An analytical framework to derive the expected precision of genomic selection

Jean-Michel Elsen

Formulae to predict the precision or accuracy of genomic estimated breeding values (GEBV) are important when modelling selection schemes. Simple versions of such formulae have been proposed in the past, based on a number of simplifying hypotheses, including absence of linkage disequilibrium and linkage between loci, a population made up of unrelated individuals, and that all genetic variability of the trait is explained by the genotyped loci. These formulae were based on approximations that were not always clear. The objective of this paper is to offer a unique framework to derive equations that predict the precision of GEBV from the size of the reference population and the heritability of and number of QTL controlling the quantitative trait. The exact formulation of the precision of GEBV involves the expectation of the inverse of a linear function of the genomic matrix, which cannot be calculated from simple algebra but can be approximated using a Taylor polynomial expansion. First order approximations performed better than the initial prediction equations published in the literature. Second order approximations produced almost perfect estimates of precision when compared to results obtained when simulating situations that agreed with the assumptions that were required to derive the precision equations. Using this proposed framework, we present several generalizations, including multi-trait genomic evaluation. Although further improvements are needed to account for the complexity of practical situations, the equations proposed here can be used to derive the precision of GEBV when comparing breeding schemes a priori.