Monte Carlo Approach To Genotype By Environment Interaction Models

Multi-location 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).

(Figure 1A) shows the case where there is no GEI, the genotype and the environment behave additively (this will be developed later) and the reaction norms are parallel. The remaining plots show different situations in which GEI occurs: divergence (Figure 1B), convergence (Figure 1C), and the most critical one, crossover interaction (Figure 1D). Crossover interactions are the most important for breeders as they imply that the choice of the best genotype is determined by the environment.

Figure 1.GEI in terms of changing mean performances across environment

Crossa ¹ pointed out that data collected in multi-location trials are intrinsically complex having three fundamental aspects: structural patterns, nonstructural noise, and relationships among genotypes, environments, and genotypes and environments considered jointly. Plant Breeders generally agree on the importance of high yield stability, but there is less accord on the most appropriate definition of "stability" and the methods to measure and to improve yield stability (Becker and Leon, 1988). Finlay et al. (2007) tested six spring wheat cultivars at five locations across Manitoba and Saskatchewan over two years to examine genotypic and environmental variation in grain, flour, dough and bread-making characteristics. They reported that the relative magnitude of the environmental contribution to wheat variance, depending on the trait (including yield), was considerably larger (14 to 89%) than the variance contribution of either genotype (0 to 33%) or G x E interaction (0 to 17%). Rodrigues, Monteiro and Lourenco ⁷ also reviewed the performance of the robust extensions of the AMMI model is assessed through a Monte Carlo simulation study where several contamination schemes are considered. Applications to two real plant datasets are also presented to illustrate the benefits of the proposed methodology, which was broadened to both animal and human genetics studies.

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 multi-location trials, (ii) to determine the efficiency of each method (AMMI, Finlay-Wilkinson, GGE and Mixed model) in detecting GEI and (iii) also to determine the adaptability and specificities of the methods.

The AMMI model combines the features of ANOVA and SVD as follows: first, the ANOVA estimates the additive main effects of the two-way data table; then the SVD is applied to the residuals from the additive ANOVA model, estimating N≤min(I-1, J-1) interaction principal components (IPCs). The model can be written as ^{5, 6}

….(1)

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;

γ_n,i and δ_n,jare the ith and jth genotype and environment IPC scores (i.e. the left and right singular vectors, scaled as unit vectors) for axis n, respectively;

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:

…..(2)

where Y is the (IXJ) two-way 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_Iβ^T_J is approximated by the product of matrices UDV^T, with U an (IXN) matrix whose columns contain the left singular vectors interactions of n, D a (NXN) diagonal matrix containing the singular values of Y^*, and V a (JXN) matrix whose columns contain the right singular vectors of Y^*

Finlay-Wilkinson Model

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 ⁴, which describes GEI as a regression line on the environmental quality. In the absence of explicit environmental information, the biological quality of an environment can be reflected in the average performance of all genotypes in that environment. The GEI part is then described by genotype-specific regression slopes on the environmental quality, and the model can be written in the following equivalent ways:

…..(4)

…..(5)

Model (5) follows from model (4) by taking μ+α_i=α’_i andβ_j + b_jβ_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’_iand 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:

…..(6)

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:

…..(7)

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 two-way 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 two-way data tables are simulated in the following way:

Create a matrix X with (NxP) data design;

(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 (NxP) data design;

(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)).