2.4.6. Additive Main Effect and Multiplicative Interaction (AMMI) Model

In the present study, multivariate stability based on the AMMI model was assessed for G×E interaction and stability analysis to predict the stability of GSR lines. The AMMI model combines the application of pooled ANOVA to evaluate the additive main effects; then factorization of a complex matrix (SVD) is applied to the total error for computing interaction principal components (IPCs). We estimated the additive main effect and AMMI model in R using the metan library [24]. As suggested by Zobel et al. [23] the base of the additive main effect and multiplicative interaction (AMMI) model was computed as follows:

$$\chi\_{ij} = \mu + \alpha\_i + \beta\_j + \sum\_{k=1}^n \lambda\_k \gamma\_{ik} \delta\_{jk} + \varepsilon\_{ij}$$

where *Yij* is the mean performance of *i*th genotype in the *j*th individual environment, *μ* is the overall mean, *α<sup>i</sup>* is the *fixed effect of the GSR line*, *β<sup>j</sup>* is the *environmental effect*, *n* is the number of IPCA kept in the AMMI model, *λ<sup>k</sup>* is the singular value for IPC axis *k*, *γik* is the *i*th genotype eigenvector value for IPC axis *k*, *δjk* is the *j*th environment eigenvector value for IPC axis *k*, and *εij* is the average residual.

### 2.4.7. Biplot Analysis

After ranking the most adoptable GSR lines with the AMMI model, a study of the sustainable phenotypic reliability of the multi-locations analysis of the biplot graphic was designed. Biplots are performance and stability-related graphs where factors of both genotypes and locations are plotted on the same axis so the inter-relationships can be visualized.

In our constructed biplots, the abscissa represents the variables that affect the values of a genotype, and the ordinate is the first interaction axis (IPCA1). The GSR line with IPCA1 close to zero will be considered a stable and "ideal" GSR line while low stability will be associated with low productivity [25,26].
