Comparing G: multivariate analysis of genetic variation in multiple populations

The additive genetic variance–covariance matrix (G) summarizes themultivariate genetic relationships among a set of traits. The geometry of Gdescribes the distribution of multivariate genetic variance, and generates geneticconstraints that bias the direction of evolution. Determining if and how the multivariategenetic variance evolves has been limited by a number of analytical challenges incomparing G-matrices. Current methods for the comparison of G typically shareseveral drawbacks: metrics that lack a direct relationship to evolutionary theory, theinability to be applied in conjunction with complex experimental designs, difficultieswith determining statistical confidence in inferred differences and an inherentlypair-wise focus. Here, we present a cohesive and general analytical framework for thecomparative analysis of G that addresses these issues, and that incorporates andextends current methods with a strong geometrical basis. We describe the application ofrandom skewers, common subspace analysis, the 4th-order genetic covariance tensor and thedecomposition of the multivariate breeders equation, all within a Bayesian framework. Weillustrate these methods using data from an artificial selection experiment on eighttraits in Drosophila serrata, where a multi-generational pedigree was availableto estimate G in each of six populations. One method, the tensor, elegantlycaptures all of the variation in genetic variance among populations, and allows theidentification of the trait combinations that differ most in genetic variance. The tensorapproach is likely to be the most generally applicable method to the comparison ofG-matrices from any sampling or experimental design.