Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T04:05:50.044Z Has data issue: false hasContentIssue false
  • English
  • Français

A Method of Optimal Scaling for Multivariate Ordinal Data and its Extensions

Published online by Cambridge University Press:  01 January 2025

Takayuki Saito and
Tatsuo Otsu
Takayuki Saito*
Affiliation:
Department of Behavioral Science, Hokkaido University
Tatsuo Otsu
Affiliation:
Fuyo Data Processing & Systems Development, Ltd.
*
Requests for reprints should be sent to Takayuki Saito, Department of Behavioral Science, Hokkaido University, Bungakubu, Kita 10 Nishi 7, Sapporo 060, JAPAN.
Get accessRights & Permissions [Opens in a new window]

Abstract

This paper develops a method of optimal scaling for multivariate ordinal data, in the framework of a generalized principal component analysis. This method yields a multidimensional configuration of items, a unidimensional scale of category weights for each item and, optionally, a multidimensional configuration of subjects. The computation is performed by alternately solving an eigenvalue problem and executing a quasi-Newton projection method. The algorithm is extended for analysis of data with mixed measurement levels or for analysis with a combined weighting of items. Numerical examples and simulations are provided. The algorithm is discussed and compared with some related methods.

Keywords

categorical dataOSMODprincipal component analysisquasi-Newton projection method

Information

Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 1 , March 1988, pp. 5 - 25
Copyright
Copyright © 1988 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

Earlier results of this research appeared in Saito and Otsu (1983). The authors would like to acknowledge the helpful comments and encouragement of the editor.

References

Bellman, R. (1970). Introduction to matrix analysis 2nd ed.,, New York: McGraw-Hill.Google Scholar
Bradley, R. A., Katti, S. K., Coons, I. J. (1962). Optimal scaling for ordered categories. Psychometrika, 27, 355–374.CrossRefGoogle Scholar
Clint, M., Jennings, A. (1970). The evaluation of eigenvalues and eigenvectors of real symmetric matrices by simultaneous iteration. The Computer Journal, 13, 76–80.CrossRefGoogle Scholar
de Leeuw, J., van Rijckevorsel, J. (1980). HOMALS & PRINCALS: Some generalizations of principal component analysis. In Diday, E., Lebart, L., Pagès, J. P., Tomasonne, R. (Eds.), Data Analysis and Informatics, Amsterdam: North Holland.Google Scholar
Gill, P. E., Murray, W. (1974). Numerical methods for constrained optimization, London: Academic Press.Google Scholar
Goldstein, H. (in press). The choice of constraints in correspondence analysis. Psychometrika.Google Scholar
Greenacre, M. J. (1984). Theory and applications of correspondence analysis, London: Academic Press.Google Scholar
Guttman, L. (1941). The quantification of a class of attributes: A theory and method of scale construction. In Horst, P. (Eds.), The prediction of personal adjustment (pp. 319–348). New York: Social Science Research Council.Google Scholar
Hayashi, C. (1952). On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view. Analysis of the Institute of Statistical Mathematics, 3, 69–98.CrossRefGoogle Scholar
Healy, M. J. R., Goldstein, H. (1976). An approach to the scaling of categorized attributes. Biometrika, 63, 219–229.CrossRefGoogle Scholar
Kruskal, J. B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.CrossRefGoogle Scholar
Kruskal, J. B., Shepard, R. N. (1974). A nonmetric variety of linear factor analysis. Psychometrika, 39, 123–157.CrossRefGoogle Scholar
Lingoes, J. C., Schönemann, P. H. (1974). Alternative measure fit for Schönemann-Carroll matrix fitting algorithm. Psychometrika, 39, 423–427.CrossRefGoogle Scholar
Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications, Toronto: University of Torronto Press.CrossRefGoogle Scholar
Nishisato, S., Arri, P. S. (1975). Nonlinear programming approach to optimal scaling of partially ordered categories. Psychometrika, 40, 525–548.CrossRefGoogle Scholar
Nishisato, S., Inukai, Y. (1972). Partially optimal scaling of items with ordered categories. Japanese Psychological Research, 14, 109–119.CrossRefGoogle Scholar
Saito, T., & Otsu, T. (1983). A method of optimal scaling for multivariate ordinal data. Hokkaido Behavioral Science Report, Series M, No. 5.Google Scholar
SAS Institute (1983). Technical Report, P-131, Cary, NC: Author.Google Scholar
Tanaka, Y. (1979). Optimal scaling for arbitrarily ordered categories. Annals of the Institute of Statistical Mathematics, 31, 115–124.CrossRefGoogle Scholar
Tanaka, Y., Kodake, K. (1980). Computational aspects of optimal scaling for ordered categories. Behaviormetrika, 7, 35–46.CrossRefGoogle Scholar
Tenenhaus, M., Young, F. W. (1985). An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis and other methods for quantifying categorical multivariate data. Psychometrika, 50, 91–119.CrossRefGoogle Scholar
Young, F. W., Takane, Y., de Leeuw, J. (1978). The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features. Psychometrika, 43, 279–281.CrossRefGoogle Scholar