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Weighted Means Are Unnecessary in Cultivar Performance Trials

Institute of Crop Science and Plant Breeding, Christian-Albrechts-University, Olshausenstrasse 40, D-24118 Kiel, Germany

^* Corresponding author (app98{at}rz.uni-kiel.d400.de).

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 ith cultivar in the jth environments is _ij = µ + _i + ß_j + _ij where µ = overall mean; _j = effect of the ith cultivar; ß_j = effect of the jth environment; and _ij = residual effect. The correlation (i) between weighted and unweighted average cultivar performances depends on = [V(_j)/V(Z_i)]^1/2 and p = correlation between _j and Z_i, where Z_i = Cov(_ij, g_j) and g_j = weight of Location _j. For any predetermined numerical magnitude (r₀) of r, r r₀ for (1 – r²₀)^–1/2 with any arbitrary value of . For r₀ = 0.90, for example, this condition reduces to r 0.90 if 2.29. In practice, the validity of 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 x environment interaction variances are heterogeneous.

Received for publication March 7, 1996.

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