The effect of the H −1 scaling factors τ and ω on the structure of H in the single-step procedure

Abstract

Abstract Background The single-step covariance matrix H combines the pedigree-based relationship matrix $${\mathbf {A}}$$ A with the more accurate information on realized relatedness of genotyped individuals represented by the genomic relationship matrix $${\mathbf {G}}$$ G . In particular, to improve convergence behavior of iterative approaches and to reduce inflation, two weights $$\tau$$ τ and $$\omega$$ ω have been introduced in the definition of $${\mathbf {H}}^{-1}$$ H-1 , which blend the inverse of a part of $${\mathbf {A}}$$ A with the inverse of $${\mathbf {G}}$$ G . Since the definition of this blending is based on the equation describing $${\mathbf {H}}^{-1}$$ H-1 , its impact on the structure of $${\mathbf {H}}$$ H is not obvious. In a joint discussion, we considered the question of the shape of $${\mathbf {H}}$$ H for non-trivial $$\tau$$ τ and $$\omega$$ ω . Results Here, we present the general matrix $${\mathbf {H}}$$ H as a function of these parameters and discuss its structure and properties. Moreover, we screen for optimal values of $$\tau$$ τ and $$\omega$$ ω with respect to predictive ability, inflation and iterations up to convergence on a well investigated, publicly available wheat data set. Conclusion Our results may help the reader to develop a better understanding for the effects of changes of $$\tau$$ τ and $$\omega$$ ω on the covariance model. In particular, we give theoretical arguments that as a general tendency, inflation will be reduced by increasing $$\tau$$ τ or by decreasing $$\omega$$ ω .