Evolutionary Biology

, Volume 35, Issue 3, pp 191–198 | Cite as

Using Correlation Proximity Graphs to Study Phenotypic Integration

  • Paul M. Magwene
Research Article


Characterizing and comparing the covariance or correlation structure of phenotypic traits lies at the heart of studies concerned with multivariate evolution. I describe an approach that represents the geometric structure of a correlation matrix as a type of proximity graph called a Correlation Proximity graph. Correlation Proximity graphs provide a compact representation of the geometric relationships inherent in correlation matrices, and these graphs have simple and intuitive properties. I demonstrate how this framework can be used to study patterns of phenotypic integration by employing this approach to compare phenotypic and additive genetic correlation matrices within and between species. I also outline a graph-based method for testing whether an inferred correlation proximity graph is one of a number of possible models that are consistent with a “soft” biological hypothesis.


Multivariate evolution Integration Modularity Graphical model Proximity graphs 


  1. Bookstein, F. L., Chernoff, B., Elder, R., Humphries, J., Smith, G., & Strauss, R. (1985). Morphometerics in evolutionary biology. Philadelphia, PA: Academy of Natural Sciences.Google Scholar
  2. Chernoff, B., & Magwene, P. M. (1999). Morphological integration: Forty years later. In: E. C. Oslon & R. C. Miller (Eds.), Morphological integration (pp. 316–360). Chicago: University of Chicago.Google Scholar
  3. Cheverud, J. M. (1982). Phenotypic, genetic, and environmental morphological integration in the cranium. Evolution, 36, 499–516.CrossRefGoogle Scholar
  4. Drton, M., & Perlman, M. D. (2004). Model selection for Gaussian concentration graphs. Biometrika, 91, 591–602.CrossRefGoogle Scholar
  5. Dunn, L. C. (1928). The effect of inbreeding on the bones of the fowl. Bulletin Storrs Agricultural Experiment Station, 152, 1–112.Google Scholar
  6. Gabriel, K. R., & Sokal, R. R. (1969). A new statistical approach to geographic variation analysis. Systematic Zoology, 18, 259–278.CrossRefGoogle Scholar
  7. Hansen, T. F. (2003). Is modularity necessary for evolvability? Remarks on the relationship between pleiotropy and evolvability. Biosystems, 69, 83–94.PubMedCrossRefGoogle Scholar
  8. Higham, N. J. (2002). Computing the nearest correlation matrix–a problem from finance. IMA Journal of Numerical Analysis, 22, 329–343.CrossRefGoogle Scholar
  9. Jaromczyk, J. W., & Toussaint, G. T. (1992). Relative neighborhood graphs and their relatives. Proceedings of the IEEE, 80, 1502–1517.CrossRefGoogle Scholar
  10. Kohn, L. A. P., & Atchley, W. R. (1988). How similar are genetic correlation structures data from mice and rats. Evolution, 42, 467–481.CrossRefGoogle Scholar
  11. Krzanowksi, W. J. (1988). Principles of multivariate analysis. Oxford: Clarendon Press.Google Scholar
  12. Lande, R., & Arnold, S. J. (1983). The measurement of selection on correlated characters. Evolution, 37, 1210–1226.CrossRefGoogle Scholar
  13. Lauritzen, S. L. (1996). Graphical models. Oxford: Oxford University Press.Google Scholar
  14. Magwene, P. M. (2001). New tools for studying integration and modularity. Evolution, 55, 1734–1745.PubMedGoogle Scholar
  15. Magwene, P. M., & Kim, J. (2004). Estimating genomic coexpression networks using first-order conditional independence. Genome Biology, 5, 1–100.CrossRefGoogle Scholar
  16. Matula, D. W., & Sokal, R. R. (1980). Properties of gabriel graphs relevant to geographic variation research and clustering of points in the plane. Geographic Analysis, 12, 205–222.Google Scholar
  17. Olson, E. C., & Miller, R. L. (1958). Morphological integration. Chicago: University of Chicago Press.Google Scholar
  18. O’Quigley, T. G. (1993). A geometric interpretation of partial correlation using spherical triangles. American Statistician, 47, 30–32.CrossRefGoogle Scholar
  19. Phillips, P. C., & Arnold, S. J. (1999). Hierarchical comparison of genetic variance-covariance matrices. I. Using the flury hierarchy. Evolution, 53, 1506–1515.CrossRefGoogle Scholar
  20. Wagner, G. P., & Altenberg, L. (1996). Complex adaptations and the evolution of evolvability. Evolution, 50, 967–976.CrossRefGoogle Scholar
  21. Wagner, G. P., Pavlicev, M., & Cheverud, J. M. (2007). The road to modularity. Nature Reviews Genetics, 8, 921–931.PubMedCrossRefGoogle Scholar
  22. Whittaker, J. (1990). Graphical models in applied mathematical multivariate statistics. Chichester: Wiley.Google Scholar
  23. Wickens, T. D. (1995). The geometry of multivariate statistics. Hilldales: Lawrence Erlbaum Associates.Google Scholar
  24. Wille, A., & Buhlmann, P. (2006). Low-order conditional independence graphs for inferring genetic networks. Statistical Applications in Genetics and Molecular Biology 5: Article 1.Google Scholar
  25. Wille, A., Zimmermann, P., Vranova, E., Furholz, A., Laule, O., Bleuler, S., Hennig, L., Prelic, A., von Rohr, P., Thiele, L., Zitzler, E., Gruissem, W., & Buhlmann, P. (2004). Sparse graphical Gaussian modeling of the isoprenoid gene network in Arabidopsis thaliana. Genome Biology, 5, R92.PubMedCrossRefGoogle Scholar
  26. Wright, S. (1932). General, group and special size factors. Genetics, 15, 603–619.Google Scholar
  27. Young, N. M., & Hallgrimsson, B. (2005). Serial homology and the evolution of mammalian limb covariation structure. Evolution, 59, 2691–2704.PubMedGoogle Scholar
  28. Zelditch, M. L. (1987). Evaluating models of developmental integration in the laboratory rat using confirmatory factor analysis. Systematic Zoology, 36, 368–380.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of BiologyDuke UniversityDurhamUSA

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