the length of a genotype .approximatest ongin to the po genotype Accurac y Gain from Model Diagnosis of the AFC ordinate (i gh Should Not Be Ov rstated ingful).the Mea vs.Stability viev of the GGE Great ac nd ma “fre e observations'”are biplot(Fig. 2)partitions this GGE into the genoty e's con claimed for m osis and identifcation of"r redi tributionoG(projectionoatofhcAEcEQ and its tively accurate"models in AMMI analysis.For ex mple contribution to ge (proiection onto the aec ordinate) Ebdon and Gauch(2002b)reported for a perennial rye This property allows identification of"ideal"genotypes(a grass(Lolium pereme L)performance dataset that a statisti large and positive contribution to g and a small contribu cal efficiency of 5.6 was achieved by using the AMMI2 tion to GE)for a given mega-environment.Many breeders model (AMMI with two IPCs),which was converted have found this application of GGE biplots to be useful. to 101844 "free observations"or a saving of $1,000,000 rlier sections,the AEC vie ot th biplot tion ng -environment, the er small by th ega-en a con p The r in the GGE bip 6 the an data H 2 the s side of the AEC ordinate fie when the g in the data is larg enough to be ing in mind hat in practice vars rather than a single one are recommended for each ness"view of the GGE biplot(Fig.3)partitions this discrim- meg nvironment Condition 2 is often false due to inating power into two components:discrimination on G practical considerations.For example,AMMIl was used (proiection to the Aec abscissa)and discrimination on Ge n mega-environment analysis and cultivar recommer projection to the AEC ordinate),whereby test environments even though AMMI2 and AMMI7 were identi ideal for selecting high-yielding and stable genotypes can be 6 dered s an con whicl renders th A omple irre 51s type ng tes nvironments that are superic th or genotype evaluatio MM use tal di MODEL DIAGNOSIS st perfo It is the f AND ACCURACY GAIN ortant and it remains a Model Diagnosis Is Useful model identified through cross We agree with Gauch (2006)that model diagnosis for predictive of future performance (Speller and dombek each dataset is useful.Many methods have been pro 1995).Therefore,model diagnosis is useful,but accuracy posed to determine how many PCs are required to fully gain from model diagnosis must not be overstated. approximate a two-way table of data,which can be used As Gauch(2006)pointed out.GED analysis is first of determine whether a biplot under-fits or over-fits the all an agricultural issue rather than a stati ical one.There data.Currently,we (Yan, Ma,an fore,it is important to understand how cultiv ars are se add reco es one kno e biplot ent o if the biplo vars on the the biplot i should be vide the data into tal and/on ealed in the iated and it is ra find a genotyne that is hest for everything (yan and division should stop when the biplot is judged as sufficient Wallace.1995).For the same reason,agr onomists always in displaying the patterns of the subset or when there are recommend a set of cultivars,rather than a single cultivar, CROP SCIENCE.VOL.47.MARCH-APRIL 2007 WWW.CROPS.ORG 651Reproduced from Crop Science. Published by Crop Science Society of America. All copyrights reserved. CROP SCIENCE, VOL. 47, MARCH–APRIL 2007 WWW.CROPS.ORG 651 In a GGE biplot, the vector length of a genotype, which is the distance from the biplot origin to the position of the genotype marker, approximates the genotype’s con￾tribution to GGE. When all environments are on the same side of the AEC ordinate (i.e., when G is large enough to be meaningful), the Mean vs. Stability view of the GGE biplot (Fig. 2) partitions this GGE into the genotype’s con￾tribution to G (projection onto the AEC abscissa) and its contribution to GE (projection onto the AEC ordinate). This property allows identifi cation of “ideal” genotypes (a large and positive contribution to G and a small contribu￾tion to GE) for a given mega-environment. Many breeders have found this application of GGE biplots to be useful. However, as illustrated in earlier sections, the AEC view of the GGE biplot is used only for genotype evaluation for a single mega-environment, where the GE is either small (a simple mega-environment) or not exploitable (a com￾plex mega-environment). The length of an environment vector in the GGE bip￾lot approximates the environment’s discriminating power. When all environments are on the same side of the AEC ordinate (i.e., when the G in the data is large enough to be meaningful), the “Discriminating power vs. Representative￾ness” view of the GGE biplot (Fig. 3) partitions this discrim￾inating power into two components: discrimination on G (projection to the AEC abscissa) and discrimination on GE (projection to the AEC ordinate), whereby test environments ideal for selecting high-yielding and stable genotypes can be identifi ed. Gauch (2006) considered E as an essential com￾ponent for environment evaluation. Although E is essential for environment evaluation for nonbreeding purposes, it is irrelevant for identifying test environments that are superior for genotype evaluation. MODEL DIAGNOSIS AND ACCURACY GAIN Model Diagnosis Is Useful We agree with Gauch (2006) that model diagnosis for each dataset is useful. Many methods have been pro￾posed to determine how many PCs are required to fully approximate a two-way table of data, which can be used to determine whether a biplot under-fi ts or over-fi ts the data. Currently, we (Yan, Ma, and Cornelius) are investi￾gating alternative methods for addressing two questions: (i) how does one know if the biplot is adequate in approxi￾mating the two-way table that is under investigation, and, (ii) what should one do if the biplot is inadequate. Briefl y, whenever the biplot is judged as inadequate, attempts should be made to divide the data into subsets based on environmental and/or genotypic groups revealed in the biplot, as demonstrated in the above example. Data sub￾division should stop when the biplot is judged as suffi cient in displaying the patterns of the subset or when there are no clear patterns (environmental or genotypic groupings) in the biplot. Accuracy Gain from Model Diagnosis Should Not Be Overstated Great accuracy gain and many “free observations” are claimed for model diagnosis and identifi cation of “predic￾tively accurate” models in AMMI analysis. For example, Ebdon and Gauch (2002b) reported for a perennial rye￾grass (Lolium perenne L.)performance dataset that a statisti￾cal effi ciency of 5.6 was achieved by using the AMMI2 model (AMMI with two IPCs), which was converted to 101 844 “free observations” or a saving of $1,000,000 (Gauch, 2006). However, this claim can be justifi ed only if all of the following conditions are met: (1) the accuracy that was achieved by the “best model” is absolutely neces￾sary; (2) the cultivar recommendations are made exactly as suggested by the “best model”; and (3) future perfor￾mances are exactly the same as expected from the cur￾rent data. However, Condition 1 is met only if adopting the best model leads to diff erent cultivar recommenda￾tions, bearing in mind that, in practice, multiple culti￾vars rather than a single one are recommended for each mega-environment. Condition 2 is often false due to practical considerations. For example, AMMI1 was used in mega-environment analysis and cultivar recommen￾dation, even though AMMI2 and AMMI7 were identi￾fi ed as the best models for two turfgrass datasets (Ebdon and Gauch, 2002b), which renders the model diagnosis completely irrelevant. Condition 3 is almost always false because genotype × year and genotype × location × year interactions are inevitable. Pertaining to Condition 3, the term “predictive success” used in AMMI analysis must be interpreted properly. There is a fundamental diff erence between predicting future performance and “predicting” past performance (cross-validation). It is the former that is important and it remains a question whether the best model identifi ed through cross-validation is truly more predictive of future performance (Sneller and Dombek, 1995). Therefore, model diagnosis is useful, but accuracy gain from model diagnosis must not be overstated. As Gauch (2006) pointed out, GED analysis is fi rst of all an agricultural issue rather than a statistical one. There￾fore, it is important to understand how cultivars are selected and recommended in the real world to have a realistic assessment about gains from model diagnosis. Breeders do not select cultivars on the basis of only a single trait (e.g., yield), because superior cultivars must meet requirements for multiple breeding objectives. Breeders do not select just one genotype with respect to a trait, because breed￾ing objectives are often negatively associated, and it is rare to fi nd a genotype that is best for everything (Yan and Wallace, 1995). For the same reason, agronomists always recommend a set of cultivars, rather than a single cultivar