A Note on the Conditioning of the H^-1 Matrix Used in Single-Step GBLUP

The single-step genomic BLUP (ssGBLUP) is used worldwide for the simultaneous genetic evaluation of genotyped and non-genotyped animals. It is easily extendible to all BLUP models by replacing the pedigree-based additive genetic relationship matrix (A) with an augmented pedigree-genomic relationship matrix (H). Theoretically, H does not introduce any artificially inflated variance. However, inflated genetic variances have been observed due to the incomparability between the genomic relationship matrix (G) and A used in H. Usually, G is blended and tuned with A22 (the block of A for genotyped animals) to improve its numerical condition and compatibility. If deflation/inflation is still needed, a common approach is weighting G-1-A22-1 in the form of τG-1-ωA22-1, added to A-1 to form H-1. In some situations, this can violate the conditional properties upon which H is built. Different ways of weighting the H-1 components (A-1, G-1, A22-1, and H-1 itself) were studied to avoid/minimise the violations of the conditional properties of H. Data were simulated on ten populations and twenty generations. Responses to weighting different components of H-1 were measured in terms of the regression of phenotypes on the estimated breeding values (the lower the slope, the higher the inflation) and the correlation between phenotypes and the estimated breeding values (predictive ability). Increasing the weight on H-1 increased the inflation. The responses to weighting G-1 were similar to those for H-1. Increasing the weight on A-1 (together with A22-1) was not influential and slightly increased the inflation. Predictive ability is a direct function of the slope of the regression line and followed similar trends. Responses to weighting G-1-A22-1 depend on the inflation/deflation of evaluations from A-1 to H-1 and the compatibility of the two matrices with the heritability used in the model. One possibility is a combination of weighting G-1-A22-1 and weighting H-1. Given recent advances in ssGBLUP, conditioning H-1 might become an interim solution from the past and then not be needed in the future.