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Nonmetric Maximum Likelihood Multidimensional Scaling from Directional Rankings of Similarities

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane  and
J. Douglas Carroll
Yoshio Takane*
Affiliation:
McGill University
J. Douglas Carroll
Affiliation:
Bell Laboratories
*
Requests for reprints should be sent to Yoshio Takane, Department of Psychology, McGill University, 1205 Docteur Penfield Ave., Montreal, Quebec H3A, 1B1, Canada.
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Abstract

A maximum likelihood estimation procedure is developed for multidimensional scaling when (dis)similarity measures are taken by ranking procedures such as the method of conditional rank orders or the method of triadic combinations. The central feature of these procedures may be termed directionality of ranking processes. That is, rank orderings are performed in a prescribed order by successive first choices. Those data have conventionally been analyzed by Shepard-Kruskal type of nonmetric multidimensional scaling procedures. We propose, as a more appropriate alternative, a maximum likelihood method specifically designed for this type of data. A broader perspective on the present approach is given, which encompasses a wide variety of experimental methods for collecting dissimilarity data including pair comparison methods (such as the method of tetrads) and the pick-M method of similarities. An example is given to illustrate various advantages of nonmetric maximum likelihood multidimensional scaling as a statistical method. At the moment the approach is limited to the case of one-mode two-way proximity data, but could be extended in a relatively straightforward way to two-mode two-way, two-mode three-way or even three-mode three-way data, under the assumption of such models as INDSCAL or the two or three-way unfolding models.

Keywords

rank ordersmodel selectionAICface perception
Type
Original Paper
Information
Psychometrika , Volume 46 , Issue 4 , December 1981 , pp. 389 - 405
Copyright
Copyright © 1981 The Psychometric Society

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Footnotes

The first author's work was supported partly by the Natural Sciences and Engineering Research Council of Canada, grant number A6394. Portions of this research were done while the first author was at Bell Laboratories. MAXSCAL-4.1, a program to perform the computations described in this paper can be obtained by writing to: Computing Information Service, Attention: Ms. Carole Scheiderman, Bell Laboratories, 600 Mountain Ave., Murray Hill, N.J. 07974. Thanks are due to Yukio Inukai, who generously let us use his stimuli in our experiment, and to Jim Ramsay for his helpful comments on an earlier draft of this paper. Confidence regions in Figures 2 and 3 were drawn by the program written by Jim Ramsay. We are also indebted to anonymous reviewers for their suggestions.

References

References Notes

Inukai, Y., Nakamura, K., Shinohara, M. A multidimensional scaling analysis of judged dissimilarities among schematic facial expressions. Bulletin of Industrial Product Research Institute, 1977, 81, 21–30 (in Japanese)Google Scholar
Takane, Y., and Carroll, J. D. On the robustness of AIC in the context of maximum likelihood multidimensional scaling. Manuscript in preparation.Google Scholar

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