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A Scalar Product Model for the Multidimensional Scaling of Choice

Published online by Cambridge University Press:  01 January 2025

Gordon G. Bechtel ,
Ledyard R Tucker  and
Wei-Ching Chang
Gordon G. Bechtel
Affiliation:
Oregon Research Institute
Ledyard R Tucker
Affiliation:
University of Illinois
Wei-Ching Chang
Affiliation:
University of Oregon and Oregon Research Institute
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Abstract

A multidimensional scaling analysis is presented for replicated layouts of pairwise choice responses. In most applications the replicates will represent individuals who respond to all pairs in some set of objects. The replicates and the objects are scaled in a joint space by means of an inner product model which assigns weights to each of the dimensions of the space. Least squares estimates of the replicates' and objects' coordinates, and of unscalability parameters, are obtained through a manipulation of the error sum of squares for fitting the model. The solution involves the reduction of a three-way least squares problem to two subproblems, one trivial and the other solvable by classical least squares matrix factorization. The analytic technique is illustrated with political preference data and is contrasted with multidimensional unfolding in the domain of preferential choice.

Keywords

Public PolicyScalar ProductStatistical TheoryJoint SpaceMultidimensional Scaling
Type
Original Paper
Information
Psychometrika , Volume 36 , Issue 4 , December 1971 , pp. 369 - 388
Copyright
Copyright © 1971 The Psychometric Society

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Footnotes

*

The present work was initiated at Oregon Research Institute under National Institute of Mental Health Grant MH 12972. It was reformulated and completed while the first author was a Visiting Research Fellow at Educational Testing Service.

†

Presently at the Department of Mathematics, University of Toronto.

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