Article Figures & Data
Figures
- Figure 1
Plot of the prediction accuracy using Pearson’s correlation for each of the 5 locations (SO, SE, PM, NM, and IP) of maize data set HEL for the proposed models MDel-GK, MDel-GB, MUCf-GK, MUCf-GB, MDe-GK, MDe-GB, MUC-GK, and MUC-GB.
- Figure 2
Plot of the prediction accuracy using Pearson’s correlation for each of the 4 environments (P-LN, P-IN, A-LN, A-IN) of maize data set USP for the proposed models MDel-GK, MDel-GB, MUCf-GK, MUCf-GB, MDe-GK, MDe-GB, MUC-GK, and MUC-GB.
- Figure 3
Plot of the prediction accuracy using Pearson’s correlation for each of the 4 environments (E1-E4) of wheat data set WHE1 for the proposed models, MDel-GK, MDel-GB, MUCf-GK, MUCf-GB, MDe-GK, MDe-GB, MUC-GK, and MUC-GB.
- Figure 4
Plot of the prediction accuracy using Pearson’s correlation for each of the 5 environments (0iFN, 2iBH, 5iBH, 5iBN, 5iFN) of wheat data set WHE5 for the proposed models MDel-GK, MDel-GB, MUCf-GK, MUCf-GB, MDe-GK, MDe-GB, MUC-GK, and MUC-GB.
Tables
- Table 1 Components of the 8 models included in this study. Each of these models is fitted with the linear kernel (GB) and the Gaussian kernel (GK)
Model General mean Fixed environmental effect Main genetic effect of line across environments Genotype × environment interaction (G×E) Random intercept of the lines Unstructured G×E Random residual MM 
ε ( MMl (ε ( MDs ge ( )ε ( MDsl ge ( ) (ε ( MDe (for each environment)ε ( MDel (for each environment) (ε ( MUC u ( ε ( MUCf u ( f ( ε ( - Table 2 MAIZE HEL data set. Estimated variance components for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs) and environment-specific variance G×E deviation model (MDe) with two kernels, GBLUP (GB) and Gaussian (GK), with l and without l, for grain yield (standard deviation in parentheses)
Variance Component* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MM- GK MM- GB MDs-GK MDs- GB MDe-GK MDe- GB 0.581 (0.02) 0.594 (0.02) 0.277 (0.02) 0.368 (0.02) 0.246 (0.02) 0.368 (0.02) 0.582 (0.02) 0.749 (0.03) 0.278 (0.02) 0.591 (0.02) 0.247 (0.02) 0.592 (0.02) 0.795 (0.1) 0.228 (0.06) 0.871 (0.11) 0.172 (0.06) 0.88 (0.12) 0.186 (0.06) 0.821 (0.1) 0.356 (0.08) 0.938 (0.1) 0.370 (0.08) 0.931 (0.1) 0.390 (0.09) — — 0.53 (0.06) 0.256 (0.03) — — — — 0.525 (0.06) 0.188 (0.03) — — — — — — 0.376 (0.1) 0.257 (0.08) — — — — 0.372 (0.1) 0.237 (0.08) — — — — 0.778 (0.19) 0.197 (0.08) — — — — 0.769 (0.2) 0.076 (0.06) — — — — 0.374 (0.08) 0.297 (0.08) — — — — 0.370 (0.09) 0.259 (0.08) — — — — 1.135 (0.2) 0.385 (0.11) — — — — 1.143 (0.21) 0.255 (0.09) — — — — 0.688 (0.16) 0.215 (0.08) — — — — 0.688 (0.16) 0.158 (0.07) 0.008 (0.01) 0.201 (0.03) 0.013 (0.01) 0.243 (0.03) 0.014 (0.01) 0.244 (0.03) — — — — — — ↵* Locations are: IP: Ipiaçú-MG, NM: Nova Mutum-MT, PM: Pato de Minas-MG, SE: Sertanópolis-PR, and SO: Sorriso-MT.
- Table 3 Maize HEL data set. Mean Pearson’s correlation (50 partitions) of each location for random cross-validation CV2, for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs), environment-specific variance G×E deviation model (MDe), multi-environment unstructured covariance models (MUC and MUCf) with two kernels, GBLUP (GB) and Gaussian kernel (GK) for grain yield with the proposed random effect l and without the random effect l (standard deviation in parentheses)
Proposed models with random effects l and f Location* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MUCf-GK MUCf-GB IP 0.596 (0.1) 0.577 (0.11) 0.802 (0.05) 0.778 (0.05) 0.808 (0.05) 0.776 (0.05) 0.809 (0.04) 0.785 (0.05) NM 0.601 (0.09) 0.569 (0.09) 0.614 (0.08) 0.589 (0.08) 0.625 (0.08) 0.588 (0.08) 0.625 (0.06) 0.605 (0.07) PM 0.643 (0.06) 0.589 (0.05) 0.776 (0.04) 0.733 (0.05) 0.774 (0.05) 0.741 (0.04) 0.775 (0.03) 0.743 (0.05) SE 0.42 (0.09) 0.372 (0.11) 0.586 (0.08) 0.544 (0.08) 0.579 (0.07) 0.548 (0.08) 0.558 (0.08) 0.522 (0.1) SO 0.544 (0.11) 0.523 (0.11) 0.673 (0.05) 0.639 (0.06) 0.671 (0.06) 0.627 (0.08) 0.66 (0.06) 0.649 (0.06) Proposed models without random effects l and f Location MM-GK MM-GB MDs-GK MDs-GB MDe-GK MDe-GB MUC-GK MUC-GB IP 0.595 (0.08) 0.51 (0.11) 0.807 (0.04) 0.683 (0.08) 0.804 (0.05) 0.678 (0.07) 0.800 (0.05) 0.669 (0.09) NM 0.601 (0.08) 0.469 (0.10) 0.627 (0.08) 0.472 (0.08) 0.616 (0.09) 0.473 (0.11) 0.632 (0.07) 0.486 (0.08) PM 0.645 (0.06) 0.584 (0.08) 0.776 (0.04) 0.697 (0.05) 0.778 (0.04) 0.693 (0.05) 0.781 (0.04) 0.693 (0.04) SE 0.427 (0.1) 0.296 (0.1) 0.591 (0.07) 0.39 (0.08) 0.592 (0.08) 0.395 (0.1) 0.572 (0.07) 0.389 (0.09) SO 0.558 (0.07) 0.396 (0.11) 0.666 (0.06) 0.466 (0.08) 0.662 (0.06) 0.468 (0.09) 0.665 (0.07) 0.463 (0.10) ↵* Locations are: IP: Ipiaçú-MG, NM: Nova Mutum-MT, PM: Pato de Minas-MG, SE: Sertanópolis-PR, and SO: Sorriso-MT.
- Table 4 Maize USP data set. Estimated variance components for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs) and environment-specific variance G×E deviation model (MDe) with two kernels, GBLUP (GB) and Gaussian kernel (GK), with l and without l for grain yield (standard deviation in parentheses)
Variance Component* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MM- GK MM- GB MDs-GK MDs- GB MDe-GK Mde- GB 0.547 (0.02) 0.548 (0.02) 0.49 (0.02) 0.503 (0.02) 0.487 (0.02) 0.503 (0.02) 0.589 (0.02) 0.854 (0.02) 0.538 (0.02) 0.834 (0.02) 0.534 (0.02) 0.833 (0.02) 0.371 (0.09) 0.175 (0.05) 0.343 (0.09) 0.164 (0.05) 0.362 (0.1) 0.165 (0.05) 1.899 (0.18) 0.214 (0.06) 2.012 (0.18) 0.209 (0.06) 2.026 (0.19) 0.206 (0.06) — — 0.091 (0.02) 0.045 (0.01) — — — — 0.077 (0.02) 0.029 (0.01) — — — — — — 0.159 (0.09) 0.055 (0.03) — — — — 0.162 (0.07) 0.031 (0.03) — — — — 0.104 (0.06) 0.048 (0.03) — — — — 0.107 (0.05) 0.034 (0.02) — — — — 0.093 (0.07) 0.046 (0.03) — — — — 0.073 (0.05) 0.038 (0.03) — — — — 0.084 (0.06) 0.063 (0.03) — — — — 0.05 (0.04) 0.052 (0.03) 0.279 (0.03) 0.336 (0.03) 0.296 (0.03) 0.349 (0.03) 0.294 (0.03) 0.349 (0.03) — — — — — — ↵* Locations are: Anhumas ideal N (A-IN), Anhumas low N (A-LN), Piracicaba ideal N (P-IN) and Piracicaba low N (P-LN)
- Table 5 Maize USP data set. Mean Pearson’s correlation (50 partitions) of each environment for random cross-validation CV2, for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs), environment-specific variance G×E deviation model (MDe), multi-environment unstructured covariance models (MUC and MUCf) with two kernels, GBLUP (GB) and Gaussian kernel (GK) for grain yield with the proposed random effect l and without the random effect l (standard deviation in parentheses)
Proposed models with random effects l and f Environment* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MUCf-GK MUCf-GB P-LN 0.525 (0.07) 0.521 (0.06) 0.545 (0.07) 0.546 (0.07) 0.549 (0.05) 0.544 (0.08) 0.563 (0.06) 0.564 (0.06) P-IN 0.575 (0.05) 0.566 (0.06) 0.593 (0.05) 0.591 (0.06) 0.594 (0.05) 0.595 (0.04) 0.592 (0.06) 0.597 (0.05) A-LN 0.493 (0.06) 0.493 (0.07) 0.508 (0.06) 0.515 (0.06) 0.509 (0.05) 0.503 (0.05) 0.526 (0.05) 0.515 (0.06) A-IN 0.603 (0.05) 0.599 (0.06) 0.627 (0.05) 0.629 (0.06) 0.631 (0.05) 0.627 (0.06) 0.630 (0.05) 0.618 (0.06) Proposed models without random effects l and f Environment MM-GK MM-GB MDs-GK MDs-GB MDe-GK MDe-GB MUC-GK MUC-GB P-LN 0.50 (0.06) 0.315 (0.06) 0.524 (0.07) 0.325 (0.06) 0.52 (0.06) 0.32 (0.05) 0.536 (0.07) 0.318 (0.06) P-IN 0.53 (0.05) 0.358 (0.06) 0.554 (0.05) 0.368 (0.05) 0.56 (0.05) 0.365 (0.06) 0.563 (0.05) 0.361 (0.07) A-LN 0.463 (0.07) 0.332 (0.07) 0.476 (0.06) 0.334 (0.07) 0.478 (0.06) 0.33 (0.07) 0.496 (0.06) 0.333 (0.06) A-IN 0.584 (0.06) 0.438 (0.07) 0.612 (0.05) 0.447 (0.05) 0.607 (0.04) 0.445 (0.06) 0.61 (0.05) 0.439 (0.06) ↵* Environments are: Anhumas ideal N (A-IN), Anhumas low N (A-LN), Piracicaba ideal N (P-IN) and Piracicaba low N (P-LN).
- Table 6 Wheat WHE1 data set. Estimated variance components for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs) and environment-specific variance G×E deviation model (MDe) with two kernels, GBLUP (GB) and Gaussian (GK), with l and without l for grain yield (standard deviation in parentheses)
Variance component* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MM- GK MM- GB MDs-GK MDs- GB MDe-GK Mde- GB 0.805 (0.03) 0.81 (0.03) 0.388 (0.02) 0.471 (0.02) 0.416 (0.02) 0.479 (0.02) 0.812 (0.03) 0.824 (0.03) 0.462 (0.03) 0.551 (0.02) 0.471 (0.02) 0.533 (0.02) 0.597 (0.11) 0.177 (0.04) 0.262 (0.14) 0.074 (0.04) 1.028 (0.2) 0.326 (0.06) 0.599 (0.1) 0.192 (0.03) 0.752 (0.14) 0.219 (0.05) 1.404 (0.17) 0.414 (0.06) — — 1.637 (0.16) 0.42 (0.05) — — — — 1.349 (0.15) 0.349 (0.04) — — — — — — 3.356 (0.44) 1.058 (0.15) — — — — 3.026 (0.39) 0.868 (0.13) — — — — 0.271 (0.16) 0.038 (0.03) — — — — 0.142 (0.07) 0.08 (0.03) — — — — 0.382 (0.24) 0.031 (0.03) — — — — 0.135 (0.07) 0.076 (0.03) — — — — 1.147 (0.24) 0.3 (0.08) — — — 0.839 (0.22) 0.217 (0.06) 0.014 (0.01) 0.024 (0.02) 0.101 (0.02) 0.107 (0.02) 0.077 (0.02) 0.09 (0.02) — — — — — ↵* Environments are 1, 2, 3, and 4.
- Table 7 WHEAT WHE1 data set. Mean Pearson’s correlation (50 partitions) of each environment for random cross-validation CV2, for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs), environment-specific variance G×E deviation model (MDe), multi-environment unstructured covariance models (MUC and MUCf) with two kernels, GBLUP (GB) and Gaussian kernel (GK) for grain yield with the proposed random effect l and without the random effect l (standard deviation in parentheses)
Proposed models with random effects l and f Environment* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MUCf-GK MUCf-GB E1 −0.052 (0.06) −0.048 (0.07) 0.458 (0.05) 0.422 (0.07) 0.455 (0.06) 0.424 (0.05) 0.616 (0.06) 0.574 (0.07) E2 0.572 (0.04) 0.572 (0.05) 0.625 (0.03) 0.626 (0.05) 0.671 (0.04) 0.668 (0.04) 0.721 (0.04) 0.726 (0.04) E3 0.486 (0.05) 0.482 (0.05) 0.50 (0.05) 0.473 (0.06) 0.558 (0.04) 0.545 (0.05) 0.703 (0.04) 0.695 (0.04) E4 0.402 (0.06) 0.399 (0.05) 0.525 (0.05) 0.501 (0.06) 0.537 (0.05) 0.516 (0.05) 0.573 (0.06) 0.543 (0.06) Proposed models without random effecst l and f Environment MM-GK MM-GB MDs-GK MDs-GB MDe-GK MDe-GB MUC-GK MUC-GB E1 −0.026 (0.06) −0.024 (0.07) 0.478 (0.06) 0.458 (0.06) 0.445 (0.07) 0.442 (0.06) 0.574 (0.08) 0.534 (0.07) E2 0.558 (0.05) 0.541 (0.05) 0.593 (0.05) 0.562 (0.04) 0.652 (0.04) 0.624 (0.04) 0.682 (0.06) 0.635 (0.05) E3 0.486 (0.06) 0.481 (0.05) 0.47 (0.06) 0.457 (0.06) 0.555 (0.05) 0.545 (0.05) 0.676 (0.04) 0.593 (0.04) E4 0.406 (0.05) 0.388 (0.06) 0.52 (0.04) 0.463 (0.05) 0.544 (0.05) 0.503 (0.05) 0.550 (0.06) 0.512 (0.07) ↵* Environments are, E1, E2, E3, and E4.
- Table 8 WHEAT WHE5 data set. Estimated variance components for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs) and environment-specific variance G×E deviation model (MDe) with two kernels, GBLUP (GB) and Gaussian kernel (GK), with l and without l (Sousa et al. 2017), for grain yield (standard deviation in parentheses)
Variance component* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MM- GK MM- GB MDs-GK MDs- GB MDe-GK Mde- GB 0.879 (0.02) 0.883 (0.02) 0.001 (0.00) 0.269 (0.03) 0.001 (0.00) 0.248 (0.03) 0.88 (0.02) 0.884 (0.02) 0.001 (0.00) 0.282 (0.03) 0.002 (0.00) 0.267 (0.03) 0.168 (0.02) 0.102 (0.01) 0.131 (0.01) 0.064 (0.03) 0.105 (0.02) 0.061 (0.03) 0.17 (0.02) 0.105 (0.01) 0.16 (0.02) 0.125 (0.02) 0.178 (0.02) 0.119 (0.02) — — 1.49 (0.03) 0.636 (0.05) — — — — 1.482 (0.04) 0.618 (0.05) — — — — — — 1.385 (0.07) 0.639 (0.07) — — — — 1.37 (0.07) 0.607 (0.06) — — — — 1.578 (0.08) 0.722 (0.08) — — — — 1.568 (0.08) 0.693 (0.08) — — — — 1.262 (0.07) 0.554 (0.06) — — — — 1.187 (0.07) 0.528 (0.06) — — — — 1.619 (0.08) 0.74 (0.08) — — — — 1.637 (0.09) 0.716 (0.07) — — — — 1.73 (0.09) 0.74 (0.08) — — — — 1.709 (0.09) 0.717 (0.08) 0.003 (0.0) 0.004 (0.0) 0.014 (0.0) 0.043 (0.02) 0.02 (0.01) 0.039 (0.02) — — — — — — ↵* Environments are described by a sequence of codes: 0i, 2i and 5i denote the number of irrigation cycles; B/F denotes whether the planting system was ‘bed’ (B) or ‘flat’ (F); N/H denotes whether planting date was normal (N) or late (H, simulating heat).
- Table 9 WHEAT WHE5 data set. Mean Pearson’s correlation (50 partitions) of each environment for random cross-validation CV2, for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs), environment-specific variance G×E deviation model (MDe), multi-environment unstructured covariance models (MUC and MUCf) with two kernels, GBLUP (GB) and Gaussian kernel (GK) for grain yield with the proposed random effect l and without the random effect l (standard deviation in parentheses)
Proposed models with random effects l and f Environment* MMl-GK MMl-GB MDsl-GK MDsl-GB Mdel-GK MDel-GB MUCf-GK MUCf-GB 0iFN 0.309 (0.05) 0.301 (0.05) 0.610 (0.04) 0.58 (0.03) 0.619 (0.04) 0.576 (0.04) 0.645 (0.05) 0.595 (0.05) 2iBN 0.186 (0.06) 0.191 (0.05) 0.495 (0.04) 0.453 (0.03) 0.502 (0.04) 0.449 (0.04) 0.498 (0.06) 0.469 (0.06) 5iBH 0.23 (0.06) 0.267 (0.05) 0.678 (0.02) 0.631 (0.04) 0.685 (0.03) 0.637 (0.03) 0.684 (0.03) 0.650 (0.04) 5iBN 0.262 (0.05) 0.256 (0.05) 0.456 (0.04) 0.430 (0.04) 0.452 (0.04) 0.406 (0.05) 0.637 (0.04) 0.618 (0.04) 5iFN 0.266 (0.05) 0.247 (0.05) 0.418 (0.05) 0.401 (0.05) 0.407 (0.05) 0.402 (0.04) 0.603 (0.05) 0.601 (0.05) Proposed models without random effects l and f Environment MM-GK MM-GB MDs-GK MDs-GB MDe-GK MDe-GB MUC-GK MUC-GB 0iFN 0.321 (0.05) 0.303 (0.05) 0.621 (0.03) 0.572 (0.03) 0.627 (0.04) 0.574 (0.03) 0.646 (0.05) 0.595 (0.05) 2iBN 0.215 (0.04) 0.211 (0.05) 0.49 (0.05) 0.451 (0.04) 0.491 (0.05) 0.459 (0.04) 0.497 (0.06) 0.470 (0.06) 5iBH 0.248 (0.06) 0.284 (0.05) 0.675 (0.02) 0.646 (0.03) 0.677 (0.03) 0.631 (0.03) 0.684 (0.03) 0.649 (0.04) 5iBN 0.255 (0.04) 0.245 (0.05) 0.452 (0.04) 0.407 (0.04) 0.440 (0.05) 0.409 (0.05) 0.635 (0.04) 0.598 (0.04) 5iFN 0.251 (0.05) 0.245 (0.05) 0.405 (0.04) 0.394 (0.04) 0.408 (0.04) 0.384 (0.04) 0.607 (0.05) 0.577 (0.05) ↵* Environments are described by a sequence of codes: 0i, 2i and 5i denote the number of irrigation cycles; B/F denotes whether the planting system was ‘bed’ (B) or ‘flat’ (F); N/H denotes whether planting date was normal (N) or late (H, simulating heat).
- Table S1. Maize HEL data set. Mean Pearson’s correlation (50 partitions) of each location for random cross-validation CV1, for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs), environment-specific variance G×E deviation model (MDe), multi-environment unstructured covariance models (MUC and MUCf) with two kernels, GBLUP (GB) and Gaussian kernel (GK) for grain yield with the proposed random effect l and without the random effect l (standard deviation in parentheses)
Proposed models with random effects l and f Location* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MUCf-GK MUCf-GB IP 0.571 (0.1) 0.439 (0.12) 0.745 (0.05) 0.644 (0.1) 0.749 (0.06) 0.634 (0.09) 0.756 (0.06) 0.659 (0.06) NM 0.503 (0.08) 0.354 (0.09) 0.532 (0.08) 0.385 (0.11) 0.525 (0.09) 0.365 (0.11) 0.537 (0.08) 0.381 (0.11) PM 0.661 (0.06) 0.574 (0.07) 0.753 (0.05) 0.682 (0.07) 0.753 (0.04) 0.685 (0.05) 0.751 (0.04) 0.685 (0.05) SE 0.347 (0.09) 0.202 (0.11) 0.505 (0.08) 0.370 (0.1) 0.513 (0.08) 0.366 (0.09) 0.489 (0.09) 0.349 (0.09) SO 0.442 (0.1) 0.287 (0.09) 0.552 (0.08) 0.402 (0.1) 0.552 (0.08) 0.395 (0.12) 0.551 (0.08) 0.39 (0.09) Proposed models without random effects l and f Location* MMl-GK MM-GB MDs-GK MDs-GB MDe-GK MDe-GB MUC-GK MUC-GB IP 0.575 (0.09) 0.426 (0.11) 0.752 (0.06) 0.607 (0.08) 0.755 (0.05) 0.618 (0.09) 0.758 (0.05) 0.641 (0.08) NM 0.506 (0.09) 0.361 (0.07) 0.54 (0.09) 0.394 (0.08) 0.538 (0.09) 0.394 (0.08) 0.545 (0.06) 0.391 (0.1) PM 0.662 (0.06) 0.533 (0.07) 0.758 (0.05) 0.662 (0.07) 0.754 (0.05) 0.671 (0.04) 0.754 (0.04) 0.669 (0.05) SE 0.346 (0.1) 0.219 (0.1) 0.527 (0.06) 0.321 (0.1) 0.524 (0.08) 0.339 (0.09) 0.505 (0.07) 0.319 (0.11) SO 0.455 (0.1) 0.293 (0.11) 0.576 (0.07) 0.376 (0.1) 0.555 (0.09) 0.383 (0.11) 0.56 (0.07) 0.377 (0.1) ↵* Locations are: IP: Ipiaçú-MG, NM: Nova Mutum-MT, PM: Pato de Minas-MG, SE: Sertanópolis-PR, and SO: Sorriso-MT.
- Table S2. Maize USP data set. Mean Pearson’s correlation (50 partitions) of each location for random cross-validation CV1, for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs), environment-specific variance G×E deviation model (MDe), multi-environment unstructured covariance models (MUC and MUCf) with two kernels, GBLUP (GB) and Gaussian kernel (GK) for grain yield with the proposed random effect l and without the random effect l (standard deviation in parentheses)
Proposed models with random effects l and f Environment* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MUCf-GK MUCf-GB P-LN 0.28 (0.06) 0.272 (0.07) 0.307 (0.06) 0.293 (0.07) 0.303 (0.06) 0.286 (0.07) 0.294 (0.07) 0.286 (0.06) P-IN 0.304 (0.06) 0.298 (0.08) 0.335 (0.08) 0.329 (0.06) 0.335 (0.07) 0.332 (0.08) 0.331 (0.08) 0.327 (0.06) A-LN 0.287 (0.07) 0.283 (0.05) 0.305 (0.08) 0.31 (0.06) 0.303 (0.06) 0.309 (0.06) 0.321 (0.08) 0.309 (0.07) A-IN 0.389 (0.07) 0.386 (0.08) 0.42 (0.07) 0.413 (0.07) 0.425 (0.07) 0.422 (0.06) 0.418 (0.05) 0.417 (0.07) Proposed models without random effects l and f Environment* MMl-GK MM-GB MDs-GK MDs-GB MDe-GK MDe-GB MUC-GK MUC-GB P-LN 0.286 (0.07) 0.278 (0.05) 0.305 (0.05) 0.289 (0.07) 0.313 (0.08) 0.295 (0.07) 0.311 (0.06) 0.30 (0.06) P-IN 0.285 (0.08) 0.313 (0.06) 0.324 (0.06) 0.332 (0.07) 0.324 (0.07) 0.33 (0.05) 0.318 (0.05) 0.341 (0.06) A-LN 0.262 (0.07) 0.292 (0.07) 0.278 (0.06) 0.313 (0.06) 0.285 (0.07) 0.308 (0.06) 0.300 (0.06) 0.318 (0.07) A-IN 0.365 (0.06) 0.391 (0.07) 0.395 (0.06) 0.415 (0.07) 0.403 (0.07) 0.417 (0.06) 0.406 (0.05) 0.424 (0.07) ↵* Environments are: Anhumas ideal N (A-IN), Anhumas low N (A-LN), Piracicaba ideal N (P-IN) and Piracicaba low N (P-LN)
- Table S3. Wheat data set WHE1. Mean Pearson’s correlation (50 partitions) of each location for random cross-validation CV1, for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs), environment-specific variance G×E deviation model (MDe), multi-environment unstructured covariance models (MUC and MUCf) with two kernels, GBLUP (GB) and Gaussian kernel (GK) for grain yield with the proposed random effect l and without the random effect l (standard deviation in parentheses)
Proposed models with random effects l and f Environment MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MUCf-GK MUCf-GB E1 0.048 (0.06) 0.054 (0.07) 0.545 (0.05) 0.512 (0.05) 0.558 (0.05) 0.510 (0.05) 0.560 (0.05) 0.515 (0.06) E2 0.397 (0.06) 0.405 (0.06) 0.49 (0.06) 0.476 (0.06) 0.48 (0.05) 0.474 (0.05) 0.472 (0.05) 0.478 (0.05) E3 0.368 (0.07) 0.373 (0.06) 0.405 (0.05) 0.366 (0.06) 0.416 (0.06) 0.399 (0.05) 0.413 (0.06) 0.386 (0.06) E4 0.341 (0.06) 0.329 (0.05) 0.472 (0.04) 0.439 (0.05) 0.467 (0.05) 0.441 (0.06) 0.464 (0.06) 0.450 (0.04) Proposed models without random effects l and f Environment MMl-GK MM-GB MDs-GK MDs-GB MDe-GK MDe-GB MUC-GK MUC-GB E1 0.066 (0.06) 0.049 (0.06) 0.544 (0.05) 0.472 (0.06) 0.539 (0.04) 0.495 (0.05) 0.571 (0.04) 0.513 (0.04) E2 0.416 (0.06) 0.414 (0.06) 0.476 (0.05) 0.475 (0.06) 0.472 (0.05) 0.464 (0.05) 0.465 (0.05) 0.454 (0.05) E3 0.377 (0.05) 0.384 (0.05) 0.397 (0.05) 0.388 (0.06) 0.423 (0.05) 0.392 (0.05) 0.405 (0.05) 0.381 (0.05) E4 0.339 (0.05) 0.339 (0.05) 0.469 (0.04) 0.437 (0.04) 0.46 (0.05) 0.416 (0.05) 0.456 (0.05) 0.418 (0.05) - Table S4. Wheat data set WHE5. Mean Pearson’s correlation (50 partitions) of each location for random cross-validation CV1, for the multi-environment models, main genotypic effect model (MM), single variance G×E deviation model (MDs), environment-specific variance G×E deviation model (MDe), multi-environment unstructured covariance models (MUC and MUCf) with two kernels, GBLUP (GB) and Gaussian kernel (GK) for grain yield with the proposed random effect l and without the random effect l (standard deviation in parentheses)
Proposed models with random effects l and f Environment* MMl-GK MMl-GB MDsl-GK MDsl-GB MDel-GK MDel-GB MUCf-GK MUCf-GB 0iFN 0.348 (0.05) 0.301 (0.05) 0.601 (0.04) 0.553 (0.03) 0.611 (0.04) 0.555 (0.04) 0.614 (0.03) 0.554 (0.03) 2iBN 0.217 (0.05) 0.201 (0.05) 0.474 (0.04) 0.431 (0.04) 0.47 (0.05) 0.439 (0.04) 0.475 (0.04) 0.448 (0.04) 5iBH 0.321 (0.05) 0.35 (0.05) 0.67 (0.03) 0.635 (0.03) 0.668 (0.03) 0.634 (0.03) 0.679 (0.03) 0.633 (0.03) 5iBN 0.163 (0.06) 0.136 (0.06) 0.399 (0.04) 0.353 (0.05) 0.395 (0.05) 0.345 (0.06) 0.401 (0.04) 0.358 (0.05) 5iFN 0.084 (0.06) 0.082 (0.06) 0.334 (0.04) 0.309 (0.04) 0.328 (0.05) 0.306 (0.05) 0.336 (0.05) 0.315 (0.04) Proposed models without random effects l and f Environment* MMl-GK MM-GB MDs-GK MDs-GB MDe-GK MDe-GB MUC-GK MUC-GB 0iFN 0.341 (0.05) 0.288 (0.05) 0.61 (0.04) 0.562 (0.03) 0.612 (0.03) 0.557 (0.03) 0.625 (0.04) 0.558 (0.04) 2iBN 0.205 (0.05) 0.216 (0.05) 0.478 (0.05) 0.439 (0.05) 0.473 (0.05) 0.436 (0.04) 0.476 (0.05) 0.429 (0.06) 5iBH 0.323 (0.04) 0.333 (0.05) 0.67 (0.02) 0.624 (0.03) 0.662 (0.03) 0.627 (0.03) 0.680 (0.03) 0.638 (0.03) 5iBN 0.171 (0.05) 0.163 (0.05) 0.397 (0.05) 0.357 (0.04) 0.405 (0.04) 0.356 (0.04) 0.407 (0.04) 0.354 (0.05) 5iFN 0.107 (0.05) 0.114 (0.06) 0.33 (0.05) 0.311 (0.04) 0.329 (0.05) 0.307 (0.05) 0.337 (0.04) 0.303 (0.04) ↵* Environments are described by a sequence of codes: 0i, 2i and 5i denote the number of irrigation; B/F denotes whether the planting system was ‘bed’ (B) or ‘flat’ (F); N/H denotes whether planting date was normal (N) or late (H, simulating heat).
- Table A1. Table A1. Phenotypic Pearson’s correlations among locations for grain yield for the four data sets HEL (maize), USP (maize), WHE1 (wheat), WHE2 (wheat). For HEL and USP maize data sets, the number in parentheses below each location’s name indicates the number of lines sown. For the two data sets in the wheat experiments (WHE1 and WHE2), the number of wheat lines is given in parentheses
HEL (452 maize lines) (Sousa et al. 2017) Location* Ipiaçú (IP) (247) Nova Mutum (NM) (330) Pato de Minas (PM) (452) Sertanópolis (SE) (367) Sorriso (SO) (330) Nova Mutum (NM) 0.46 — — — — Pato de Minas (PM) 0.51 0.44 — — — Sertanópolis (SE) 0.29 0.36 0.30 — — Sorriso (SO) 0.43 0.48 0.39 0.38 — USP (739 maize lines) (Sousa et al. 2017) Environment Piracicaba-LN (P-LN) (731) Piracicaba-IN (P-IN) (732) Anhumas-LN (A-LN) (731) Anhumas-IN (L-IN) (737) Piracicaba-IN (P-LN) 0.54 — — — Anhumas-LN (P-IN) 0.31 0.35 — — Anhumas-IN (A-IN) 0.43 0.47 0.47 — WHE1 (599 wheat lines) Location* E1 E2 E3 E4 E2 −0.19 — — — E3 −0.19 0.661 — — E4 −0.12 0.411 0.388 — WHE5 (807 wheat lines) Location* 0iFN 2iBN 5iBH 5iBN 5iFN 2iBN 0.166 — — — — 5iBH 0.30 −0.033 — — — 5iBN −0.10 0.122 −0.091 — — 5iFN −0.01 0.035 0.023 0.546 — ↵* Locations in HEL data set are: IP: Ipiaçú-MG, NM: Nova Mutum-MT, PM: Pato de Minas-MG, SE: Sertanópolis-PR, and SO: Sorriso-MT. Locations in USP data set are: IN = ideal Nitrogen; LN = low nitrogen. In WHE5 data set, environments are described by a sequence of codes: 0i, 2i and 5i denote the number of irrigations; B/F denotes whether the planting system was ‘bed’ (B) or ‘flat’ (F); N/H denotes whether planting date was normal (N) or late (H, simulating heat).
