PS1-7. Contemporary Parent Substitution: A Simple Approach for Constructing Numerator Relationship Matrix with Unknown Parents.

Abstract Legarra et al. (2015) introduced metafounders (MF) to represent groups of unknown parents as pseudo-animals at the base of the pedigree. In this framework, the relationships between MFs are defined using a relationship matrix (Γ), with diagonal elements reflecting the inbreeding level of the group and off-diagonal elements representing genetic relatedness among groups. The MF approach correctly models relationships among animals sharing unknown-parent ancestry through the relationship between MFs. However, by setting the relationship between the first MF and subsequent MFs as zero, the approach implicitly assumes MFs, and consequently animals with unknown parents, have zero relationship to animals with a complete pedigree. This causes inbreeding to be underestimated for animals whose unknown parent is related to their known parent, a bias that grows with selection intensity and pedigree depth. In a previous report, we showed that biases in inbreeding coefficients can be reduced by allowing an indefinite Γ. Nonetheless, indefiniteness limits its use in single-step genomic evaluations, where positive definiteness of the numerator relationship matrix is required for genomic information to propagate correctly to non-genotyped animals. Motivated by the average numerator relationship matrix of Henderson (1988), I propose an alternative approach: Contemporary Parent Substitution (CPS). Instead of introducing pseudo-animals, CPS replaces each unknown parent with real animals in the pedigree according to the frequencies with which they appear as parents within the corresponding contemporary group. In doing so, CPS approximates the relationship of each unknown parent by the weighted average relationship of contemporary known parents to all other animals in the pedigree. To evaluate CPS, an intensively selected population consisting of 25 sires and 5,000 dams per generation was simulated over 11 generations with 10 replicates. Missing pedigree information was introduced by randomly removing 20% of sire records and 40% of dam records. Results suggest that, compared with the MF approach, CPS yields less biased and more precise estimates of inbreeding coefficients. Importantly, CPS preserves the positive definiteness of the resulting numerator relationship matrix. Its inverse can be constructed using modified Henderson rules, and matrix–vector products involving the inverse can be computed efficiently. These properties make CPS both theoretically and computationally suitable for integrating genomic information, with potential applications in inbreeding management and single-step genomic evaluations. For image description, please refer to the figure legend and surrounding text.