factor analysis or Tom Swift and his electric factor analysis machine. The American Statistician, 1967, 21(12), 17–21CrossRefGoogle Scholar
Arnold, J. B. A multidimensional scaling of semantic distance. Unpublished manuscript, Saint Mary's College, 1970.Google Scholar
Attneave, F. Dimensions of similarity. American Journal of Psychology, 1950, 63, 516–556CrossRefGoogle ScholarPubMed
Attneave, F. Perception and related areas. In Koch, S. (Eds.), Psychology: a study of a science Vol. 4. New York: McGraw-Hill. 1962, 619–659Google Scholar
Cross, D. V. Metric properties of multidimensional stimulus generalization. In Mostofsky, D. I. (Ed.), Stimulus generalization. Stanford, 1965. Pp. 72–93.Google Scholar
Fillenbaum, S. F., & Rapoport, A. Paper presented at the Advanced Research Seminar on Scaling and Measurement, Newport Beach, Calif., June, 1969.Google Scholar
Householder, A. S., Landahl, H. D. Mathematical biophysics of the central nervous system, 1945, Bloomington, Ind.: Principia PressCrossRefGoogle Scholar
Hyman, R. & Well, A. Judgments of similarity and spatial models. Perception and Psychophysics, 1967, 2, 233–248CrossRefGoogle Scholar
Hyman, R. & Well, A. Perceptual separability and spatial models. Perception and Psychophysics, 1968, 3, 161–166CrossRefGoogle Scholar
Klahr, D. A Monte Carlo investigation of the statistical significance of Kruskal's nonmetric scaling procedure. Psychometrika, 1969, 3, 319–330CrossRefGoogle Scholar
Kruskal, J. B. Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika, 1964, 29, 1–28 (a)CrossRefGoogle Scholar
Kruskal, J. B. Nonmetric multidimensional scaling: a numerical method. Psychometrika, 1964, 29, 115–130 (b)CrossRefGoogle Scholar
Richardson, M. W. Multi-dimensional psychophysics. Psychological Bulletin, 1938, 35, 659–659 (Abstract)Google Scholar
Shepard, R. N. The analysis of proximities: multidimensional scaling with an unknown distance function: I. Psychometrika, 1962, 27, 125–140 (a)CrossRefGoogle Scholar
Shepard, R. N. The analysis of proximities: multidimensional scaling with an unknown distance function: II. Psychometrika, 1962, 27, 219–246 (b)CrossRefGoogle Scholar
Shepard, R. N. Attention and the metric structure of the stimulus space. Journal of Mathematical Psychology, 1964, 1, 54–87CrossRefGoogle Scholar
Shepard, R. N. Some principles and prospects for the spatial representation of behavioral science data. Paper presented at the Advanced Research Seminar on Scaling and Measurement, Newport Beach, Calif., June, 1969.Google Scholar
Shepard, R. N. & Carroll, J. D. Parametric representation of nonlinear data structures. In Krishnaian, P. R. (Eds.), Multivariate analysis. New York: Academic Press. 1966, 561–592Google Scholar
Sherman, C. R. & 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
Spence, I. Multidimensional scaling: An empirical and theoretical investigation. Unpublished doctoral dissertation, University of Toronto, September, 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(2), 122–126CrossRefGoogle 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 multidimensional scaling. Behavioral Science, 1968, 13, 343–344CrossRefGoogle Scholar
Young, F. W. Polynomial conjoint analysis of similarities: a model for constructing polynomial conjoint measurement algorithms. Chapel Hill, N. C.: L. L. Thurstone Psychometric Laboratory Report No. 74, 1969. (a)Google Scholar
Young, F. W. Polynomial conjoint analysis of similarities: definitions for a specific algorithm. Chapel Hill, N. C.: L. L. Thurstone Psychometric Laboratory Report No. 76, 1969. (b)Google Scholar
Young, F. W. Polynomial conjoint analysis of similarities: operations of a specific algorithm. Chapel Hill, N. C.:L. L. Thurstone Psychometric Laboratory Report No. 77, 1969. (c)Google Scholar
Young, F. W. Nonmetric multidimensional scaling: recovery of metric information. Psychometrika, 1970, 35, 455–473CrossRefGoogle Scholar