Cross‐Validation in AMMI and GGE Models: A Comparison of Methods

Abstract

Corresponding author (Hans-Peter.Piepho@ uni-hohenheim.de). Assigned to Associate Editor Manjit Kang.Abbreviations: AMMI, additive main effects and multiplicative interaction; CV, cross-validation; EM, expectation maximization; GCV, generalized cross-validation criterion; GEI, genotype ´ environment interaction; GGE, genotype and genotype ´ environment interaction; MET, multi-environment trial; MSED, mean squared error of differences; MSEPD, mean squared error of predicting differences; n, number of multiplicative terms retained in the AMMI or the GGE model; N, true number of multiplicative terms; RCBD, randomized complete block design; rIBD, resolvable incomplete block design; SVD, singular value decomposition; TMORR, true model order recovery rate.

Estimating Broad‐Sense Heritability with Unbalanced Data from Agricultural Cultivar Trials

“…is the coefficient matrix of the mixed-model equations (McLean et al, 1991), and C 22 has the same dimension as Z¢R −1 Z + G −1 . Data were analyzed using two mixed models fitted by restricted maximum likelihood method using ASREML-R 3.0 (Gilmour et al, 2009) andSAS 9.4 (SAS Institute, 2013). For practical reasons (i.e., computational burden), both mixedmodel analyses are performed in two stages and thus in the framework of a multistage analysis, as will be further detailed below, and for each of the four datasets separately.…”

“…Moreover, if field trial designs are analyzed with spatial models (e.g., incomplete block designs, georeferenced data) and exploit relationship data via kinship matrices, neither the residuals nor the genotype effects are independent, even for balanced data. Furthermore, data may be analyzed via additive main effect and multiplicative interaction (AMMI) or genotype and genotype ´ environment interaction (GGE) models (Gauch, 1992;Hadasch et al, 2017), and/or there may be more than one genotypic main effect due to the use of so-called check cultivars. It is not clear how the alternative H 2 estimators perform in such scenarios.…”

“…Furthermore, data may be analyzed via additive main effect and multiplicative interaction (AMMI) or genotype and genotype ´ environment interaction (GGE) models (Gauch, 1992;Hadasch et al, 2017), and/or there may be more than one genotypic main effect due to the use of so-called check cultivars. In other MET, datasets may display a much higher degree of unbalancedness due to a completely different trial design and/or genotype set for each environment, respectively.…”

Constrained AMMI Model: Application to Polish Winter Wheat Post‐Registration Data

“…The exhaustive leave‐one‐out cross‐validation procedure (Geisser, 1993, Paderewski and Rodrigues, 2014; Hadasch et al, 2017) was adapted and generalized to evaluate the prediction ability of AMMI and C‐AMMI. Because the algorithm divides the original sample into a training set and a validation set in all possible ways, the model ratings given by the cross‐validation could be treated as the total population sampling, and the validation should be treated as the final mark.…”

“…Thus, the dataset in hand is the complete GE matrix and, at the same time, one value is not used for the modeling. The leave‐one‐out cross‐validation procedure was conducted, and the root mean square prediction differences (RMSPDs) (Gauch and Zobel, 1988, 1990; Paderewski, 2013; Paderewski and Rodrigues, 2014; Hadasch et al, 2017) were calculated, meaning the difference between the original cell value, validation set, and model estimation value was calculated for each cell separately, one by one, and the square root of mean squares is the RMSPD value. A higher RMSPD represents a lower ability for model prediction.…”