Let us consider a single trait measured from the same individual at multiple environments as separate traits. We want to analyze such multi-environmental trial of a single trait using the multivariate mixed model. Let us consider l different environments/locations so that the vector y₁ contain n observations from the first environment, y₂ that of the second environment, and y_l the observations from the l^{t h} environment. Then the multi-environmental mixed linear model for l locations can be written as: y i = X i β i + Z i u i + ϵ i , i = 1 , 2 . . , l (1) Here β_i is a vector of fixed effects associated with environment i, u_i is a vector of random additive genetic effects associated with the environment i (note that the genotype by environment interaction effects are confounded with the main genetic effect and may differ between locations), ϵ_i is a vector of error terms associated with the environment. Moreover, X_i and Z_i are known incidence matrices for the fixed effects and the random effects for the location i, respectively. In our study we considered the phenotypic observations from three locations so i = 1, 2, 3. Thus β = [ β 1 ′ , β 2 ′ , β 3 ′ ] ′, u = [u^′₁, u^′₂, u^′₃]′, ϵ = [ϵ^′₁, ϵ^′₂, ϵ^′₃]′ and y contains the phenotypic observation from the locations y₁, y₁, y₃. Then mixed model equation (MME) for the model (1) is: X ′ R - 1 X X ′ R - 1 Z Z ′ R - 1 X Z ′ R - 1 Z + G - 1 β u = X ′ R - 1 y Z ′ R - 1 y . (2) Here, R and G are covariance matrices associated with the vector ϵ of errors and vector u of random effects. If R₀ (of order 3 × 3) is the residual covariance for the three locations then R can be calculated as R = R₀ ⊗ I ((here ‘⊗’ is the Kronecker product of two matrices and I is the identity matrix). Similarly, the genetic covariance matrix G can be calculated as G = G₀ ⊗ K. Here K is the additive genomic relationship matrix which was calculated based on the available marker information following the first approach of VanRaden method [23] and G₀ is a 3 × 3 genomic covariance matrix.

For the Bayesian inference using model (1) one need to specify the conditional distribution for the data (y) and prior distribution for the unknown parameters. So the conditional distribution of data y, given the parameters assumed to follow a multivariate normal distribution: y | β , u , R 0 ∼ N ( X β + Z u , R 0 ⊗ I ) . (3) The additive genetic effects (u_i’s) were assigned multivariate normal distributions with a mean vector of zeros, 0, as: u | G 0 , K ∼ N ( 0 , G 0 ⊗ K ) , (4) and the errors (ϵ_i’s) were assumed to follow, ϵ | R 0 ∼ N ( 0 , R 0 ⊗ I ) , (5) where I is an identity matrix. In Bayesian analysis fixed effects also have a prior and here β was assigned a vague, large-variance Gaussian prior distribution.

In order to study the effect of different homogeneous and heterogeneous residual covariance structures to the GEBV estimation accuracy, we considered following structures for R₀. 1) The first-order ante dependence (ANT1) covariance structure, which allows unequal variances over different locations and unequal correlations and covariances among different locations, 2) The unstructured (US) covariance structure, which allows unequal variances over different locations and unequal covariance between different locations, 3) The diagonal homogeneous covariance (IDV) structure, which allows a constant variance across all locations and 4) The diagonal heterogeneous covariance (IDH) structure, which allows different variances across different locations. The ANT1 structure requires l + (l − 1) parameters (l is the number of locations) to be estimated, whereas the US requires the estimation of l(l − 1)/2 parameters (see [24] for more details). Additionally, the inverse of the ANT1 covariance structure is tri-diagonal (only three diagonals are non zero) and has less parameters to be estimated than the US covariance structure when l is greater than three. However, for the genomic covariance matrix (G₀), we assumed a priori the unstructured covariance structure in all the cases.

In order to model the genotype by environment interaction we considered the following mixed model: y = X β + Z u + W v + ϵ (6)

β_i is a vector of fixed effects (in this case including grand mean and location effects), u_i is a vector of random additive genetic effects, v is the vector of random genotype by environment interaction effects, ϵ is a vector of error terms associated with the locations. Moreover, X and Z are known incidence matrices for the fixed effects and the random additive genetic effects, respectively. Here, the dimension of X is (l × n) × (1 + l) (1 for the grand mean, l is the number of locations and n is the number of lines) and Z is (l × n) × (n). Finally, W is the incidence matrix which relates the lines to different locations. In our study we considered three locations (1, 2, 3) thus: ϵ = [ϵ^′₁, ϵ^′₂, ϵ^′₃]′, which are independently normally distributed with mean zero and variance σ e 2 (ϵ i ∼ N ( 0 , σ e 2 I )), and y = [y^′₁, y^′₂, y^′₃]′. Moreover u | K ∼ N ( 0 , σ g 2 K ), v | K ∼ N ( 0 , σ v 2 I ⊗ K ). Here K is the additive genomic relationship matrix calculated using the available marker information and I is a 3 × 3 identity matrix. With this model the GEBVs were estimated as u + v_l (here v_l is location specific interaction effect). Model 2 can be seen as a special case of model 1 with R₀ being IDV structure and G₀ having identical diagonals and identical off-diagonals (thus two parameters to be estimated). This model is commonly known as the compound symmetry model [25]. Additionally, with model 2 it is also possible to model the heterogeneous residual covariance structure (IDH).

We also performed univariate analysis using the phenotypic information from a single location and for that we considered the following model: y = X β + Z u + ϵ (7) with u | K ∼ N ( 0 , σ g 2 K ) and ϵ ∼ N ( 0 , σ e 2 I ). Here y is the phenotypic information from a single location and K is the additive genomic relationship matrix calculated using the available marker information. Moreover, with model 3 the dimension of the incidence matrix X is n × 1 (1 for the grand mean) and for Z is n × n (n is the number of lines).

In order to compare the different models we used two real dataset having the phenotypic information from three different locations.

Rice dataset: This dataset is publicly available at http://www.ricediversity.org/data/ and consists of 413 diverse accessions of O. sativa [26]) collected from 82 different countries. The accessions were genotyped with single nucleotide polymorphism (SNP) markers and 36 901 SNPs were available for the analysis after excluding markers with minor allele frequency (MAF) ≤ 0.05 and missing values ≥ 20%. [26] measured the trait flowering time in three different locations. The first location (ARK) was in Stuttgart, Arkansas, USA, the second one in in Aberdeen (ABR) and the third location was Faridpur (FAD), Bangladesh (see [26] for more details). Out of the 413 lines, phenotypic informations were missing for 42 lines in all three environments and we did not consider those for the final analysis. So we analyzed a subset of 371 lines in this study.

Maize dataset: This data set consists a total of 504 double-haploid maize lines and the phenotype as well as the genotype information which are all made publicly available [15]. Three traits, yield (Yield), anthesis-silking interval (ASI) and plant height (PH) were measured in three rain fed environments called E1, E2 and E3 (see [15] for more details about the experimental design). This dataset was genotyped using genotyping-by-sequencing (GBS) method and after filtering for the minor allele frequency, around 158 281 SNPs were available for the analysis. One of the main problem with GBS is the large proportion of missing genotypes and in order to cope with that, while calculating the additive genomic relationship matrix, Crossa et al [15] modified the method of VanRaden [23] to account for the missing genotypes (see [15] for more details). In this study we analyzed the trait Yield (which was already standardized to unit variance). This dataset consist of markers whose MAF was ≥ 0.05 and the markers that had maximum of ≤ 20% missing values. For both data sets (Rice and Maize) pedigree information was not available.

We applied five fold cross validation [27] in order to estimate the prediction abilities of different mixed models using the real datasets. Prediction abilities were calculated as the Pearson correlations between the observed and predicted phenotypes (GEBV). We repeated the five fold CV procedure 10 times and the prediction ability estimates were averaged over to produce a single estimate. In five fold CV, we used 80% of the data as the training set and the remaining 20% as the validation set (due to computational challenges we only considered five fold cross validation (80/20), however it might be interesting to estimate the prediction abilities using other combination like 70/30, 60/40, and 50/50 by reducing the training population size). We used the same training and validation sets in each analysis. For the cross validation using models to tackle G×E effects (model 1 & 2), following [13] we used two different approaches, with the first approach, in the validation set we included the lines from all three environments and obtained genomic prediction abilities. In the second approach we selected the lines from only a single environment into the validation set (note that the phenotype information from the other two environments were included in the training set) and calculated the prediction ability for that single environment. The second approach was intended to mimic the situation where the breeder has the phenotype from two environments available and a value missing in the third environment. Hereafter we refer the first approach as multiple environment cross validation (M_CV) procedure and the latter one as single environment cross validation (S_CV) procedure. Finally, we also applied the CV to estimated the prediction abilities using the univariate model (model 3) assuming normal distributed random residuals and refer as RND.

In order to compare the prediction abilities we estimated the Bayesian and traditional GBLUP using all three models. For the Bayesian analysis we used the R package ’MCMCglmm’ [28], which is based on Markov Chain Monte Carlo (MCMC) sampling methods. We considered a total MCMC chain of length 10 000 iterations with a burning period of 3 000 iterations for the Bayesian inference with the multivariate as well as the univariate model and calculated the posterior mode of the distribution. In multivariate GBLUP estimation using MCMCglmm package we assigned inverse-Wishart with a diagonal scaling matrix (the diagonal elements were the univariate variance components estimate corresponding to each location) as the prior distribution for the random genetic (G₀) and residual (R₀) covariance matrices between the three locations. The traditional multivariate GBLUP estimation was performed using the recently published R package ’sommer’ [29]. Using MCMCglmm we were able to estimate parameters from model 1 using all the four residual covariance structures, but with sommer we were able to consider only the IDH and US covariance structures.

Our univariate analysis showed strong genomic correlation between the environments and the genomic correlation based on univariate and multivariate analysis are shown in Table 1 along with the SNP heritability estimates. Here the narrow-sense SNP-heritabilities (h²) were estimated as: h² = σ g 2 / ( σ g 2 + σ e 2), here σ g 2 and σ e 2 are the genomic and residual variances, respectively. It is not a common practice to estimate the genomic correlation between locations based on univariate model by considering a single environment, but for the comparison point we present the results here. The genomic correlation between the environments were higher for the multivariate approach compared to the univariate model. Due to the strong genomic correlation between the environments. For both M_CV and S_CV cross validation procedures using model 1, the US, IDH and ANT1 residual covariance structures showed similar prediction abilities, whereas the prediction ability of IDV covariance structure was lower than the other residual structures. Overall, model 1 showed better prediction abilities than model 2, mainly due to the strong genomic correlation between the environments. Table 2 summarizes the prediction abilities for the rice dataset using different CV procedures with different models.

The maize dataset showed strong G×E interactions (less genomic correlation between the environments (Table 1)) as compared to the rice dataset. Similar to the rice dataset the US, IDH and ANT1 residual covariance structures showed similar prediction abilities. Unlike the rice dataset model 1 & 2 gave the same prediction abilities for both M_CV and S_CV procedures. We believe that this is mainly due to the strong genotype by environment interaction (low genomic correlation between the environments). Table 3 summarizes the results based on different models and CV procedures. The prediction ability based on Bayesian approach was better for the single environment cross validation (S_CV) procedure than the frequentist method. Unlike the rice dataset here we did not find any improvement in prediction ability with S_CV (model 1 & 2) procedure, mainly due to the moderate genomic correlation between the environments.