Derivation of prediction error variance for non-genotyped individuals in genomic selection

Genomic selection has transformed plant and animal breeding by enabling accurate prediction of genetic merit using DNA markers; however, comprehensive genotyping of all selection candidates remains economically prohibitive for most breeding programs. While breeding programs must decide which subset of individuals to genotype within budget constraints, current approaches rely primarily on experience-based decisions rather than quantitative frameworks. We present explicit mathematical derivations for prediction error variance (PEV) in non-genotyped individuals under mixed model equations, providing a theoretical foundation for evaluating genotyping strategies prospectively. The approach derives PEV expressions for non-genotyped selection candidates under different relationship matrix structures, including pedigree-based, genomic, and hybrid single-step methodologies that combine both information sources. The derivations accommodate complex breeding program structures with historical training populations containing both genotypes and phenotypes alongside contemporary selection candidates with only pedigree information. Using Schur complement methods applied to partitioned mixed model equations, the framework enables calculation of prediction uncertainty without requiring actual phenotypic data from selection candidates. The expressions simplify under different information scenarios, from cases with complete phenotypic data to situations where only relationship information is available. The method was validated through simulations across six scenarios with populations ranging from 180 to 15,500 individuals, confirming numerical equivalence with direct matrix inversion while demonstrating computational and memory advantages that increase with population size. Although genomic relationship matrix operations dominate the complexity, matrix decomposition techniques, including Cholesky factorization and APY methodology, can improve efficiency. The mathematical framework provides quantitative tools for transitioning from experience-based to mathematically-informed genotyping decisions, with applications extending to any field requiring prospective quantification of prediction uncertainty under resource constraints.