Quadratic Components of Covariance of Inbred Relatives and Their Estimation in Naturally Self-Pollinated Species
Several expressions for the covariances of inbred relatives, under assumptions of equilibrium genotypic frequencies in the initial population, no linkage and no epistasis, have been presented in the literature. These expressions in multiple-allelic cases involve five quadratic components that must be estimated if certain genetic variances and covariances are to be estimated. Four of the five components in one parameterization, the D-model, are involved in a linear dependency and, therefore, nonestimable in experiments in which only selfed progenies are evaluated. If all five components are to be estimated in a self-pollinated species, selfed progeny must be augmented with crossed progeny or with selfed progeny derived from crosses among individuals in the base population. A mating scheme that could be implemented with a minimum of crossing in a selfpollinated species is proposed, along with formulas for the covariances of relatives involved. Two alternative parameterizations, the C-model and Q-model, are considered. In the C-model σ2A and σ2D are not estimable, but the total genetic variance at panmixia σ2G0 = σ2A + σ2Dis estimable. The Q-model, derived by subdividing σ2A into three particular functions of contrasts of genotypic values, allows three subcomponents of σ2A0 ago to be estimated. The models reduced to four parameters are adequate if only covariances of relatives under self-fertilization need to be estimated, but reduction to four parameters is neither adequate nor possible under some other inbreeding system, mixture of inbreeding systems, or mixture of selfing and crossing.
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