Methods for detecting 'missing' dimensions in genetic covariance matrices

Blows and Hoffmann (2005) and others have suggested that low levels of genetic variation in some dimensions of an additive genetic variance-covariance matrix (G) will be detectable as eigenvalues approaching zero. These dimensions may be indicative of trade-offs (Blows and Hoffmann 2005) or, potentially, of "holes" in fitness landscapes (Dochtermann et al. 2023). However, because the estimation of G typically constrains variances to be positive, and matrices to therefore be positive definite, "approaching zero" is challenging to statistically define. It is, therefore, not currently clear how to statistically identify dimensions with effectively zero variances. Being able to identify dimensions of G with variances approaching zero would improve our ability to understand trade-offs which, typically and inappropriately, focus on the signs of bivariate correlations (Houle 1991). Viable approaches would also allow better understanding of the structure of G and the evolutionary processes that have shaped G across taxa. Mezey and Houle (2005) and Kirkpatrick and Lofsvold (1992) used matrix rank to determine if there were dimensions lacking variation. Rank is the number of eigenvalues for a matrix that are greater than zero. If a Gs rank is less than its dimensionality, this would be evidence that there are dimensions without variation. However, most contemporary analysis procedures force estimates of G to be positive semi-definite. Consequently, estimated Gs will not have eigenvalues of 0 or negative. Therefore, the estimated eigenvalues on their own will not allow the identification of absolute constraints (sensu (Houle 2001)) or more generally dimensions lacking variation. One alternative would be to compare the eigenvalues of observed matrices to those of random matrices and, if there were dimensions with less variation than expected by chance, this comparison could be used to identify dimensions in which evolution is constrained. If trade-offs or landscape holes result in dimensions of G that have minimal variance, we would expect that the associated eigenvalues exhibit less variation than expected by chance. Here, I tested whether the distribution of eigenvalues from randomly generated correlated matrices can be used for such testing.