Genomic Best Linear Prediction (GBLUP) and reaction norm models

Mixed linear models were used as a baseline comparison of prediction accuracies with PLS models. The models used considered the random main effects of markers (G model), the random main effects of markers and EC (G+W model), and the random main effects of markers, EC, and the interactions between them (G+W+GW model).

The G model constituted of a standard GBLUP model for the mean performance of genotypes within each set of environments, using the following model:

where μ is the overall mean, gi is the genotypic random effect of the i^th line expressed as a regression on marker covariates of the form: gi=∑m=1pximbm, where xim is the genotype of the i^th line at the m^th marker, and bm is the effect of the m^th marker. Marker effects are considered as IID draws from normal distributions of the form bm∼N(0,σb2).

The vector g=Xb contains the genomic values of all the lines, and follows a multivariate normal density with null mean and covariance matrix Cov(g)=Gσg2, where G is a genomic relationship matrix whose entries are given by G=XXT/p_.

As previously reported by Jarquín et al. (2014), it is possible to model the environmental effects with a random regression on the EC that describes the environmental conditions faced by each genotype, that is: wij=∑q=1QWijqγq, where Wijq is the value of the q^th EC evaluated in the ij^th environment × genotype combination, γq is the main effect of the corresponding EC, and Q is the total number of EC. Again, we consider the effects of the EC as IID draws from normal densities, γq∼N(0,σγ2). The vector w=Wγ follows a multivariate normal density with null mean and a covariance matrix proportional to Ω whose entries are computed the same way as those of the G matrix but using EC instead of markers. This covariance structure describes the similarity among environmental conditions. Then, the model becomes:

This model also includes a marker × EC interaction term, where the covariance of the interaction is modeled by the Hadamard product of ZgGZgT and Ω, denoted as [ZgGZgT]∘Ω, where Zg is an incidence matrix for the vector of additive genetic effects. This model extends Equation (4) as follows:

with w∼N(0,Ωσw2), g∼N(0,Gσg2), gw∼N(0,[ZgGZgT]∘Ωσgw2), ε∼N(0,σε2).