In order to benefit from the theoretical advantage of the BLUP animal model, an across-herd evaluation system was developed which reduces computation cost for a routine application to a large horse population (80 000 horses and 4000 stallions). The model used for the log of annual earnings includes an individual additive genetic component, permanent environmental and herd × dam effects considered as random and sex age × year subclass effects as fixed. This herd × dam effect is caused by the difference between the maternal and paternal components and represents the common environmental conditions shared by progeny of the same mother. The variance-covariance matrix includes the complete relationship matrix between all the animals whatever the kind of animal: recorded or non-recorded; parent or non-parent horses; males or females. The system of mixed-model equations is solved by using a particular structure of the inverse of the relationship matrix. Animals with no progeny and mares are absorbed. The resulting system (fixed effects and stallions) is solved by iteration. An approximation of the coefficient of determination is then calculated by using matrix approximations. This method is of very great practical interest in horse populations, where rigorous selection plans are rare. © 1988.
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