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Optimal Scaling Methods for Graphical Display of Multivariate Data

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COMPSTAT
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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.

    MATH  Google Scholar 

  • Burt, C. (1950). The factorial analysis of qualitative data. British Journal of Psychology, 3, 166–185.

    Google Scholar 

  • 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.

    Google Scholar 

  • Carroll, J.D., Green, P.E., & Schaffer, C.M. (1986). Interpoint distances comparisons in correspondence analysis. Journal of Marketing Research, 23, 271–280.

    Article  Google Scholar 

  • Coombs, C.H., Dawes, R.M., & Tversky, A. (1970). Mathematical psychology: An elementary introduction. Englewood Cliffs, NJ: Prentice-Hall.

    MATH  Google Scholar 

  • Eckart, C. & Young, G. (1936). The approximation of one matrix by another of lower rank. Psychometrika, 1, 211–218.

    Article  MATH  Google Scholar 

  • Fisher, R.A. (1938). Statistical methods for research workers. Edinburgh: Oliver & Boyd.

    MATH  Google Scholar 

  • Fisher, R.A. (1940). The precision of discriminant functions. Annals of Eugenics, 10, 422–429.

    Google Scholar 

  • Gabriel, K.R. (1971). The biplot graphic display of matrices with application to principal components analysis. Biometrika, 58, 453–467.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • Gifi, A. (1990). Nonlinear multivariate analysis. Chichester: John Wiley & Sons.

    MATH  Google Scholar 

  • Gower, J.C. (1966). Some distance properties of latent roots and vector methods used in multivariate analysis. Biometrika, 53, 325–338.

    MathSciNet  MATH  Google Scholar 

  • Gower, J.C. & Hand, D.J. (1996). Biplots. London: Chapman & Hall.

    MATH  Google Scholar 

  • Greenacre, M.J. (1984). Theory and applications of correspondence analysis. London: Academic Press.

    MATH  Google Scholar 

  • Greenacre, M.J. (1989). The Caroll-Green-Schaffer scaling in correspondence analysis: A theoretical and empirical appraisal. Journal of Marketing Research, 26, 358–365.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Kruskal, J.B. (1964). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • 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.

    Google Scholar 

  • Kruskal, J.B. & Shepard, R.N. (1974). A nonmetric variety of linear factor analysis. Psychometrika, 39, 123–157.

    Article  MathSciNet  MATH  Google Scholar 

  • Kruskal, J.B. & Wish, M. (1978). Multidimensional scaling. Newbury Park, CA: Sage.

    Google Scholar 

  • Meulman, J.J. (1986). A distance approach to nonlinear multivariate analysis. Leiden: DSWO Press.

    Google Scholar 

  • Meulman, J.J. (1992). The integration of multidimensional scaling and multivariate analysis with optimal transformations of the variables. Psychometrika, 57, 539–565.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Nishisato, S. (1980). Analysis of categorical data: Dual scaling and its applications. Toronto: University of Toronto Press.

    MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Stewart, G.W. (1993). On the early history of the singular value decomposition. SIAM Review, 35, 551–566.

    Article  MathSciNet  MATH  Google Scholar 

  • Schmidt (1907). Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil. Entwicklung willkürlichen Funktionen nach System vorgeschriebener. Mathematische Annalen, 63, 433–476.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

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© 1998 Springer-Verlag Berlin Heidelberg

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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

  • Print ISBN: 978-3-7908-1131-5

  • Online ISBN: 978-3-662-01131-7

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