Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-07T15:14:09.095Z Has data issue: false hasContentIssue false
  • English
  • Français

Alternative Weights and Invariant Parameters in Optimal Scaling

Published online by Cambridge University Press:  01 January 2025

Roderick P. McDonald
Roderick P. McDonald*
Affiliation:
Macquarie University
*
Requests for reprints should be addressed to Roderick P. McDonald, School of Education, Macquarie University, North Ryde, N.S.W. 2113, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Under conditions that are commonly satisfied in optimal scaling problems, arbitrary sets of optimal weights can be obtained by choices of generalized inverse procedures. A simple relationship holds between these and the corresponding invariant item scores. The case of optimal scaling originally treated by Guttman [1941] yields a restricted form of multicategory factor analysis. It is suggested that the invariant parameters of optimal scaling should be interpreted, according to the principles of latent trait theory, rather than the arbitrary weights.

Keywords

optimal scalingeignevalueseigenvectors
Type
Original Paper
Information
Psychometrika , Volume 48 , Issue 3 , September 1983 , pp. 377 - 391
Copyright
Copyright © 1983 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.)

Footnotes

This paper benefits from a number of suggestions and comments made by Professors M. J. R. Healy, H. Goldstein, and S. Nishisato, and by Mr. C. Fraser, to whom grateful acknowledgments are due. The author is solely responsible for the final form of the paper, including of course such errors as may remain in it.

This research was partly supported by Grant No. A6346 from the Natural Sciences and Engineering Research Council of Canada.

References

Benzécri, J. P. et al. L'Analyse des données, Paris: Dunod, 1973.Google Scholar
Burt, C. The factorial analysis of qualitative data. British Journal of Psychology (Statistical Section), 1950, 3, 166–185.CrossRefGoogle Scholar
de Leeuw, J. Generalized eigenvalue problems with positive semi-definite matrices. Psychometrika, 1982, 47, 87–93.CrossRefGoogle Scholar
Graybill, F. A. Matrix algebra with statistical applications, Calif.: Wadsworth, 1969.Google Scholar
Guttman, L. The quantification of a class of attributes. In Horst, P. (Ed.) The prediction of personal adjustment. Social Science Research Council, 1941.Google Scholar
Guttman, L. The principal components of scale analysis. In Stouffer, S. S. et al (Eds.), Measurement and Prediction, Princeton: University Press, 1950.Google Scholar
Hayashi, C., Higuchi, I., & Kamazawa, T. Joho shori to tokei suri [Information processing and statistical mathematics], Chapter 6 by Hayashi, C., Tokyo: Sangyo Tosho Press, 1970.Google Scholar
Healy, M. J. R. & Goldstein, H. An approach to the scaling of categorical attributes. Biometrika, 1976, 63, 219–229.CrossRefGoogle Scholar
McDonald, R. P. Nonlinear factor analysis. Psychometric Monograph No. 15, 1977a.Google Scholar
McDonald, R. P. Numerical methods for polynomial models in nonlinear factor analysis. Psychometrika, 1967, 32, 77–112.CrossRefGoogle Scholar
McDonald, R. P. Factor interaction in nonlinear factor analysis. British Journal of Mathematical and Statistical Psychology, 1967, 20, 209–215.CrossRefGoogle ScholarPubMed
McDonald, R. P. A unified treatment of the weighting problem. Psychometrika, 1968, 33, 351–381.CrossRefGoogle ScholarPubMed
McDonald, R. P. The common factor analysis of multicategory data. British Journal of Mathematical and Statistical Psychology, 1969, 22, 165–175.CrossRefGoogle Scholar
McDonald, R. P. Three common factor models for groups of variables. Psychometrika, 1970, 35, 111–128.CrossRefGoogle Scholar
McDonald, R. P. The simultaneous estimation of factor loadings and scores. British Journal of Mathematical and Statistical Psychology, 1979, 32, 212–228.CrossRefGoogle Scholar
McDonald, R. P. A simple comprehensive model for the analysis of covariance structures: some remarks on applications. British Journal of Mathematical and Statistical Psychology, 1980, 33, 161–183.CrossRefGoogle Scholar
McDonald, R. P. The dimensionality of tests and items. British Journal of Mathematical and Statistical Psychology, 1981, 34, 100–117.CrossRefGoogle Scholar
McDonald, R. P. Fitting latent trait models. In Spearritt (Ed.) The improvement of measurement in psychology and education, Australian Council for Educational Research, 1982.Google Scholar
McDonald, R. P. Exploratory and confirmatory nonlinear factor analysis. Lord Festschrift (in press).Google Scholar
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
Nishisato, S. Analysis of categorical data: dual scaling and its applications, Toronto: University of Toronto Press, 1980.CrossRefGoogle Scholar
Rao, C. R. & Mitra, B. K. Generalized inverse of matrices and its applications, New York: Wiley, 1971.Google Scholar
Shine, L. C. A note on McDonald's generalization of principal components analysis. Psychometrika, 1972, 37, 99–101.CrossRefGoogle Scholar
Torii, Y. Some generalizations of optimal scaling. Unpublished Ph.D. thesis. University of Toronto, 1977.Google Scholar
Whittle, P. On principal components and least-square methods of factor analysis. Skandinavisk Aktuarietidskrift, 1952, 35, 223–239.Google Scholar