Traiti stands for a vector of traits of the person shown in the photograph.
This treatment needs to simultaneously estimate the genotypic vector of traits y and z, expressed as (uj t1),.., uj tT), vj t1),.., vj tT)).
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).
Heritability is then estimated by maximizing the log-likelihood of the data
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.
Then, trait phenotypic values of individuals in a purebred population can be modeled as (1) where y is the vector of trait phenotypic values, β is a vector of non-genetic fixed effects, μ is the QTL substitution effect, u is the vector of the sum of additive effects of all RQTL, e is a vector of residuals, and X, Q and Z are known incidence matrices.
These models can be described by the following general model: 1 where y is the vector of trait phenotypes, X is a known incidence matrix that relates the vector of non-genetic, "fixed" effects to y, T= M− E(M), M is a matrix of marker covariates, α is the vector of random, partial-regression coefficients of the marker covariates, and e is a vector of residuals.
If P1 is the ng × nf matrix of probabilities corresponding to the first QTL chosen, for a multitrait analysis the model (2) becomes u g = (I t ⊗ P 1 ) a 1 t + u p and (4) becomes u g = (I t ⊗ P 1 ) a 1 t + (I t ⊗ P − 1 ) a − 1 + u p, where a1 t is the vector of trait-by-founder effect sizes for the first QTL.
The multivariate version of Lande's equation is (3) where, for n traits, ′ is the vector of mean trait values (with ′ denoting transposition) and ′ is the multivariate selection gradient, which points in the direction of steepest ascent on the fitness landscape.
Given that the allometric change of one trait is not only regulated by the underlying genes, but also by physiology-and metabolism-related characteristics that contain the influences of both genes and environments [7], we model the genotypic vector of a trait with the phenotypic value of a second allometrically related trait.
Evolution where multiple traits co-vary can be studied using the multivariate response equation Δ z ¯ = G P − 1 S, where Δ z ¯ is the vector of mean trait responses, G is the additive genetic variance-covariance matrix, P is the phenotypic variance-covariance matrix, and S is the vector of selection differentials [ 5].
