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932Table 4. The mean, median and the variance ( sb) of the proportionof variation explained by 19 target sets from the analysis of adataset with 40 genotypes by 9 environments (subset of theCIMMYT ISWYN data for 1967–1968).Target set 2 4 6 8 10 12 14 16 18 20Mean 30.2 29.0 31.8 32.4 32.5 32.3 34.5 31.6 31.6 30.7Median 30.4 31.4 28.7 31.1 31.3 30.6 37.4 30.3 33.4 31.02sb 559 255 160 89 121 66 81 50 18 14Target set 22 24 26 28 30 32 34 36 38Mean 30.6 27.8 27.4 31.1 28.5 26.6 23.1 19.8 14.4Median 31.2 28.5 28.0 32.3 29.5 26.2 26.5 20.0 15.92sb 39 19 18 27 38 20 48 51 30On average, regressing the target sets on Complement indexvector accounted for 33%. This indicates a dramatic reduction in thesums of squares compared to values obtained from using the Selfindex vector. The apparent differences between the Self index vector2and the Complement index vector in terms of ˆR values illustrate theeffect of correlations and optimism in the sums of squares of AMMI2models. If we compare the ˆR values based on the Complement index2vector with “expected” ˆR value expressed as ratio of the degrees offreedom of the fi tting sum of squares with (g - 1) df to the degreesof freedom of interaction sum of squares ((t - 1)(b - 1)) df then theComplement index vector can be described as a useful index vector.Optimum target setWe can identify the size of the target set which gives the best prediction.This is achieved by varying the sizes of the target sets and estimating2the ˆR for each target set. Once again, using a subset of the CIMMYTdataset (40 genotypes by 9 environments by 3 replications), targetset sizes were chosen so that complement set sizes ranged from 2 to38 genotypes. For example, a target set of size 2 genotypes involvedthe use of two randomly selected genotypes to calculate the residualinteraction matrix and the Complement index vector estimated fromthe remaining 38 genotypes. Using the 40 genotypes, 19 target setsconsisting of 2, 4, ..., 38 genotypes were created. For each targetset, 10 random samples of different combinations of genotypes were2 2generated. The mean, median and variance of ˆR ( sb) values foreach target set was estimated and the results presented in Table 4.The results show that the proportion of the variation explainedis very variable for target sets of sizes less than eight genotypes as it2is indicated by high sbvalues. For target sets of such few genotypesthe regression fi ts were unduly infl uenced by infl uential or dominant