4. Conclusions

3.2. Real data sets

(Trait-Env).

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

CorP SE MSEP SE CorP SE MSEP SE

DTHD\_Bed2IR 0.876 0.008 8.117 0.692 0.875 0.009 10.636 0.882 GNDVI\_Bed2IR 0.848 0.008 0.000 0.000 0.009 0.022 0.103 0.006 GRYLD\_Bed2IR 0.639 0.014 0.055 0.002 0.463 0.015 0.161 0.007 PTHT\_Bed2IR 0.658 0.014 22.527 0.841 0.566 0.020 25.798 0.895 DTHD\_Bed5IR 0.873 0.007 13.074 0.733 0.845 0.010 15.312 0.508

GRYLD\_Bed5IR 0.178 0.023 0.251 0.008 0.175 0.020 0.336 0.014 PTHT\_Bed5IR 0.076 0.016 24.064 0.620 0.245 0.023 20.831 0.721 DTHD\_Drip 0.915 0.005 4.514 0.201 0.895 0.006 3.321 0.224 GNDVI\_Drip 0.681 0.012 0.000 0.000 0.262 0.022 0.123 0.008 GRYLD\_Drip 0.653 0.011 0.126 0.005 0.638 0.011 0.144 0.005 PTHT\_Drip 0.658 0.019 21.565 0.531 0.602 0.012 21.306 0.728 Average 0.651 0.013 7.858 0.303 0.462 0.016 8.191 0.334 Yield\_EBU 0.320 0.019 0.789 0.018 0.365 0.018 0.731 0.017 ASI\_EBU 0.501 0.016 0.396 0.012 0.510 0.015 0.391 0.012 PH\_EBU 0.308 0.025 0.015 0.003 0.305 0.011 0.010 0.000 Yield\_KAK 0.402 0.022 0.446 0.020 0.416 0.020 0.438 0.019

Wheat GNDVI\_Bed5IR 0.758 0.019 0.000 0.000 0.496 0.023 0.219 0.011

Maize ASI\_KAK 0.389 0.015 0.936 0.043 0.423 0.018 0.822 0.029

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

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

PH\_KAK 0.462 0.025 0.011 0.001 0.369 0.022 0.013 0.001 Yield\_KTI 0.276 0.015 0.848 0.022 0.318 0.018 0.825 0.024 ASI\_KTI 0.290 0.018 0.607 0.018 0.280 0.020 0.614 0.019 PH\_KTI 0.460 0.017 0.019 0.001 0.443 0.017 0.020 0.001 Average 0.379 0.019 0.452 0.015 0.381 0.018 0.429 0.014

Data set Trait\_Env M2 M3

28 Physical Methods for Stimulation of Plant and Mushroom Development

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 in Eq. (8).
