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 $\mathit{l}$ 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	0.28 (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 $\mathit{l}$ and f
Environment*	MM$\mathit{l}$-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)