Abstract
Various methods are discussed that emphasize the graphical representation of results from the analysis of multivariate data. Specifically, qualitative (or categorical) multivariate data will be considered, where observed variables are only partially known, and dealt with by using an optimal scaling framework that replaces nominal and ordinal variables by optimally quantified variables. The approach to graphical display that is advocated is closely related to techniques of multidimensional scaling, and involves the choice of particular standardizations and metrics to be used.
“Many measurement models in the behavioral sciences are based on geometric representations of the observed behavior. Frequently this geometric representation is a one-dimensional scale but it need not be, and multidimensional representations are becoming more common. The points on these scales or in these spaces may represent individuals or stimuli or both, and the relations among the points reflect the observations according to some rule.” (Coombs, Dawes & Tversky, 1970, p. 32).
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References
Benzécri, J.-P. (1992). Correspondence analysis handbook. New York: Marcel Dekkker.
Burt, C. (1950). The factorial analysis of qualitative data. British Journal of Psychology, 3, 166–185.
Carroll, J.D. (1972). Individual differences and multidimensional scaling, In: Multidimensional scaling: Theory and applications in the behavioral sciences (ed. R.N. Shepard, A.K. Romney & S.B. Nerlove), Vol. 1, 105–155. New York and London: Seminar Press.
Carroll, J.D., Green, P.E., & Schaffer, C.M. (1986). Interpoint distances comparisons in correspondence analysis. Journal of Marketing Research, 23, 271–280.
Coombs, C.H., Dawes, R.M., & Tversky, A. (1970). Mathematical psychology: An elementary introduction. Englewood Cliffs, NJ: Prentice-Hall.
Eckart, C. & Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, 1, 211–218.
Fisher, R.A. (1938). Statistical methods for research workers. Edinburgh: Oliver & Boyd.
Fisher, R.A. (1940). The precision of discriminant functions. Annals of Eugenics, 10, 422–429.
Gabriel, K.R. (1971). The biplot graphic display of matrices with application to principal components analysis. Biometrika, 58, 453–467.
Gabriel, K.R. & Odoroff G. (1986). Some diagnoses of models by 3-D biplots. In: Multidimensional data analysis (ed. J. de Leeuw, W.J. Heiser, J.J. Meulman & F. Critchley), 91–111. Leiden: DSWO Press.
Gifi, A. (1990). Nonlinear multivariate analysis. Chichester: John Wiley & Sons.
Gower, J.C. (1966). Some distance properties of latent roots and vector methods used in multivariate analysis. Biometrika, 53, 325–338.
Gower, J.C. & Hand, D.J. (1996). Biplots. London: Chapman & Hall.
Greenacre, M.J. (1984). Theory and applications of correspondence analysis. London: Academic Press.
Greenacre, M.J. (1989). The Caroll-Green-Schaffer scaling in correspondence analysis: A theoretical and empirical appraisal. Journal of Marketing Research, 26, 358–365.
Guttman, L. (1941). The quantification of a class of attributes: a theory and method of scale construction. In: The prediction of personal adjustment (ed. P. Horst et al.), 319–348. New York: Social Science Research Council.
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, 2, 93–96.
Heiser, W.J. (1987). Joint ordination of species and sites: the unfolding technique. In: Developments in numerical ecology (ed. P. Legendre & L. Legendre), 189–221. New York: Springer.
Kruskal, J.B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–28.
Kruskal, J.B. (1965). Analysis of factorial experiments by estimating monotone transformations of the data. Journal of the Royal Statistical Society Series B, 27, 251–263.
Kruskal, J.B. (1978): Factor analysis and principal components analysis: bilinear methods. In: International encyclopedia of statistics (ed. W.H. Kruskal & J.M. Tanur), 307–330. New York: The Free Press.
Kruskal, J.B. & Shepard, R.N. (1974). A nonmetric variety of linear factor analysis. Psychometrika, 39, 123–157.
Kruskal, J.B. & Wish, M. (1978). Multidimensional scaling. Newbury Park, CA: Sage.
Meulman, J.J. (1986). A distance approach to nonlinear multivariate analysis. Leiden: DSWO Press.
Meulman, J.J. (1992). The integration of multidimensional scaling and multivariate analysis with optimal transformations of the variables. Psychometrika, 57, 539–565.
Meulman, J.J., (1998). A distance-based biplot for multidimensional scaling of multivariate data. In: Data science, classification, and related methods (ed. C. Hayashi, N. Ohsumi & Y. Baba), 506–517. Tokyo: Springer Verlag.
Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications. Toronto: University of Toronto Press.
Rietveld, W.J., Boon, M.E. & Meulman, J.J. (1997). Seasonal fluctuations in the cervical smear detection rates for (pre)malignant changes and for infections. Diagnostic Cytopathology, 17, 452–455.
Stewart, G.W. (1993). On the early history of the singular value decomposition. SIAM Review, 35, 551–566.
Schmidt (1907). Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil. Entwicklung willkürlichen Funktionen nach System vorgeschriebener. Mathematische Annalen, 63, 433–476.
Tucker, L. R (1960). Intra-individual and inter-individual multidimensionality. In: Psychological scaling: theory and applications (ed. H. Gulliksen & S. Messick), 155–167. New York: Wiley.
Van der Ham, Th., Meulman, J.J., Van Strien, D.C. & Van Engeland, H. (1997). Empirically based subgrouping of eating disorders in adolescents: a longitudinal perspective. British Journal of Psychiatry, 170, 363–368.
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Meulman, J.J. (1998). Optimal Scaling Methods for Graphical Display of Multivariate Data. In: Payne, R., Green, P. (eds) COMPSTAT. Physica, Heidelberg. https://doi.org/10.1007/978-3-662-01131-7_6
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DOI: https://doi.org/10.1007/978-3-662-01131-7_6
Publisher Name: Physica, Heidelberg
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