COMPARISON OF QUANTITATIVE GENETIC PARAMETERS.

Shaw (1991) has described a maximum-likelihood procedure for comparing genetic covariance matrices between populations. While her efforts to expand the procedures available for quantitative genetic analyses are laudable, her discussion is incomplete in several respects. For example, consider her criticism of Dietz's matrix permutation test (Dietz, 1983) in favor of her own maximum-likelihood approach. Shaw contends (p. 149) that a "further problem with the matrix permutation test is that its null hypothesis, that the corresponding elements of two matrices are independent (Dietz, 1983), is inappropriate here." Shaw goes on to say that this "is not the hypothesis of interest" and she concludes that the "natural null hypothesis of interest is that the matrices (or some defined subset of the matrices) are the same for both groups." Clearly, one comparison of interest is to test the hypothesis that genetic covariance matrices are equal between populations or species and Shaw's procedure has merit in this case. However, other meaningful hypotheses should also be considered. Suppose one wants to compare covariance matrices for two species that differ greatly in body size, e.g., rats and mice (Kohn and Atchley, 1988; Atchley et al., 1992). The a priori expectation is that covariance matrices for the two species will differ simply because the variance (or covariance) commonly scales with the mean(s). Hence, rejecting the null hypothesis of equal genetic covariance matrices simply because the taxa differ in overall size tells us nothing of biological or evolutionary interest. While it is trivial to adjust different groups to a common mean vector, it is a significant problem to assess how size-adjustment for one or more variables affects the covariances among a suite of characters. Similarly, it can be difficult to find a single transformation that will eliminate scaling between variance and mean. Our experience is that a simple log-transformation is often too severe and will actually yield a negative association between variance and mean (e.g., Cowley et al., 1986, p. 552). Using a simple two variable example we can expand the question about homogeneity of covariance matrices to a set of specific statistical hypotheses. First, G and G2 might differ in the relative magnitude of variances, resulting in differences in inflation or contraction of the respective bivariate ellipses for the populations. Second, G l and G2 might differ in the sign and/or magnitude of the covariances, rather than in the variances. In this case, differences would exist in the orientation of the respective bivariate ellipses. Indeed, this approach can be expanded to a hierarchy of hypothesis tests about covariance structures (Flury, 1987), including the equality of covariance matrices for k groups (H ), proportionality of all k covariance matrices (H2), a common principal component structure for the k matrices (H3), the first several principal components are common between the k covariance matrices (H4), and the k covariance matrices are arbitrary (random) with respect to one another (H5). Notice that this hierarchical set of hypotheses differs from the set of nested comparisons Shaw advocates. Shaw (1991, p. 148) proposed that when the null hypothesis of equality of genetic covariance matrices is rejected, then one might want to explore whether or not the additive genetic variances and/or covariances for some subset of traits might be equal. Shaw suggested that in this way, an investigator might be able to isolate elements in which two genetic covariance matrices differ. In the hierarchical system of hypothesis testing we have outlined above, Shaw's maximumlikelihood comparison is a test of H, while Dietz's matrix permutation comparison is a test of H5. Whereas Shaw suggests that H, is the only hypothesis of interest to evolutionary biologists (p. 149), rejection of H leads naturally to evolutionary questions about the similarity of pattern in genetic covariance matrices. In our opinion, Shaw's condemnation of Dietz's test is misleading since if one fails to reject the null hypothesis of randomness inherent in Dietz's test (H5), how can H, be tenable? On the other hand, the information content of the matrix permutation test is very limited. For example, when H5 is rejected with Dietz's test, one cannot ascertain if the covariance matrices are equal, proportional or merely similar in pattem to one another. There is also an issue of statistical power which is different for Shaw's maximum-likelihood test and for the permutation test. In Shaw's test, one might have low statistical power and fail to reject the null hypothesis because the sample size is small. Small sample size would result in the individual elements of the covariance matrices being estimated poorly. With the permutation test, there is an additional issue conceming statistical power. The permutation test does not consider the precision with which the elements of the covariance matrices are estimated. Hence, applying the permutation test to matrices with poorly-estimated elements produces meaningless results. Even when one has a matrix with well-estimated elements, the matrix permutation test will have low statistical power if it is applied to small matrices (less than about five by five) since the number of possible permutations (n!) is small.