Evolutionary Biology

, Volume 41, Issue 2, pp 336–350 | Cite as

Comparing Covariance Matrices by Relative Eigenanalysis, with Applications to Organismal Biology

Tools and Techniques
First Online:
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Abstract

Most biologists are familiar with principal component analysis as an ordination tool for questions about within-group or between-group variation in systems of quantitative traits, and with multivariate analysis of variance as a tool for one useful description of the latter in the context of the former. Less familiar is the mathematical approach of relative eigenanalysis of which both of these are special cases: computing linear combinations for which two variance–covariance patterns have maximal ratios of variance. After reviewing this common algebraic–geometric core, we demonstrate the effectiveness of this exploratory approach in studies of developmental canalization and the identification of divergent and stabilizing selection. We further outline a strategy for statistical classification when group differences in variance dominate over differences in group averages.

Keywords

Classification Covariance matrix Developmental canalization Morphometrics Natural selection Principal component analysis 
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Notes

Acknowledgments

This research was supported by the Focus of Excellence grant “Biometrics of EvoDevo” from the Faculty of Life Sciences, University of Vienna, to Philipp Mitteroecker, and Grant DEB-1019583 to Fred Bookstein and Joseph Felsenstein from the National Sciences Foundation of the United States. We thank Katharina Puschnig for drawing the skull used in Figs. 6 and 8.

References

  1. Anderson, T. W. (1958). An introduction to multivariate statistical analysis. New York: Wiley.Google Scholar
  2. Anderson, T. W. (1963). Asymptotic theory for principal component analysis. The Annals of Mathematical Statistics, 34, 122–148.CrossRefGoogle Scholar
  3. Atchley, W., & Rutledge, J. (1980). Genetic components of size and shape. I. Dynamics of components of phenotypic variability and covariablity during ontogeny in the laboratory rat. Evolution, 34, 1161–1173.CrossRefGoogle Scholar
  4. Badyaev, A. V., & Foresman, K. R. (2004). Evolution of morphological integration. I. Functional units channel stress-induced variation in shrew mandibles. The American Naturalist, 163, 868–879.PubMedCrossRefGoogle Scholar
  5. Bookstein, F. L. (1991). Morphometric tools for landmark data: Geometry and biology. Cambridge: Cambridge University Press.Google Scholar
  6. Bookstein, F. L. (2014). Measuring and reasoning: Numerical inferences in the sciences. Cambridge: Cambridge University Press.Google Scholar
  7. Bookstein, F.L., Connor, P.D., Huggins, J.E., Barr, H. M., Pimentel, K. D., & Streissguth, A. P. (2007). Many infants prenatally exposed to high levels of alcohol show one particular anomaly of the corpus callosum. Alcoholism: Clinical and Experimental Research, 31, 868–879.CrossRefGoogle Scholar
  8. Bookstein, F. L., Streissguth, A. P., Sampson, P. D., Connor, P. D., & Bar, H. M. (2002). Corpus callosum shape and neuropsychological deficits in adult males with heavy fetal alcohol exposure. Neuroimage, 15, 233–251.PubMedCrossRefGoogle Scholar
  9. Cheverud, J. M. (1988). A comparison of genetic and phenotypic correlations. Evolution, 42, 958–968.CrossRefGoogle Scholar
  10. Coquerelle, M., Bookstein, F. L., Braga, J., Halazonetis, D. J., Weber, G. W., & Mitteroecker, P. (2011). Sexual dimorphism of the human mandible and its association with dental development. American Journal of Physical Anthropology, 145, 192–202.PubMedCrossRefGoogle Scholar
  11. Debat, V., & David, P. (2001). Mapping phenotypes: Canalization, plasticity and developmental stability. Trends in Ecology & Evolution, 16, 555–561.CrossRefGoogle Scholar
  12. Falconer, D. S., & Mackay, T. F. C. (1996). Introduction to quantitative genetics. Essex: Longman.Google Scholar
  13. Felsenstein, J. (1985). Phylogenies and the comparative method. American Naturalist, 125, 1–15.CrossRefGoogle Scholar
  14. Felsenstein, J. (1988). Phylogenies and quantitative characters. Annual Review of Ecology, Evolution, and Systematics, 19, 445–471.CrossRefGoogle Scholar
  15. Flury, B. N. (1983). Some relations between the comparison of covariance matrices and principal component analysis. Computational Statistics & Data Analysis, 1, 97–109.CrossRefGoogle Scholar
  16. Flury, B. N. (1985). Analysis of linear combinations with extreme ratios of variance. Journal of the American Statistical Association, 80, 915–922.CrossRefGoogle Scholar
  17. Förstner W., & Moonen, B. (1999). A metric for covariance matrices. In: F. Krumm, V. S. Schwarze (Eds.), Quo vadis geodesia ...?, Festschrift for Erik W. Grafarend on the occasion of his 60th birthday. Stuttgart: Stuttgart University.Google Scholar
  18. Gibson, G., & Wagner, G. (2000). Canalization in evolutionary genetics: A stabilizing theory? Bioessays, 22, 372–380.Google Scholar
  19. Hallgrimsson, B., Brown, J. J., Ford-Hutchinson, A. F., Sheets, H. D., Zelditch, M. L., & Jirik, F. R. (2006). The brachymorph mouse and the developmental-genetic basis for canalization and morphological integration. Evolution & Development, 8, 61–73.CrossRefGoogle Scholar
  20. Hallgrimsson, B., & Hall, B. K. (2005). Variation: A central concept in biology. New York: Elsevier Academic Press.Google Scholar
  21. Hallgrimsson, B., Willmore, K., & Hall, B. K. (2002). Canalization, developmental stability, and morphological integration in primate limbs. American Journal of Physical Anthropology Supplement, 35, 131–158.CrossRefGoogle Scholar
  22. Hansen, T. F., & Houle, D. (2008). Measuring and comparing evolvability and constraint in multivariate characters. Journal of Evolutionary Biology, 21, 1201–1219.PubMedCrossRefGoogle Scholar
  23. Houle, D., & Fierst, J. (2013). Properties of spontaneous mutational variance and covariance for wing size and shape in Drosophila melanogaster. Evolution, 67, 1116–1130.PubMedCrossRefGoogle Scholar
  24. Howells, W. W. (1996). Howells craniometric data on the internet. American Journal of Physical Anthropology, 101, 441–442.PubMedCrossRefGoogle Scholar
  25. Huttegger, S., & Mitteroecker, P. (2011). Invariance and meaningfulness in phenotype spaces. Evolutionary Biology, 38, 335–352.CrossRefGoogle Scholar
  26. Klingenberg, C. P., Debat, V., & Roff, D. A. (2010). Quantitative genetics of shape in cricket wings: Developmental integration in a functional structure. Evolution, 64, 2935–2951.PubMedGoogle Scholar
  27. Koots, K. R., & Gibson, J. P. (1996). Realized sampling variances of estimates of genetic parameters and the difference between genetic and phenotypic correlations. Genetics, 143:1409–1416.PubMedCentralPubMedGoogle Scholar
  28. Lande, R. (1979). Quantitative genetic analysis of multivariate evolution, applied to brain: Body size allometry. Evolution, 33, 402–416.CrossRefGoogle Scholar
  29. Manly, B. F. J., & Rayner, J. C. W. (1987). The comparison of sample covariance matrices using likelihood ratio tests. Biometrika, 74, 841–847.CrossRefGoogle Scholar
  30. Mardia, K. V., Kent, J. T., & Bibby, J. M. (1979). Multivariate analysis. London: Academic Press.Google Scholar
  31. Martin, G., Chapuis, E., & Goudet, J. (2008). Multivariate QST-FST comparisons: A neutrality test for the evolution of the g matrix in structured populations. Genetics, 180, 2135–2149.PubMedCentralPubMedCrossRefGoogle Scholar
  32. Mitteroecker, P. (2009). The developmental basis of variational modularity: Insights from quantitative genetics, morphometrics, and developmental biology. Evolutionary Biology, 36, 377–385.CrossRefGoogle Scholar
  33. Mitteroecker, P., & Bookstein, F. L. (2009). The ontogenetic trajectory of the phenotypic covariance matrix, with examples from craniofacial shape in rats and humans. Evolution, 63, 727–737.PubMedCrossRefGoogle Scholar
  34. Mitteroecker, P., & Bookstein, F. L. (2011). Classification, linear discrimination, and the visualization of selection gradients in modern morphometrics. Evolutionary Biology, 38, 100–114.CrossRefGoogle Scholar
  35. Mitteroecker, P., Gunz, P., Neubauer, S., & Müller, G. B. (2012). How to explore morphological integration in human evolution and development? Evolutionary Biology, 39, 536–553.CrossRefGoogle Scholar
  36. Mitteroecker, P., & Huttegger, S. (2009). The concept of morphospaces in evolutionary and developmental biology: Mathematics and metaphors. Biological Theory, 4, 54–67.CrossRefGoogle Scholar
  37. Morrison, D. F. (1976). Multivariate statistical methods. New York: McGraw-Hill.Google Scholar
  38. Nonaka, K., & Nakata, M. (1984). Genetic variation and craniofacial growth in inbred rats. Journal of Craniofacial Genetics and Developmental Biology, 4, 271–302.PubMedGoogle Scholar
  39. Philipps, P. C., & Arnold, S. J. (1989). Visualizing multivariate selection. Evolution, 43, 1209–1222.CrossRefGoogle Scholar
  40. Rao, C. R. (1948). The utilization of multiple measurements in problems of biological classification. Journal of the Royal Statistical Society. Series B, 10, 159–203.Google Scholar
  41. Roff, D. (1995). The estimation of genetic correlations from phenotypic correlations: A test of Cheverud’s conjecture. Heredity, 74, 481–490.CrossRefGoogle Scholar
  42. Roff, D. (2000). The evolution of the G matrix: Selection or drift? Heredity (Edinb), 84, 135–142.CrossRefGoogle Scholar
  43. Smith, S. T. (2005). Covariance, subspace, and intrinsic Cramer-Rao bounds. IEEE Transactions on Signal Processing, 53, 1610–1630.CrossRefGoogle Scholar
  44. Tanner, J. M. (1963). Regulation of growth in size in mammals. Nature, 199, 845–850.PubMedCrossRefGoogle Scholar
  45. Tyler, D. E., Critchley, F., Dümbgen, L., & Oja, H. (2009). Invariant co-ordinate selection. Journal of the Royal Statistical Society: Series B, 71, 549–592.CrossRefGoogle Scholar
  46. Zelditch, M. L., Bookstein, F. L., & Lundrigan, B. (1992). Ontogeny of integrated skull growth in the cotton rat Sigmodon fulviventer. Evolution, 46, 1164–1180.CrossRefGoogle Scholar
  47. Zelditch, M. L., Lundrigan, B. L., & Garland, T. (2004). Developmental regulation of skull morphology. I. Ontogenetic dynamics of variance. Evolution & Development, 6, 194–206.CrossRefGoogle Scholar
  48. Zelditch, M. L., Mezey, J. G., Sheets, H. D., Lundrigan, B. L., & Garland, T. (2006). Developmental regulation of skull morphology II: Ontogenetic dynamics of covariance. Evolution & Development, 8, 46–60.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of AnthropologyUniversity of ViennaViennaAustria
  2. 2.Department of StatisticsUniversity of WashingtonSeattleUSA
  3. 3.Department of Theoretical BiologyUniversity of ViennaViennaAustria

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