Abstract
This paper presents an overview of an approach to the quantitative analysis of qualitative data with theoretical and methodological explanations of the two cornerstones of the approach, Alternating Least Squares and Optimal Scaling. Using these two principles, my colleagues and I have extended a variety of analysis procedures originally proposed for quantitative (interval or ratio) data to qualitative (nominal or ordinal) data, including additivity analysis and analysis of variance; multiple and canonical regression; principal components; common factor and three mode factor analysis; and multidimensional scaling. The approach has two advantages: (a) If a least squares procedure is known for analyzing quantitative data, it can be extended to qualitative data; and (b) the resulting algorithm will be convergent. Three completely worked through examples of the additivity analysis procedure and the steps involved in the regression procedures are presented.
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Tenenhaus, M. Principal components analysis of qualitative variables. Report No. 175/1981. Jouy-en-Josas, France, Centre d'Enseignement Superieur des Affaires, 1981.
Tenenhaus, M. Principal components analysis of qualitative variables. Les Cahiers de Recherche No. 175/1981. Jouy-en-Josas, France, CESA, 1981.
de Leeuw, J. A normalized cone regression approach to alternating least squares algorithms. Unpublished note, University of Leiden, 1977b.
de Leeuw, J., & van Rijkevorsel, J. How to use HOMALS 3. A program for principal components analysis of mixed data which uses the alternating least squares method. Unpublished mimeo, Leiden University, 1976.
Young, F. W., Null, C. H., & De Soete, G. The general Euclidean Model. 1981 (in preparation).
References
Benzecri, J. P.L'analyse des donnees—Tome II: Correspondances Dunod, Paris, 1973.
Benzecri, J. P., Histoire et Prehistoire de l'analyse des donnees; l'analyse des correspondence.Les Cahiers de l'Analyse des Donnees (Volume II), Paris, 1977.
Bock, R. D. Methods and applications of optimal scaling. Psychometric Laboratory Report #25, University of North Carolina, 1960.
Burt, C. The factorial analysis of qualitative data.British Journal of Psychology, Statistical Section, 1950,3, 166–185.
Burt, C. Scale analysis and factor analysis.British Journal of Statistical Psychology, 1953,6, 5–24.
Carroll, J. D., & Chang, J. J. Analysis of individual differences in multi-dimensional scaling via ann-way generalization of “Eckart-Young” decomposition.Psychometrika, 1970,35, 283–319.
Coombs, C. H.A theory of Data. New York: Wiley, 1964.
de Leeuw, J.Canonical analysis of categorical data. University of Leiden, The Netherlands, 1973.
de Leeuw, J. Normalized cone regression. Leiden, The Netherlands: University of Leiden, Data Theory, mimeographed paper, 1975.
de Leeuw, J. Correctness of Kruskal's algorithms for monotone regression with ties.Psychometrika, 1977a,42, 141–144.
de Leeuw, J., Young, F. W., & Takane, Y. Additive structure in qualitative data: An alternating least squares method with optimal scaling features.Psychometrika, 1976,41, 471–503.
Fisher, R.Statistical methods for research workers. (10th ed.) Edinburgh: Oliver and Boyd, 1938.
Gifi, A.Nonlinear multivariate analysis (preliminary version). University of Leiden, Data Theory Department. 1981.
Guttman, L. The quantification of a class of attributes: A theory and method of scale construction. In P. Horst (Ed.)The prediction of personal adjustment. New York: Social Science Research Council, 1941.
Guttman, L. A note on Sir Cyril Burt's “Factorial Analysis of Qualitative Data,”The British Journal of Statistical Psychology, 1953,7, 1–4.
Hageman, L. A., & Porsching, T. A. Aspects of nonlinear block successive over-relaxation.SIAM Journal of Numerical Analysis, 1975,12, 316–335.
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.
Horan, C. B. Multidimensional scaling: Combining observations when individuals have different perceptual structures.Psychometrika, 1969,34, 139–165.
Kruskal, J. B. Nonmetric multidimensional scaling.Psychometrika, 1964,29, 1–27, 115–129.
Kruskal, J. B. Analysis of factorial experiments by estimating monotone transformations of the data.Journal of the Royal Statistical Society, Series B, 1965,27, 251–263.
Kruskal, J. B., & Carroll, J. D. Geometric models and badness-of-fit functions. In P. R. Krishnaiah (Ed.),Multivariate analysis (Vol. 2). New York: Academic Press, 1969.
Mardia, K. V., Kent, J. T., & Bibby, J. M.Multivariate analysis. London: Academic Press, 1979.
Nishisato, S.Analysis of categorical data: Dual scaling and its applications. University of Toronto Press, 1980.
Roskam, E. E.Metric analysis of ordinal data in psychology. Voorschoten, Holland: VAM, 1968.
Saito, T.Quantification of categorical data by using the generalized variance. Soken Kiyo, Nippon UNIVAC Sogo Kenkyn-Sho, 61–80, 1973.
Sands, R., & Young, F. W. Component models for three-way data: An alternating least squares algorithm with optimal scaling features.Psychometrika, 1980,45, 39–67.
Saporta, G. Liaisons entre plusieurs ensembles de variables et codages de donnes qualitatives. These de Doctorat de 3eme cycle, Paris, 1975.
Takane, Y., Young, F. W., & de Leeuw, J. Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features.Psychometrika, 1977,42, 7–67.
Takane, Y., Young, F. W., & de Leeuw, J. An individual differences additive model: An alternating least squares method with optimal scaling features.Psychometrika, 1980,45, 183–209.
Torgerson, W. S.Theory and methods of scaling. New York: Wiley, 1958.
Wold, H., & Lyttkens, E. Nonlinear iterative partial least squares (NIPALS) estimation procedures.Bulletin ISI, 1969,43, 29–47.
Young, F. W. A model for polynomial conjoint analysis algorithms. In R. N. Shepard, A. K. Romney, & S. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences. New York: Academic Press, 1972.
Young, F. W. Methods for describing ordinal data with cardinal models.Journal of Mathematical Psychology, 1975a,12, 416–436.
Young, F. W. An asymmetric Euclidian model for multi-process asymmetric data. U.S.-Japan Seminar on Multidimensional Scaling, 1975b.
Young, F. W., de Leeuw, J., & Takane, Y. Multiple (and canonical) regression with a mix of qualitative and quantitative variables: An alternating least squares method with optimal scaling features.Psychometrika, 1976,41, 505–529.
Young, F. W., & Lewyckyj, R.ALSCAL Users Guide. Carrboro, NC; Data Analysis and Theory, 1979.
Young, F. W., & Lewyckyj, R. The ALSCAL procedure. In SAS Supplemental Library User's Guide, Reinhardt, P. (Ed.). SAS Institute, Raleigh, NC, 1980.
Young, F. W., & Null, C. H. Multidimensional scaling of nominal data: The recovery of metric information with ALSCAL.Psychometrika, 1978,43, 367–379.
Young, F. W., Takane, Y., & de Leeuw, J. The principal components of mixed measurement level data; An alternating least squares method with optimal scaling features.Psychometrika, 1978,43, 279–282.
Young, F. W., Takane, Y., & Lewyckyj, R. ALSCAL: A nonmetric multidimensional scaling program with several individual differences options.Behavioral Research Methods and Instrumentation, 1978,10, 451–453.
Young, F. W., Takane, Y., & Lewyckyj, R. ALSCAL: A multidimensional scaling package with several individual differences options.American Statistician, 1980,34, 117–118.
Yule, G. U.An introduction to the theory of statistics. London: Griffin, 1910.
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Young, F.W. Quantitative analysis of qualitative data. Psychometrika 46, 357–388 (1981). https://doi.org/10.1007/BF02293796
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DOI: https://doi.org/10.1007/BF02293796