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.