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Wheat data set. Mean correlation (for 50 random partitions) for single-environment and genome × environment interaction (G × E) models with genomic best linear unbiased prediction (GBLUP), reproducing kernel Hilbert space (RKHS) empirical Bayes (EB), and RKHS kernel averaging (KA) methods with standard deviation in parentheses.

 
Single-environment model
Environment GBLUP RKHS EB RKHS KA Percentage change
RKHS EB vs. GBLUP RKHS KA vs. GBLUP RKHS EB vs. RKHS KA
G × E model
Pair Sample correlation Single GBLUP-G × E
RKHS EB-G × E
RKHS KA-G × E
Percentage change
Mean Percentage change† Mean Percentage change‡ Mean Percentage change§ RKHS EB-G × E vs. GBLUP-G × E RKHS KA-G × E vs. GBLUP-G × E RKHS EB-G × E vs. RKHS KA-G × E
1 0.500 (0.056) 0.577 (0.043) 0.556 (0.052) 15 11 4
2 0.474 (0.048) 0.477 (0.056) 0.482 0.050) 1 2 −1
3 0.370 (0.056) 0.422 (0.053) 0.401 (0.055) 14 8 5
4 0.447 (0.047) 0.511 (0.044) 0.522 (0.040) 14 17 −2
1–2 −0.019 1 0.509 (0.045) 2 0.582 (0.040) 1 0.581 (0.055) 4 14 14 0
2 0.476 (0.055) 0 0.475 (0.048) 0 0.480 (0.052) 0 0 1 −1
1–3 −0.1934 1 0.501 (0.045) 0 0.585 (0.041) 1 0.566 (0.057) 2 17 13 3
3 0.361 (0.056) −3 0.415 (0.053) −2 0.399 (0.053) −1 15 11 4
1–4 −0.123 1 0.505 (0.045) 1 0.585 (0.042) 1 0.577 (0.054) 4 16 14 1
4 0.431 (0.055) −4 0.502 (0.045) −2 0.511 (0.044) −2 16 19 −2
2–3 0.661 2 0.650 (0.033) 37 0.681 (0.032) 43 0.689 (0.032) 43 5 6 −1
3 0.576 (0.035) 36 0.674 (0.036) 60 0.673 (0.035) 68 17 17 0
2–4 0.441 2 0.542 (0.039) 14 0.523 (0.055) 10 0.527 (0.042) 9 −3 −3 −1
4 0.493 (0.049) 10 0.551 (0.040) 8 0.567 (0.037) 9 12 15 −3
3–4 0.387 3 0.461 (0.046) 25 0.498 (0.051) 18 0.503 (0.043) 25 8 9 −1
4 0.498 (0.050) 11 0.563 (0.042) 10 0.558 (0.034) 7 13 12 1
1–2–3–4 1 0.445 (0.054) −11 0.458 (0.060) −21 0.438 (0.063) −21 3 −2 5
2 0.600 (0.041) 27 0.644 (0.047) 35 0.661 (0.043) 37 7 10 −3
3 0.523 (0.043) 41 0.586 (0.046) 39 0.565 (0.039) 41 12 8 4
4 0.489 (0.043) 9 0.543 (0.044) 6 0.573 (0.042) 10 11 17 −5
2–3–4 2 0.639 (0.040) 35 0.678 (0.040) 42 0.678 (0.036) 41 6 6 0
3 0.590 (0.04) 59 0.654 (0.038) 55 0.655 (0.040) 3 11 11 0
4 0.537 (0.05) 20 0.550 (0.055) 8 0.555 (0.043) 6 2 33 −1
Percentage change between GBLUP-G × E vs. GBLUP single environment.
Percentage change between RKHS EB-G × E vs. RKHS EB single environment.
§Percentage change between RKHS KA-G × E vs. RKHS KA single environment.



View Full Table | Close Full ViewTable 2.

Maize data set. Average correlation (for 50 random partitions) for single-environment and G × E models with genomic best linear unbiased prediction (GBLUP), reproducing kernel Hilbert space (RKHS) empirical Bayes (EB), and RKHS kernel averaging (KA) methods with standard deviation in parentheses.

 
Single-environment model
Environment GBLUP RKHS EB RKHS KA Percentage change
RKHS EB vs. GBLUP RKHS KA vs. GBLUP RKHS EB vs. RKHS KA
G × E model
Pair Sample correlation Single GBLUP-G × E
RKHS EB-G × E
RKHS KA-G × E
Percentage change
Mean Percentage change† Mean Percentage change‡ Mean Percentage change§ RKHS EB-G × E vs. GBLUP-G × E RKHS KA-G × E vs. GBLUP-G × E RKHS EB-G × E vs. RKHS KA-G × E
1 0.558 (0.038) 0.583 (0.042) 0.577 (0.052) 4 3 1
2 0.507 (0.049) 0.542 (0.056) 0.535 (0.057) 7 6 1
3 0.508 (0.051) 0.568 (0.044) 0.552 (0.050) 12 9 3
1–2 0.388 1 0.600 (0.042) 8 0.623 (0.046) 7 0.626 (0.043) 8 4 4 0
2 0.572 (0.053) 13 0.590 (0.049) 5 0.583 (0.052) 9 3 2 1
1–3 0.262 1 0.572 (0.046) 3 0.594 (0.043) 2 0.603 (0.044) 5 4 6 −1
3 0.518 (0.050) 2 0.579 (0.048) 2 0.577 (0.046) 5 12 11 0
2–3 0.153 2 0.520 (0.064) 3 0.536 (0.048) −1 0.535 (0.057) 0 3 3 1
3 0.502 (0.046) −1 0.564 (0.050) −1 0.560 (0.0471) 1 12 9 1
1–2–3 1 0.618 (0.042) 11 0.630 (0.043) 8 0.648 (0.037) 12 2 5 −3
2 0.547 (0.053) 8 0.566 (0.043) 4 0.564 (0.067) 5 3 3 0
3 0.519 (0.044) 2 0.556 (0.055) −2 0.576 (0.047) 4 7 11 −3
Percentage change between GBLUP-G × E vs. GBLUP single environment.
Percentage change between RKHS EB-G × E vs. RKHS EB single environment.
§Percentage change between RKHS KA-G × E vs. RKHS KA single environment.



View Full Table | Close Full ViewTable B1.

Wheat data set with four environments (1–4). Average variance component across partitions for residual, marker main effect, and marker-specific effects for single-environment (standard deviation in parenthesis) (genomic best linear unbiased prediction [GBLUP], reproducing kernel Hilbert space [RKHS] empirical Bayes [EB], and RKHS kernel averaging [KA]), and genotype × environment interaction (G × E) models (GBLUP-G × E, RKHS EB-G × E, and RKHS KA-G × E). The variance components associated with the three kernels for RKHS KA-G × E are averaged. Sample and estimated correlations between pair of environments.

 
Single-environment model
Environment GBLUP
RKHS EB
RKHS KA
Residual Main effect Residual Main effects Residual Main effects
G × E model
Combination Sample correlation Single GBLUP-G × E
RKHS EB-G × E
RKHS KA-G × E
Residual Main effect Specific effect Estimated correlation Residual Main effect Specific effect Estimated correlation Residual Main effect Specific effect Estimated correlation
1 0.525 (0.0405) 0.567 (0.0599) 0.309 (0.0546) 0.856 (0.0770) 0.277 (0.0615) 1.121 (0.0966)
2 0.578 (0.0420) 0.483 (0.0449) 0.571 (0.0557) 2.090 (0.6776) 0.411 (0.0358) 1.250 (0.1252)
3 0.601 (0.0511) 0.504 (0.0716) 0.267 (0.0526) 0.801 (0.0.0913) 0.302 (0.0366) 1.220 (0.1371)
4 0.600 (0.0419) 0.469 (0.0474) 0.377 (0.0601) 0.792 (0.0687) 0.338 (0.0395) 1.091 (0.0820)
1–2 −0.019 1 0.570 (0.0297) 0.030 (0.0022) 0.205 (0.0235) 0.037 0.464 (0.0418) 0.103 (0.0184) 0.614 (0.077) 0.050 0.380 (0.0464) 0.118 (0.0049) 0.784 (0.0592) 0.082
2 0.191 (0.0232) 2.620 (0.9138) 1.110 (0.1547)
1–3 −0.1934 1 0.582 (0.0294) 0.027 (0.0022) 0.202 (0.0229) 0.033 0.333 (0.0466) 0.088 (0.0338) 0.780 (0.0669) 0.070 0.304 (0.0411) 0.095 (0.0023) 0.851 (0.0538) 0.070
3 0.201 (0.0232) 0.942 (0.1424) 1.000 (0.1012)
1–4 −0.123 1 0.572 (0.0286) 0.025 (0.0023) 0.212 (0.0221) 0.03 0.344 (0.0482) 0.076 (0.0244) 0.750 (0.0653) 0.060 0.330 (0.0390) 0.106 (0.0038) 0.828 (0.0567) 0.082
4 0.193 (0.0236) 0.800 (0.08950) 0.857 (0.0607)
2–3 0.661 2 0.495 (0.0224) 0.282 (0.0254) 0.040 (0.0033) 0.345 0.368 (0.0196) 0.115 (0.1550) 0.181 (0.0422) 0.680 0.291 (0.0201) 1.060 (0.2008) 0.192 (0.0082) 0.685
3 0.043 (0.0037) 0.116 (0.0156) 0.210 (0.0089)
2–4 0.441 2 0.575 (0.0263) 0.140 (0.0219) 0.076 (0.0114) 0.18 0.481 (0.0389) 0.611 (0.1606) 0.859 (0.8322) 0.377 0.290 (0.0294) 1.100 (0.1734) 0.538 (0.0717) 0.577
4 0.076 (0.0121) 0.256 (0.0960) 0.511 (0.0367)
3–4 0.387 3 0.596 (0.0261) 0.150 (0.0249) 0.071 (0.0096) 0.185 0.418 (0.0359) 0.520 (0.0577) 0.317 (0.0815) 0.424 0.395 (0.0343) 0.605 (0.0892) 0.344 (0.0236) 0.450
4 0.066 (0.0101) 0.260 (0.0566) 0.344 (0.0207)
1–2–3–4 1 0.571 (0.0177) 0.139 (0.0148) 0.324 (0.0291) 0.445 (0.0320) 0.814 (0.2997) 0.895 (0.0992) 0.394 (0.0284) 0.477 (0.1229) 1.300 (0.2541)
2 0.049 (0.0071) 0.529 (0.2920) 0.35 (0.0492)
3 0.046 (0.0074) 0.247 (0.0704) 0.341 (0.0345)
4 0.080 (0.0162) 0.353 (0.0534) 0.426 (0.0332)



View Full Table | Close Full ViewTable B2.

Maize data set with three environments (1-3). Average variance component for residual, marker main effect, and marker-specific effects for single-environment model (standard deviation in parenthesis) (GBLUP, RKHS EB, and RKHS KA) and for G × E models (GBLUP- G × E, RKHS EB-G × E, and RKHS KA-G × E). The variance components associated with the three kernels for RHKS KA are averaged. Sample and estimated correlations are between pair of environments.

 
Single-environment model
Environment GBLUP
RKHS EB
RKHS KA
Residual Main effect Residual Main effect Residual Main effect
G × E model
Combination Sample correlation Single GBLUP-G × E
RKHS EB-G × E
RKHS KA-G × E
Residual Main effect Specific effect Estimated correlation Residual Main effect Specific effect Estimated correlation Residual Main effect Specific effect Estimated correlation
1 0.443 (0.0396) 0.522 (0.0536) 0.326 (0.0388) 0.873 (0.0915) 0.232 (0.0162) 1.080 (0.1090)
2 0.390 (0.0401) 0.723 (0.0913) 0.269 (0.0377) 1.100 (0.0635) 0.214 (0.0168) 1.430 (0.1399)
3 0.500 (0.0325) 0.526 (0.0615) 0.267 (0.0269) 0.801 (0.0573) 0.235 (0.0176) 0.990 (0.0677)
1-2 0.388 1 0.415 (0.0305) 0.377 (0.0639) 0.180 (0.0292) 0.371 0.290 (0.0296) 0.533 (0.0444) 0.380 (0.1059) 0.423 0.234 (0.0230) 0.599 (0.0907) 0.443 (0.0275) 0.423
2 0.271 (0.0509) 0.488 (0.0578) 0.668 (0.0649)
1-3 0.262 1 0.489 (0.0275) 0.200 (0.0457) 0.248 (0.0411) 0.212 0.306 (0.0330) 0.290 (0.0608) 0.571 (0.1067) 0.263 0.248 (0.0261) 0.271 (0.0239) 0.706 (0.0662) 0.232
3 0.285 (0.0538) 0.448 (0.0610) 0.593 (0.0324)
2-3 0.153 2 0.491 (0.0366) 0.105 (0.0213) 0.437 (0.0617) 0.104 0.252 (0.0263) 0.124 (0.0280) 0.960 (0.1052) 0.105 0.231 (0.0223) 0.135 (0.077) 1.334 (0.1637) 0.100
3 0.392 (0.0698) 0.664 (0.0547) 0.736 (0.0406)
1-2-3 1 0.472 (0.0177) 0.230 (0.0148) 0.193 (0.0291) 0.279 (0.0320) 0.274 (0.02097) 0.576 (0.0992) 0.242 (0.0284) 0.300 (0.1229) 0.592 (0.2541)
2 0.313 (0.0071) 0.721 (0.2920) 0.720 (0.0492)
3 0.343 (0.0074) 0.539 (0.0704) 0.539 (0.0345)