Regression with qualitative and quantitative variables: An alternating least squares method with optimal scaling features Authors
Received: 01 December 1975 Revised: 20 April 1976 DOI:
Cite this article as: Young, F.W., de Leeuw, J. & Takane, Y. Psychometrika (1976) 41: 505. doi:10.1007/BF02296972 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.
Key words linear model canonical analysis measurement multivariate data survey data successive block algorithm
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.
Yoshio Takane can be reached at the Department of Psychology, University of Tokyo, Tokyo, Japan.
Carroll, J. Douglas. Personal communication, 1976.
de Leeuw, J.
Canonical analysis of categorical data. Unpublished doctoral dissertation, University of Leiden, The Netherlands, 1973.
de Leeuw, J.
The linear nonmetric model (Research Note RN003-69). Leiden, The Netherlands: University of Leiden, Data Theory, 1969.
de Leeuw, J.
Nonmetric discriminant analysis (Research Note RN006-68). Leiden, The Netherlands: University of Leiden, Data Theory, 1968.
de Leeuw, J.
Normalized cone regression. Unpublished paper, University of Leiden, Data Theory, The Netherlands, 1975.
Spatial analysis of Senate voting patterns. Unpublished master's thesis, University of North Carolina, 1974.
Linear programming computational procedures for ordinal regression. Unpublished manuscript, Graduate School of Business, Stanford University, 1974. References
Carroll, J. D. Individual differences and multidimensional scaling. In R. N. Shepard, A. K. Romney, & S. Nerlove (Eds.),
Multidimensional scaling: Theory and applications in the behavioral sciences. Seminar Press, New York, 1972.
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.
Cliff, N. & Young, F. W. On the relation between unidimensional judgments and multidimensional scaling.
Organizational Behavior and Human Performance
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.
Statistical methods for research workers (10th ed.). Edinburgh: Oliver and Boyd, 1938.
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.
Hotelling, H. The most predictable criterion.
Journal of Educational Psychology, 1935, 26, 139–142.
Lingoes, J. C.
The Guttman-Lingoes nonmetric program series. Ann Arbor, Michigan: Mathesis Press, 1973.
Shepard, R. N. & Kruskal, J. B. A nonmetric variety of linear factor analysis.
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.
Wold, H. & Lyttkens, E. (Eds.) Nonlinear iterative partial least squares (NIPALS) estimation procedures (group report).
Bulletin of the International Statistical Institute, 1969, 43, 29–51.
Young, F. W. A model for polynomial conjoint analysis algorithms. In R. N. Shepard, A. K. Romney, & S. Nerlove (Eds.),
Multidimensional scaling, Vol. 1. New York: Seminar Press, 1972.
Young, F. W. Methods for describing ordinal data with cardinal models.
Journal of Mathematical Psychology
CrossRef Copyright information
© Psychometric Society 1976