Where y is a vector of trait residuals from (2), SNP is a vector of SNP genotypes and e is a vector of random residuals.
Furthermore, Dantcheva et al. [9] bring to the fore that, under certain conditions, a biometric system may be composed of only a vector of different soft biometric traits.
The model can be expressed as: y = 1 μ + Z a + e where y is a vector of phenotypes for the traits; μ is the overall mean; Z is a M y × M design matrix linking the animals to the records, where M (M y ) is the number of bulls (with records); a is a M × 1 vector of genetic effects of the animals and e is the random residual vector.
We calculated three vectors of selection gradients: a vector of selection gradients on female traits (β f); a vector of selection gradients on male traits (β m); and then, by combining β f and β m, a vector of selection gradients for both female and male traits (β bs).
The objective of any linear selection index, whether phenotypic or genomic, is to predict the net genetic merit H = w ′ a, where a ′ = [ a 1 a 2... a t ] (t = number of traits) is a vector of true breeding values for an individual and w ′ = [ w 1 w 2... w t ] is a vector of economic weights.
To estimate within-sex phenotypic covariances among traits, we ran multivariate equivalents of the models represented by Equation 1, where y now represents a matrix of phenotypic observations of all traits measured within each sex and μ is a vector of means for each phenotypic trait.
Applying Robertson's second theorem of natural selection, the response in y ¯ will equal the additive genetic covariance between the vector of traits and fitness (Δ y ¯ = σ A (y, w ¯ ) ) (Rausher 1992; Walsh and Blows 2009).
This treatment needs to simultaneously estimate the genotypic vector of traits y and z, expressed as (uj t1),.., uj tT), vj t1),.., vj tT)).
If we place the effects for all environments or traits in a single vector (as a row vector) for line i, u g i T = ∑ k = 1 c ∑ j = 1 r k − 1 q i k j T A k j + u p i T, where A kj is a nf × t matrix of sizes of effects for nf founder alleles and t environments or traits and u pi is a vector of polygenic effects for line i for all environments or traits.
where y is a vector of individual tree observations for a trait, b is a vector of fixed effects, u is a vector of random effects, e is a vector of random residuals, and X and Z correspond to design matrices relating the observations in y to the fixed and random effects in b and u, respectively.
