On the eigenvalue distribution of genetic and phenotypic dispersion matrices: Evidence for a nonrandom organization of quantitative character variation

A quantitative genetic model of random pleiotropy is introduced as reference model for detecting the kind and degree of organization in quantitative genetic variation. In this model the genetic dispersion matrix takes the form of G = BB^T, where B is a general, real, Gaussian random matrix. The eigenvalue density of the corresponding ensemble of random matrices (_G) is considered. The first two moments are derived for variance-covariance matrices G as well as for correlation matrices R, and an approximate expression of the density function is given. The eigenvalue distribution of all empirical correlation matrices deviates from that of a random pleiotropy model by a very large leading eigenvalue associated with a size factor. However the frequency-distribution of the remaining eigenvalues shows only minor deviations in mammalian skeletal data. A prevalence of intermediate eigenvalues in insect data may be caused by the inclusion of many functionally unrelated characters. Hence two kinds of deviations from random organization have been found: a mammal like and an insect like organization. It is concluded that functionally related characters are on the average more tightly correlated than by chance (= mammal like organization), while functionally unrelated characters appear to be less correlated than by random pleiotropy (insect like organization).