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 $\mathit{l}$ and without the random effect $\mathit{l}$ (standard deviation in parentheses).

	Proposed models with random effects $\mathit{l}$ and f
Environment*	MM$\mathit{l}$-GK	MM$\mathit{l}$-GB	MDs$\mathit{l}$-GK	MDs$\mathit{l}$-GB	MDe$\mathit{l}$-GK	MDe$\mathit{l}$-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 $\mathit{l}$ 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).