Reduced Animal Models Fitting Only Equations for Phenotyped Animals

Reduced models are equivalent models to the full model that enable reduction in the computational demand for solving the problem, here, mixed model equations for estimating breeding values of selection candidates. Since phenotyped animals provide data to the model, the aim of this study was to reduce animal models to those equations corresponding to phenotyped animals. Non-phenotyped ancestral animals have normally been included in analyses as they facilitate formation of the inverse numerator relationship matrix. However, a reduced model can exclude those animals and obtain identical solutions for the breeding values of the animals of interest. Solutions corresponding to non-phenotyped animals can be back-solved from the solutions of phenotyped animals and specific blocks of the inverted relationship matrix. This idea was extended to other forms of animal model and the results from each reduced model (and back-solving) were identical to the results from the corresponding full model. Previous studies have been mainly focused on reduced animal models that absorb equations corresponding to non-parents and solve equations only for parents of phenotyped animals. These two types of reduced animal model can be combined to formulate only equations corresponding to phenotyped parents of phenotyped progeny.