Comparing G matrices: are common principal components informative?

Common principal components (CPC) analysis is a technique for assessing whether variance-covariance matrices from different populations have similar structure. One potential application is to compare additive genetic variance-covariance matrices, G. In this article, the conditions under which G matrices are expected to have common PCs are derived for a two-locus, two-allele model and the model of constrained pleiotropy. The theory demonstrates that whether G matrices are expected to have common PCs is largely determined by whether pleiotropic effects have a modular organization. If two (or more) populations have modules and these modules have the same direction, the G matrices have a common PC, regardless of allele frequencies. In the absence of modules, common PCs exist only for very restricted combinations of allele frequencies. Together, these two results imply that, when populations are evolving, common PCs are expected only when the populations have modules in common. These results have two implications: (1) In general, G matrices will not have common PCs, and (2) when they do, these PCs indicate common modular organization. The interpretation of common PCs identified for estimates of G matrices is discussed in light of these results.