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Piecewise Method of Reciprocal Averages for Dual Scaling of Multiple-Choice Data

Published online by Cambridge University Press:  01 January 2025

Shizuhiko Nishisato  and
Wen-Jenn Sheu
Shizuhiko Nishisato*
Affiliation:
The Ontario Institute for Studies in Education The University of Toronto
Wen-Jenn Sheu
Affiliation:
The Ontario Institute for Studies in Education The University of Toronto
*
Requests for reprints should be sent to S. Nishisato, The Ontario Institute for Studies in Education, 252 Bloor Street West, Toronto, Ontario, Canada MSS 1V6.
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Abstract

The proposed method handles the classical method of reciprocal averages (MRA) in a piecewise (item-by-item) mode, whereby one can deal with smaller matrices and attain faster convergence to a solution than the MRA. A new concept “the principle of constant proportionality” is introduced to provide an interesting interpretation for scaling multiple-choice data á la Guttman. A small example is presented for discussion of the technique.

Keywords

dual scalingGuttman weightingmultiple-choice dataprinciple of constant proportionalityreciprocal averaging algorithm
Type
Original Paper
Information
Psychometrika , Volume 45 , Issue 4 , December 1980 , pp. 467 - 478
Copyright
Copyright © 1980 The Psychometric Society

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Footnotes

This study was supported by the Natural Sciences and Engineering Research Council Canada Grant (No. A4581) to S. Nishisato. The authors are indebted to reviewers for valuable comments.

References

Reference Notes

Bock, R. D. Methods and applications of optimal scaling, 1960, Chapel Hill, NC: The University of North Carolina Psychometric Laboratory.Google Scholar
de Leeuw, J. Canonical analysis of categorical data, 1973, The Netherlands: University of Leiden.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

References

Fisher, R. A. The precision of discriminant functions. Annals of Eugenics, 1940, 10, 422–429.CrossRefGoogle Scholar
Fox, L. An introduction to numerical linear algebra, 1964, Oxford: Oxford University Press.Google Scholar
Gabriel, K. R. The biplot graphical display of matrices with applications to principal component analysis. Biometrika, 1971, 58, 453–467.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. The principal components of scale analysis. In Stouffer, S. A., Guttman, L., Suchman, E. A., Lazarsfeld, P. F., Star, S. A., Clausen, J. A. (Eds.), Measurement and prediction, 1950, Princeton: Princeton University Press.Google Scholar
Hill, M. O. Correspondence analysis: A neglected multivariate method. Applied Statistics, 1974, 23, 340–354.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
Horst, P. Relations among m sets of measures. Psychometrika, 1961, 26, 129–149.CrossRefGoogle Scholar
Lingoes, J. C. Simultaneous linear regression: An IBM 7090 program for analyzing metric/nonmetric or linear/nonlinear data. Behavioral Science, 1964, 9, 87–88.Google Scholar
Lord, F. M. Some relations between Guttman's principal components of scale analysis and other psychometric theory. Psychometrika, 1958, 23, 291–296.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, 189–223.CrossRefGoogle Scholar
McDonald, R. P. A unified treatment of the weighting problem. Psychometrika, 1968, 33, 351–381.CrossRefGoogle ScholarPubMed
McDonald, R. P., Torii, Y., & Nishisato, S. Some results on proper eigenvalues and eigenvectors with applications to scaling. Psychometrika, 1979, 44, 211–227.CrossRefGoogle Scholar
Mosier, C. I. Machine methods in scaling by reciprocal averages. Endicath, New York: International Business Corporation. 1946, 35–39.Google Scholar
Nishisato, S. Dual (optimal) scaling and its history. Mathematical Sciences, 1979, 190, 76–83 (in Japanese)Google Scholar
Nishisato, S. Analysis of categorical data: Dual scaling and its applications, 1980, Toronto: University of Toronto Press.CrossRefGoogle Scholar
Richardson, M., & Kuder, G. F. Making a rating scale that measures. Personnel Journal, 1933, 12, 36–40.Google Scholar