Genomic Breeding Value Estimation in Wheat

We performed genomic breeding value estimation (GEBV) and hybrid prediction with wheat data, and the results were compared to other genomic selection and mixed model software, including rrBLUP [13 ], ASReml [21 ], regress (used by synbreed as well) [17 ,18 (link)], EMMREML [19 ], MCMCglmm [15 (link)], and BGLR [16 (link)]. We used the wheat data contained in the R package BGLR consisting of 599 inbred lines genotyped with 1279 diversity array technology (DArT) markers [16 (link)]. Phenotypic data consisted of grain yield (GY) for the 599 lines from the historical CIMMYT's Global Wheat Program evaluated in four mega-environments.
From the 599 wheat lines, 179,101 distinct single crosses can be performed. Kinship-based BLUP prediction for the 599 lines were obtained using rrBLUP (ridge regression), ASReml (average information), regress (Newton-Raphson), EMMREML (modified EMMA), BGLR (using the Reproducing kernel Hilbert space [RKHS] kernel), MCMCglmm (Gibbs sampling) and the three algorithms implemented in sommer (AI, EM, and EMMA). Similarity among BLUPs using all software was performed in R and displayed in tables and figures [26 ]. The genomic estimated breeding values (GEBV) for each of the 599 inbred lines was used to predict the performance of possible crosses as the average among the breeding value of the parental lines. The mixed model fitted has the form:
$y=\mathit{X}\beta +\mathit{Z}u+\epsilon$
with variance:
$V\left(y\right)=V\left(\mathit{Z}u+\epsilon \right)={\mathbf{Z}\mathbf{G}\mathbf{Z}}^{\mathrm{\prime }}+\mathit{R}$
and the mixed model equations for this model are:
${\left[\begin{array}{cc}\mathit{X}\mathrm{\prime }{\mathit{R}}^{-1}\mathit{X}& \mathit{X}\mathrm{\prime }{\mathit{R}}^{-1}\mathit{Z}\\ \mathit{Z}\mathrm{\prime }{\mathit{R}}^{-1}\mathit{X}& {\mathit{Z}}^{\mathrm{\prime }}{\mathit{R}}^{-1}\mathit{Z}+{\mathit{G}}^{-1}\end{array}\right]}^{-1}\left[\begin{array}{c}\mathit{X}\prime {\mathit{R}}^{-1}y\\ \mathit{Z}\prime {\mathit{R}}^{-1}y\end{array}\right]=\left[\begin{array}{c}\beta \\ u\end{array}\right]$
Here, G = Kσ²_u, is the variance covariance matrix of the random effect u, from a multivariate normal distribution u ~ MVN(0, Kσ²_u), K being, in the genomics context, the additive or genomic relationship matrix (A or A_g). X and Z are incidence matrices for fixed and random effects respectively, and R is the matrix for residuals (here Iσ²_e). A mixed model with a single variance component other than the error (σ²_e) can be used to estimate the genetic variance (σ²_u) along with genotype BLUPs to exploit the genetic relationships between individuals coded in K (A). The genomic relationship matrix was constructed according to VanRaden where K = ZZ’/2Σp_i(1-p_i) [27 (link)]. Genotype BLUPs were calculated and considered equal to the GEBV and these were used to predict the performance of the 179,101 possible crosses as the average of parental genomic breeding values. We fitted this model using the sommer package by specifying the incidence and variance-covariance matrices and using the three algorithms implemented (AI, EM, EMMA). In addition, a five-fold cross validation was performed to calculate the predictive correlation for grain yield in the four mega environments available for the wheat data using the sommer package. In addition, heritability was estimated as h² = σ²_u / σ²_u + σ²_e.