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Psychometrika

, Volume 56, Issue 2, pp 197–212 | Cite as

Simple structure in component analysis techniques for mixtures of qualitative and quantitative variables

  • Henk A. L. Kiers
Article

Abstract

Several methods have been developed for the analysis of a mixture of qualitative and quantitative variables, and one, called PCAMIX, includes ordinary principal component analysis (PCA) and multiple correspondence analysis (MCA) as special cases. The present paper proposes several techniques for simple structure rotation of a PCAMIX solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for PCA. In addition, a recently developed technique for the analysis of mixtures of qualitative and quantitative variables, called INDOMIX, is shown to construct component scores (without rotational freedom) maximizing the quartimax criterion over all possible sets of component scores. A numerical example is used to illustrate the implication that when used for qualitative variables, INDOMIX provides axes that discriminate between the observation units better than do those generated from MCA.

Key words

multiple correspondence analysis INDSCAL varimax quartimax orthomax discrimination between objects 

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References

  1. Carroll, J. D., & Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition.Psychometrika, 35, 283–319.Google Scholar
  2. Clarkson, D. B., & Jennrich, R. I. (1988). Quartic rotation criteria and algorithms.Psychometrika, 53, 251–259.Google Scholar
  3. Crawford, C. B., & Ferguson, G. A. (1970). A general rotation criterion and its use in orthogonal rotation.Psychometrika, 35, 321–332.Google Scholar
  4. de Leeuw, J. (1973).Canonical analysis of categorical data. Unpublished doctoral dissertation, University of Leiden.Google Scholar
  5. de Leeuw, J., & Pruzansky, S. (1978). A new computational method to fit the weighted Euclidean distance model.Psychometrika, 43, 479–490.Google Scholar
  6. de Leeuw, J., & van Rijckevorsel, J. L. A. (1980). HOMALS and PRINCALS, some generalizations of principal components analysis. In E. Diday et al. (Eds.),Data analysis and informatics II (pp. 231–242). Amsterdam: Elsevier Science Publishers.Google Scholar
  7. Escofier, B. (1979). Traitement simultané de variables qualitatives et quantitatives en analyse factorielle [Simultaneous treatment of qualitative and quantitative variables in factor analysis].Cahiers de l'Analyse des Données, 4, 137–146.Google Scholar
  8. Ferguson, G. A. (1954). The concept of parsimony in factor analysis.Psychometrika, 19, 281–290.Google Scholar
  9. Gifi, A. (1990).Nonlinear multivariate analysis. New York: Wiley.Google Scholar
  10. Harman, H. H. (1976).Modern factor analysis (3rd ed.). Chicago: University of Chicago Press.Google Scholar
  11. Hartigan, J. A. (1975).Clustering algorithms. New York: Wiley.Google Scholar
  12. Jennrich, R. I. (1970). Orthogonal rotation algorithms.Psychometrika, 35, 229–235.Google Scholar
  13. Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis.Psychometrika, 23, 187–200.Google Scholar
  14. Kiers, H. A. L. (1988). Principal components analysis on a mixture of quantitative and qualitative data based on generalized correlation coefficients. In M. G. H. Jansen & W. H. van Schuur (Eds.),The many faces of multivariate analysis, Vol. I, Proceedings of the SMABS-88 Conference in Groningen (pp. 67–81). Groningen: Rion.Google Scholar
  15. Kiers, H. A. L. (1989a). A computational short-cut for INDSCAL with orthonormality constraints on positive semi-definite matrices of low rank.Computational Statistics Quarterly, 2, 119–135.Google Scholar
  16. Kiers, H. A. L. (1989b). INDSCAL for the analysis of categorical data. In R. Coppi & S. Bolasco (Eds.),Multiway data analysis (pp. 155–167). Amsterdam: Elsevier Science Publishers.Google Scholar
  17. Kiers, H. A. L. (1989c).Three-way methods for the analysis of qualitative and quantitative two-way data. Leiden: DSWO Press.Google Scholar
  18. Kiers, H. A. L. (1990). Majorization as a tool for optimizing a class of matrix functions,Psychometrika, 55, 417–428.Google Scholar
  19. Nishisato, S. (1980).Analysis of categorical data: Dual scaling and its applications. Toronto: University Press.Google Scholar
  20. Nishisato, S., & Sheu, W.-J. (1980). Piecewise method of reciprocal averages for dual scaling of multiple-choice data.Psychometrika, 45, 467–478.Google Scholar
  21. Saporta, G. (1976). Quelques applications des opérateurs d'Escoufier au traitement des variables qualitatives [Several Escoufier operators for the treatment of qualitative variables].Statistique et Analyse des Données, 1, 38–46.Google Scholar
  22. ten Berge, J. M. F. (1983). A generalization of Kristof's theorem on the trace of certain matrix products.Psychometrika, 48, 519–523.Google Scholar
  23. ten Berge, J. M. F. (1984). A joint treatment of varimax rotation and the problem of diagonalizing symmetric matrices simultaneously in the least-squares sense.Psychometrika, 49, 347–358.Google Scholar
  24. ten Berge, J. M. F., Knol, D. L., & Kiers, H. A. L. (1988). A treatment of the orthomax rotation family in terms of diagonalization, and a re-examination of a singular value approach to varimax rotation.Computational Statistics Quarterly, 3, 207–217.Google Scholar
  25. Tenenhaus, M., & Young, F. W. (1985). An analysis and synthesis of multiple correspondence analysis, optimal scaling, dual scaling, homogeneity analysis and other methods for quantifying categorical multivariate data.Psychometrika, 50, 91–119.Google Scholar

Copyright information

© The Psychometric Society 1991

Authors and Affiliations

  • Henk A. L. Kiers
    • 1
  1. 1.University of GroningenThe Netherlands

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