A BLUP derivation of the multivariate breeder’s equation, with an elucidation of errors in BLUP variance estimates, and a prediction method for inbred populations

Abstract

The multivariate breeder's equation, Lande (1979), was derived from the Price equation Price (1970,1972). Here, I present a derivation based on the BLUP (best linear unbiased predictions) equations in matrix form, first given in summation form by Henderson (1950). The derivation makes use of a comparison with the known form of the multivariate breeder's equation, and it is therefore not an independent derivation. The alternative derivation does, however, clarify why and to which extent the variances of BLUP predictions of random effects are underestimated. The BLUP random effects can in fact be used for prediction of phenotypic responses by use of the Robertson-Price identity, Robertson (1966), Price (1970,1972). The BLUP derivation also leads to a prediction method for populations with inbreeding, i.e., where the additive genetic relationship matrix departs from a unity matrix, and where the multivariate breeder's equation therefore will fail.