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Nonmetric Individual Differences Multidimensional Scaling: An Alternating Least Squares Method with Optimal Scaling Features

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane ,
Forrest W. Young  and
Jan de Leeuw
Yoshio Takane
Affiliation:
University of North Carolina
Forrest W. Young*
Affiliation:
University of North Carolina
Jan de Leeuw
Affiliation:
University of Leiden
*
Requests for reprints should be sent to Forrest W. Young, Psychometric Lab, University of North Carolina, Davie Hall 013 A, Chapel Hill, North Carolina 27514.
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Abstract

A new procedure is discussed which fits either the weighted or simple Euclidian model to data that may (a) be defined at either the nominal, ordinal, interval or ratio levels of measurement; (b) have missing observations; (c) be symmetric or asymmetric; (d) be conditional or unconditional; (e) be replicated or unreplicated; and (f) be continuous or discrete. Various special cases of the procedure include the most commonly used individual differences multidimensional scaling models, the familiar nonmetric multidimensional scaling model, and several other previously undiscussed variants.

The procedure optimizes the fit of the model directly to the data (not to scalar products determined from the data) by an alternating least squares procedure which is convergent, very quick, and relatively free from local minimum problems.

The procedure is evaluated via both Monte Carlo and empirical data. It is found to be robust in the face of measurement error, capable of recovering the true underlying configuration in the Monte Carlo situation, and capable of obtaining structures equivalent to those obtained by other less general procedures in the empirical situation.

Keywords

Euclidian modelINDSCALmeasurementsimilaritiesdata analysissimilarities dataquantificationsuccessive block algorithm
Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 1 , March 1977 , pp. 7 - 67
Copyright
Copyright © 1977 The Psychometric Society

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Footnotes

This project was supported in part by Research Grant No. MH10006 and Research Grant No. MH26504, awarded by the National Institute of Mental Health, DHEW. We wish to thank Robert F. Baker, J. Douglas Carroll, Joseph Kruskal, and Amnon Rapoport for comments on an earlier draft of this paper. Portions of the research reported here were presented to the spring meeting of the Psychometric Society, 1975. ALSCAL, a program to perform the computations discussed in this paper, may be obtained from any of the authors.

Jan de Leeuw is currently at Datatheorie, Central Rekeninstituut, Wassenaarseweg 80, Leiden, The Netherlands. Yoshio Takane can be reached at the Department of Psychology, University of Tokyo, Tokyo, Japan.

References

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