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A short history of statistical association: From correlation to correspondence analysis to copulas

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  • Cuadras, Carles M.
  • Greenacre, Michael

Abstract

We present in three parts different concepts of correlation and statistical association, with some historical notes, starting with Galton’s notion of correlation, subsequently improved by Pearson. Continuing in this first part, we discuss the correlation ratio, the intraclass correlation, multiple correlation, and redundancy analysis. Throughout we use the classic data set of Galton on the heights of parents and their children. In the second part we explain how these same data can be studied from a multivariate viewpoint, using canonical correlation analysis, Procrustes correlation and simple/multiple correspondence analysis. For correspondence analysis, we use the same data as categorized by Galton into intervals of heights for the parents and their children. In this part we also make an incursion into the continuous form of correspondence analysis. The third part is dedicated to bivariate distributions, where we give the main results of bivariate distributions with given marginals, commenting on the correlations of Spearman and Kendall. Seeing that a bivariate distribution can be generated using a copula, we fit Galton’s data to two copulas: the Gaussian copula and the copula which has the best fit.

Suggested Citation

  • Cuadras, Carles M. & Greenacre, Michael, 2022. "A short history of statistical association: From correlation to correspondence analysis to copulas," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    • Handle: RePEc:eee:jmvana:v:188:y:2022:i:c:s0047259x21001755
    DOI: 10.1016/j.jmva.2021.104901
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    References listed on IDEAS

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    1. Wachsmuth A. & Wilkinson L. & Dallal G.E., 2003. "Galtons Bend: A Previously Undiscovered Nonlinearity in Galtons Family Stature Regression Data," The American Statistician, American Statistical Association, vol. 57, pages 190-192, August.
    2. Niels Waller, 2011. "The Geometry of Enhancement in Multiple Regression," Psychometrika, Springer;The Psychometric Society, vol. 76(4), pages 634-649, October.
    3. Carles M. Cuadras & Walter Diaz & Sonia Salvo-Garrido, 2020. "Two generalized bivariate FGM distributions and rank reduction," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(23), pages 5639-5665, December.
    4. J. Gower, 1975. "Generalized procrustes analysis," Psychometrika, Springer;The Psychometric Society, vol. 40(1), pages 33-51, March.
    5. John Aitchison & Michael Greenacre, 2002. "Biplots of compositional data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 51(4), pages 375-392, October.
    6. Cuadras, Carles M., 2015. "Contributions to the diagonal expansion of a bivariate copula with continuous extensions," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 28-44.
    7. Papadatos, Nickos & Xifara, Tatiana, 2013. "A simple method for obtaining the maximal correlation coefficient and related characterizations," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 102-114.
    8. Cuadras, C. M., 2002. "On the Covariance between Functions," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 19-27, April.
    9. Arnold Wollenberg, 1977. "Redundancy analysis an alternative for canonical correlation analysis," Psychometrika, Springer;The Psychometric Society, vol. 42(2), pages 207-219, June.
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