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

Authors: LEGARRA, ANDRES - Council On Dairy Cattle Breeding; Van Raden, Paul; MANTYSAARI, ESA - Natural Resources Institute Finland (LUKE); BERMANN, MATIAS - University Of Georgia; SULLIVAN, PETER - Collaborator

Submitted to: Journal of Dairy Science
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 10/17/2025
Publication Date: 2/1/2026
Citation: Legarra, A., Van Raden, P.M., Mantysaari, E., Bermann, M., Sullivan, P. 2026. Scalar methods to deregress and split genomic predictions, and associated behavior of simple regressions, for later use in combined prediction and validations. Journal of Dairy Science. 109(2):1727–1741. https://doi.org/10.3168/jds.2025-26859.
DOI: https://doi.org/10.3168/jds.2025-26859

Interpretive Summary: Prediction models often combine information using advanced statistical methods and matrix algebra. Simpler scalar formulas are useful in quantifying and understanding the sources of information from published evaluations, with little programming required. Deregression procedures allow splitting accumulated information into independent pieces such as from sequential intervals of time or from parents and progeny. Our derivation and presentation help understanding common methods that scientists use to validate their predictions.

Technical Abstract: Scalar deregressions are used in dairy cattle to post-process genetic evaluations into early and late, or separate, pieces of information. These separate pieces of information are in turn used e.g. 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 (pseudo-phenotypes) and equivalent record contributions (pseudo-number 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 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 are 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.