Demand-Driven Technologies for Sustainable Maize ... - IITA

91Method for estimating Complement index vectorSince there are almost always strong indications that the nature ofthe residuals after fi tting the additive terms inevitably produces amatrix, which tends to be correlated, sums of squares from AMMImodels will tend to be optimistic. In this section, we therefore proposean alternative index vector unrelated to the genotype vectors of theresidual interaction matrix. Using analysis of variance notation, it iscustomary to write a non-additive model asyij=μ+ gi+ ej+ wij+ εij ij(5)where wij ijthe genotype by environment interaction term is afunction of two factors. Given that we have an environment indexwhich represents the environment variation ( θj), we can calculatethe regression dependence of the genotype-environment interactionterms by regressing wijon θjfor each genotype, obtaining regressioncoeffi cient β . Effectively, we are fi tting the modeliw= βiθj+ δij(6)ijijwhere δij ijrepresents the random component. By combining equation(5) and (6) the full model can be rewritten as an additive multiplicativemixture.y = μ + g + e + β θ + δ(7)ij ijijMany other models are special cases of this general model and themodel is also equivalent to the AMMI models of Gauch (1988). Theenvironment index of the AMMI model (θ j) shall be referred to asSelf index vector. The Self index vector (θ s)is the fi rst eigenvector ofthe matrix ZZ’. The regression of the terms of Z on θ sis the AMMI1model. Since Self index vector is correlated with highly correlatedterms of the matrix Z, we propose an alternative index vector referredto as the Complement index vector (θ c).The Complement index vector is defi ned as a vector of values estimatedby splitting the total number of genotypes into two subgroups (targetset). One subgroup constitutes the main data being analysed and theenvironment index θ cis estimated as the fi rst eigenvector from theremaining number of genotypes. By this approach we estimate anenvironment index completely unrelated to the genotypes whose GEIare analysed.Comparison between Self index vector and Complementindex vectorThe performance of the two index vectors discussed above wasassessed using a model with a single multiplicative interaction term (7).The data used for the assessment were grain yields from a set of springijij ij