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Genomic-Enabled Prediction Kernel Models with Random Intercepts for Multi-environment Trials

View ORCID ProfileJaime Cuevas, View ORCID ProfileItalo Granato, View ORCID ProfileRoberto Fritsche-Neto, Osval A. Montesinos-Lopez, Juan Burgueño, Massaine Bandeira e Sousa and View ORCID ProfileJosé Crossa
G3: Genes, Genomes, Genetics April 1, 2018 vol. 8 no. 4 1347-1365; https://doi.org/10.1534/g3.117.300454
Jaime Cuevas
Universidad de Quintana Roo, Chetumal, Quintana Roo, México
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Italo Granato
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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Roberto Fritsche-Neto
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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Osval A. Montesinos-Lopez
Facultad de Telemática, Universidad de Colima, CP 28040 Colima, Edo. de Colima, México
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Juan Burgueño
Biometrics and Statistics Unit, International Maize and Wheat Improvement Center (CIMMYT). Apdo. Postal 6-641, 06600 México DF, México
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Massaine Bandeira e Sousa
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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José Crossa
Biometrics and Statistics Unit, International Maize and Wheat Improvement Center (CIMMYT). Apdo. Postal 6-641, 06600 México DF, México
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  • For correspondence: j.crossa@cgiar.org
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  • Figure 1
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    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.

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    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.

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    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.

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    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

  • Figures
  • 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)
    ModelGeneral meanFixed environmental effectMain genetic effect of line across environmentsGenotype × environment interaction (G×E)Random intercept of the linesUnstructured G×ERandom residual
    MMEmbedded ImageEmbedded ImageEmbedded Imageε (Embedded Image
    MMlEmbedded ImageEmbedded ImageEmbedded ImageEmbedded Image (Embedded Imageε (Embedded Image
    MDsEmbedded ImageEmbedded ImageEmbedded Imagege (Embedded Image)ε (Embedded Image
    MDslEmbedded ImageEmbedded ImageEmbedded Imagege (Embedded Image)Embedded Image (Embedded Imageε (Embedded Image
    MDeEmbedded ImageEmbedded ImageEmbedded ImageEmbedded Image (Embedded Imagefor each environment)ε (Embedded Image
    MDelEmbedded ImageEmbedded ImageEmbedded ImageEmbedded Image (Embedded Imagefor each environment)Embedded Image (Embedded Imageε (Embedded Image
    MUCEmbedded ImageEmbedded Imageu (Embedded Imageε (Embedded Image
    MUCfEmbedded ImageEmbedded Imageu (Embedded Imagef (Embedded Imageε (Embedded Image
  • 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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMM- GKMM- GBMDs-GKMDs- GBMDe-GKMDe- GB
    Embedded Image0.581 (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)
    Embedded Image0.795 (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)
    Embedded Image——0.53 (0.06)0.256 (0.03)————0.525 (0.06)0.188 (0.03)——
    Embedded Image————0.376 (0.1)0.257 (0.08)————0.372 (0.1)0.237 (0.08)
    Embedded Image————0.778 (0.19)0.197 (0.08)————0.769 (0.2)0.076 (0.06)
    Embedded Image————0.374 (0.08)0.297 (0.08)————0.370 (0.09)0.259 (0.08)
    Embedded Image————1.135 (0.2)0.385 (0.11)————1.143 (0.21)0.255 (0.09)
    Embedded Image————0.688 (0.16)0.215 (0.08)————0.688 (0.16)0.158 (0.07)
    Embedded Image0.008 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMUCf-GKMUCf-GB
    IP0.596 (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)
    NM0.601 (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)
    PM0.643 (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)
    SE0.42 (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)
    SO0.544 (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
    LocationMM-GKMM-GBMDs-GKMDs-GBMDe-GKMDe-GBMUC-GKMUC-GB
    IP0.595 (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)
    NM0.601 (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)
    PM0.645 (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)
    SE0.427 (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)
    SO0.558 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMM- GKMM- GBMDs-GKMDs- GBMDe-GKMde- GB
    Embedded Image0.547 (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)
    Embedded Image0.371 (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)
    Embedded Image——0.091 (0.02)0.045 (0.01)————0.077 (0.02)0.029 (0.01)——
    Embedded Image————0.159 (0.09)0.055 (0.03)————0.162 (0.07)0.031 (0.03)
    Embedded Image————0.104 (0.06)0.048 (0.03)————0.107 (0.05)0.034 (0.02)
    Embedded Image————0.093 (0.07)0.046 (0.03)————0.073 (0.05)0.038 (0.03)
    Embedded Image————0.084 (0.06)0.063 (0.03)————0.05 (0.04)0.052 (0.03)
    Embedded Image0.279 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMUCf-GK MUCf-GB
    P-LN0.525 (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-IN0.575 (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-LN0.493 (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-IN0.603 (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
    EnvironmentMM-GKMM-GBMDs-GKMDs-GBMDe-GKMDe-GBMUC-GK MUC-GB
    P-LN0.50 (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-IN0.53 (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-LN0.463 (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-IN0.584 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMM- GKMM- GBMDs-GKMDs- GBMDe-GKMde- GB
    Embedded Image0.805 (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)
    Embedded Image0.597 (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)
    Embedded Image——1.637 (0.16)0.42 (0.05)————1.349 (0.15)0.349 (0.04)——
    Embedded Image————3.356 (0.44)1.058 (0.15)————3.026 (0.39)0.868 (0.13)
    Embedded Image————0.271 (0.16)0.038 (0.03)————0.142 (0.07)0.08 (0.03)
    Embedded Image————0.382 (0.24)0.031 (0.03)————0.135 (0.07)0.076 (0.03)
    Embedded Image————1.147 (0.24)0.3 (0.08)———0.839 (0.22)0.217 (0.06)
    Embedded Image0.014 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMUCf-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)
    E20.572 (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)
    E30.486 (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)
    E40.402 (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
    EnvironmentMM-GKMM-GBMDs-GKMDs-GBMDe-GKMDe-GBMUC-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)
    E20.558 (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)
    E30.486 (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)
    E40.406 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMM- GKMM- GBMDs-GKMDs- GBMDe-GKMde- GB
    Embedded Image0.879 (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)
    Embedded Image0.168 (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)
    Embedded Image——1.49 (0.03)0.636 (0.05)————1.482 (0.04)0.618 (0.05)——
    Embedded Image————1.385 (0.07)0.639 (0.07)————1.37 (0.07)0.607 (0.06)
    Embedded Image————1.578 (0.08)0.722 (0.08)————1.568 (0.08)0.693 (0.08)
    Embedded Image————1.262 (0.07)0.554 (0.06)————1.187 (0.07)0.528 (0.06)
    Embedded Image————1.619 (0.08)0.74 (0.08)————1.637 (0.09)0.716 (0.07)
    Embedded Image————1.73 (0.09)0.74 (0.08)————1.709 (0.09)0.717 (0.08)
    Embedded Image0.003 (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-GKMMl-GBMDsl-GKMDsl-GBMdel-GKMDel-GBMUCf-GK MUCf-GB
    0iFN0.309 (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)
    2iBN0.186 (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)
    5iBH0.23 (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)
    5iBN0.262 (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)
    5iFN0.266 (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
    EnvironmentMM-GKMM-GBMDs-GKMDs-GBMDe-GKMDe-GBMUC-GK MUC-GB
    0iFN0.321 (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)
    2iBN0.215 (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)
    5iBH0.248 (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)
    5iBN0.255 (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)
    5iFN0.251 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMUCf-GK MUCf-GB
    IP0.571 (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)
    NM0.503 (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)
    PM0.661 (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)
    SE0.347 (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)
    SO0.442 (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-GKMM-GBMDs-GKMDs-GBMDe-GKMDe-GBMUC-GK MUC-GB
    IP0.575 (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)
    NM0.506 (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)
    PM0.662 (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)
    SE0.346 (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)
    SO0.455 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMUCf-GK MUCf-GB
    P-LN0.28 (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-IN0.304 (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-LN0.287 (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-IN0.389 (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-GKMM-GBMDs-GKMDs-GBMDe-GKMDe-GBMUC-GK MUC-GB
    P-LN0.286 (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-IN0.285 (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-LN0.262 (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-IN0.365 (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
    EnvironmentMMl-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMUCf-GK MUCf-GB
    E10.048 (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)
    E20.397 (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)
    E30.368 (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)
    E40.341 (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
    EnvironmentMMl-GKMM-GBMDs-GKMDs-GBMDe-GKMDe-GBMUC-GK MUC-GB
    E10.066 (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)
    E20.416 (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)
    E30.377 (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)
    E40.339 (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-GKMMl-GBMDsl-GKMDsl-GBMDel-GKMDel-GBMUCf-GK MUCf-GB
    0iFN0.348 (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)
    2iBN0.217 (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)
    5iBH0.321 (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)
    5iBN0.163 (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)
    5iFN0.084 (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-GKMM-GBMDs-GKMDs-GBMDe-GKMDe-GBMUC-GK MUC-GB
    0iFN0.341 (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)
    2iBN0.205 (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)
    5iBH0.323 (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)
    5iBN0.171 (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)
    5iFN0.107 (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)0.510.44———
    Sertanópolis (SE)0.290.360.30——
    Sorriso (SO)0.430.480.390.38—
    USP (739 maize lines) (Sousa et al. 2017)
    EnvironmentPiracicaba-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)0.310.35——
    Anhumas-IN (A-IN)0.430.470.47—
    WHE1 (599 wheat lines)
    Location*E1E2E3E4
    E2−0.19———
    E3−0.190.661——
    E4−0.120.4110.388—
    WHE5 (807 wheat lines)
    Location*0iFN2iBN5iBH5iBN5iFN
    2iBN0.166————
    5iBH0.30−0.033———
    5iBN−0.100.122−0.091——
    5iFN−0.010.0350.0230.546—
    • ↵* 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).

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Volume 8 Issue 4, April 2018

G3: Genes|Genomes|Genetics: 8 (4)

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Genomic Selection
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Genomic-Enabled Prediction Kernel Models with Random Intercepts for Multi-environment Trials

View ORCID ProfileJaime Cuevas, View ORCID ProfileItalo Granato, View ORCID ProfileRoberto Fritsche-Neto, Osval A. Montesinos-Lopez, Juan Burgueño, Massaine Bandeira e Sousa and View ORCID ProfileJosé Crossa
G3: Genes, Genomes, Genetics April 1, 2018 vol. 8 no. 4 1347-1365; https://doi.org/10.1534/g3.117.300454
Jaime Cuevas
Universidad de Quintana Roo, Chetumal, Quintana Roo, México
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Italo Granato
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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Roberto Fritsche-Neto
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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Osval A. Montesinos-Lopez
Facultad de Telemática, Universidad de Colima, CP 28040 Colima, Edo. de Colima, México
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Juan Burgueño
Biometrics and Statistics Unit, International Maize and Wheat Improvement Center (CIMMYT). Apdo. Postal 6-641, 06600 México DF, México
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Massaine Bandeira e Sousa
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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José Crossa
Biometrics and Statistics Unit, International Maize and Wheat Improvement Center (CIMMYT). Apdo. Postal 6-641, 06600 México DF, México
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Citation

Genomic-Enabled Prediction Kernel Models with Random Intercepts for Multi-environment Trials

View ORCID ProfileJaime Cuevas, View ORCID ProfileItalo Granato, View ORCID ProfileRoberto Fritsche-Neto, Osval A. Montesinos-Lopez, Juan Burgueño, Massaine Bandeira e Sousa and View ORCID ProfileJosé Crossa
G3: Genes, Genomes, Genetics April 1, 2018 vol. 8 no. 4 1347-1365; https://doi.org/10.1534/g3.117.300454
Jaime Cuevas
Universidad de Quintana Roo, Chetumal, Quintana Roo, México
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  • ORCID record for Jaime Cuevas
Italo Granato
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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  • ORCID record for Italo Granato
Roberto Fritsche-Neto
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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  • ORCID record for Roberto Fritsche-Neto
Osval A. Montesinos-Lopez
Facultad de Telemática, Universidad de Colima, CP 28040 Colima, Edo. de Colima, México
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Juan Burgueño
Biometrics and Statistics Unit, International Maize and Wheat Improvement Center (CIMMYT). Apdo. Postal 6-641, 06600 México DF, México
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Massaine Bandeira e Sousa
Department of Genetics, Luiz de Queiroz College of Agriculture, University of São Paulo, Piracicaba, São Paulo, Brazil
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José Crossa
Biometrics and Statistics Unit, International Maize and Wheat Improvement Center (CIMMYT). Apdo. Postal 6-641, 06600 México DF, México
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  • ORCID record for José Crossa
  • For correspondence: j.crossa@cgiar.org

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  • A Bayesian Decision Theory Approach for Genomic Selection
  • Optimising Genomic Selection in Wheat: Effect of Marker Density, Population Size and Population Structure on Prediction Accuracy
  • BGGE: A New Package for Genomic-Enabled Prediction Incorporating Genotype × Environment Interaction Models
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