The authors have declared that no competing interests exist.
Understanding the implication of GenotypebyEnvironment (GXE) interaction structure is an important consideration in plant breeding programs. Traditional statistical analyses of yield trials provide little or no insight into the particular pattern or structure of the GXE interaction. In this study, efforts were made to solve these problems under different level of data occurrence. We employed the simulation process of Monte Carlo in generating since use of a reallife data may pose a serious difficulty. In this paper, we simulated for two data Types of Balance and Unbalance designs with different Levels of generations (3X3, 7X7, 10X10, and 3X7, 7X3, 7X10, 10X7 , , respectively). We therefore check the performance of GXE interaction on four different models (AMMI, FW, GGE and Mixed model), and also their stability and adaptability. The findings revealed that, when the assumption was maintained, AMMI outperformed FinlayWilkinson model, GGE Biplot model and Mixed model.
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Multilocation trials play an important role in plant breeding and agronomic research. A number of parametric statistical procedures have been developed over the years to analyze genotype by environment interaction and especially yield stability over environments. A number of different approaches have been used to describe the performance of genotypes over environments. Therefore, the function that described the phenotypic performance of a genotype in relation to an environmental characterization is called the "norm of reaction" (Griffiths et al., 1996).
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The general aim of this study is to determine which of these models best suit GEI using Monte Carlo simulated data. The specific objectives are: (i) to compare the various statistical methods and determine the most suitable parametric procedure that best describe genotype performance under multilocation trials, (ii) to determine the efficiency of each method (AMMI, FinlayWilkinson, GGE and Mixed model) in detecting GEI and (iii) also to determine the adaptability and specificities of the methods.
A combined analysis of variance procedure is the most common method used to identify the existence of GEI from replicated multilocation trials. If the GEI variance is found to be significant, one or more of the various methods for measuring the stability of genotypes can be used to identify the stable genotype (s). A wide range of methods is available for the analysis of GEI and can be broadly classified into four groups: the analysis of components of variance, stability analysis, multivariate methods and qualitative methods.
The methods to be adopted in this study are suitable for the plant breeders in estimating Genotype by Environment Interaction (GEI) parameters. The methods are as follows;
The AMMI model combines the features of ANOVA and SVD as follows: first, the ANOVA estimates the additive main effects of the twoway data table; then the SVD is applied to the residuals from the additive ANOVA model, estimating N≤min(I1, J1) interaction principal components (IPCs). The model can be written as
where y_{ijk} is the phenotypic trait (yield or some other quantitative trait of interest) of the ith genotype in the jth environment for replicate k; model
μ is the grand mean;
α_{i} are the genotype deviations from μ;
β_{i} are the environment deviations from μ;
𝞴_{n} is the singular value of the IPC analysis axis n;
ρ_{i,j} is the residual containing all multiplicative terms not included in the model;
e_{ijk} is the experimental error; and N is the number of principal components retained in the model.
In matrix formulation the AMMI model can be written as:
where Y is the (IXJ) twoway table of genotypic means across environments. The interaction part of the model Y^{*}=Y_{I }1^{T}_{J }μ  α_{I }1^{T}_{J}  1_{I}β^{T}_{J }is approximated by the product of matrices
A more attractive alternative is to extend the additive model:
by incorporating terms that explain as much as possible of the GEI. A popular strategy in plant breeding is that proposed by Finlay and Wilkinson
Model (5) follows from model (4) by taking μ+α_{i}_{=}α’_{i} andβ_{j }+ b_{j}β_{i}= (1+b_{j}) β_{j }= b_{t}^{’ }β_{j} Model (5) is easier to interpret because it looks as a set of regression lines; each genotype has a linear reaction norm with intercept α’_{i}and slope b’_{i}. The explanatory environmental variable in these reaction norms is simply the environmental main effect β_{j}. Model (4) shows more clearly how GEI is captured by a regression on the environmental main effect, with the hope that as much as possible of the GEI signal will be retained by the term b_{t} β_{j}. Note that in model (5) the average value of b’is 1, meaning that b’ > 1 for genotypes with a higher than average sensitivity, and b’ > 1 for genotypes that are less sensitive than average.
Plant breeders are interested in the total genetic variation and not exclusively in the GEI part. For that reason, it is useful to have a modification of model (1) that considers the joint effects of the genotypic main effect and the GEI as a sum of interpretation procedures hold as for model (1). Because genotypic scores now describe genotypic main effects G and GEI together, this type of model is also known as the "GGE model" and the Biplots are called "GGE Biplots" (Yan et al., 2000). The model reads:
In GGE, the result of SVD is often presented in a "Biplot illustration". Its approximate overall performance (G + GEI).
The REML/BLUP method allows the consideration of different structures of variance and covariance for the genotypes by environments effects, which makes the model more realistic. For the GEI evaluation by mixed model, the following statistical model was used:
Where, y is the vector of observed data; α is the vector of genotype effects (assumed as random); β is the vector of block effects within each environment (assumed as fixed); β is the vector of GEI effect (assumed as random); and Ԑ is the error vector (random). The uppercase letters represent the matrices of incidence for the referred effects. The distribution of the random effects were:
We simulate twoway data tables for balanced and unbalanced design with 3 replications each, where the interaction is explained by two multiplicative terms (i.e. two IPCs; k = 2 components to be retained). Without loss of generality, the twoway data tables are simulated in the following way:
Create a matrix X with
(3x3) data design, where n = 3 rows (Genotypes) and p = 3 columns (Environments)
(7x7) data design, where n = 7 rows (Genotypes) and p = 7 columns (Environments).
(10x10) data design, where n = 10 rows (Genotypes) and p = 10 columns (Environments).
with observations drawn from a Unif (0, 0.5) distribution.
Do the SVD of X and obtain the matrices U, V and D, containing, respectively, the left and right singular vectors and the singular values of X;
Simulate the grand mean, the genotypic and environmental main effects, considering: μ ~ N(15,3) α ~ N(5,1) and β ~ N(8,2) (Rodrigues et al.(2015)).
Create a matrix X with
(3x7)data design, where n = 3 rows (Genotypes) and p = 7 columns (Environments)
(7x3)data design, where n = 7 rows (Genotypes) and p = 3 columns (Environments).
(7x10) data design, where n = 7 rows (Genotypes) and p = 10 columns (Environments).
(10x7) data design, where n = 10 rows (Genotypes) and p = 7 columns (Environments).
with observations drawn from a Unif (0, 0.5) distribution.
Do the SVD of X and obtain the matrices U, V and D, containing, respectively, the left and right singular vectors and the singular values of X;
Simulate the grand mean, the genotypic and environmental main effects, considering: μ ~ N(15,3) α ~ N(5,1) and β ~ N(8,2) (Rodrigues et al.(2015)).
Comparison of stability of different models using different stability parameters
(







Design  Mean  ASV  Rank  b_{t}  Rank  IPCs  Rank  σ_{Ԑ}^{2}  Rank 
3X3  18.73  16.80  2  0.8375  2  98.5%  1  1.919  1 
7X7  24.18  6.08  1  1.6375  1  79.7%  2  28.29  2 
10X10  23.70  3.86  3  0.7419  3  67.5%  3  25.57  3 
The biplot analysis system showing in
Therefore, it was observed that the closer the concentric circles to the center point, the more adaptable the models. Similarly, in the second plot, the closer the model to the thick blue arrow line, the more adaptable the model. It can be deduced that from the balance design simulated data, AMMI model is more stable and better adaptable.
(







Design  Mean  ASV  Rank  b_{t}  Rank  IPCs  Rank  σ_{Ԑ}^{2}  Rank 
3X7  23.15  23.19  2  0.7079  4  94.5%  1  30.42  3 
7X3  24.5  3.17  1  4.4698  1  62.3%  4  47.78  4 
10X7  22.83  4.34  3  1.0957  3  81.9%  2  30.18  2 
7X10  21.90  2.43  4  1.4761  2  72.5%  3  28.19  1 
In the same vein, the biplot analysis system showing in
In this study, efforts were made to solve these problems under different level of data occurrence. We employed the simulation process of Monte Carlo in generating since use of a reallife data may pose a serious difficulty.
In this research work, we simulated for two data Types of balance and unbalance designs with different Levels of generations (3X3, 7X7, 10X10 and 3X7, 7X3, 7X10, 10X7 respectively).
The findings revealed that, when the assumption was maintained, AMMI outperformed FinlayWilkinson model, GGE Biplot model and Mixed model. We therefore check the performance of GXEinteraction on four different models (AMMI, FW, GGE and Mixed model), and also their stability and adaptability.
Finally, the study has clearly shown that the four models considered detects the GXE interaction effect in a different way. We were able to evaluate and described GXE interaction performance by their stability and adaptability using multilocation trials. Also, this study confirmed the suitability of AMMI in detecting GXE when the assumptions are maintained or kept. That is, when outlier is not influential, AMMI can be used. (
Balance  RMSE  MSE  Abs. Bias  
Data Design  AMMI  FW  GGE  Mixed Model  AMMI  FW  GGE  Mixed Model  AMMI  FW  GGE  Mixed Model 
3X3 Data  1.1312  1.2218  1.7874  1.1374  0.0370  1.9194  1.9190  1.2938  0.6319  4.4565  2.5617  0.7907 
7X7 Data  2.7233  4.9308  4.7120  4.3430  18.2120  26.8717  28.2920  22.2025  0.3931  3.0206  2.3156  2.4673 
10X10 Data  2.9672  4.8729  4.7044  4.1288  23.4850  25.4414  25.5710  23.1311  0.2982  3.6605  2.1024  1.8547 
Unbalance  RMSE  MSE  Abs. Bias  
Data Design  AMMI  FW  GGE  Mixed Model  AMMI  FW  GGE  Mixed Model  AMMI  FW  GGE  Mixed Model 
3X7 Data  4.0414  5.8680  4.7957  4.5036  27.1070  38.0586  30.4240  22.9984  0.9037  4.8829  3.1856  2.7243 
7X3Data  3.6666  6.4907  6.4199  5.6436  39.1170  54.1660  47.7760  41.2155  0.8199  5.6584  1.9236  2.5613 
10X7Data  2.1601  4.7352  4.9967  5.6436  24.2270  24.7819  28.1930  24.9669  0.2600  3.6762  3.2005  1.7961 
7X10 Data  3.0695  5.2520  5.1482  5.6436  27.8110  29.5536  30.1800  28.5039  0.3695  4.4930  3.2565  1.9173 