Weighted Means Are Unnecessary in Cultivar Performance Trials

The mean performance of a particular cultivar across environments is commonly estimated by the arithmetic (unweighted) mean. However, if variances are heterogeneous across environments, weighted means may be better estimators. The yield of the i th cultivar in the j th environments is χ ij = μ + α i + β j + τ ij where μ = overall mean; α j = effect of the i th cultivar; β j = effect of the j th environment; and τ ij = residual effect. The correlation ( r ) between weighted and unweighted average cultivar performances depends on κ = [V(κ j )/V(Z i )] 1/2 and ρ = correlation between α j and Ζ i , where Ζ i = Cov(τ ij , g j ) and g j = weight of Location j . For any predetermined numerical magnitude ( r 0 ) of r , r ≥ r 0 for κ ≥ (1 − r 2 0 ) −1/2 with any arbitrary value of ρ. For r 0 = 0.90, for example, this condition reduces to r ≥ 0.90 if κ ≥ 2.29. In practice, the validity of κ ≥ 2.29 and its consequence r ≥ 0.90 will be realized in almost all relevant situations. This result was clearly demonstrated by analyses of extensive data sets from the official German registration trials for several agronomic crops. The theoretical investigations, therefore, lead to the recommendation to use the arithmetic mean in estimating average cultivar performance, even if error and cultivar × environment interaction variances are heterogeneous.