Hierarchical comparison of genetic variance-covariance matrices. I. Using the Flury hierarchy

The comparison of additive genetic variance-covariance matrices (G-matrices ) is an increasingly popular exercise in evolutionary biology because the e volution of the C-matrix is central to the issue of persistence of genetic constraints and to the use of dynamic models in an evolutionary time frame. The comparison of G-matrices is a nontrivial statistical problem because f amily structure induces nonindependence among the elements in each matrix. Past solutions to the problem of G-matrix comparison have dealt with this p roblem, with varying success, but have tested a single null hypothesis (mat rix equality or matrix dissimilarity). Because matrices can differ in many ways, several hypotheses are of interest in matrix comparisons. Flury (1988 ) has provided an approach to matrix comparison in which a variety of hypot heses are tested, including the two extreme hypotheses prevalent in the evo lutionary literature. The hypotheses are arranged in a hierarchy and involv e comparisons of both the principal components (eigenvectors) and eigenvalu es of the matrix. We adapt Flury's hierarchy of tests to the problem of com paring G-matrices by using randomization testing to account for nonindepend ence induced by family structure. Software has been developed for carrying out this analysis for both genetic and phenotypic data. The method is illus trated with a garter snake test case.