3. Results

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

<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.

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.

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

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

[3] for this reason those interested in more details of this data set see this publication.

this reason those interested in more details of this data set see this publication.

<sup>t</sup> ¼ diag 0ð :9; 0:8; 0:9),

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>

24 Physical Methods for Stimulation of Plant and Mushroom Development

D<sup>1</sup>=<sup>2</sup>

2.7.2. Real wheat data set

2.7.3. Real maize data set

2.8. Assessing prediction accuracy

The results are presented in two sections. The first section presents the results of the simulated data set, while the second the results with the real data sets.

#### 3.1. Simulated data sets

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, respectively (Table 2).


Scenario Trait\_Env M2 M3

CorP SE MSEP SE CorP SE MSEP SE

http://dx.doi.org/10.5772/intechopen.71521

A Bayesian Multiple-Trait and Multiple-Environment Model Using the Matrix Normal Distribution

 0.495 0.042 0.782 0.052 0.483 0.043 0.800 0.056 0.569 0.028 0.693 0.050 0.534 0.035 0.731 0.055 0.621 0.028 0.589 0.038 0.596 0.033 0.619 0.044 0.467 0.043 0.814 0.044 0.449 0.044 0.850 0.043

 0.572 0.034 0.548 0.035 0.534 0.035 0.597 0.034 0.498 0.040 0.975 0.060 0.486 0.035 0.984 0.060 0.535 0.035 0.812 0.051 0.520 0.032 0.824 0.054 0.631 0.034 0.638 0.043 0.604 0.029 0.674 0.044 Average 0.540 0.036 0.727 0.046 0.516 0.036 0.758 0.049 0.403 0.052 0.805 0.055 0.405 0.050 0.807 0.056 0.537 0.029 0.666 0.047 0.510 0.035 0.688 0.049 0.567 0.031 0.595 0.040 0.555 0.032 0.608 0.042

 0.432 0.041 0.722 0.043 0.406 0.043 0.749 0.048 0.509 0.034 0.554 0.037 0.503 0.035 0.564 0.035 0.416 0.043 1.025 0.056 0.413 0.040 1.024 0.055 0.487 0.033 0.791 0.042 0.488 0.034 0.784 0.045 0.588 0.037 0.625 0.040 0.589 0.032 0.630 0.038 Average 0.482 0.039 0.742 0.046 0.474 0.039 0.751 0.047 0.370 0.054 0.798 0.057 0.369 0.052 0.802 0.057 0.512 0.028 0.635 0.043 0.485 0.033 0.654 0.043 0.521 0.034 0.587 0.040 0.511 0.034 0.596 0.041

S4 22 0.471 0.040 0.689 0.040 0.440 0.041 0.740 0.046

S5 12 0.399 0.051 0.899 0.051 0.397 0.053 0.907 0.053

S6 12 0.367 0.052 0.945 0.057 0.364 0.055 0.948 0.060

CorP: average of Pearson correlation obtained across all trait-environment combination; SE: standard error; MSEP: mean square error of prediction. S4: scenario with high correlation (0.85); S5 the scenario with medium correlation (0.5); S6: scenario with low correlation (0.25). The values of this table correspond to the simulations done with 100 lines in each environment. In

Table 2. Comparison in terms of prediction accuracy of models M2 and M3 under scenarios S4, S5 and S6.

bold are the best predictions of each row (Trait-Env).

 0.412 0.040 0.759 0.045 0.382 0.042 0.776 0.047 0.449 0.034 0.576 0.036 0.466 0.034 0.568 0.034 0.379 0.045 1.013 0.053 0.374 0.042 1.016 0.051 0.462 0.032 0.759 0.038 0.463 0.033 0.751 0.039 0.542 0.039 0.618 0.040 0.558 0.035 0.610 0.036 Average 0.446 0.040 0.743 0.045 0.441 0.040 0.747 0.045

CorP: average of Pearson correlation; SE: standard error, MSEP: mean square error of prediction. S1: scenario with high correlation (0.85); S2: scenario with medium correlation (0.5); S3: scenario with low correlation (0.25). The values of this table correspond to the simulations done with 80 lines in each environment. In bold are the best predictions of each row (Trait-Env).

Table 1. Comparison in terms of prediction accuracy of models M2 and M3 under scenarios S1, S2 and S3.

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


Scenario Trait\_Env M2 M3

Physical Methods for Stimulation of Plant and Mushroom Development

CorP SE MSEP SE CorP SE MSEP SE

 0.401 0.052 0.693 0.050 0.375 0.048 0.723 0.050 0.481 0.044 0.561 0.033 0.434 0.044 0.605 0.035 0.563 0.042 0.494 0.033 0.530 0.043 0.522 0.033 0.408 0.037 0.658 0.045 0.343 0.041 0.715 0.046

 0.506 0.042 0.580 0.049 0.420 0.049 0.642 0.049 0.595 0.030 0.528 0.033 0.570 0.034 0.535 0.034 0.473 0.043 0.565 0.036 0.461 0.039 0.582 0.039 0.629 0.031 0.424 0.027 0.619 0.036 0.441 0.033 Average 0.505 0.041 0.572 0.040 0.461 0.043 0.610 0.042 0.349 0.054 0.748 0.057 0.302 0.052 0.750 0.055 0.486 0.044 0.571 0.030 0.447 0.042 0.603 0.031 0.503 0.044 0.588 0.033 0.508 0.045 0.579 0.031

 0.476 0.049 0.664 0.047 0.407 0.053 0.726 0.057 0.415 0.044 0.626 0.059 0.368 0.048 0.651 0.061 0.599 0.028 0.548 0.043 0.566 0.030 0.537 0.037 0.373 0.051 0.719 0.058 0.374 0.048 0.723 0.057 0.565 0.034 0.530 0.043 0.584 0.037 0.513 0.047 Average 0.448 0.044 0.632 0.046 0.413 0.045 0.646 0.046 0.326 0.054 0.764 0.055 0.297 0.053 0.777 0.055 0.480 0.043 0.588 0.030 0.443 0.041 0.616 0.030 0.446 0.045 0.657 0.035 0.465 0.047 0.629 0.030

S1 22 0.485 0.049 0.648 0.049 0.393 0.053 0.728 0.056

S2 12 0.384 0.037 0.590 0.045 0.335 0.038 0.602 0.040

S3 12 0.404 0.038 0.545 0.045 0.391 0.038 0.553 0.039

CorP: average of Pearson correlation; SE: standard error, MSEP: mean square error of prediction. S1: scenario with high correlation (0.85); S2: scenario with medium correlation (0.5); S3: scenario with low correlation (0.25). The values of this table correspond to the simulations done with 80 lines in each environment. In bold are the best predictions of each row

Table 1. Comparison in terms of prediction accuracy of models M2 and M3 under scenarios S1, S2 and S3.

(Trait-Env).

 0.470 0.047 0.661 0.045 0.402 0.050 0.721 0.055 0.343 0.045 0.630 0.062 0.311 0.048 0.648 0.064 0.567 0.035 0.598 0.048 0.552 0.030 0.592 0.042 0.327 0.054 0.832 0.067 0.324 0.052 0.831 0.067 0.498 0.034 0.615 0.055 0.522 0.036 0.584 0.056 Average 0.429 0.044 0.654 0.049 0.412 0.044 0.661 0.049

> CorP: average of Pearson correlation obtained across all trait-environment combination; SE: standard error; MSEP: mean square error of prediction. S4: scenario with high correlation (0.85); S5 the scenario with medium correlation (0.5); S6: scenario with low correlation (0.25). The values of this table correspond to the simulations done with 100 lines in each environment. In bold are the best predictions of each row (Trait-Env).

Table 2. Comparison in terms of prediction accuracy of models M2 and M3 under scenarios S4, S5 and S6.

#### 3.2. Real data sets

In Table 3 we can observe that in the wheat data set the best predictions were observed under the proposed improved BMTME model (M2), since in all trait-environment combinations was better model M2 in terms of Pearson correlation and in 10 out of 12 was the better in terms of MSEP than model M3 (that ignore the correlation between traits and between environments). However, in the maize data set the best predictions were observed under model M3, since in 5 out of 9 trait-environment combinations this model was superior to model M2, however there is not a great superiority of the results under model M3 regarded to model M2. This results obtained in the maize data set are in agreement with the correlation study performed since this data set has a very low genetic correlation between

A Bayesian Multiple-Trait and Multiple-Environment Model Using the Matrix Normal Distribution

http://dx.doi.org/10.5772/intechopen.71521

29

According to the results observed with the simulated data sets (Tables 1 and 2) and real data sets (Table 3) there is evidence that the larger the correlation between traits (genetic and residual) and environments (genetic) the better the performance of the proposed improved BMTME (M2) model with regard to the uncorrelated multiple-trait and multiple-environment model (M3), which means that when the there is considerable correlation between traits and

In this paper we proposed an improved version of the Bayesian multiple-trait multipleenvironment (BMTME) model of Montesinos-López et al. [3] that was derived using the matrix normal distribution. The advantage of the proposed model (M2) is that it is more efficient in terms of time of implementation since this improved version works using as rows the genotypes by environment combinations in place of using as rows the combination of traits, genotypes and environments which allows a more practical implementation of the Gibbs sampler in terms of time of implementation. Another, improvement of the BMTME model is that now allows unstructured covariance matrix for modeling environments in place of only a diagonal matrix as the original BMTME model. We compared the extended model (M2) with an uncorrelated multiple-trait and multiple-environment model (M3) that ignores the general correlation between traits (genetic and residual) and between environments and we found that the proposed improved BMTME model (M2) outperforms model (M3) in all the scenarios under study with simulation, however the larger the correlation between traits and between environments the better the performance in terms of prediction accuracy of the improved BMTME model. Additionally, we provided all full conditionals required for the implementation of the improved BMTME model (see Gibbs sampler section and Appendix A). However, we are aware that more empirical evidence with real and simulated data is needed to support our findings, and for this reason, we encourage researcher to implement our proposed improved model and compare with models that ignore the correlation between traits and between environments like the model M3 given

A. Derivation of full conditionals of the improved BMTME model under

between environments this help to increase prediction accuracy.

traits and between environments.

4. Conclusions

in Eq. (8).

the matrix normal distribution

Full conditional distribution for vec(β)


CorP: average of Pearson correlation obtained across all trait-environment combination; SE: standard error; MSEP: mean square error of prediction. Trait\_Env means trait-environment combination. In bold are the best predictions of each row (Trait-Env).

Table 3. Comparison in terms of prediction accuracy of models M2 and M3 using the two real data sets.

model M3, since in 5 out of 9 trait-environment combinations this model was superior to model M2, however there is not a great superiority of the results under model M3 regarded to model M2. This results obtained in the maize data set are in agreement with the correlation study performed since this data set has a very low genetic correlation between traits and between environments.

According to the results observed with the simulated data sets (Tables 1 and 2) and real data sets (Table 3) there is evidence that the larger the correlation between traits (genetic and residual) and environments (genetic) the better the performance of the proposed improved BMTME (M2) model with regard to the uncorrelated multiple-trait and multiple-environment model (M3), which means that when the there is considerable correlation between traits and between environments this help to increase prediction accuracy.
