Measuring Evolutionary Constraints Through the Dimensionality of the Phenotype: Adjusted Bootstrap Method to Estimate Rank of Phenotypic Covariance Matrices

Abstract

The potential and direction of phenotypic evolution is constrained by the distribution of genetic variation for the traits as described by the phenotypic (P) and genetic covariance matrices (G). The rank of the covariance matrix reflects the number of independent variational dimensions of the phenotype. Covariance matrices with less than full rank indicate lack of variation in some directions of the phenotype space and thus are an indication of absolute evolutionary constraints. Because selection acts upon phenotypic variation, the rank of P represents the upper limit of the dimensionality in G, relevant for selection response. The limitations of current methods to estimate matrix rank motivated us to analyze and adjust a bootstrap method and evaluate its performance by simulation. The results show that the modified bootstrap method (ABRE) gives reliable and rather conservative rank estimates when the sample size is sufficient for the number of variables studied (the sample size is at least five-fold the number of variables). Applying the method to various datasets suggests high phenotypic dimensionality in all cases. The analysis thus provides no evidence for absolute evolutionary constraints.

Appendix 1

The Comparison of the Results of Rank Estimation by Factor Analysis (FA) and ABRE

Fifteen subsets of six traits have been randomly drawn from the lateral dataset s of Drosophila wing data (A) and tamarin cranial data (B). The trait subsets 1L-15L designate the traits drawn from the left side, 1R-15R those on the right side. Both sides were drawn independently, thus, e.g., 2L doesn’t correspond to 2R trait combination. For each trait combination, the rank was estimated by FA and ABRE. Whereas FA requires multiple measurements on all individuals, ABRE was run in several versions, in which measurement error was estimated on different proportion of the total sample size. As the choice of individuals included in subsample can affect the estimate of error, we repeatedly drew 30 subsamples of the same size and the rank estimate was averaged across them (hence decimal values). ‘no conv.’ refers to the trait combinations in which FA did not reach convergence. Note that the results are consistent between FA and ABRE. The effect of estimating measurement error on low number of individuals in ABRE can be seen for 10 individuals, but is minimal for these phenotypic examples (see Table 2).

Table 2