Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-30T14:54:39.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Scaling of Paired Comparison and Rank Order Data: An Alternative to Guttman's Formulation

Published online by Cambridge University Press:  01 January 2025

Shizuhiko Nishisato
Shizuhiko Nishisato*
Affiliation:
The Ontario Institute for Studies in Education
*
Requests for reprints should be sent to Shizuhiko Nishisato, Department of Measurement, Evaluation and Computer Applications, The Ontario Institute for Studies in Education, 252 Bloor Street West, Toronto, Ontario, Canada, M5S 1V6.
Get access Rights & Permissions [Opens in a new window]

Abstract

A formulation, which is different from Guttman's is presented. The two formulations are both called the optimal scaling approach, and are proven to provide identical scale values. The proposed formulation has at least two advantages over Guttman's. Namely, (i) the former serves to clarify close relations of the optimal scaling approach to those of Slater and the vector model of preferential choice, and (ii) in addition to the stimulus scale values, it provides scores for the subjects, which indicate the degrees of response consistency (transitivity), relative to the optimum solution. The method is assumption-free and capable of multidimensional analysis.

Keywords

Guttman's modelSlater's modelthe vector model of preferential choiceranking and individual differences

Information

Type
Original Paper
Information
Psychometrika , Volume 43 , Issue 2 , June 1978 , pp. 263 - 271
Copyright
Copyright © 1978 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.)

Article purchase

Temporarily unavailable

Footnotes

This study was partly supported by the National Research Council Grant (No. A4581) to S. Nishisato. The author is indebted to Dr. Bert F. Green, Jr., Mr. Tomoichi Ishizuka, and anonymous reviewers for their valuable comments on an earlier draft.

References

Reference Notes

Carroll, J. D. & Chang, J. J. Nonparametric multidimensional analysis of paired comparisons data. Paper presented at the joint meeting of the Psychometric and Psychonomic Societies, Niagara Falls, October 1964.Google Scholar
Carroll, J. D. & Chang, J. J. How to use MDPREF, a computer program for multidimensional analysis of preference data. Unpublished report, Bell Laboratories, 1968.Google Scholar
Nishisato, S. Optimal scaling as applied to different forms of data, 1976, Toronto: Department of Measurement and Evaluation, The Ontario Institute for Studies in Education.Google Scholar
de Leeuw, J. Canonical analysis of categorical data, 1973, The Netherlands: University of Leiden.Google Scholar

References

Bechtel, G. G., Tucker, L. R., & Chang, W. A scalar product model for the multidimensional scaling of choice. Psychometrika, 1971, 36, 369–387.CrossRefGoogle Scholar
Carroll, J. D. Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K. & Nerlove, S. B. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences. (Vol. I), 1972, New York: Seminar Press.Google Scholar
Fisher, R. A. The precision of discriminant functions. Annals of Eugenics, 1940, 10, 422–429.CrossRefGoogle Scholar
Gower, J. C. Some distance properties of latent root and vector methods in multivariate analysis. Biometrika, 1966, 53, 325–338.CrossRefGoogle Scholar
Guttman, L. The quantification of a class of attributes: A theory and method of scale construction. In Horst, P. (Eds.), The prediction of personal adjustment. New York: Social Science Research Council. 1941, 319–348.Google Scholar
Guttman, L. An approach for quantifying paired comparisons and rank order. Annals of Mathematical Statistics, 1946, 17, 144–163.CrossRefGoogle Scholar
Hayashi, C. Multidimensional quantification of the data obtained by the method of paired comparison. Annals of the Institute of Statistical Mathematics, the Twentieth Anniversary Volume, 1964, 16, 231–245.CrossRefGoogle Scholar
Hayashi, C. Note on multidimensional quantification of data obtained by paired comparison. Annals of the Institute of Statistical Mathematics, 1967, 19, 363–365.CrossRefGoogle Scholar
Hirschfeld, H. O. A connection between correlation and contingency. Cambridge Philosophical Society Proceedings, 1935, 31, 520–524.CrossRefGoogle Scholar
Horst, P. Measuring complex attitudes. Journal of Social Psychology, 1935, 6, 369–374.CrossRefGoogle Scholar
Maung, K. Measurement of association in contingency table with special reference to the pigmentation of hair and eye colours of Scottish school children. Annals of Eugenics, 1941, 11, 189–123.CrossRefGoogle Scholar
Richardson, M., & Kuder, G. F. Making a rating scale that measures. Personnel Journal, 1933, 12, 36–40.Google Scholar
Slater, P. The analysis of personal preferences. The British Journal of Statistical Psychology, 1960, 13, 119–135.CrossRefGoogle Scholar
Tucker, L. R. Intra-individual and inter-individual multidimensionality. In Gulliksen, H. and Messick, S. (Eds.), Psychological scaling, 1960, New York: Wiley.Google Scholar