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Interpreting canonical correlation analysis through biplots of structure correlations and weights

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Abstract

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.

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References

  • Adelman, I., & Morris, C. T. (1967).Society, politics & economic development. Baltimore: John Hopkins Press.

    Google Scholar 

  • Adelman, I., Geier, M., & Morris, C. T. (1969). Instruments and goals in economic development.American Economic Review—Papers and Prodeedings, 59, 409–426.

    Google Scholar 

  • Anderson, T. W. (1951). Estimating linear restrictions on regression coefficients for multivariate normal distributions.Annals of mathematical Statistics, 22, 327–351.

    Google Scholar 

  • Anderson, T. W. (1984).An introduction to multivariate statistical analysis (2nd. ed.). New York: Wiley.

    Google Scholar 

  • Bartlett, M. S. (1938). Further aspects of the theory of multiple regression.Proceedings of the Cambridge Philosophical Society, 34, 33–40.

    Google Scholar 

  • Bock, R. D. (1975).Multivariate statistical methods in behavioral research, New York: McGraw-Hill.

    Google Scholar 

  • Brillinger, D. R. (1981).Time series: Data analysis and theory. San Francisco: Holden-Day.

    Google Scholar 

  • Browne, M. W. (1979). The maximum-likelihood solution in inter-battery factor analysis.British Journal of Mathematical and Statistical Psychology, 32, 75–86.

    Google Scholar 

  • Cailliez, F., & Pagès, J. P. (1976).Introduction à l'analyse des données [Introduction to data analysis]. Paris: SMASH.

    Google Scholar 

  • Corsten, L. C. A. (1976). Matrix approximation, a key to application of multivariate methods.Proceedings of the 9th International Biometric Conference (pp. 61–77). Raleigh: The Biometric Society.

    Google Scholar 

  • Davies, P. T., & Tso, M.K-S. (1982). Procedures for reduced-rank regression.Applied Statistics, 31, 244–255.

    Google Scholar 

  • de Leeuw, J., Mooijaart, A., & van der Leeden, R. (1985).Fixed factor score models with linear restrictions (Technical Report RR-85-06). Leiden: Department of Datatheory.

    Google Scholar 

  • DeSarbo, W. S. (1981). Canonical/redundancy factoring analysis.Psychometrika, 46, 307–329.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Gabriel, K. R. (1982). Biplot. In S. Kotz & N. L. Johnson (Eds.),Encyclopedia of statistical sciences (Vol. 1, pp. 263–271). New York: Wiley.

    Google Scholar 

  • Gifi, A. (1981).Nonlinear multivariate analysis. Leiden: Department of Datatheory.

    Google Scholar 

  • Gittins, R. (1985).Canonical analysis. A review with applications in ecology. Berlin: Springer-Verlag.

    Google Scholar 

  • Gower, J. C., & Harding, S. (1988). Nonlinear biplots.Biometrika, 75, 445–455.

    Google Scholar 

  • Greenacre, M. J. (1984).Theory and applications of correspondence analysis. New York: Wiley.

    Google Scholar 

  • Haber, M., & Gabriel, K. R. (1976).Weighted least squares approximation of matrices and its application to canonical correlations and biplot display (Technical Report). Rochester: University of Rochester, Department of Statistics.

    Google Scholar 

  • Horst, P. (1961). Relations amongm sets of measures.Psychometrika, 26, 129–150.

    Google Scholar 

  • Israëls, A. Z. (1984). Redundancy analysis for qualitative variables.Psychometrika, 49, 331–346.

    Google Scholar 

  • Israëls, A. Z. (1987).Eigenvalue techniques for qualitative data. Leiden: DSWO Press.

    Google Scholar 

  • Izenman, A. J. (1975). Reduced-rank regression for the multivariate linear model.Journal of Multivariate Analysis, 5, 248–264.

    Google Scholar 

  • Jöreskog, K. G., & Sörbom, D. (1983).LISREL VI Users Guide. Uppsala: Department of Statistics.

    Google Scholar 

  • Kendall, M. (1975).Multivariate analysis. London: Griffin.

    Google Scholar 

  • McKeon, J. J. (1966). Canonical analysis: Some relations between canonical correlation, factor analysis, discriminant function analysis, and scaling theory.Psychometrika Monograph No. 13.

  • Meredith, W. (1964). Canonical correlations with fallible data.Psychometrika, 29, 55–65.

    Google Scholar 

  • Montgomery, D. C., & Peck, E. A. (1982).Introduction to linearregression analysis. New York: Wiley.

    Google Scholar 

  • Rao, C. R. (1975).Linear statistical inference and its applications. New York: Wiley.

    Google Scholar 

  • Rao, C. R. (1979). Separation theorems for singular values of matrices and their applications in multivariate analysis.Journal of Multivariate analysis, 9, 362–377.

    Google Scholar 

  • Rao, C. R., (1980). Matrix approximations and reduction of dimensionality in multivariate statistical analysis. In P. R. Krishnaiah (Ed.),Multivariate Analysis V (pp. 3–22). Amsterdam: North-Holland.

    Google Scholar 

  • Rencher, A. C. (1988). On the use of correlations to interpret canonical functions.Biometrika, 75, 363–365.

    Google Scholar 

  • Röhr, M. (1987).Kanonische Korrelationsanalyse [Canonical correlation analysis]. Berlin: Akademie-Verlag.

    Google Scholar 

  • Roy, S. N., & Whittlesey, J. (1952). Canonical partial correlations.Annals of Mathematical Statistics, 23, 642.

    Google Scholar 

  • Sabatier, R., Jan, Y., & Escoufier, Y. (1984). Approximations d'applications lineaires et analyse en composantes principales [Approximations of linear operators and principal components analysis]. In E. Diday et al. (Eds),Data analysis and informatics III (pp. 569–580). Amsterdam: North Holland.

    Google Scholar 

  • Saris, W., & Stronkhorst, H. (1984).Causal modelling in nonexperimental research. Amsterdam: Sociometric Research Foundation.

    Google Scholar 

  • Thompson, B. (1984).Canonical correlation analysis: Uses and interpretation, Beverly Hills: Sage.

    Google Scholar 

  • Thorndike, R. M., & Weiss, D. J. (1973). A study of the stability of canonical correlations and canonical components.Educational Psychological Measurement, 3, 123–134.

    Google Scholar 

  • Timm, N. H., & Carlson, J. E. (1976). Part and bipartial canonical correlation analysis.Psychometrika, 41, 159–176.

    Google Scholar 

  • Tso, M. K-S. (1981). Reduced-rank regression and canonical analysis.Journal of the Royal Statistical Society, Series B,43, 183–189.

    Google Scholar 

  • Tucker, L. R., (1958). An inter-battery method of factor analysis.Psychometrika, 23, 111–136.

    Google Scholar 

  • van den Wollenberg, A. L. (1977). Redundancy analysis. An alternative for canonical correlation analysis.Psychometrika, 42, 207–219.

    Google Scholar 

  • van der Burg, E., & de Leeuw, J. (1983). Nonlinear canonical correlation.British Journal of Mathematical and Statistical Psychology, 36, 54–80.

    Google Scholar 

  • van der Geer, J. P. (1986).Introduction to linear multivariate data analysis, Vol. 1. Leiden: DSWO Press.

    Google Scholar 

  • Velu, R. P., Reinsel, G. C., & Wichern, D. W. (1986). Reduced rank models for multiple time series.Biometrika, 73, 105–118.

    Google Scholar 

  • Wold, H. (1982). Soft modeling: The basic design and some extensions. In K. G. Jöreskog & H. Wold (Eds.),Systems under indirect observation, Vol. 2 (pp. 1–54). Amsterdam: North-Holland.

    Google Scholar 

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I am indebted to L. C. A. Corsten, K. R. Gabriel, A. Z. Israëls, J. M. F. ten Berge, H. A. L. Kiers, J. de Bree and A. A. M. Jansen for comments on the manuscript.

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ter Braak, C.J.F. Interpreting canonical correlation analysis through biplots of structure correlations and weights. Psychometrika 55, 519–531 (1990). https://doi.org/10.1007/BF02294765

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