Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-07T18:06:19.219Z Has data issue: false hasContentIssue false
  • English
  • Français

A Quasi-Nonmetric Method for Multidimensional Scaling Via an Extended Euclidean Model

Published online by Cambridge University Press:  01 January 2025

Suzanne Winsberg  and
J. Douglas Carroll
Suzanne Winsberg
Affiliation:
IRCAM
J. Douglas Carroll*
Affiliation:
AT&T Bell Laboratories
*
Requests for reprints should be sent to J. Douglas Carroll, Bell Laboratories 2C-553,600 Mountain Ave., Murray Hill, NJ 07974.
Get access Rights & Permissions [Opens in a new window]

Abstract

An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the “standard” (Two-Way) MDS model. In this Extended Two-Way Euclidean model the n stimuli (or other objects) are assumed to be characterized by coordinates on R common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between object i and object j then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity, sj, can be thought of as the sum of squares of coordinates on those dimensions specific to object i, all of which have nonzero coordinates only for object i. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)

We further assume that δij =F(dij) +eij where δij is the proximity value (e.g., similarity or dissimilarity) of objects i and j, dij is the extended Euclidean distance defined above, while eij is an error term assumed i.i.d. N(0, σ2). F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition to R common) dimensions.

This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.

Keywords

multidimensional scalingmonotone splinespecific dimensionsmaximum likelihood estimation
Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 2 , June 1989 , pp. 217 - 229
Copyright
Copyright © 1989 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.)

References

Aikake, H. (1974). A new look at the statistical model. IEEE Transactions on Automatic Control, 48, 575–595.Google Scholar
Bentler, P. M., Weeks, D. G. (1978). Restricted multidimensional scaling methods. Journal of Mathematical Psychology, 17, 138–151.CrossRefGoogle Scholar
Carroll, J. D. (1983, July). A “common dimensions” model for multidimensional scaling of proximity data ans its implications for nonmetric analysis. Paper presented at Workshop on Nonmetric Data Analysis, Institut de Statistique de Paris, Universite Peerre et Marie Curie, France.Google Scholar
Carroll, J. D., Winsberg, S. (1986). Maximum likelihood procedures for metric and quasi-nonmetric fitting of an extended INDSCAL model assuming both common and specific dimensions. In de Leeuw, J., Heiser, W., Meulman, J., Critchley, F. (Eds.), Multidimensional Data Analysis [Abstract], Leiden: DWSO Press.Google Scholar
de Boor, C. (1978). A practical guide to splines, New York: Springer-Verlag.CrossRefGoogle Scholar
de Leeuw, J., Heiser, W. J. (1980). Multidimensional scaling with restrictions on the configuration. In Krishniah, P. R. (Eds.), Multivariate analysis, Vol. 5, Amsterdam: North Holland.Google Scholar
Kruskal, J. B. (1964). Multidimensional scaling by oplinuzing goodness of fit to a nonmetric hypothesis. Psychometrika, 49, 1–27.CrossRefGoogle Scholar
Kruskal, J. B., Wish, M. (1978). Multidimensional scaling, Beverly Hills: Sage.CrossRefGoogle Scholar
Ramsay, (1978). Multiscale I Manual.Google Scholar
Ramsay, (1982). Multiscale II Manual.Google Scholar
Schwarz, G. (1978). Estimating the dimensions of a model. The Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Takane, Y., Sergent, J. (1983). Multidimensional scaling models for reaction times and some different judgements. Psychometrika, 48, 329–424.CrossRefGoogle Scholar
Tversky, A., & Hutchinson, J. W. (1986). Nearest neighbor analysis of psychological spaces. Journal of Mathematical Psychology.Google Scholar
Weeks, D. G., Bentler, P. M. (1982). Restricted multidimensional scaling models for asymmetric proximities. Psychometrika, 47, 201–208.CrossRefGoogle Scholar
Weinberg, S.L., Carroll, J. D., & Cohen, H. S. (1984). Confidence regions for INDSCAL using the jackknife and bootstrap techniques. Psychometrika.CrossRefGoogle Scholar
Winsberg, S., Ramsay, J. O. (1980). Monotonic transformations to additivity using splines. Biometrika, 67, 669–674.CrossRefGoogle Scholar
Winsberg, S., Ramsay, J. O. (1981). Analysis of pairwise preference data using integrated B-splines. Psychometrika, 46, 171–186.CrossRefGoogle Scholar
Winsberg, S., Ramsay, J. O. (1984). Monotone spline transformations for dimension reduction. Psychometrika, 48, 575–595.CrossRefGoogle Scholar
Wish, M. (1971). Individual differences in perceptions and preferences among nations. In King, C. W., Tigert, D. (Eds.), Attitude research reaches new heights, Chicago: American Marketing Association.Google Scholar