Abstract
This paper describes a computational method for weighted euclidean distance scaling which combines aspects of an “analytic” solution with an approach using loss functions. We justify this new method by giving a simplified treatment of the algebraic properties of a transformed version of the weighted distance model. The new algorithm is much faster than INDSCAL yet less arbitrary than other “analytic” procedures. The procedure, which we call SUMSCAL (subjectivemetricscaling), gives essentially the same solutions as INDSCAL for two moderate-size data sets tested.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Reference notes
Bloxom, B. Individual differences in multidimensional scaling (Res. Bull. 68–45). Princeton, New Jersey: Educational Testing Service, 1968.
Carroll, J. D., & Chang, J. J.IDIOSCAL: A generalization of INDSCAL allowing IDIOsyncratic reference systems as well as an analytic approximation to INDSCAL. Paper presented at the Spring Meeting of the Psychometric Society, March 30–31, 1972, Princeton, New Jersey.
Cohen, H. S.Three-mode rotation to approximate INDSCAL structure. Paper presented at the annual meeting of the Psychometric Society, Palo Alto, California, March, 1974.
de Leeuw, J.An initial estimate for INDSCAL. Unpublished paper, Bell Telephone Laboratories, Murray Hill, June, 1974.
Harshman, R. A. Foundations of the PARAFAC procedure: Models and conditions for an “explanatory” multi-model factor analysis.UCLA Working Papers in Phonetics, 1970,16, 1–84.
Harshman, R. A. PARAFAC2; mathematical and technical notes.UCLA Working Papers in Phonetics, 1972,22, 30–44.
Heiser, W. J.Individual differences scaling (Report MT 001-75). Leiden, Holland; Psychology Department, University of Leiden, 1975.
Pruzansky, S.SINDSCAL: A Computer Program for Individual Differences in Multidimensional Scaling, Unpublished manuscript, Bell Laboratories, September 1975.
References
Bhimasankaram, P. Simultaneous reduction of several Hermitian forms.Sankhya, A, 1971,33, 417–422.
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, 283–320.
de Leeuw, J., & Heiser, W. J. Convergence of correction matrix algorithms for multidimensional scaling. In J. Lingoes (Ed.),Geometric representations of relational data. Ann Arbor: Mathesis Press, 1977.
Greenstadt, J. The determination of the characteristic roots of a matrix by the Jacobi method. In A. Ralston & H. Wilf (Eds.),Mathematical Methods for Digital Computers, (Vol. I) New York: Wiley, 1960, pp. 84–91.
Helm, C. E., A multidimensional ratio scaling analysis of perceived color relations.Journal of the Optical Society of America, 1964,54, 256–262.
Horan, C. B., Multidimensional scaling: Combining observations when individuals have different perceptual structures.Psychometrika, 1969,34, 139–165.
Jacobi, C. G. J., Über ein leichtes Verfahren, die in der Theorie der Säkularstörungen vorkommenden Gleichungen numerisch aufzulösen.J. Reine Agnew. Math, 1846,30, 51–95.
Jöreskog, K., A general approach to confirmatory maximum likelihood factor analysis.Psychometrika, 1969,34, 183–202.
Kruskal, J. B. More factors than subjects, tests and treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling.Psychometrika, 1976,41, 281–293.
Kruskal, J. B., & Carroll, J. D. Geometric Models and Badness-of-Fit Functions. In P. R. Krishnaiah (Ed.),Multivariate Analysis, (Vol. II) New York: Academic Press, 1969, pp. 639–671.
MacCallum, R. C. Transformation of a three-mode multidimensional scaling solution to INDSCAL form.Psychometrika, 1976,41, 385–400.
Schönemann, P. H. An algebraic solution for a class of subjective metrics models.Psychometrika, 1972,37, 441–451.
Takane, Y., Young, F., & De Leeuw, J. Non-metric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features.Psychometrika, 1977,42, 7–67.
Torgerson, W. S.Theory and methods of scaling, New York: Wiley, 1958.
Wish, M., & Carroll, J. D. Applications of Individual Differences Scaling to Studies of Human Perception and Judgement. In E. C. Carterette & M. P. Friedman (Eds.),Handbook of Perception, (Vol. II)Psychophysical Judgment and Measurement, New York: Academic Press, Inc., 1974, pp 449–491.
Author information
Authors and Affiliations
Additional information
Comments by J. Douglas Carroll and J. B. Kruskal have been very helpful in preparing this paper.
Rights and permissions
About this article
Cite this article
de Leeuw, J., Pruzansky, S. A new computational method to fit the weighted euclidean distance model. Psychometrika 43, 479–490 (1978). https://doi.org/10.1007/BF02293809
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02293809