Properties of components of covariance of inbred relatives and their estimates in a maize population

Summary Properties of three parameterizations, denoted as the C-model, D-model and Q-model, for covariances of inbred relatives under assumptions of no linkage or epistasis are explored and compared. Additive variance in an inbred population with inbreeding coefficient F, ² _AF =(1+F) ² _A where ² _A is additive variance in a panmictic population, if Q-model parameters Q _xx and Q _xy are both zero. Conditions sufficient for this to hold are presented in terms of gene frequencies and dominance contrasts (homozygotes vs. heterozygotes). Some other properties and potential uses of estimates of components in the models are also discussed. Estimates of components in the D-model and Q-model were calculated from a maize (Zea mays L.) study from which estimates of components in the C-model were previously published. Of particular interest were the covariance (Q _xy ) of effects of alleles at complete homozygosity with inbreeding depression effects, the covariance (D ₁) of additive effects at panmixia with inbreeding depression effects and the within-locus variance (D ₂, alias Q _xx ) of inbreeding depression effects. Estimates of Q _xy , D ₁, and D ₂ were small and nonsignificant in most cases. For ear height in the second year of the study, D ₂ appeared to be a major component. In some cases, results were obtained which had contradictory implications (negative D ₂ coupled with positive Q _xy or D ₁, and positive D ₂ coupled with negative ² _D ). A negative estimate of one or the other of ² _D or ² _A was obtained in one of the two within-year analyses for every character. Problems in getting realistic results were thought to be owing to excessive multicollinearity among the coefficients of the components in the expectations of the covariances of the kinds of relatives included in the study. Implications for future studies of this kind are discussed.Journal Article No. 87-3-14 of the Kentucky Agricultural Experiment Station published with the approval of the Director