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Appendix 1 Proof that the PEV is equal to the covariance between PBV () and the residual environmental effect (e), and thus that the covariance between phenotype (y) and true breeding value (a) is equal to the covariance between phenotype and PBV. If we have one observation per individual (so Z = I), no fixed effects and μ = 0, then As If fitness w is a function of y, then also
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