WEIGHTED MEANS ARE UNNECESSARY IN CULTIVAR PERFORMANCE TRIALS

The mean performance of a particular cultivar across environments is c ommonly estimated by the arithmetic (unweighted) mean. However, if var iances are heterogeneous across environments, weighted means may be be tter estimators. The yield of the ith cultivar in the jth environments is x(ij) = mu + alpha(i) + beta(j) + tau(ij) where mu = overall mean; alpha(i) = effect of the ith cultivar; beta(j) = effect of the jth en vironment; and tau(ij) = residual effect. The correlation (r) between weighted and unweighted average cultivar performances depends on k = [ V(alpha(i))/ V(z(i))](1/2) and p = correlation between alpha(i) and z( i), where z(i) = Cov(tau(ij), g(j)) and g(j) = weight of Location j, F or any predetermined numerical magnitude (r(0)) of r, r greater than o r equal to r(0) for k greater than or equal to (1 - r(0)(2))(-1/2) wit h any arbitrary value of p. For r(0) = 0.90, for example, this conditi on reduces to r greater than or equal to 0.90 if k greater than or equ al to 2.29. In practice, the validity of k greater than or equal to 2. 29 and its consequence r greater than or equal to 0.00 will be realize d in almost all relevant situations. This result was dearly demonstrat ed by analyses of extensive data sets from the official German registr ation trials for several agronomic crops. The theoretical investigatio ns, therefore, lead to the recommendation to use the arithinetic mean in estimating average cultivar performance, even if error and cultivar x environment interaction variances are heterogeneous.