A BLUP derivation of the multivariate breeder's equation, with an elucidation of errors in BLUP variance estimates, and a prediction method for inbred populations

Abstract

The multivariate breeder's equation Lande (1979) was derived from the Price equation Price (1970, 1972). Here, I present a derivation based on the BLUP (best linear unbiased predictions) equations in matrix form, first given in summation form by Henderson (1950). The derivation makes use of a comparison with the known form of the multivariate breeder's equation, and it is therefore not an independent derivation. The alternative derivation does, however, clarify why and to which extent the variances of BLUP predictions of random effects are underestimated. The BLUP random effects can in fact be used for prediction of phenotypic responses by use of the Robertson-Price identity, Robertson (1966), Price (1970, 1972. The BLUP derivation also leads to a prediction method for populations with inbreeding, i.e., where the additive genetic relationship matrix departs from a unity matrix, and where the multivariate breeder's equation therefore will fail.

Dynamical BLUP modeling of reaction norm evolution, accommodating changing environments, overlapping generations, and multivariate data

“…Theorems 1, 2, and 3 were verified in simulations with use of 𝑨 𝑡 = 𝑰 𝑛 , and population sizes down to 𝑛 = 2, while Theorem 4 was verified by use of 𝑨 𝑡 ≠ 𝑰 𝑛 , and all heritability values close to one. Theorem 5 was verified by simulations in Ergon (2022c). Simulation results with 𝑨 𝑡 ≠ 𝑰 𝑛 were found by use of a constant and possibly unrealistic additive genetic relationship matrix, but these results still serve the purpose of showing that the BLUP and GRAD results…”

“…( 2) was derived from a multivariate version of Eq. (1), which requires several assumptions, as detailed in Ergon (2019Ergon ( ,2022c):…”

“…A proof of Theorem 5 is given in Ergon (2022c), although then relying on a comparison with the multivariate breeder's equation. Here, Theorem 1 makes it into an independent proof.…”

“…Here, we will see why and to which extent that is necessary in order to obtain the correct incremental changes in mean reaction norm parameter values from generation to generation, and that these changes may be found from the well-known Roberson-Price identity (Robertson, 1966;Price, 1970) applied on the random effects. This also applies to non-plastic organisms, where the mean reaction norms degenerate into mean phenotypic trait values (Ergon, 2022c).…”

Dynamical BLUP modeling of reaction norm evolution, accommodating changing environments, overlapping generations, and multivariate data

“…Second, we need to see how the multivariate breeder's equation (Lande, 1979; Lande & Arnold, 1983),$$$$ \Delta {\overline{\boldsymbol{z}}}_t=\boldsymbol{G}{\boldsymbol{P}}^{-1}\mathit{\operatorname{cov}}\left({w}_{i,t},{\boldsymbol{z}}_{i,t}\right)=\boldsymbol{G}{\boldsymbol{\beta}}_t, $$$$where $$$ {\boldsymbol{\beta}}_t $$$ is the selection gradient, which can be applied on the parameters in a reaction norm model. Equation () was derived from a multivariate version of Equation (), which requires several assumptions, as detailed in Ergon (2019, 2022c): The vector $$$ {\boldsymbol{z}}_{i,t} $$$ of individual phenotypic traits is the sum of independent additive genetic effects $$$ {\boldsymbol{x}}_{i,t} $$$ and nonadditive environmental and genetic effects $$$ {\boldsymbol{e}}_{i,t} $$$, that is, $$$ {\boldsymbol{z}}_{i,t}={\boldsymbol{x}}_{i,t}+{\boldsymbol{e}}_{i,t} $$$. The nonadditive effects $$$ {\boldsymbol{e}}_{i,t} $$$ are zero mean, independent, and identically distributed (iid) random variables. There are no expected fitness‐weighted changes in the individual additive genetic effects …”