2. Material and methods

#### 2.1. Matrix normal distribution

The matrix normal distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. According with Rowe [4] the n p matrix normal distribution can be derived as a special case of the np-variate Multivariate Normal distribution when the covariance matrix is separable. A np-dimensional vector x is distributed according to multivariate normal distribution with np-dimensional mean μ and np np covariance matrix Ω if its probability density function is given by

A Bayesian Multiple-Trait and Multiple-Environment Model Using the Matrix Normal Distribution http://dx.doi.org/10.5772/intechopen.71521 21

$$P(\mathbf{x}|\boldsymbol{\mu}, \boldsymbol{\Omega}) = \left(2\pi\right)^{-\frac{np}{2}} |\boldsymbol{\Omega}|^{-\frac{1}{2}} e^{-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^{T}\boldsymbol{\Omega}^{-1}(\mathbf{x}-\boldsymbol{\mu})} \tag{1}$$

When the covariance matrix Ω is separable, that is, is one of the form Ω =Σ ⊗ Φ, where ⊗ is the Kronecker product which multiplies every entry of its first matrix argument by its entire second matrix argument, Eq. (1) becomes

$$p(\mathbf{X}|\mathbf{M}, \Sigma, \mathbf{O}) = (2\pi)^{-\frac{np}{2}} |\Sigma|^{-\frac{n}{2}} |\mathbf{O}|^{-\frac{p}{2}} e^{-\frac{1}{2}tr\left[\Sigma^{-1}(\mathbf{X}-\mathbf{M})^{T}\Phi^{-1}(\mathbf{X}-\mathbf{M})\right]} \tag{2}$$

upon using the following matrix identities

$$|\mathfrak{Q}| = |\Sigma \otimes \mathfrak{P}| = |\Sigma| |\mathfrak{Q}|\tag{3}$$

and

linked directly to the use of statistical models, since the process of selection of candidate individuals is done using statistical models. However, most of the models currently used in genomic selection are univariate models mostly for continuous phenotypes, which not exploit the existing correlation between traits when the selection of individuals (genotypes or animals) is done with the purpose to improve simultaneously multiple-traits. The advantage of jointly modeling multiple-traits compared to analyzing each trait separately, is that the inference process appropriately accounts for the correlation among the traits, which helps to increase prediction accuracy, statistical power, parameter estimation accuracy, and reduce trait selection bias [1, 2]. For this reason, there is a great interest of plant and animal scientist to develop appropriate genomic selection models for multiple-traits and multiple-environments to take advantage of this correlation and to improve the prediction accuracy in the selection of

For this reason, in this chapter we propose an improved version of the Bayesian multiple-trait, multiple-environment (BMTME) model proposed by Montesinos-López et al. [3] that is appropriate for correlated multiple-traits and multiple-environments but instead of building this model using the multivariate normal distribution we propose to build it using the matrix normal distribution which should avoid that the number of rows of the datasets grows

Also, the BMTME model was improved adding a general covariance structure for the genetic covariance of environments in place of assuming a diagonal matrix as the original BMTME model. Additionally, in this chapter we compare the improved model in terms of prediction accuracy and time of implementation with the original BMTME model of Montesinos-López et al. [3] and with a multiple-trait and multiple-environment model where it is ignored the correlation between traits and between environments. Our hypothesis is that the improved model should be similar in terms of prediction accuracy, but considerably faster in terms of time of implementation with regard to the original BMTME of Montesinos-López et al. [3] and a little better in terms of prediction accuracy that a multiple-trait and multiple-environment model that ignore the correlation between traits and environments. Also, we propose to implement the proposed model with simulated and real data sets. Our results suggest that the construction and implementation of the proposed model should be of great help for breeding scientist and programs since will help to select candidate genotypes early in time

The matrix normal distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. According with Rowe [4] the n p matrix normal distribution can be derived as a special case of the np-variate Multivariate Normal distribution when the covariance matrix is separable. A np-dimensional vector x is distributed according to multivariate normal distribution with np-dimensional mean μ and

np np covariance matrix Ω if its probability density function is given by

candidate individuals.

with more accuracy.

2. Material and methods

2.1. Matrix normal distribution

proportional to the number of traits under study.

20 Physical Methods for Stimulation of Plant and Mushroom Development

$$(\mathbf{x} - \boldsymbol{\mu})^T (\boldsymbol{\Sigma} \otimes \boldsymbol{\Phi})^{-1} (\mathbf{x} - \boldsymbol{\mu}) = \operatorname{tr} \left[ \boldsymbol{\Sigma}^{-1} (\mathbf{X} - \mathbf{M})^T \boldsymbol{\Phi}^{-1} (\mathbf{X} - \mathbf{M}) \right] \tag{4}$$

where X and M are matrix of dimension n � p such that x = vec(X) and μ = vec(M), with vec is the vec operator that stacks the columns of its matrix argument from left to right into a single vector, tr(.) is the trace operator which gives the sum of the diagonal elements of a square matrix argument.

Then, according with Rowe [4] the density function given in Eq. (2) correspond to a random variable that follows a n � p matrix normal distribution and it is denoted as

$$\mathbf{X}|\mathbf{M}, \boldsymbol{\Sigma}, \boldsymbol{\Phi} \sim \text{MN}\_{n \times p}(\mathbf{M}, \boldsymbol{\Phi}, \boldsymbol{\Sigma}) \tag{5}$$

where (M, Σ, Φ) parametrize the above distribution with M ∈ R<sup>n</sup> � <sup>p</sup> , and Σ and Φ are positive defined matrix of dimension n � n and p � p, respectively. The matrices Σ and Φ are commonly referred to as the within and between covariance matrices. Sometimes they are referred to as the right and left covariance matrices [4].

Some useful properties of the matrix normal distribution are: the mean and model is equal to E(X| M, Σ, Φ) = M and the variance var(vec(X)| M, Σ, Φ) = Σ ⊗ Φ, which can be found by integration and differentiation. Since X follows a Matrix Normal distribution, the conditional and marginal distributions of any row or column subset are Multivariate Normal distributions [4].

#### 2.2. Univariate model with genotype by environment interaction (M1)

First, for each trait we considered the following univariate linear mixed model:

$$y\_{i\dot{\jmath}} = E\_i + \mathbf{g}\_{\dot{\jmath}} + \mathbf{g}E\_{i\dot{\jmath}} + \mathbf{e}\_{i\dot{\jmath}} \tag{6}$$

were yij represents the normal response from the jth line in the ith environment (i = 1, 2, …, I, j = 1, 2, …, J). For illustration purposes, we will use I = 3. Ei represents the fixed effect of the ith environment and is assumed as a fixed effect, gj represents the random effect of the genomic effect of jth line, with g ¼ gj ,…, gJ <sup>T</sup> � <sup>N</sup> <sup>0</sup>, <sup>σ</sup><sup>2</sup> <sup>1</sup>Gg , Gg is of order <sup>J</sup>� <sup>J</sup> and represents the Genomic Relationship Matrix (GRM) and is calculated using the VanRaden method [5] as Gg <sup>¼</sup> WW<sup>T</sup> <sup>p</sup> , with W as the matrix of marker of order J� p. gEij is the random interaction term between the genomic effect of the <sup>j</sup>th line and the <sup>i</sup>th environment where gE <sup>¼</sup> gE11;…; gEIJ <sup>T</sup> � <sup>N</sup> <sup>0</sup>, <sup>σ</sup><sup>2</sup> <sup>2</sup>I<sup>I</sup> <sup>⊗</sup> <sup>G</sup> and eij is a random error term associated with the <sup>j</sup>th line in the <sup>i</sup>th environment distributed as N(0, σ<sup>2</sup> ). As previously mentioned, this model was used for each of the l = 1,…, L traits, where L denotes the number of traits under study.

#### 2.3. Multivariate correlated model with multiple-trait and multiple-environment (M2)

To account for the correlation between traits, all of the L traits given in Eq. (6) should be jointly modeled in a whole multiple-trait, multiple-environment mixed model as the following:

$$\mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\_1\mathbf{b}\_1 + \mathbf{Z}\_2\mathbf{b}\_2 + \mathbf{e} \tag{7}$$

2.5. Gibbs sampler

and A.3).

diagnostics.

traits and between environments:

gET <sup>¼</sup> gET111; …; gETIJL <sup>T</sup>

2.7. Experimental data sets

2.7.1. Simulate data sets

In order to produce posterior means for all relevant model parameters, below we outline the exact Gibbs sampler procedure that we proposed for estimating the parameters of interest. The

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Step 2. Simulate b<sup>h</sup> for h = 1, 2, according to the normal distribution given in Appendix A (A.2

Step 6. Return to step 1 or terminate when chain length is adequate to meet convergence

2.6. Multivariate uncorrelated model with multiple-trait and multiple-environment (M3)

To compare the model given in Eq. (7) we considered also model M3 (Eq. 6) that consists of using the following multi-trait, multi-environment model that ignore the correlation between

where yijl represents the normal response from the jth line in the ith environment for trait l (i = 1, 2, …, I, j = 1, 2, …, J, l = 1, …, L). Tl represents the fixed effect of the lth trait, TEil is the fixed interaction term between the lth trait and the ith environment, gTjl represents the random effect

gETijl is the three-way interaction of genotype j, the ith environment and the lth trait, with

For testing the proposed models and methods we simulated multiple-trait and multipleenvironment data using model in Eq. (7). We studied six scenarios depending of the parameters used. For the first scenario (S1) we used the following parameters: three environments, three traits, 80 genotypes, 1 replication for environment-trait-genotype combination. We assumed that β<sup>T</sup> = [15, 12, 7, 14, 10, 9, 13, 11, 8], where the first three beta coefficients belong to traits 1, 2 and 3 in environment 1, the second three values for the three traits in environment 2 and the last three for environment 3. We assumed that the genomic relationship matrix is

<sup>22</sup>I<sup>I</sup> ⊗ G ⊗ I<sup>L</sup>

of the interaction of genotype <sup>j</sup> and the <sup>l</sup>th trait, with gT <sup>¼</sup> gT11;…; gTJL <sup>T</sup>

� <sup>N</sup> <sup>0</sup>, <sup>σ</sup><sup>2</sup>

with the jth line in the ith environment distributed as N(0, σ<sup>2</sup>

yijl ¼ Ei þ gj þ Tl þ gEij þ TEil þ gTjl þ gETijl þ eijl (9)

and eijl is a random error term associated

).

� <sup>N</sup> <sup>0</sup>, <sup>σ</sup><sup>2</sup>

<sup>11</sup>G ⊗ I<sup>L</sup> ,

ordering of draws is somewhat arbitrary; however, we suggest the following order: Step 1. Simulate β according to the normal distribution given in Appendix A (A.1).

Step 3. Simulate Σ<sup>t</sup> according to the IW distribution given in Appendix A (A.4). Step 4. Simulate Σ<sup>E</sup> according to the IW distribution given in Appendix A (A.5). Step 5. Simulate R<sup>e</sup> according to the IW distribution given in Appendix A (A.6).

where Y is of order n � L, with n = I � J, X is of order n � I, β is of order I � L and contains the beta coefficients of the environment by trait combinations, Z<sup>1</sup> is of order n � J, Z<sup>2</sup> is of order n � IJ, b<sup>1</sup> is of order J � L and follows a normal matrix distribution MNJ� <sup>L</sup>(0, Gg,Σt), b<sup>2</sup> is of order IJ� L with a normal matrix distribution b2� MNIJ � <sup>L</sup>(0,Σ<sup>E</sup> ⊗ Gg,Σt) and e is of order n � L with a normal matrix distribution e � MNn � <sup>L</sup>(0,In, Re), where Σ<sup>t</sup> is the genetic covariance matrix between traits and it is assumed unstructured (or general), ⊗ denotes a Kronecker product, Σ<sup>E</sup> is assumed as a general matrix of order I � I, R<sup>e</sup> is the residual general covariance matrix between traits. It is important to point out that the trait � environment (T � E) interaction term is included in the fixed effects, while the trait � genotype (T � G) interaction term is included in the random effect b<sup>1</sup> and the three-way (T � G � E) interaction term is included in b2.

#### 2.4. Joint posterior density and prior specification

In this section, we provide the joint posterior density and prior specification for the improved BMTME model. Assuming independent prior distributions for β, Σt, ΣE, and Re, the joint posterior density of the parameter vector becomes:

$$P\left(\mathcal{B}, \mathbf{b}\_{1}, \mathbf{b}\_{2}, \boldsymbol{\Sigma}\_{t}, \boldsymbol{\Sigma}\_{\mathcal{E}}, \mathbf{R}\_{\epsilon}\right) \quad \propto P\left(\mathbf{Y}|\mathcal{B}, \mathbf{b}\_{1}, \mathbf{b}\_{2}, \mathbf{R}\_{\epsilon}\right) P\left(\mathcal{B}\right) P(\mathbf{b}\_{1}|\boldsymbol{\Sigma}\_{t}) P(\mathbf{b}\_{2}|\boldsymbol{\Sigma}\_{t}, \boldsymbol{\Sigma}\_{\mathcal{E}}) P(\boldsymbol{\Sigma}\_{t}) P(\boldsymbol{\Sigma}\_{t}) P(\mathbf{R}\_{\epsilon}) \tag{8}$$

where P(β), P(Σt), P(ΣE) and P(Re) denote the density prior distributions of β, Σt, ΣE, and Re, respectively. Specifically, we are assuming an Inverse-Wishart (IW) for Ω<sup>v</sup> with shape parameter κ and scale matrix parameter B, and is denoted by Ωv� IW(κ,B), with density function given by <sup>P</sup>ð Þ <sup>Ω</sup><sup>v</sup> <sup>∝</sup>j j <sup>B</sup> <sup>κ</sup> <sup>2</sup> j j <sup>Ω</sup><sup>v</sup> �κþpþ<sup>1</sup> <sup>2</sup> exp � <sup>1</sup> <sup>2</sup> tr <sup>B</sup>Ω�<sup>1</sup> v , κ > 0, B, Ω<sup>v</sup> both are positive definite matrices. For the remaining parameters we are assuming the following prior distributions: � MNn� <sup>p</sup>(β0,II,IL), b1|Σt� MNJ � <sup>L</sup>(0, Gg,Σt),Σt� IW(ν<sup>t</sup> + L � 1, St), b2|Σt,ΣE� MNIJ � <sup>L</sup>(0,Σ<sup>E</sup> ⊗ Gg,Σt), ΣE� IW(ν<sup>E</sup> + I � 1, SE), and Re� IW(ν<sup>e</sup> + L � 1, Se). Next we combine the joint posterior density of the parameter vector with the priors to obtain the full conditional distribution for parameters β, b1, b2, Σt, Re. All full conditionals, as well as details of their derivations, are given in Appendix A.

#### 2.5. Gibbs sampler

effect of jth line, with g ¼ gj

� <sup>N</sup> <sup>0</sup>, <sup>σ</sup><sup>2</sup>

distributed as N(0, σ<sup>2</sup>

,…, gJ <sup>T</sup>

22 Physical Methods for Stimulation of Plant and Mushroom Development

traits, where L denotes the number of traits under study.

� <sup>N</sup> <sup>0</sup>, <sup>σ</sup><sup>2</sup>

<sup>1</sup>Gg

mic Relationship Matrix (GRM) and is calculated using the VanRaden method [5] as Gg <sup>¼</sup> WW<sup>T</sup>

with W as the matrix of marker of order J� p. gEij is the random interaction term between the genomic effect of the jth line and the ith environment where gE ¼ gE11;…; gEIJ

2.3. Multivariate correlated model with multiple-trait and multiple-environment (M2)

To account for the correlation between traits, all of the L traits given in Eq. (6) should be jointly modeled in a whole multiple-trait, multiple-environment mixed model as the following:

where Y is of order n � L, with n = I � J, X is of order n � I, β is of order I � L and contains the beta coefficients of the environment by trait combinations, Z<sup>1</sup> is of order n � J, Z<sup>2</sup> is of order n � IJ, b<sup>1</sup> is of order J � L and follows a normal matrix distribution MNJ� <sup>L</sup>(0, Gg,Σt), b<sup>2</sup> is of order IJ� L with a normal matrix distribution b2� MNIJ � <sup>L</sup>(0,Σ<sup>E</sup> ⊗ Gg,Σt) and e is of order n � L with a normal matrix distribution e � MNn � <sup>L</sup>(0,In, Re), where Σ<sup>t</sup> is the genetic covariance matrix between traits and it is assumed unstructured (or general), ⊗ denotes a Kronecker product, Σ<sup>E</sup> is assumed as a general matrix of order I � I, R<sup>e</sup> is the residual general covariance matrix between traits. It is important to point out that the trait � environment (T � E) interaction term is included in the fixed effects, while the trait � genotype (T � G) interaction term is included in the random

In this section, we provide the joint posterior density and prior specification for the improved BMTME model. Assuming independent prior distributions for β, Σt, ΣE, and Re, the joint

P β

where P(β), P(Σt), P(ΣE) and P(Re) denote the density prior distributions of β, Σt, ΣE, and Re, respectively. Specifically, we are assuming an Inverse-Wishart (IW) for Ω<sup>v</sup> with shape parameter κ and scale matrix parameter B, and is denoted by Ωv� IW(κ,B), with density function given by

remaining parameters we are assuming the following prior distributions: � MNn� <sup>p</sup>(β0,II,IL), b1|Σt� MNJ � <sup>L</sup>(0, Gg,Σt),Σt� IW(ν<sup>t</sup> + L � 1, St), b2|Σt,ΣE� MNIJ � <sup>L</sup>(0,Σ<sup>E</sup> ⊗ Gg,Σt), ΣE� IW(ν<sup>E</sup> + I � 1, SE), and Re� IW(ν<sup>e</sup> + L � 1, Se). Next we combine the joint posterior density of the parameter vector with the priors to obtain the full conditional distribution for parameters β, b1, b2, Σt, Re.

All full conditionals, as well as details of their derivations, are given in Appendix A.

effect b<sup>1</sup> and the three-way (T � G � E) interaction term is included in b2.

 

<sup>2</sup> tr <sup>B</sup>Ω�<sup>1</sup> v

2.4. Joint posterior density and prior specification

posterior density of the parameter vector becomes:

∝P Y β, b1, b2, Re

<sup>2</sup> exp � <sup>1</sup>

P β, b1, b2, Σt, ΣE, R<sup>e</sup>

<sup>2</sup> j j <sup>Ω</sup><sup>v</sup> �κþpþ<sup>1</sup>

<sup>P</sup>ð Þ <sup>Ω</sup><sup>v</sup> <sup>∝</sup>j j <sup>B</sup> <sup>κ</sup>

<sup>2</sup>I<sup>I</sup> <sup>⊗</sup> <sup>G</sup> and eij is a random error term associated with the <sup>j</sup>th line in the <sup>i</sup>th environment

). As previously mentioned, this model was used for each of the l = 1,…, L

Y ¼ Xβ þ Z1b<sup>1</sup> þ Z2b<sup>2</sup> þ e (7)

<sup>P</sup> <sup>b</sup><sup>1</sup> <sup>Σ</sup><sup>t</sup> j ÞP b<sup>2</sup> <sup>Σ</sup><sup>t</sup> <sup>ð</sup> <sup>ð</sup> j Þ , <sup>Σ</sup><sup>E</sup> <sup>P</sup>ð Þ <sup>Σ</sup><sup>t</sup> <sup>P</sup>ð Þ <sup>Σ</sup><sup>E</sup> P Rð Þ<sup>e</sup> (8)

, κ > 0, B, Ω<sup>v</sup> both are positive definite matrices. For the

, Gg is of order <sup>J</sup>� <sup>J</sup> and represents the Geno-

<sup>p</sup> ,

<sup>T</sup>

In order to produce posterior means for all relevant model parameters, below we outline the exact Gibbs sampler procedure that we proposed for estimating the parameters of interest. The ordering of draws is somewhat arbitrary; however, we suggest the following order:

Step 1. Simulate β according to the normal distribution given in Appendix A (A.1).

Step 2. Simulate b<sup>h</sup> for h = 1, 2, according to the normal distribution given in Appendix A (A.2 and A.3).

Step 3. Simulate Σ<sup>t</sup> according to the IW distribution given in Appendix A (A.4).

Step 4. Simulate Σ<sup>E</sup> according to the IW distribution given in Appendix A (A.5).

Step 5. Simulate R<sup>e</sup> according to the IW distribution given in Appendix A (A.6).

Step 6. Return to step 1 or terminate when chain length is adequate to meet convergence diagnostics.

#### 2.6. Multivariate uncorrelated model with multiple-trait and multiple-environment (M3)

To compare the model given in Eq. (7) we considered also model M3 (Eq. 6) that consists of using the following multi-trait, multi-environment model that ignore the correlation between traits and between environments:

$$y\_{ijl} = E\_i + g\_j + T\_l + gE\_{ij} + TE\_{il} + gT\_{jl} + gET\_{ijl} + e\_{i\parallel} \tag{9}$$

where yijl represents the normal response from the jth line in the ith environment for trait l (i = 1, 2, …, I, j = 1, 2, …, J, l = 1, …, L). Tl represents the fixed effect of the lth trait, TEil is the fixed interaction term between the lth trait and the ith environment, gTjl represents the random effect of the interaction of genotype <sup>j</sup> and the <sup>l</sup>th trait, with gT <sup>¼</sup> gT11;…; gTJL <sup>T</sup> � <sup>N</sup> <sup>0</sup>, <sup>σ</sup><sup>2</sup> <sup>11</sup>G ⊗ I<sup>L</sup> , gETijl is the three-way interaction of genotype j, the ith environment and the lth trait, with gET <sup>¼</sup> gET111; …; gETIJL <sup>T</sup> � <sup>N</sup> <sup>0</sup>, <sup>σ</sup><sup>2</sup> <sup>22</sup>I<sup>I</sup> ⊗ G ⊗ I<sup>L</sup> and eijl is a random error term associated with the jth line in the ith environment distributed as N(0, σ<sup>2</sup> ).

#### 2.7. Experimental data sets

#### 2.7.1. Simulate data sets

For testing the proposed models and methods we simulated multiple-trait and multipleenvironment data using model in Eq. (7). We studied six scenarios depending of the parameters used. For the first scenario (S1) we used the following parameters: three environments, three traits, 80 genotypes, 1 replication for environment-trait-genotype combination. We assumed that β<sup>T</sup> = [15, 12, 7, 14, 10, 9, 13, 11, 8], where the first three beta coefficients belong to traits 1, 2 and 3 in environment 1, the second three values for the three traits in environment 2 and the last three for environment 3. We assumed that the genomic relationship matrix is

known and is equal to G<sup>g</sup> = 0.3I<sup>80</sup> + 0.7J80, where I<sup>80</sup> is an identity matrix of order 80 and J<sup>80</sup> is a matrix of order 80 � 80 of ones. Therefore, the total number of observations is 3 � 80 � 3 � 1 = 720, that is, 240 for each trait. Since a covariance matrix can be expressed in terms of a correlation matrix (Rr) and a standard deviation matrix (D<sup>1</sup>=<sup>2</sup> <sup>r</sup> <sup>Þ</sup> as: <sup>Σ</sup><sup>r</sup> <sup>¼</sup> <sup>D</sup><sup>1</sup>=<sup>2</sup> <sup>r</sup> RrD<sup>1</sup>=<sup>2</sup> <sup>r</sup> , with r = t, E,e, where r = t represent the genetic covariance between traits, r = E represents the genetic covariance matrix between environments and r = e, represents the residual covariance matrix between traits. For the three covariance matrices (r = t, E,e) in this scenario we used Rr = 0.15I<sup>3</sup> + 0.85J3, where J<sup>3</sup> is a matrix of order 3x3 of ones, and D<sup>1</sup>=<sup>2</sup> <sup>t</sup> ¼ diag 0ð :9; 0:8; 0:9), D<sup>1</sup>=<sup>2</sup> <sup>E</sup> <sup>¼</sup> diag 0<sup>ð</sup> :5; <sup>0</sup>:65; <sup>0</sup>:75) and <sup>D</sup><sup>1</sup>=<sup>2</sup> <sup>e</sup> <sup>¼</sup> diag 6<sup>ð</sup> ; <sup>0</sup>:43; <sup>0</sup>:33). For the second scenario (S2) we used exactly the same set of parameters defined in S1 except that for the correlation matrix now we assumed that the pair of correlations between traits and between environments is equal to 0.5, that is, Rr = 0.5I<sup>3</sup> + 0.5J3, while the third scenario (S3) also is exactly as S1 with the exception that Rr = 0.75I<sup>3</sup> + 0.25J3, that is, the pair of correlations between traits and between environments is equal to 0.25. These three set of correlation matrices given in S1, S2 and S3 were proposed in order to study the performance of the methods proposed in the context of high correlation (S1), medium (S2) and low correlation (S3) between traits (genetic and residual) and between environments. Other 3 scenarios were studied: scenario 4 (S4) is exactly as scenario S1 but in place of 80 lines were used 100 lines, scenario 5 (S5) was exactly as scenario S2 but with 100 lines and the last scenario (S6) was exactly as scenario S3 but using 100 lines in place of 80.

where lines were evaluated in some environments for the traits of interest; however, some lines were missing in all traits in the other environments, this cross-validation scheme is called CV1. Under this cross-validation, we assigned 80% of the lines to the trn set and the remaining 20% to the tst set. We used the Pearson correlation and mean square error of prediction (MSEP) to compare the predictive performance of the proposed models. Models with Pearson correlation closet to one indicated better predictions, while under the MSEP values closed to zero are better in terms of prediction accuracy. It is important to point out that model M2 was implemented with R code done for the authors implementing the Gibbs sampler given above

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The results are presented in two sections. The first section presents the results of the simulated

In Table 1, under scenario S1 we can observe that the proposed model M2 was the best in terms of prediction accuracy (with Pearson correlation and MSEP) since in the 9 traitenvironment combinations model M2 (improved BMTME model) was better than model M3 (uncorrelated multiple-trait multiple-environment). In average in terms of Pearson correlation the model M2 was better than the model M3 by 8.72%, while in terms of MSEP model M2 was better than model M3 in average by 6.24%. Under scenario S2, in terms of Pearson correlation model M2 was better in 7 out of 9 trait-environment combinations and in 6 out of 9 traitenvironment combination in terms of MSEP. In terms of Pearson correlation model M2 was better than M3 in average by 7.76%, while in terms of MSEP was better by 2.27% in average (Table 1). While under scenario S3 also model M2 was better than model M3, since in 7 out of 9 trait-environment was the best, while under MSEP model M2 was better than M3 in 5 out of 9 trait-environment combination, however, in average model M2 was better than model M3 by

3.98 and 1.028% in terms of Pearson correlation and MSEP, respectively (Table 1).

In Table 2, under scenario S4 model M2 was the best in terms of prediction accuracy (with Pearson correlation and MSEP) since in the 9 trait-environment combinations was better than model M3. In average in terms of Pearson correlation and MSEP model M2 was better than model M3 by 4.4 and 4.1%, respectively. Also, under scenario S5, in terms of Pearson correlation and MSEP, model M2 was better than model M3 in 7 out of 9 and in 6 out of 9 trait-environment combinations, respectively. Model M2 was better than M3 in average by 1.6% in terms of Pearson correlation and by 1.2% in average in terms of MSEP (Table 2). While under scenario S6 also model M2 was better than model M3 in terms of Pearson correlation, since in 7 out of 9 trait-environment was the best, while under MSEP model M2 was better than M3 in 5 out of 9 trait-environment combination, however, in average model M2 was better than model M3 by 1.6 and 1.02% in terms of Pearson correlation and MSEP,

for this model, while model M3 was implemented in the R package BGLR [7].

data set, while the second the results with the real data sets.

3. Results

3.1. Simulated data sets

respectively (Table 2).

#### 2.7.2. Real wheat data set

Here, we present the information on the first real data set used for implementing the proposed models. This real data set composed of 250 wheat lines that were extracted from a large set of 39 yield trials grown during the 2013–2014 crop season in Ciudad Obregon, Sonora, Mexico [6]. The trials under study were days to heading (DTHD), grain yield (GRYLD), plant height (PTHT) and the green normalized difference vegetation index (GNDVI), each of these traits were evaluated in three environments (Bed2IR, Bed5IR and Drip). The marker information used after editing was 12,083 markers. This data set was also used by Montesinos-López et al. [3] for this reason those interested in more details of this data set see this publication.

#### 2.7.3. Real maize data set

The second real data set used for implementing the proposed models is composed of 309 double-haploid maize lines. Traits available in this data set include grain yield (Yield), anthesis-silking interval (ASI), and plant height (PH); each of these traits were evaluated in three optimum rainfed environments (EBU, KAT, and KTI). The marker information used after editing was 12,083 markers. Also, this data set was also used by Montesinos-López et al. [3] for this reason those interested in more details of this data set see this publication.

#### 2.8. Assessing prediction accuracy

For assessing prediction accuracy for the simulated and real data sets a 20 training (trn)-testing (tst) random partitions were implemented under a cross-validation that mimicked a situation where lines were evaluated in some environments for the traits of interest; however, some lines were missing in all traits in the other environments, this cross-validation scheme is called CV1. Under this cross-validation, we assigned 80% of the lines to the trn set and the remaining 20% to the tst set. We used the Pearson correlation and mean square error of prediction (MSEP) to compare the predictive performance of the proposed models. Models with Pearson correlation closet to one indicated better predictions, while under the MSEP values closed to zero are better in terms of prediction accuracy. It is important to point out that model M2 was implemented with R code done for the authors implementing the Gibbs sampler given above for this model, while model M3 was implemented in the R package BGLR [7].
