Estimation of phenotypic variance components from an inbred population part 1 elaboration of the model

Chevalet, C., 1976: Estimation of phenotypic variance components from an inbred population part 1 elaboration of the model. Annales de Genetique et de Selection Animale 8(2): 181-206

From the theoretical expressions of genetic variances and covariances among related individuals, a general statistical model may be derived for the estimation of variance components from data collected in an inbred flock. The general expression of the variance, covariance matrix of phenotypes is: .**GRAPHIC**. where (dk) are matrices of kinship, and identity coefficients, and (.theta.k) are the scalar components of variance to be estimated. The condition for existence of a set of sufficient and independent statistics is that, for any 2 sets .theta. and .theta.' of parameters, the identity: V(.theta.)V(.theta.') = V(.theta.')V(.theta.) holds. Except in the case of additive genic contributions, there is no general and efficient estimation. In all other cases a precise structuration of the mating scheme is necessary to obtain, at least, a partial reduction of information into independent statistics. Conditions are defined about the usual hierarchical design, that allow some simplifications in the numerical computations, and give efficient estimations. In these conditions are involved the values taken by identity coefficients, the distribution of these values within and between families; a partial equilibrium of data is also needed. Even in the additive case, such conditions are required, for the computations to be tractable. Uses of maximum likelihood estimators, and of minimum variance quadratic unbiased estimators are investigated and discussed.