, Volume 166, Issue 4, pp 961–971 | Cite as

Size correction in biology: how reliable are approaches based on (common) principal component analysis?

Population ecology - Original Paper


Morphological traits typically scale with the overall body size of an organism. A meaningful comparison of trait values among individuals or populations that differ in size therefore requires size correction. A frequently applied size correction method involves subjecting the set of n morphological traits of interest to (common) principal component analysis [(C)PCA], and treating the first principal component [(C)PC1] as a latent size variable. The remaining variation (PC2–PCn) is considered size-independent and interpreted biologically. I here analyze simulated data and natural datasets to demonstrate that this (C)PCA-based size correction generates systematic statistical artifacts. Artifacts arise even when all traits are tightly correlated with overall size, and they are particularly strong when the magnitude of variance is heterogeneous among the traits, and when the traits under study are few. (C)PCA-based approaches are therefore inappropriate for size correction and should be abandoned in favor of methods using univariate general linear models with an adequate independent body size metric as covariate. As I demonstrate, (C)PC1 extracted from a subset of traits, not themselves subjected to size correction, can provide such a size metric.


Bias Body size Morphology Multivariate statistics Shape 



This paper benefited greatly from comments and suggestions by F. James Rohlf, Ben Bolker, Thom DeWitt, Dan Bolnick, Dolph Schluter, Pedro Peres-Neto, and an anonymous reviewer. Dan Bolnick kindly shared his stickleback data, and Philadelphia Airport provided electricity and a rocking chair. Funding was provided by an Ambizione fellowship from the Swiss National Science Foundation (grant PZ00P3_126391/1), and by the Research Fund of the University of Basel.

Supplementary material

442_2011_1934_MOESM1_ESM.doc (44 kb)
Supplementary material 1 (DOC 44 kb)


  1. Albrecht GH, Gelvin BR, Hartman SE (1993) Ratios as a size adjustment in morphometrics. Am J Phys Anthropol 91:441–468PubMedCrossRefGoogle Scholar
  2. Badyaev AV, Hill GE (2000) The evolution of sexual dimorphism in the house finch. I. Population divergence in morphological covariance structure. Evolution 54:1784–1794PubMedGoogle Scholar
  3. Berner D, Adams DC, Grandchamp A-C, Hendry AP (2008) Natural selection drives patterns of lake-stream divergence in stickleback foraging morphology. J Evol Biol 21:1653–1665PubMedCrossRefGoogle Scholar
  4. Berner D, Stutz WE, Bolnick DI (2010) Foraging trait (co)variances in stickleback evolve deterministically and do not predict trajectories of adaptive diversification. Evolution 64:2265–2277PubMedGoogle Scholar
  5. Bolnick DI (2004) Can intraspecific competition drive disruptive selection? An experimental test in natural populations of sticklebacks. Evolution 58:608–618PubMedGoogle Scholar
  6. Bolnick DI, Lau OL (2008) Predictable patterns of disruptive selection in stickleback in postglacial lakes. Am Nat 172:1–11PubMedCrossRefGoogle Scholar
  7. Burnaby TP (1966) Growth-invariant discriminant functions and generalized distances. Biometrics 22:96–110CrossRefGoogle Scholar
  8. Cotton S, Fowler K, Pomiankowski A (2004) Condition dependence of sexual ornament size and variation in the stalk-eyed fly Cyrtodiopsis dalmanni (Diptera : Diopsidae). Evolution 58:1038–1046PubMedGoogle Scholar
  9. Darlington RB, Smulders TV (2001) Problems with residual analysis. Anim Behav 62:599–602CrossRefGoogle Scholar
  10. Flury BD (1988) Common principal components and related multivariate models. Wiley, New YorkGoogle Scholar
  11. Fox CW, Wolf JB (2006) Evolutionary genetics: concepts and case studies. Oxford University Press, New YorkGoogle Scholar
  12. Garcia-Berthou E (2001) On the misuse of residuals in ecology: testing regression residuals vs. the analysis of covariance. J Anim Ecol 70:708–711CrossRefGoogle Scholar
  13. Hansen TF, Houle D (2008) Measuring and comparing evolvability and constraint in multivariate characters. J Evol Biol 21:1201–1219PubMedCrossRefGoogle Scholar
  14. Holtmeier CL (2001) Heterochrony, maternal effects, and phenotypic variation among sympatric pupfishes. Evolution 55:330–338PubMedGoogle Scholar
  15. Houle D (1992) Comparing evolvability and variability of quantitative traits. Genetics 130:195–204PubMedGoogle Scholar
  16. Hoverman JT, Relyea RA (2008) Temporal environmental variation and phenotypic plasticity: a mechanism underlying priority effects. Oikos 117:23–32CrossRefGoogle Scholar
  17. Hulsey CD, Mims MC, Streelman JT (2007) Do constructional constraints influence cichlid craniofacial diversification? Proc R Soc Lond B 274:1867–1875CrossRefGoogle Scholar
  18. Humphries JM, Bookstein FL, Chernoff B, Smith GR, Elder RL, Poss SG (1981) Multivariate discrimination by shape in relation to size. Syst Zool 30:291–308CrossRefGoogle Scholar
  19. Jockusch EL (1997) Geographic variation and phenotypic plasticity of number of trunk vertebrae in slender salamanders, Batrachoseps (Caudata : Plethodontidae). Evolution 51:1966–1982CrossRefGoogle Scholar
  20. Jolicoeur P (1963) The multivariate generalization of the allometry equation. Biometrics 19:497–499CrossRefGoogle Scholar
  21. Jungers WL, Falsetti AB, Wall CE (1995) Shape, relative size, and size-adjustments in morphometrics. Am J Phys Anthropol 38:137–161CrossRefGoogle Scholar
  22. Klingenberg CP (1996) Multivariate allometry. In: Marcus LF, Corti M, Loy A, Naylor GJP, Slice DE (eds) Advances in morphometrics. Plenum, New York, pp 23–49Google Scholar
  23. Lande R, Arnold SJ (1983) The measurement of selection on correlated characters. Evolution 37:1210–1226CrossRefGoogle Scholar
  24. Leon-Garcia A (2008) Probability, statistics, and random processes for electrical engineering, 3rd edn. Pearson Education, Upper Saddle RiverGoogle Scholar
  25. Losos JB, Jackman TR, Larson A, de Queiroz K, Rodriguez-Schettino L (1998) Contingency and determinism in replicated adaptive radiations of island lizards. Science 279:2115–2118PubMedCrossRefGoogle Scholar
  26. Lynch M, Walsh B (1998) Genetics and analysis of quantitative traits. Sinauer, SunderlandGoogle Scholar
  27. McCoy MW, Bolker BM (2008) Trait-mediated interactions: influence of prey size, density and experience. J Anim Ecol 77:478–486PubMedCrossRefGoogle Scholar
  28. McCoy MW, Bolker BM, Osenberg CW, Miner BG, Vonesh JR (2006) Size correction: comparing morphological traits among populations and environments. Oecologia 148:547–554PubMedCrossRefGoogle Scholar
  29. McGuigan K, Franklin CE, Moritz C, Blows MW (2003) Adaptation of rainbow fish to lake and stream habitats. Evolution 57:104–118PubMedGoogle Scholar
  30. Merilä J, Björklund M (1999) Population divergence and morphometric integration in the greenfinch (Carduelis chloris)—evolution against the trajectory of least resistance? J Evol Biol 12:103–112CrossRefGoogle Scholar
  31. Mosimann JE, James FC (1979) New statistical methods for allometry with applications to Florida red-winged blackbirds. Evolution 33:444–459CrossRefGoogle Scholar
  32. Phillips PC, Arnold SJ (1999) Hierarchical comparisons of genetic variance-covariance matrices. I. Using the Flury hierarchy. Evolution 53:1506–1515CrossRefGoogle Scholar
  33. Pimentel RA (1979) Morphometrics. Kendall/Hunt, DubuqueGoogle Scholar
  34. R Development Core Team (2009) R: a language and environment for statistical computing. R Foundation for Statistical Computing, AustriaGoogle Scholar
  35. Reist JD (1985) An empirical evaluation of several univariate methods that adjust for size variation in morphometric data. Can J Zool 63:1429–1439CrossRefGoogle Scholar
  36. Reist JD (1986) An empirical evaluation of coefficients used in residual and allometric adjustment of size covariation. Can J Zool 64:1363–1368CrossRefGoogle Scholar
  37. Revell LJ, Johnson MA, Schulte JA, Kolbe JJ, Losos JB (2007) A phylogenetic test for adaptive convergence in rock-dwelling lizards. Evolution 61:2898–2912PubMedCrossRefGoogle Scholar
  38. Robinson BW, Schluter D (2000) Natural selection and the evolution of adaptive genetic variation in northern freshwater fishes. In: Mousseau A, Sinervo B, Endler JA (eds) Adaptive genetic variation in the wild. Oxford University Press, New York, pp 65–94Google Scholar
  39. Robinson BW, Wilson DS (1994) Character release and displacement in fishes—a neglected literature. Am Nat 144:596–627CrossRefGoogle Scholar
  40. Rohlf FJ, Bookstein FL (1987) A comment on shearing as a method for size correction. Syst Zool 36:356–367CrossRefGoogle Scholar
  41. Schluter D (1996) Adaptive radiation along genetic lines of least resistance. Evolution 50:1766–1774CrossRefGoogle Scholar
  42. Schluter D, McPhail JD (1992) Ecological character displacement and speciation in sticklebacks. Am Nat 140:85–108PubMedCrossRefGoogle Scholar
  43. Sneath RR, Sokal RR (1973) Numerical taxonomy. Freeman, San FranciscoGoogle Scholar
  44. Sprent P (1972) The mathematics of size and shape. Biometrics 28:23–37PubMedCrossRefGoogle Scholar
  45. Touchon JC, Warkentin KM (2010) Short- and long-term effects of the abiotic egg environment on viability, development and vulnerability to predators of a neotropical anuran. Funct Ecol 24:566–575CrossRefGoogle Scholar
  46. Zelditch ML, Swiderski DL, Sheets HD, Fink WL (2004) Geometric morphometrics for biologists. Elsevier, LondonGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Zoological InstituteUniversity of BaselBaselSwitzerland

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