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Maximum Likelihood Estimation in Multidimensional Scaling

Published online by Cambridge University Press:  01 January 2025

J. O. Ramsay
J. O. Ramsay*
Affiliation:
McGill University
*
Requests for reprints should be sent to Professor J. O. Ramsay, Department of Psychology, McGill University, Montreal, Quebec, Canada H3A 1B1
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Abstract

A variety of distributional assumptions for dissimilarity judgments are considered, with the lognormal distribution being favored for most situations. An implicit equation is discussed for the maximum likelihood estimation of the configuration with or without individual weighting of dimensions. A technique for solving this equation is described and a number of examples offered to indicate its performance in practice. The estimation of a power transformation of dissimilarity is also considered. A number of likelihood ratio hypothesis tests are discussed and a small Monte Carlo experiment described to illustrate the behavior of the test of dimensionality in small samples.

Keywords

lognormalhypothesis testingimplicit equation

Information

Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 2 , June 1977 , pp. 241 - 266
Copyright
Copyright © 1977 The Psychometric Society

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Footnotes

The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.

References

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, 283–319.CrossRefGoogle Scholar
Guttman, L. A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 1968, 33, 469–506.CrossRefGoogle Scholar
Jöreskog, K.G., & Goldberger, A.S. Factor analysis by generalized least squares. Psychometrika, 1972, 37, 243–260.CrossRefGoogle Scholar
Kruskal, J.B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964, 28, 1–27.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
Marquardt, D.W. An algorithm for least squares estimation of non-linear parameters. SIAM Journal of Applied Mathematics, 1963, 11, 431–441.CrossRefGoogle Scholar
Novick, M.R., & Jackson, P.H. Statistical methods for educational and psychological research, 1974, New York: McGraw-Hill.Google Scholar
Ramsay, J.O. Economical method of analyzing perceived color differences. Journal of the Optical Society of America, 1968, 58, 19–22.CrossRefGoogle ScholarPubMed
Ramsay, J.O. Some statistical considerations in multidimensional scaling. Psychometrika, 1969, 34, 167–182.CrossRefGoogle Scholar
Ramsay, J.O. Solving implicit equations in psychometric data analysis. Psychometrika, 1975, 40, 337–360.CrossRefGoogle Scholar
Roskam, E.E., & Lingoes, J.C. MINISSA-I: A Fortran IV(G) program for the smallest space analysis of square symmetric matrices. Behavioral Sciences, 1970, 15, 204–205.Google Scholar
Shepard, R.N. Representation of structure in similarity data: problems and prospects. Psychometrika, 1974, 39, 373–424.CrossRefGoogle Scholar
Stevens, S.S. On the psychophysical law. Psychological Review, 1957, 64, 153–181.CrossRefGoogle ScholarPubMed
Torgerson, W.S. Theory and Methods of scaling, 1958, New York: Wiley.Google Scholar
Wish, M., & Carroll, J.D. Applications of individual differences scaling to studies of human perception and judgment. In Carterette, E.C. & Friedman, M.P. (Eds.), Handbook of Perception. Volume II: Psychophysical Judgment and Measurement, 1974, New York: Academic Press.Google Scholar