View Full Table | Close Full ViewTable 1.

Description of the pedigree (P) and genomic (M) models used for the analysis.†

 
Model
Covariance structure
No. Code R0 G0P G0M
Pedigree
1 PI-I I I
2 PD-I I D
3 PFA-I I FA(2)
4 PI-D D I
5 PD-D D D
6 PFA-D D FA(2)
7 PI-UN UN I
8 PD-UN UN D
9 PFA-UN UN FA(2)
Marker
10 MI-I I I
11 MD-I I D
12 MFA-I I FA(2)
13 MI-D D I
14 MD-D D D
15 MFA-D D FA(2)
16 MI-UN UN I
17 MD-UN UN D
18 MFA-UN UN FA(2)
Pedigree + marker
19 PMI-I I I I
20 PMD-I I D D
21 PMFA-I I FA(2) FA(2)
22 PMI-D D I I
23 PMD-D D D D
24 PMFA-D D FA(2) FA(2)
25 PMI-UN UN I I
26 PMD-UN UN D D
27 PMFA-UN UN FA(2) FA(2)
I = Identity (imposes independence and homoscedasticity, one parameter); D = Diagonal (imposes independence but allow heterogeneous variances, four parameters); FA(2) = two-common factor model (imposes heterogeneous variances and correlation between environments, 10 parameters); UN = unstructured (imposes heterogeneous variances and correlation between environments, 10 parameters).



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Estimates of residual and genetic covariance matrices (correlation in the upper diagonal), variances and covariances in the diagonal and lower diagonal components of each matrix, respectively, derived from a full-data analysis of models using different covariance structures for four environments (1–4).

 
R0
G0P
G0M
Model† Env.‡ 1 2 3 4 1 2 3 4 1 2 3 4
PFA-UN (Model 9) 1 0.558 0.064 −0.248 0.021 0.573 −0.139 −0.131 −0.316
2 0.035 0.556 0.390 0.250 −0.078 0.553 0.965 0.575
3 −0.131 0.205 0.498 0.163 −0.080 0.583 0.660 0.593
4 0.011 0.134 0.082 0.517 −0.185 0.331 0.373 0.598
MFA-UN (Model 18) 1 0.526 0.153 −0.195 0.116 2.998 −0.258 −0.2080 −0.479
2 0.084 0.573 0.484 0.259 −0.727 2.634 0.925 0.623
3 −0.112 0.292 0.634 0.189 −0.547 2.280 2.305 0.648
4 0.065 0.152 0.117 0.602 −1.290 1.572 1.529 2.414
PMFA-UN (Model 27) 1 0.433 0.146 −0.220 0.183 2.665 −0.306 −0.344 −0.613 0.179 0.139 −0.011 −0.146
2 0.068 0.499 0.390 0.225 −0.695 1.930 0.888 0.686 0.027 0.204 0.968 0.425
3 −0.099 0.189 0.469 0.114 −0.670 1.471 1.422 0.771 −0.003 0.268 0.377 0.441
4 0.082 0.108 0.053 0.462 −1.222 1.165 1.124 1.494 −0.036 0.112 0.159 0.343
Model 9: PFA-UN = factor analytic-pedigree for G0P and unstructured for R0; Model 18: MFA-UN = factor analytic-genomic for G0M and unstructured for R0; Model 27: PMFA-UN = factor analytic-pedigree for G0P, factor analytic-genomic for G0M and unstructured for R0.
Env. = Environment.



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Mean correlations from 10-fold cross-validation between the predicted and the observed values of genotypes for all environments, for individual Environments 1, 2, 3, and 4, and for Environments 2, 3, 4 together for 18 different models (Models 1–18) for two different cross-validation schemes (CV1 and CV2), each with 10-fold. For CV1 and CV2, the best predicted model among Models 1–9 and among Models 10–18 are underlined.

 
Model no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Model code† PI-I PD-I PFA-I PI-D PD-D PFA-D PI-UN PD-UN PFA-UN MI-I MD-I MFA-I MI-D MD-D MFA-D MI-UN MD-UN MFA-UN
CV1
Environment 1 0.444 0.443 0.437 0.444 0.443 0.433 0.428 0.444 0.447 0.534 0.534 0.532 0.534 0.534 0.531 0.536 0.541 0.550
Environment 2 0.416 0.416 0.410 0.416 0.416 0.406 0.365 0.350 0.427 0.496 0.495 0.467 0.496 0.495 0.463 0.466 0.450 0.490
Environment 3 0.387 0.386 0.389 0.387 0.386 0.388 0.331 0.317 0.397 0.387 0.386 0.404 0.387 0.387 0.401 0.338 0.332 0.406
Environment 4 0.441 0.441 0.432 0.441 0.442 0.427 0.402 0.409 0.435 0.438 0.438 0.433 0.438 0.437 0.428 0.420 0.417 0.441
Environments E2–E4 0.414 0.413 0.409 0.414 0.413 0.405 0.365 0.355 0.418 0.439 0.438 0.432 0.439 0.439 0.428 0.408 0.397 0.445
Environments E1–E4 0.420 0.420 0.415 0.420 0.418 0.408 0.378 0.376 0.423 0.464 0.464 0.457 0.465 0.465 0.452 0.442 0.434 0.473
CV2
Environment 1 0.449 0.449 0.460 0.449 0.447 0.454 0.437 0.451 0.458 0.550 0.550 0.552 0.550 0.550 0.553 0.556 0.560 0.558
Environment 2 0.422 0.421 0.623 0.422 0.421 0.626 0.397 0.382 0.619 0.502 0.500 0.609 0.502 0.500 0.611 0.511 0.490 0.606
Environment 3 0.374 0.374 0.633 0.374 0.374 0.635 0.354 0.336 0.607 0.392 0.391 0.581 0.392 0.390 0.585 0.384 0.368 0.556
Environment 4 0.452 0.452 0.533 0.452 0.452 0.529 0.425 0.429 0.527 0.453 0.452 0.513 0.453 0.452 0.510 0.445 0.439 0.506
Environments E2–E4 0.414 0.413 0.596 0.413 0.410 0.596 0.393 0.381 0.584 0.454 0.453 0.571 0.455 0.453 0.571 0.452 0.435 0.559
Environments E1–E4 0.420 0.419 0.565 0.419 0.416 0.564 0.401 0.397 0.555 0.475 0.473 0.564 0.475 0.473 0.565 0.475 0.462 0.556
Model code: Model 1: PI-I = identity-pedigree for G0P and identity for R0; Model 2: PD-I = diagonal-pedigree for G0P and identity for R0; Model 3: PFA-I = factor analytic-pedigree for G0P and identity for R0; Model 4: PI-D = identity-pedigree for G0P and diagonal for R0; Model 5: PD-D = diagonal-pedigree for G0P and diagonal for R0; Model 6: PFA-D = factor analytic-pedigree for G0P and diagonal for R0; Model 7: PI-UN = identity-pedigree for G0P and unstructured for R0; Model 8: PD-UN = diagonal-pedigree for G0P and unstructured for R0; Model 9: PFA-UN = factor analytic-pedigree for G0P and unstructured for R0; Model 10: MI-I = identity-genomic for G0M and identity for R0; Model 11: MD-I = diagonal-genomic for G0M and identity for R0; Model 12: MFA-I = factor analytic-genomic for G0M and identity for R0; Model 13: MI-D = identity-genomic for G0M and diagonal for R0; Model 14; MD-D = diagonal-genomic for G0M and diagonal for R0; Model 15: MFA-D = factor analytic-genomic for G0M and diagonal for R0; Model 16: MI-UN = identity-genomic for G0M and unstructured for R0; Model 17: MD-UN = diagonal-genomic for G0M and unstructured for R0; Model 18: MFA-UN = factor analytic-genomic for G0M and unstructured for R0.



View Full Table | Close Full ViewTable 4.

Mean correlations from 10-fold cross-validation between the predicted and the observed values of genotypes for all environments, for individual Environments 1, 2, 3, and 4, and for Environments 2, 3, 4 together for nine different models including AP and AM simultaneously (Models 19–27) for two different cross-validation schemes (CV1 and CV2), each with 10-fold. For CV1 and CV2, the best predicted model among Models 19–27 are underlined.

 
Model no. 19 20 21 22 23 24 25 26 27
Model code† PMI-I PMD-I PMFA-I PMI-D PMD-D PMFA-D PMI-UN PMD-UN PMFA-UN
CV1
Environment 1 0.562 0.559 0.542 0.563 0.560 0.551 0.560 0.564 0.560
Environment 2 0.502 0.504 0.469 0.502 0.503 0.482 0.469 0.458 0.495
Environment 3 0.432 0.433 0.432 0.433 0.433 0.432 0.380 0.374 0.423
Environment 4 0.479 0.481 0.444 0.479 0.480 0.453 0.457 0.462 0.462
Environments E2–E4 0.471 0.474 0.445 0.471 0.472 0.454 0.437 0.431 0.459
Environments E1–E4 0.494 0.496 0.4722 0.494 0.495 0.480 0.469 0.464 0.486
CV2
Environment 1 0.572 0.570 0.580 0.573 0.570 0.580 0.574 0.577 0.574
Environment 2 0.507 0.505 0.646 0.507 0.502 0.646 0.512 0.497 0.642
Environment 3 0.428 0.424 0.631 0.427 0.424 0.632 0.414 0.400 0.604
Environment 4 0.495 0.496 0.548 0.495 0.496 0.545 0.481 0.487 0.544
Environments E2–E4 0.480 0.479 0.610 0.480 0.477 0.609 0.475 0.465 0.599
Environments E1–E4 0.502 0.500 0.603 0.502 0.498 0.602 0.498 0.4897 0.592
Model code: Model 19: PMI-I = identity-pedigree (G0P), identity-genomic (G0M) with identity for R0; Model 20: PMD-I = diagonal-pedigree (G0P), diagonal-genomic for (G0M) with identity for R0; Model 21: PMFA-I = factor analytic-pedigree (G0P), factor analytic-genomic for (G0M) with identity for R0; Model 22: PMI-D = identity-pedigree (GP), identity-genomic (G0M) with diagonal for R0; Model 23: PMD-D = diagonal-pedigree (G0P), diagonal-genomic for (G0M) with diagonal for R0; Model 24: PMFA-D = factor analytic-pedigree (G0P), factor analytic-genomic for (G0M) with diagonal for R0; Model 25: PMI-UN = identity-pedigree (G0P), identity-genomic (G0M) with unstructured for R0; Model 26: PMD-UN = diagonal-pedigree (G0P), diagonal-genomic for (G0M) with unstructured for R0; Model 27: PMFA-UN = factor analytic-pedigree (G0P), factor analytic-genomic for (G0M) with unstructured for R0.



View Full Table | Close Full ViewTable C1.

Example of one-fold Cross-validation Scheme 1 (CV1) and Cross-validation Scheme 2 (CV2). The numbers in the table represent the actual standardized grain yield data of 599 genotypes in four environments (E1–E4). Missing genotypes are denoted as dots.

 
CV1
CV2
Genotype E1 E2 E3 E4 E1 E2 E3 E4
1 1.7 −1.7 −1.9 0.1 1.7 −1.7 −1.9 0.1
2 . . . . . . 0.3 −1.7
3 0.3 −0.6 −0.8 −1.1 0.3 −0.6 −0.8 −1.1
4 0.8 0.1 0.6 0.6 0.8 0.1 0.6 0.6
5 1.0 −0.3 1.6 −0.1 1.0 −0.3 1.6 −0.1
6 2.3 0.6 0.1 0.7 2.3 0.6 . .
7 . . . . 0.6 −0.3 0.1 0.0
8 0.6 −0.4 −0.7 1.0 0.6 −0.4 −0.7 1.0
9 −1.0 −1.8 −1.9 −1.5 . −1.8 . −1.5
10 −1.1 −1.6 −2.0 −0.6 −1.1 −1.6 −2.0 −0.6
11 1.7 −0.3 −0.2 0.3 1.7 −0.3 −0.2 0.3
12 . . . . 0.9 −0.2 −0.4 0.7
13 1.2 −0.2 0.8 −0.2 . −0.2 0.8 .
14 0.3 −0.7 −0.4 0.0 0.3 −0.7 −0.4 0.0
15 0.8 −0.6 −0.7 −0.2 0.8 −0.6 −0.7 −0.2
589 −0.8 0.7 0.7 2.6 −0.8 0.7 0.7 2.6
590 −0.9 0.1 1.1 2.8 −0.9 0.1 1.1 2.8
591 . . . . −1.1 . 2.0 .
592 −1.0 1.1 1.9 1.7 −1.0 1.1 1.9 1.7
593 −1.1 0.6 2.4 1.2 −1.1 0.6 2.4 1.2
594 . . . . −0.5 . . 1.8
595 −1.2 1.4 1.6 1.7 −1.2 1.4 1.6 1.7
596 −1.1 0.1 2.1 0.6 −1.1 0.1 2.1 0.6
597 −1.2 0.5 2.0 1.8 . 0.5 . 1.8
598 . . . . −0.9 0.5 1.7 2.7
599 0.2 −0.5 −0.3 0.4 0.2 −0.5 −0.3 0.4