The additive genetic (co)variance structures of individual sire across the range of the environmental covariate (Γ) was defined as

where G is the estimated (co)variance matrix between the intercept and slope terms and Φ is a matrix containing a column of “1” (intercept) and the environmental covariate. Heritability at each single value m of ENV (hm2) was calculated as

where Γmm is the mth value of the diagonal of Γ, and σba2, σli2, and σet2 are as defined above. Phenotypic predicted values at the population were calculated as rm=α+βφ1m, while phenotypic predicted values at the sire level (i.e., reaction norms) were calculated as rmn=(α+α0n)×1(β+a1nφ1m where a_0 and a_1 were as in model 2, r_m is the prediction at value m at the population level, and rlm is the prediction at value m for individual n. Genotyped sires were ranked based on their genomic estimated breeding values for the intercept a_0 and slope a_1 in terms of the first-order random regression model. Twenty genotyped sires showing a_0 were labeled as intHi while those showing lowest values were labeled intLo. Similarly, the twenty genotyped sires showing the highest a_1 were labeled as sloHi and the sires showing the lowest values were labeled as sloLo.

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