Scalar methods to deregress and split genomic predictions, and associated behavior of simple regressions, for later use in combined prediction and validations

Scalar deregressions are used in dairy cattle to postprocess genetic evaluations into early and late, or separate, pieces of information. These separate pieces of information are in turn used, for example, to include foreign information or for validation of the genomic evaluation procedure. Here we detail the scalar algebra to separate "partial" evaluations from "whole," with associated breeding values and reliabilities, into equivalent deregressed proofs (pseudophenotypes) and equivalent record contributions (pseudonumber of observed phenotypes). We start from basic principles, and we show several derivations leading to same expressions. The final expressions are similar, but not equal, to other expressions found in the literature. We moreover investigate its use in genetic evaluations and in validations. In evaluation, the new expressions guarantee scalar reversibility (we obtain "whole" evaluations back from deregressed proofs and equivalent record contributions) but not necessarily reversibility of the whole system of equations, in which case matrix-based evaluations would be preferred. In validation, we derive weights (which are very similar but not identical to commonly used weights). We also observe that by construction, the expected value of the regression of the deregressed proof on early predictions is 1, given that early proof is subtracted from late proof, no matter the scale. These derivations help to understand why existing methods work and they may serve as principles to conceive more accurate deregression procedures. We illustrate use of the math with population (large-scale evaluation) examples and with scalar (single-bull) examples.