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Estimation of Heritability Across the Range of the Environmental Covariate
This protocol is extracted from research article:
Genetic Parameters for Tolerance to Heat Stress in Crossbred Swine Carcass Traits
Front Genet, Feb 4, 2021;

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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 $σb⁢a2$, $σl⁢i2$, and $σe⁢t2$ 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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