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

Background The single-step covariance matrix H combines the pedigree-based relationship matrix 𝐀 with the more accurate information on realized relatedness of genotyped individuals represented by the genomic relationship matrix 𝐆 . In particular, to improve convergence behavior of iterative approaches and to reduce inflation, two weights τ and ω have been introduced in the definition of 𝐇^-1 , which blend the inverse of a part of 𝐀 with the inverse of 𝐆 . Since the definition of this blending is based on the equation describing 𝐇^-1 , its impact on the structure of 𝐇 is not obvious. In a joint discussion, we considered the question of the shape of 𝐇 for non-trivial τ and ω . Results Here, we present the general matrix 𝐇 as a function of these parameters and discuss its structure and properties. Moreover, we screen for optimal values of τ and 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 τ and ω on the covariance model. In particular, we give theoretical arguments that as a general tendency, inflation will be reduced by increasing τ or by decreasing ω .