Testing multiplicative terms in AMMI and GGE models for multienvironment trials with replicates.

Genotype × Environment (G×E) interaction and stability performance were investigated on paddy yield of eighteen rice genotypes and twelve locations using two well renowned statistical models; genotype main effect and G×E Biplot analysis (GGE) and additive main effects and multiplicative interaction (AMMI) analysis. The aim of this study was to elucidate the performance of some advance rice lines/genotypes at multiple locations in multi environment trials (METs) using GGE biplot and AMMI analyses. The results of GGE biplot and AMMI analyses performed over the data of paddy yield at multiple locations of two years 2014 and 2015 indicated that G×E interaction plays a crucial role in determining the performance of genetic material in METs. The results declare that GGE and AMMI not only provide easy and affective evaluation of genotypes into environment interactions in a number of locations but also a comprehensive understanding of the variability of the target locations. AMMI analyses for data of both years indicated that RRI 7 was the highest priority selected genotype for six locations, NIAB 1175 for four and RRI 3 for three locations. Dhokri and Kala Shah Kaku were the highest yielding, while Faisalabad and Dhokri were the most stable environments in 2014. Likewise, Faisalabad and PARC Islamabad were the highest yielding as well as most stable environments in 2015. Basmati 515 and PS 2 were the most favorable genotypes in 2014 and 2015, respectively for their high paddy yield and stability at all locations. The results further suggested that both models were useful and presented similar interpretations about MET data. Key words: Genotype main effect and G × E biplot analysis (GGE), additive main effects and multiplicative interaction (AMMI) analysis, rice, fine type, multiple locations.

The additive main effects and multiplicative interaction (AMMI) model has been proposed for the analysis of genotype–environmental data. For plant breeding, the recovery of pattern might be considered to be the principal objective of analysis. However, some problems still remain with the analysis, notably in selecting the number of multiplicative components in the model. Methods based on distributional assumptions do not have a sound methodological basis, while existing data-based approaches do not optimize the cross-validation process. This paper first summarizes the AMMI model and outlines the available methodology for selecting the number of multiplicative components to include in it. Then two new methods are described that are based on a full “leave-one-out” procedure optimizing the cross-validation process. Both methods are illustrated and compared on some unstructured multivariate data. Finally, their applications to analysis of genotype × environment interaction (GEI) are demonstrated on experimental grain yield data. Conclusions of the study are that the “leave-one-out” procedure is preferable in practice to either distributional F-test or cross-validation randomization methods, and of the two “leave-one-out” procedures the Eastment-Krzanowski method exhibits the greater parsimony and stability.

Yield trials frequently have both significant main effects and a significant genotype X environment (GE) interaction. Traditional statistical analyses are not always effective with this data structure: the usual analysis of variance (ANOVA), having a merely additive model, identifies the GE interaction as a source but does not analyze it; principal components analysis (PCA), on the other hand is a multiplicative model and hence contains no sources for additive genotype or environment main effects; and linear regression (LR) analysis is able to effectively analyze interaction terms only where the pattern fits a specific regression model. The consequence of fitting inappropriate statistical models to yield trial data is that the interaction may be declared nonsignificant, although a more appropriate analysis would find agronomically important and statistically significant patterns in the interaction. Therefore, agronomists and plant breeders may fail to perceive important interaction effects. This paper compares the above three traditional models with the additive main effects and multiplicative interaction (AMMI) Model, in an analysis of a soybean [Glycine max (L.) Merr.] yield trial. ANOVA fails to detect a significant interaction component, PCA fails to identify and separate the significant genotype and environment main effects, and LR accounts for only a small portion of the interaction sum of squares. On the other hand, AMMI analysis reveals a highly significant interaction component that has clear agronomic meaning. Since ANOVA, PCA, and LR are sub‐cases of the more complete AMMI model, AMMI offers a more appropriate first statistical analysis of yield trials that may have a genotype X environment interaction. AMMI analysis can then be used to diagnose whether or not a specific sub‐case provides a more appropriate analysis. AMMI has no specific experimental design requirements, except for a two‐way data structure.

<ns2:p>The genotype (G) by environment (E) interaction (GEI) determines the stability of maize grain yield in multi-environment trials (METs). This study evaluated the high-yielding and promising maize genotypes over years and locations by the additive main effects and multiplicative interaction (AMMI) model. The grain yield of 13 spring maize genotypes was evaluated for two consecutive years (2012–2013) when planted in six and eight ecological environments, respectively, using a randomized complete block design (RCBD) with three replications. The AMMI model explained 77.49 and 75.57% of total observed genotypic variation, respectively. A comprehensive analysis of variances showed a highly significant impact of environment, genotype and genotype × environment (GE) interaction on grain yield (P &lt; 0.01). The AMMI model analysis of variance showed that the environment contributed the most to variations in grain yield (55.58 and 72.50% of the total variation, respectively), followed by GE interaction (24.61 and 10.71% of the total variation, respectively) and genotype (3.01 and 3.01% of the total variation, respectively). Among the interaction effects, first interaction principal component axis (IPCA1), IPCA2 and IPCA3 explained the vast majority of genetic and environmental interaction information. Two years of experimental data showed that the genotype with high yield and stability was G4 (Zhongdi175) while G3 (C807) and G8 (LY10) of poor yield and unstable. The check genotype G6 (Nongda108) had good stability and general high-yielding. The best and worst discriminative environments for each of the locations in 2012–2013 were XT (Xingtai) and LH (Longhua), WA (Wuan) and PQ (Pingquan), respectively.</ns2:p>

ABSTRACTWhen a multienvironment trial (MET) is established across several locations and years, the interaction is referred to as a three‐way array Three‐way interaction can be studied by means of three‐way principal components analysis. In this study, the three‐way principal components analysis is adapted to the sites regression model (three‐way SREG). The three‐way SREG with location and year combines the effects of genotype, genotype × location, genotype × year, and genotype × location × year. The objective of this study is to show how the three‐way SREG can be put to practical use in agriculture and breeding. We utilized two wheat (Triticum aestivum L) data sets that have already been used for fitting a three‐way additive main effects and multiplicative interaction model. One data set had genotype (25) × location (4) × sowing times (4) and eight attributes, and the other data set included genotype (20) × irrigation level × year on grain yield. The three‐way SREG applied simultaneously to eight attributes facilitates the interpretation of genotypic performance for all traits in specific locations and across locations for a selected sowing time. Results of the three‐way SREG for both data sets show the different response patterns of genotypes for locations and sowing dates (Data Set 1), as well as genotypic responses across irrigation levels in different years (Data Set 2). Using Data Set 1, we show that fitting a three‐way data structure to a three‐way SREG model is more effective for detecting important interaction patterns than using the two‐way SREG.

The additive main effects and multiplicative interaction (AMMI) model has emerged as a powerful analytical tool for genotype x environment studies. The objective of the present study was to assess its value in quantitative trait locus (QTL) mapping. This was done through the analysis of a large two-way table of genotype-by-environment data of barley (Hordeum vulgare L.) grain yields, where the genotypes constituted a genetic population suitable for mapping studies. Grain yield data of 150 doubled haploid lines derived from the 'Steptoe' x 'Morex' cross, and the two parental lines, were taken by the North American Barley Genome Mapping Project (NABGMP) at 16 environments throughout the barley production areas of the USA and Canada. Four regions of the genome were responsible for most of the differential genotypic expression across environments. They accounted for approximately 50% of the genotypic main effect and 30% of the genotype x environment interaction (GE) sums of squares. The magnitude and sign of AMMI scores for genotypes and sites facilitate inferences about specific interactions. The parallel use of classification (cluster analysis of environments) and ordination (principal component analysis of GE matrix) techniques allowed most of the variation present in the genotype x environment matrix to be summarized in just a few dimensions, specifically four QTLs showing differential adaptation to four clusters of environments. Thus, AMMI genotypic scores, when the genotypes constituted a population suitable for QTL mapping, could provide an adequate way of resolving the magnitude and nature of QTL x environment interactions.

Bread wheat is one of the most represented field crops whose level and stability of yield is very important for the food security in Republic Serbia. In the paper was investigated stability of yield expression of 15 winter bread wheat genotypes in different agroecological conditions of Serbia, using the Additive main effects and multiplicative interaction (AMMI) model and GGE-biplot method of analysis. Aim of investigation was to determine which of applied analysis is superior in identification of the most desirable genotypes for cultivation in given environments. Analysis of variance showed that genotype and genotype-environment (G×E) interaction represent highly significant sources of variability in expression of grain yield. AMMI and GGE analyses were point out similar results and an indisputable conclusion is that multienvironment trials, besides routine usage of analysis of variance, must be analized with one of this two models, which combine analysis of variance and PCA analysis. AMMI analysis is simpler for interpretation and closer to the concept of view of the agronomical trial, while GGE analysis is more complex and gives more precise interpretation of "which-won-where", i .e. for defining of narrowly adapted genotypes in given environments. Thus, G11 as the genotype with highest average yield is narrow adapted to the environment Sombor and can be recommended, as well as in the environment 2 (Kruševac), while genotypes with modest requests (G2) rather can be recommended for cultivation in the environment 1 (Kragujevac), which is characterized by less fertile soil and a smaller amount and uneven distribution of precipitation.

SUMMARYAnalysing the stability and adaptation of cultivars to different environments is always necessary before recommending them for planting on large areas. Additive main effects and multiplicative interaction (AMMI) models have been used to analyse genotype-by-environment interactions (G × E). AMMI models require data with homogeneous variance, normal errors and additive effects. However, agronomic data do not always conform to these statistical assumptions. The objective of the present study was to analyse G × E interactions for severity and incidence of grey leaf spot, a foliar disease in maize caused by Cercospora zeae-maydis, using a generalized AMMI model. Data were collected and evaluated for 36 maize cultivars from experiments carried out in nine Brazilian regions in 2010/11 by the Empresa Brasileira de Pesquisa Agropecuária (EMBRAPA – Milho e Sorgo). Only two of three stable genotypes defined by a quasi-likelihood model with a logistic link function could be recommended for their desirable agronomic characteristics. Four growing locations in which the genotypes were stable were identified, but in only one of these was stability associated with very severe grey leaf spot disease. Cultivars adapted to specific locations with low percentage disease severity were also identified.

The objective of this study was to assess genotype by environment interaction for seed yield in rapeseed cultivars grown in West Poland by the additive main effects and multiplicative interaction model. The study comprised 25 rapeseed genotypes (15 F1 CMS ogura hybrids, their parental lines and two varieties: Californium and Hercules F1), analyzed in five localities through field trials arranged in a randomized complete block design, with four replicates. Seed yield of the tested genotypes varied from 15.9 to 80.99 dt/ha throughout the five environments/localities, with an average of 39.69 dt/ha. In the variance analysis, 69.82 % of the total yield variation was explained by environment, 13.67 % by differences between genotypes, and 8.15 % by genotype by environment interaction. Seed yield is highly influenced by environmental factors. Due to high influence of the environment on yield high adaptability of the genome is required.

Yield stability, alongside high yield potential and broad adaptation to various agroclimatic environments, is a key objective for rice breeders aiming to ensure food security. This study aimed to explore the most suitable and stable Boro rice genotypes for Bangladesh. Ten Boro rice genotypes underwent testing in four environments during the 2022 Boro season to investigate genotype-environment interaction (GEI) and yield stability performance. The experiment utilized three replications of a completely randomized block design. Yield stability performance was assessed through combined analysis and the additive main effects and multiplicative interaction (AMMI) model. The combined ANOVA revealed that the environment explained 10.23%, while GEI accounted for 9.17%, and the genotypes captured 80.60% of the variance, significantly impacting grain yield. Significance was observed in the environment, genotype main effects, and GEI. Analysis indicated that BRRI dhan 68 yielded the highest (6,754 kg·ha-1) and BRRI dhan 88 the lowest (5,620 kg·ha-1) among the investigated genotypes. In addition, genotypes BRRI dhan 84, BRRI dhan 81, and BRRI dhan 67 exhibited the highest grain yields. The Rangpur environment demonstrated considerable stability across the four environments with a high mean value of grain yield (7,206 kg·ha-1). Therefore, the AMMI model emerges as a valuable tool for identifying the most suitable and stable Boro rice genotypes with high-yielding potential across various regions in Bangladesh, as well as under diverse conditions.

Linseed (Linum usitatissimum L.) is one of the most prominent industrial oilseed crops cultivated for both seed and fiber. Lack of stable genotypes across the linseed production area is one of the problems. Thirteen linseed genotypes were planted in randomized complete block design with three replications at six linseed major growing agro-ecologies of North, central and Southeastern Ethiopia (Werabe, D\Markos, Welkite, Holeta, Kulumsa and Adiet) in 2021/2022 cropping season. With the objectives of determining the effects of GEI, on oil yield of linseed and identifying better performing and well adapted linseed genotypes than the control variety, and to prepare for registration and release of selected high oil yielding genotypes in the different linseed agro-environment conditions of Ethiopia. The oil yield subjected to the combined analysis of variance showed a highly significant (p&lt;0.01) effect of genotype, location, and genotype x location interactions (GLI). Similarly the combined AMMI ANOVA for oil yield revealed that there were highly significant differences among genotypes, locations and genotype by location interactions and accounted 22.11%, 31.40% and 46.49% of the total variations respectively. The highest percentages of environmental variations are an indication that environment is the major factor that influences the yield performance of linseed oil yield in Ethiopia. In addition, the first two IPCAs were significant and accounted for 80.77% of the total interactions sum squares. Six stability measures viz Eberhart and Russell analysis (bi and S2di), Additive Main Effect and Multiplicative Interaction (AMMI) model, AMMI Stability Value (ASV),Yield Stability Index (YSI),Genotype Main Effect and Genotype by Environment Interaction Effect (GGE) bi plot analysis Model were used to evaluate the stable genotypes across the testing locations. Genotypes 10097(G2), 10103 (G3) and 239716 (G7) were more stable by Eberhart and Russell analysis and AMMI Stability Value. Genotypes 10103 (G3) and 208360 (G8) were more stable by Yield Stability Index. Genotypes 208360 (G8) and 10103 (G3) were selected as better genotypes that appeared in the five and four locations by AMMI analysis, respectively. According to one-year data, the six locations are grouped into one mega environment for linseed production with one winning genotype and genotype 208360 (G8) was an ideal genotype, while location A diet was an ideal environment by GGE analysis. Genotypes 208360 (G8), 234005 (G4)and 10103 (G3) are the three of the best performing genotypes than the other genotypes and control variety (Berene) in oil yield across locations. Therefore, those three highest oil yielder genotypes have a potential to be registered in Ethiopia. However, this trail need to be repeated for one more season, and or three of the best performing genotypes will be verified along with the check on farmers' fields for release.

Analysis of T-RFLP data using analysis of variance and ordination methods: A comparative study

Genotype x environment interaction (GEI) of 42 rice genotypes tested over nine seasons was analyzed to identify stable resistance to blast disease incited by Magnaporthe oryzae. The genotypes were raised in uniform blast nursery in a randomized complete block design with three replications. The GEI was analyzed following the regression models as well as additive main effects and multiplicative interaction (AMMI) model. AMMI analysis of variance revealed that the first two interaction principal component axes (IPCA) explained 37.28 and 33.47% of the interaction effects in 14.63 and 14.02% of interaction degrees of freedom, respectively and rest of the five IPCAs were noisy. Integrating biplot display and genotypic stability statistics enabled five groupings of genotypes based on similarities in their performance across environments. The biplot generated using the environment and genotype scores for the first two IPCAs revealed the positioning of the five host genotype groups (HG) into four sectors. HG-1 constituting of 28 genotypes exhibiting low stability index (Di values), low IPCA-1 as well as IPCA-2 scores and low mean disease scores across seasons of testing, were identified as possessing stable resistance to the disease. Although, both regression and AMMI models were equally potential in partitioning of GEI, AMMI analysis and the biplot display were more informative in differentiating genotype response over environments, describing specific and non-specific resistance of genotypes, identifying most discriminating environments and thus could be useful to plant pathologists as well as breeders in supporting breeding program decisions. Key words: Additive main effects and multiplicative interaction (AMMI) model, rice blast disease, Magnaporthe oryzae, regression model, stable resistance.

To improve multi-environmental trial (MET) analysis, a compound method—which combines factor analytic (FA) model with additive main effect and multiplicative interaction (AMMI) and genotype main effect plus genotype-by-environment interaction (GGE) biplot—was conducted in this study. The diameter at breast height of 36 open-pollinated (OP) families of Pinus taeda at six sites in South China was used as a raw dataset. The best linear unbiased prediction (BLUP) data of all individual trees in each site was obtained by fitting the spatial effects with the FA method from raw data. The raw data and BLUP data were analyzed and compared by using the AMMI and GGE biplot. BLUP results showed that the six sites were heterogeneous and spatial variation could be effectively fitted by spatial analysis with the FA method. AMMI analysis identified that two datasets had highly significant effects on the site, family, and their interactions, while BLUP data had a smaller residual error, but higher variation explaining ability and more credible stability than raw data. GGE biplot results revealed that raw data and BLUP data had different results in mega-environment delineation, test-environment evaluation, and genotype evaluation. In addition, BLUP data results were more reasonable due to the stronger analytical ability of the first two principal components. Our study suggests that the compound method combing the FA method with the AMMI and GGE biplot could improve the analysis result of MET data in Pinus teada as it was more reliable than direct AMMI and GGE biplot analysis on raw data.

Plant breeders utilize the additive main effects and multiplicative interaction (AMMI) model for analyzing yield data from multi‐environment trials (METs) to visualize interaction patterns between genotypes and environments. AMMI‐based selection indexes, such as the weighted average of absolute scores (WAAS) and the weighted average of absolute scores combining yield (WAASY), guide breeders in identifying superior varieties within METs. Despite being powerful, the frequentist approach of AMMI model and its derived indices presents challenges for identifying genotypes and environments, causing significant genotype‐by‐environment (G × E) interactions. This study built upon the Bayesian AMMI framework to allow to perform inferences on AMMI‐based selection indexes. The Bayesian versions of WAAS and WAASY (Bayesian weighted average of absolute scores and Bayesian weighted average of absolute scores combining yield) were compared with the frequentist approach. A novel stability measure (SM), using Mahalanobis distance, was also proposed and integrated with yield performance into a graphical tool called the stability Mahalanobis trait (SMT) plot. Nine maize genotypes evaluated for grain yield across 20 environments were analyzed. The B‐WAAS, B‐WAASY, and SM indexes provided informative statistical inference through posterior distribution and credible intervals (highest posterior density [HPD]). HPD intervals allowed grouping similar genotypes based on stability and performance, offering reliable information for selection and recommendation. The SMT plot allows a direct comparison to an ideal scenario of high stability, facilitating the identification of genotypes aligned with breeding goals by analyzing the four quadrants. Genotypes in quadrant IV, exhibiting both high yield and high stability, are particularly valuable for breeding programs.