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An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis and other methods for quantifying categorical multivariate data. (English) Zbl 0585.62104
It is shown that all the four approaches, viz. reciprocal averages, analysis of variance, principal component analysis and canonical analysis are equivalent for multiple correspondence analysis for scaling of categorical data.
The most interesting part of the paper, however, is the introduction of the ’duality diagram’ and the role it plays in the understanding of the geometry of multiple correspondence analysis and its synthesis. The duality diagram has indeed played a pivotal role in the development of this area of data analysis in the French literature.
Reviewer: J.S.Murty

62H25 Factor analysis and principal components; correspondence analysis
62P15 Applications of statistics to psychology
62-07 Data analysis (statistics) (MSC2010)
Full Text: DOI
[1] Baker, F. B. (1960). Univac scientific computer program for scaling of psychological inventories by the method of reciprocal averages CPA 22.Behavioral Science, 5, 268–269.
[2] Benzécri, J. P. (1973).L’analyse des données: T. 2, l’analyse des correspondances [Data Analysis: T. 2, Correspondence analysis]. Paris: Dunod. · Zbl 0297.62039
[3] Benzécri, J. P. (1977a). Historie et préhistoire de l’analyse des données: l’analyse des correspondances [History and prehistory of data analysis: correspondence analysis].Les Cahiers de l’Analyse des Données, 2, 9–53.
[4] Benzécri, J. P. (1977b). Sur l’analyse des tableaux binaires associés à une correspondance multiple [The analysis of boolean tables associated with a multiple correspondence].Les Cahiers de l’Analyse des Données, 2, 55–71.
[5] Bock, R. D. (1960).Methods and applications of optimal scaling (Rep. No. 25). Chapel Hill: University of North Carolina.
[6] Bouroche, J. M., Saporta, G., & Tenenhaus, M. (1975, August).Generalized canonical analysis of qualitative data. Paper presented at the U.S.-Japan Seminar on Theory, Methods and Applications of Multidimensional Scaling and Related Techniques, San Diego: University of California.
[7] Burt, C. (1950). The factorial analysis of qualitative data.British Journal of Psychology, 3, 166–185.
[8] Burt, C. (1953). Scale analysis and factor analysis.British Journal of Statistical Psychology, 6, 5–23.
[9] Cailliez, F., & Pagès, J. P. (1976).Introduction à l’analyse des données [Introduction to data analysis]. Paris: Smash.
[10] Carroll, J. D. (1968). Generalization of canonical correlation analysis to three or more sets of variables.Proceedings of the 76th Annual Convention of the American Psychological Association, 3, 227–228.
[11] Cazes, P. (1972).Etude du dédoublement d’un tableau en analyse des correspondances [Analysis of a table and its complementary in correspondence analysis]. Unpublished manuscript, Université Pierre et Marie Curie, Laboratoire de Statistique Mathématique, Paris.
[12] Cazes, P., Baumerder, A., Bonnefous, S., & Pagès, J. P. (1977). Codage et analyse des tableaux logiques. Introduction à la pratique des variables qualitatives [Scaling and analysis of binary tables. Introduction to the practice of qualitative variables].Cahiers du Bureau Universitaire de Recherche Opérationnelle (Série Recherche, Cahier No. 27) Paris: Institut de Statistique des Universités de Paris, Université Pierre et Marie Curie.
[13] Daudin, J. J., & Trecourt, P. (1980). Analyse factorielle des correspondances et modèle log-linéaire. Comparaison des deux méthodes sur un exemple [Correspondence analysis and Log-linear model. Comparison of both models on an example].Revue de Statistique Appliquée, 28, 5–24.
[14] de Leeuw, J. (1973).Canonical analysis of categorical data. Unpublished doctoral dissertation, University of Leiden, Leiden, The Netherlands.
[15] Dempster, A. P. (1969).Elements of continuous multivariate analysis. Reading, MA: Addison-Wesley. · Zbl 0197.44904
[16] Eckart, C., & Young, C. (1936). The approximation of one matrix by another of lower rank.Psychometrika, 1, 211–218. · JFM 62.1075.02
[17] Escofier, B. (1979a). Une représentation des variables dans l’analyse des correspondances multiples [Representation of variables in multiple correspondence analysis].Revue de Statistique Appliquée, 27, 37–47.
[18] Escofier, B. (1979b). Traitement simultané de variables qualitatives et quantitatives en analyse factorielle [Simultaneous treatment of qualitative and quantitative variables in factor analysis].Les Cahiers de l’Analyse des Données, 4, 137–146.
[19] Fisher, R. A. (1940). The precision of discriminant functions.Annals of Eugenics, 10, 422–429. · Zbl 0063.01384
[20] Gifi, A. (1981).Nonlinear multivariate analysis. Leiden, The Netherlands: University of Leiden, Afdeling Datatheorie. · Zbl 0697.62048
[21] Greenacre, M. J. (1984).Theory and applications of correspondence analysis. London: Academic Press. · Zbl 0555.62005
[22] Guttman, L. (1941). The quantification of a class of attributes: A theory and method of scale construction. In P. Horst et al. (Eds.),The prediction of personal adjustment. (pp. 319–348). New York: Social Science Research Council.
[23] Guttman, L. (1950). The principal components of scale analysis. In S. A. Stouffer, L. Guttman, E. A. Suchman, P. F. Lazarsfeld, S. A. Star, & J. A. Clausen.Measurement and prediction. Princeton: Princeton University Press.
[24] Guttman, L. (1953). A note on Sir Cyril Burt’s factorial analysis of qualitative data.British Journal of Statistical Psychology, 6, 1–4.
[25] Guttman, L. (1959). Metricizing rank-ordered or unordered data for a linear factor analysis.Sankhya, 21, 257–268. · Zbl 0090.11501
[26] Hayashi, C. (1950). On the quantification of qualitative data from the mathematico-statistical point of view.Annals of the Institute of Statistical Mathematics, 2 (No. 1), 35–47. · Zbl 0041.26003
[27] Hayashi, C. (1952). On the prediction of phenomena from qualitative data and the quantification of qualitative data from the mathematico-statistical point of view.Annals of the Institute of Statistical Mathematics, 3 (No. 2), 69–98. · Zbl 0049.09902
[28] Hayashi, C. (1954). Multidimensional quantification–with applications to analysis of social phenomena.Annals of the Institute of Statistical Mathematics, 5 (No. 2), 121–143. · Zbl 0056.38003
[29] Healy, M. J. R., & Goldstein, H. (1976). An approach to the scaling of categorized attributes.Biometrika, 63, 219–229. · Zbl 0328.62034
[30] Hill, M. O. (1973). Reciprocal averaging: An eigenvector method of ordination.Journal of Ecology, 61, 237–251.
[31] Hill, M. O., & Smith, J. E. (1976). Principal component analysis of taxonomic data with multi-state discrete characters.Taxonomy, 25, 249–255.
[32] Hirshfield, H. O. (1935). A connection between correlation and contingency.Cambridge Philosophical Society Proceedings, 31, 520–524. · JFM 61.1304.01
[33] Horst, P. (1935). Measuring complex attitudes.Journal of Social Psychology, 6, 369–374.
[34] Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components.Journal of Educational Psychology, 24, 417–441, 498–520. · JFM 59.1182.04
[35] Kettenring, J. R. (1971). canonical analysis of several sets of variables.Biometrika, 58, 433–451. · Zbl 0225.62072
[36] Lauro, N. C., & Decarli, A. (1982). Correspondence analysis and log-linear models in multiway contingency tables study: Some remarks on experimental data.Metron, 40, 213–234.
[37] Lebart, L. (1975). L’orientation du dépouillement de certaines enquêtes par l’analyse des correspondances multiples [The orientation of the analysis of some surveys by multiple correspondence analysis].Consommation, 2, 73–96.
[38] Lebart, L., & Fénelon, J. P. (1971).Statistique et informatique appliquées [Applied statistics and informatics]. Paris: Dunod.
[39] Lebart, L., Morineau, A. & Tabard, N. (1977).Techniques de la description statistique [Statistical description technics]. Paris: Dunod.
[40] Lebart, L., Morineau, A., & Warwick, K. M. (1984).Multivariate descriptive analysis: Correspondence analysis and related techniques for large matrics. New York: Wiley-Interscience. · Zbl 0658.62069
[41] Leclerc, A. (1980) Quelques propriétés optimales en analyse de données en terme de corrélation entre variables [Some optimal properties in data analysis in term of correlation between variables].Mathématique et Sciences Humaines, 18, 51–67. · Zbl 0484.62002
[42] Levine, J. H. (1979). Joint space analysis of ”pick any” data: Analysis of choices from an unconstrained set of alternatives.Psychometrika, 44, 85–92.
[43] Lingoes, J. C. (1963).Multivariate analysis of contingencies: An IBM 7090 program for analyzing metric/non-metric or linear/non-linear data. [Computer program]. Ann Arbor, MI: University of Michigan Computing Center. (Computational Report, 2, 1–24.)
[44] Lingoes, J. C. (1964). Simultaneous linear regression: An IBM 7090 program for analyzing metric/non-metric or linear/non-linear data.Behavioral Science, 9, 87–88.
[45] Lingoes, J. C. (1968). The multivariate analysis of qualitative data.Multivariate Behavioral Research, 3, 61–94.
[46] Lingoes, J. C. (1972). A general survey of the Guttman-Lingoes nonmetric program series. In R. N. Shepard, A. K. Romney, & S. Nerlove (Eds.),Multidimensional scaling: Theory and applications in the behavioral sciences, Vol. 1: Theory (pp. 49–68). New York: Seminar Press.
[47] Lingoes, J. C. (1973).The Guttman-Lingoes nonmetric program series. Ann Arbor: Mathesis Press.
[48] Lingoes, J. C. (1977).Geometric representations of relational data: Readings in multidimensional scaling. Ann Arbor: Mathesis Press.
[49] Mardia, K. V., Kent, J. T., & Bibby, J. M., (1979).Multivariate Analysis. London: Academic Press. · Zbl 0432.62029
[50] Masson, M. (1974). Processus linéaires et analyse des données non linéaires [Linear processes and non-linear data analysis.] unpublished doctoral dissertation, Université Pierre et Marie Curie, Paris.
[51] Masson, M. (1980).Méthodologies générales de traitement statistique de l’information de masse. [General methodologies for the statistical treatment of large information]. Paris: Cedic-Fernand Nathan.
[52] McDonald, R. P. (1968). A unified treatment of the weighting problem.Psychometrika, 33, 351–381. · Zbl 0255.62052
[53] McKeon, J. J. (1966). Canonical analysis: Some relations between canonical correlation, factor analysis, discriminant function analysis and scaling theory. [Monograph No. 13].Psychometrika.
[54] Mosier, C. I. (1946). Machine methods in scaling by reciprocal averages.Proceedings of Research Forum (pp. 35–39). New York: IBM Corporation.
[55] Mosteller, F. (1949). A theory of scalogram analysis using noncumulative types of items. (Report No. 9). Cambridge: Harvard University, Laboratory of Social Relations.
[56] Nishisato, S. (1972).Optimal scaling and its generalizations. I (Methods. Measurement and evaluation of Categorical Data Technical Report No. 1) Toronto: Department of Measurement and Evaluation, the Ontario Institute for Studies in Education.
[57] Nishisato, S. (1973).Optimal scaling and its generalizations. II (Applications. Measurement and Evaluation of Categorical Data Technical Report No. 2). Toronto: Department of Measurement and Evaluation, the Ontario Institute for Studies in Education.
[58] Nishisato, S. (1976).Optimal scaling as applied to different forms of data (Measurement and Evaluation of Categorical Data Technical Report No. 4). Toronto: Department of Measurement and Evaluation, the Ontario Institute for Studies in Education.
[59] Nishisato, S. (1978).Multidimensional Scaling: A historical sketch and bibliography (Tech. Rep.). Toronto: Department of Measurement, Evaluation and Computer Applications, the Ontario Institute for Studies in Education.
[60] Nishisato, S. (1979). Dual Scaling and its variants.New Directions for Testing and Measurement, 4, 1–12.
[61] Nishisato, S. (1980).Analysis of categorical data: Dual Scaling and its applications. Toronto: University of Toronto Press. · Zbl 0487.62001
[62] Nishisato, S. (1982).Shitsuteki Data no Suryoka: Sotsui Shakudoho to sono Oyo. Tokyo: Asakura Shoten.
[63] Nishisato, S., & Inukai, Y. (1972). Partially optimal scaling of items with ordered categories.Japanese Psychological Research, 14, 109–119.
[64] Nishisato, S., & Leong, K. S. (1975).OPSCAL: A FORTRAN IV Program for analysis of qualitative data by optimal scaling (Measurement and Evaluation of Categorical Data Technical Report No. 3). Toronto: Department of Measurement and Evaluation, The Ontario Institute for Studies in Education.
[65] Nishisato, S., Sheu, W. J. (1980). Piecewise method of reciprocal averages for dual scaling of multiple-choice data.Psychometrika, 45, 467–478. · Zbl 0481.62049
[66] Rao, C. R. (1964). The use and interpretation of principal component analysis in applied research.Sankhya, series A,26, 329–358. · Zbl 0137.37207
[67] Richardson, M., & Kuder, G. F. (1933). Making a rating scale that measures.Personnel Journal, 12, 36–40.
[68] Saito, T. (1973). Quantification of categorical data by using the generalized variance.Soken Kiyo, Nippon UNIVAC Sogo Kenkyu-sho, Inc., 61–80.
[69] Saporta, G. (1975). Liaison entre plusieurs ensembles de variables et codage de données qualitatives [Relationship between several sets of variables and scaling of qualitative data]. Unpublished doctoral dissertation, Université Pierre et Marie Curie, Paris.
[70] Saporta, G. (1980, June).About some remarkable properties of generalized canonical analysis. Paper presented at the second European meeting of the Psychometric Society, Groningen, The Netherlands.
[71] Shiba, S. (1965). A method for scoring multicategory items.Japanese Psychological Research, 7, 75–79.
[72] Tenenhaus, M. (1977). Analyse en composantes principales d’un ensemble de variables nominales ou numériques [Principal component analysis of a set of nominal or numerical variables].Revue de Statistique Appliquée, 25, 39–56.
[73] Torgerson, W. S. (1958).Theory and methods of scaling. New York: Wiley.
[74] Van Rijckevorsel, J., & de Leeuw, J. (1978).An outline to HOMALS-1. Leiden: University of Leiden.
[75] Van Rijckevorsel, J., & de Leeuw, J. (1979).An outline to PRINCALS. Leiden: University of Leiden.
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