There is more than one way to skin a G matrix

Because of its importance in directing evolutionary trajectories, there has been considerable interest in comparing variation among genetic variancecovariance (G) matrices. Numerous statistical approaches have been suggested but no general analysis of the relationship among these methods has previously been published. In this study, we used data from a half-sib experiment and simulations to explore the results of applying eight tests (T method, modified Mantel test, Bartletts test, Flury hierarchy, jackknife-manova, jackknife-eigenvalue test, random skewers, selection skewers). Whereas a randomization approach produced acceptable estimates, those from a bootstrap were typically unacceptable and we recommend randomization as the preferred method. All methods except the jackknife-eigenvalue test gave similar results although a fine-scale analysis suggested that the former group can be subdivided into two or possibly three groups, hierarchical tests, skewers and the rest (jackknife-manova, modified Mantel, T method, probably Bartletts). An advantage of the jackknife methods is that they permit tests of association with other factors, such as in this case, temperature and sex. We recommend applying all the tests described in this article, with the exception of the T method, and provide R functions for this purpose.