Statistical Science

Elliptical Insights: Understanding Statistical Methods through Elliptical Geometry

Michael Friendly, Georges Monette, and John Fox

Full-text: Open access


Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the virtues of the ellipse and her higher-dimensional cousins for both these purposes in a variety of contexts, including linear models, multivariate linear models and mixed-effect models. We emphasize the strong relationships among statistical methods, matrix-algebraic solutions and geometry that can often be easily understood in terms of ellipses.

Article information

Statist. Sci. Volume 28, Number 1 (2013), 1-39.

First available in Project Euclid: 29 January 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Added-variable plots Bayesian estimation concentration ellipse data ellipse discriminant analysis Francis Galton hypothesis-error plots kissing ellipsoids measurement error mixed-effect models multivariate meta-analysis regression paradoxes ridge regression statistical geometry


Friendly, Michael; Monette, Georges; Fox, John. Elliptical Insights: Understanding Statistical Methods through Elliptical Geometry. Statist. Sci. 28 (2013), no. 1, 1--39. doi:10.1214/12-STS402.

Export citation


  • Alker, H. R. (1969). A typology of ecological fallacies. In Social Ecology (M. Dogan and S. Rokkam, eds.) 69–86. MIT Press, Cambridge, MA.
  • Anderson, E. (1935). The irises of the Gaspé peninsula. Bulletin of the American Iris Society 35 2–5.
  • Antczak-Bouckoms, A., Joshipura, K., Burdick, E. and Tulloch, J. F. (1993). Meta-analysis of surgical versus non-surgical methods of treatment for periodontal disease. J. Clin. Periodontol. 20 259–268.
  • Beaton, A. E. Jr (1964). The use of special matrix operators in statistical calculus. Ph.D. thesis, Harvard Univ., ProQuest LLC, Ann Arbor, MI.
  • Belsley, D. A., Kuh, E. and Welsch, R. E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. Wiley, New York.
  • Berkey, C. S., Hoaglin, D. C., Antczak-Bouckoms, A., Mosteller, F. and Colditz, G. A. (1998). Meta-analysis of multiple outcomes by regression with random effects. Stat. Med. 17 2537–2550.
  • Boyer, C. B. (1991). Apollonius of Perga. In A History of Mathematics, 2nd ed. 156–157. Wiley, New York.
  • Bravais, A. (1846). Analyse mathématique sur les probabilités des erreurs de situation d’un point. Mémoires Présentés Par Divers Savants à L’Académie Royale des Sciences de l’Institut de France 9 255–332.
  • Bryant, P. (1984). Geometry, statistics, probability: Variations on a common theme. Amer. Statist. 38 38–48.
  • Bryk, A. S. and Raudenbush, S. W. (1992). Hierarchical Linear Models: Applications and Data Analysis Methods. Sage, Thousand Oaks, CA.
  • Campbell, N. A. and Atchley, W. R. (1981). The geometry of canonical variate analysis. Systematic Zoology 30 268–280.
  • Cramér, H. (1946). Mathematical Methods of Statistics. Princeton Mathematical Series 9. Princeton Univ. Press, Princeton, NJ.
  • Dempster, A. P. (1969). Elements of Continuous Multivariate Analysis. Addison-Wesley, Reading, MA.
  • Denis, D. (2001). The origins of correlation and regression: Francis Galton or Auguste Bravais and the error theorists. History and Philosophy of Psychology Bulletin 13 36–44.
  • Diez-Roux, A. V. (1998). Bringing context back into epidemiology: Variables and fallacies in multilevel analysis. Am. J. Public Health 88 216–222.
  • Fisher, R. A. (1936). The use of multiple measurements in taxonomic problems. Annals of Eugenics 8 379–388.
  • Fox, J. (2008). Applied Regression Analysis and Generalized Linear Models, 2nd ed. Sage, Thousand Oaks, CA.
  • Fox, J. and Suschnigg, C. (1989). A note on gender and the prestige of occupations. Canadian Journal of Sociology 14 353–360.
  • Fox, J. and Weisberg, S. (2011). An R Companion to Applied Regression, 2nd ed. Sage, Thousand Oaks, CA.
  • Friendly, M. (1991). SAS System for Statistical Graphics, 1st ed. SAS Institute, Cary, NC.
  • Friendly, M. (2007a). A.-M. Guerry’s Moral statistics of France: Challenges for multivariable spatial analysis. Statist. Sci. 22 368–399.
  • Friendly, M. (2007b). HE plots for multivariate linear models. J. Comput. Graph. Statist. 16 421–444.
  • Friendly, M. (2013). The generalized ridge trace plot: Visualizing bias and precision. J. Comput. Graph. Statist. 22. To appear.
  • Friendly, M., Monette, G. and Fox, J. (2012). Supplement to “Elliptical Insights: Understanding Statistical Methods Through Elliptical Geometry.” DOI:10.1214/12-STS402SUPP.
  • Galton, F. (1886). Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute 15 246–263.
  • Galton, F. (1889). Natural Inheritance. Macmillan, London.
  • Gasparrini, A. (2012). MVMETA: multivariate meta-analysis and meta-regression. R package version 0.2.4.
  • Guerry, A. M. (1833). Essai sur la statistique morale de la France. Crochard, Paris. [English translation: Hugh P. Whitt and Victor W. Reinking, Edwin Mellen Press, Lewiston, NY (2002).]
  • Henderson, C. R. (1975). Best linear unbiased estimation and prediction under a selection model. Biometrics 31 423–448.
  • Hoerl, A. E. and Kennard, R. W. (1970a). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12 55–67.
  • Hoerl, A. E. and Kennard, R. W. (1970b). Ridge regression: Applications to nonorthogonal problems. Technometrics 12 69–82. [Correction: 12 723.]
  • Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology 24 417–441.
  • Jackson, D., Riley, R. and White, I. R. (2011). Multivariate meta-analysis: Potential and promise. Stat. Med. 30 2481–2498.
  • Kramer, G. H. (1983). The ecological fallacy revisited: Aggregate-versus individual-level findings on economics and elections, and sociotropic voting. The American Political Science Review 77 92–111.
  • Laird, N. M. and Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics 38 963–974.
  • Lichtman, A. J. (1974). Correlation, regression, and the ecological fallacy: A critique. The Journal of Interdisciplinary History 4 417–433.
  • Longley, J. W. (1967). An appraisal of least squares programs for the electronic computer from the point of view of the user. J. Amer. Statist. Assoc. 62 819–841.
  • Marquardt, D. W. (1970). Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation. Technometrics 12 591–612.
  • Monette, G. (1990). Geometry of multiple regression and interactive 3-D graphics. In Modern Methods of Data Analysis, Chapter 5 (J. Fox and S. Long, eds.) 209–256. Sage, Beverly Hills, CA.
  • Nam, I.-S., Mengersen, K. and Garthwaite, P. (2003). Multivariate meta-analysis. Stat. Med. 22 2309–2333.
  • Pearson, K. (1896). Contributions to the mathematical theory of evolution—III, regression, heredity and panmixia. Philosophical Transactions of the Royal Society of London 187 253–318.
  • Pearson, K. (1901). On lines and planes of closest fit to systems of points in space. Philosophical Magazine 6 559–572.
  • Pearson, K. (1920). Notes on the history of correlation. Biometrika 13 25–45.
  • Raudenbush, S. W. and Bryk, A. S. (2002). Hierarchical Linear Models: Applications and Data Analysis Methods, 2nd ed. Sage, Newbury Park, CA.
  • Riley, M. W. (1963). Special problems of sociological analysis. In Sociological Research I: A Case Approach (M. W. Riley, ed.) 700–725. Harcourt, Brace, and World, New York.
  • Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review 15 351–357.
  • Robinson, G. K. (1991). That BLUP is a good thing: The estimation of random effects. Statist. Sci. 6 15–32.
  • Saville, D. and Wood, G. (1991). Statistical Methods: The Geometric Approach. Springer Texts in Statistics. Springer, New York.
  • Simpson, E. H. (1951). The interpretation of interaction in contingency tables. J. Roy. Statist. Soc. Ser. B. 13 238–241.
  • Speed, T. (1991). That BLUP is a good thing: The estimation of random effects: Comment. Statist. Sci. 6 42–44.
  • Stigler, S. M. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900. The Belknap Press of Harvard Univ. Press, Cambridge, MA.
  • Timm, N. H. (1975). Multivariate Analysis with Applications in Education and Psychology. Brooks/Cole Publishing Co., Monterey, CA.
  • Velleman, P. F. and Welsh, R. E. (1981). Efficient computing of regression diagnostics. Amer. Statist. 35 234–242.
  • von Humboldt, A. (1811). Essai Politique sur le Royaume de la Nouvelle-Espagne. (Political Essay on the Kingdom of New I: Founded on Astronomical Observations, and Trigonometrical and Barometrical Measurements), Vol. 1. Riley, New York.
  • Wickens, T. D. (1995). The Geometry of Multivariate Statistics. Lawrence Erlbaum Associates, Hillsdale, NJ.

Supplemental materials

  • Supplementary material: Supplementary materials for Elliptical insights: Understanding statistical methods through elliptical geometry. The supplementary materials include SAS and R scripts to generate all of the figures for this article. Several 3D movies are also included to show phenomena better than can be rendered in static print images. A new R package, gellipsoid, provides computational support for the theory described in Appendix A.1. These are also available at and described in [Friendly, Monette, Fox (2012)].