Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-11T09:17:23.634Z Has data issue: false hasContentIssue false
  • English
  • Français

Regression with Qualitative and Quantitative Variables: An Alternating Least Squares Method with Optimal Scaling Features

Published online by Cambridge University Press:  01 January 2025

Forrest W. Young ,
Jan de Leeuw  and
Yoshio Takane
Forrest W. Young*
Affiliation:
University of North Carolina
Jan de Leeuw
Affiliation:
Rijksuniversiteit Te Leiden
Yoshio Takane
Affiliation:
University of North Carolina
*
Requests for reprints should be sent to Forrest W. Young, Psychometric Laboratory, University of North Carolina, Davie Hall 013 A, Chapel Hill, North Carolina 27514.
Get access Rights & Permissions [Opens in a new window]

Abstract

A method is discussed which extends canonical regression analysis to the situation where the variables may be measured at a variety of levels (nominal, ordinal, or interval), and where they may be either continuous or discrete. There is no restriction on the mix of measurement characteristics (i.e., some variables may be discrete-ordinal, others continuous-nominal, and yet others discrete-interval). The method, which is purely descriptive, scales the observations on each variable, within the restriction imposed by the variable's measurement characteristics, so that the canonical correlation is maximal. The alternating least squares algorithm is discussed. Several examples are presented. It is concluded that the method is very robust. Inferential aspects of the method are not discussed.

Keywords

linear modelcanonical analysismeasurementmultivariate datasurvey datasuccessive block algorithm
Type
Original Paper
Information
Psychometrika , Volume 41 , Issue 4 , December 1976 , pp. 505 - 529
Copyright
Copyright © 1976 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 research was partially supported by grants MH10006 and MH26504 from the National Institute of Mental Health to the Psychometric Laboratory of the University of North Carolina. Portions of the research reported here were presented to the spring meeting of the Psychometric Society, 1975. We wish to thank John B. Carroll and Elliot M. Cramer for their critical evaluations of an earlier draft of this report, and Jack Hoadley and John B. Carroll for letting us use their data. Copies of the paper may be obtained from the first author.

Jan de Leeuw is currently at Datatheorie, Central Rekeninstituut, Wassenaarseweg 80, Leiden, The Netherlands. Yoshio Takane can be reached at the Department of Psychology, University of Tokyo, Tokyo, Japan.

References

Reference Notes

Carroll, J. Douglas. Personal communication, 1976.Google Scholar
de Leeuw, J. Canonical analysis of categorical data, 1973, The Netherlands: University of Leiden.Google Scholar
de Leeuw, J. The linear nonmetric model, 1969, Leiden, The Netherlands: University of Leiden, Data Theory.Google Scholar
de Leeuw, J. Nonmetric discriminant analysis, 1968, Leiden, The Netherlands: University of Leiden, Data Theory.Google Scholar
de Leeuw, J. Normalized cone regression, 1975, The Netherlands: University of Leiden, Data Theory.Google Scholar
Hoadley, J. Spatial analysis of Senate voting patterns. Unpublished master's thesis, University of North Carolina, 1974.Google Scholar
Srinivasan, V. Linear programming computational procedures for ordinal regression. Unpublished manuscript, Graduate School of Business, Stanford University, 1974.Google Scholar

References

Carroll, J. D. Individual differences and multidimensional scaling. In Shepard, R. N., Romney, A. K., & Nerlove, S. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences, 1972, New York: Seminar Press.Google Scholar
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, 283–319.CrossRefGoogle Scholar
Cliff, N. & Young, F. W. On the relation between unidimensional judgments and multidimensional scaling. Organizational Behavior and Human Performance, 1968, 3, 207–208.CrossRefGoogle Scholar
de Leeuw, J., Young, F. W., & Takane, Y. Additive structure in qualitative data: An alternating least squares method with optimal scaling features. Psychometrika, 1976, 471–503.CrossRefGoogle Scholar
Fisher, R. Statistical methods for research workers, 10th ed., Edinburgh: Oliver and Boyd, 1938.Google Scholar
Hayashi, C. On the quantification of qualitative data from the mathematico: Statistical point of view. Annals of the Institute of Statistical Mathematics, 1950, 2, 35–47.CrossRefGoogle Scholar
Hotelling, H. The most predictable criterion. Journal of Educational Psychology, 1935, 26, 139–142.CrossRefGoogle Scholar
Lingoes, J. C. The Guttman-Lingoes nonmetric program series, 1973, Ann Arbor, Michigan: Mathesis Press.Google Scholar
Shepard, R. N. & Kruskal, J. B. A nonmetric variety of linear factor analysis. Psychometrika, 1974, 39, 123–157.Google Scholar
Takane, Y., Young, F. W., & de Leeuw, J. Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, in press.Google Scholar
Wold, H. & Lyttkens, E. Nonlinear iterative partial least squares (NIPALS) estimation procedures (group report). Bulletin of the International Statistical Institute, 1969, 43, 29–51.Google Scholar
Young, F. W. A model for polynomial conjoint analysis algorithms. In Shepard, R. N., Romney, A. K., & Nerlove, S. (Eds.), Multidimensional scaling, Vol. 1, 1972, New York: Seminar Press.Google Scholar
Young, F. W. Methods for describing ordinal data with cardinal models. Journal of Mathematical Psychology, 1975, 12, 416–436.CrossRefGoogle Scholar