Prediction error variance and restricted maximum likelihood estimation for animal model with relationship grouping

Genetic evaluation using animal model with relationship grouping has been shown to be feasible. However, algorithms were unavailable for prediction error variance and REML estimation of variance components. This paper shows that prediction error variance of an estimable function of the total merit of additive genetic and group effects is a simple function of a generalized inverse of the coefficient matrix for a transformed mixed model equation or of the inverse of the coefficient matrix when it is restricted to full rank. The REML algorithms, using the transformed equation, having slightly more complicated expressions than usual but could be more feasible computationally. Formulae for prediction error variance apply in general. The REML algorithms are extended to an animal model with an arbitrary number of random factors and can be extended to estimate covariance components.