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.