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166 Crop Breeding and Applied Biotechnology, v. 2, n. 2, p. 161-170, 2002Considering the observed high correlation between Bˆand B ( r = 0.74 = 0. 86) it can be said that in thisexample the error associated with a decision based on Binstead of on Bˆ would not be great. It is justifiedbecause, under an adequate choice of the number ofmultiplicative terms (or singular axes), a low magnitudecorrelation between Bˆ and B is related to thepredominance of noise in the B matrix. Thus, in thissituation, it may be wrong to interpret the residues ŝ ijas reliable estimates of the true SCA (s ij).Based on the Bˆ matrix the following crosses wereoutstanding among the more favorable combinations:6x8, 3x6, 4x9 and 4x5 (female x male or vice-versa).Also, note that in the case of complete diallels andunder the assumption of no reciprocal effects, the SCAprediction matrix is also symmetric. Further, the SCAof each parent with itself (s jj) is estimated, allowinginferences on varietal heterosis (Cruz and Vencovsky,1989). The majority of these values were negative,indicating that this type of cross is not adequate forhybrid extraction. The parents identified by thenumbers 8, 6 and 7, in this order, should show greatervarietal heterosis than those of the numbers 9 and 2.Further note that, in spite of the arbitrariness of theassignment of zero values for the main diagonalelements of B, the analysis was able to predict thedistinct reaction of the parents when crossed withthemselves. This opens the perspective of predictingthe SCA from unavailable crosses, avoiding the needto assess all possible single crosses.The possibility of predicting the hybrid performanceof non-realized crosses comes from the fact that theanalysis deals with the interaction (SCA) not only as aproperty of each parental combination, but also in anintegrated manner, as a phenomenon determinedtogether by rows and columns of the B matrix.However, if the major interest is for this type ofprediction, it is recommended that one proceedsinteractively according to Gauch (1992). Furthermore,to predict the genotypic responses it is necessary totake into account the main effects associated with eachparent. Charcosset et al. (1993) used models withmultiplicative interaction terms (AMMI analysis) witha similar purpose of predicting the performance of linehybrids in maize, when the assessment of all the singlecrosses were not possible. The authors concluded thatthe models offer the possibility of improving predictionefficiency, without supplementary measurements. Theproblem could also be treated by the mixed modelsapproach, obtaining the BLUP (best linear unbiasedpredictor) of the s ijof missing hybrids (Bernardo, 1996;André, 1999), followed by the SVD adjustment of thesepredictions.The coordinates of each parent, used to construct thegraphics, are summarized in Table 2. A SAS programto execute the analysis is freely available from theweb-address: www.agro.ufg.br.The relationships commented upon may be seen bylooking at the corresponding biplot. Figure 1, althoughbringing all the biplot-AMMI3 information, does notallow a perfect assessment of the relative distancesamong the parents because it is a three dimensionalgraph projected on the paper plane. Thus, for a morerigorous inspection, it is convenient to project the pointsof the 3-D space in at least two complementary planes,for example, IPCA1 vs. IPCA2 and IPCA1 vs. IPCA3(Figure 2). Therefore, the closeness between two points(genotypes) must always be confirmed by inspectionin these principal planes.Figures 1 and 2 summarize the potential of thedifferent parental combinations to produce hybrids.These graphics do not explain the whole SCA value.However, according to the AMMI analysis concept(as in PCA), the true pattern of interaction (SCA) isTable 1. The partition of SS SCA(specific combining ability sum of the squares) by singular decomposition, proportionretained in each singular axis and accumulated percentage up to k th axis, in a diallel AMMI analysis.IPCA k SS IPCAk SS IPCAk /SS SCA % Accumulated1 2978.8753 0.3251 32.50532 2006.2649 0.2189 54.39753 1810.3817 0.1975 74.15234 1500.2918 0.1637 90.52345 678.5611 0.0740 97.92786 55.5711 0.0170 99.62547 32.8437 0.0036 99.98388 1.4862 0.0002 100.0000Total 9164.2760 1.0000 -2002, Brazilian Society of Plant Breeding
Crop Breeding and Applied Biotechnology, v. 2, n. 2, p. 161-170, 2002167described mainly by the first singular axes. Each axisis a linear combination of the rows and columns(vectors) which form the B matrix. As all these vectorscontain ŝij values, the axes represent SCA linearfunctions constructed with all the matrix data. Finally,as the successive axes capture ever less pattern andmore noise, discarding the latter becomes necessaryto avoid the incorporation of substantial errors in thedescription of the interaction.Conscious of these concepts and observing the pairsof points corresponding to each female and malecombination, the proximity between F6/M3, F6/M8,F4/M9 and F4/M5 can be depicted. This indicatesthat such combinations are promising for single crossproduction in this set of parents. A strongcharacteristic in all of them is the high magnitudeand same sign coordinates for the genotypes of eachpair. On the other hand, it can be deduced that thegenotypes with good GCA should have lowmagnitude coordinates (small interaction) as is thecase of parents 1 and 7 (Figure 1). These lines areshown to have high GCA in Griffing analysis (1956).Among the combinations with lower potential forhybrid production are: F2/M9, F6/M5 and F4/M8,all showing great distance among the respective F/Mpoints in the Figures 1 and 2. Further, the low potentialis also observed for crosses among the same genotype(F6/M6, F9/M9, etc.), which always show points inopposite position. It is interesting to note that theF3xM5 cross although shown as fairly favorable inTable 2. Coordinates of the parents for the three firstprincipal axes (IPCA1, IPCA2 and IPCA3) of a biplotgraphic display, in a diallel AMMI analysis.Point Female/Male IPCA 1 IPCA 2 IPCA 31 F1 2.3197 1.4001 -1.25792 F2 -3.0311 -2.8749 -0.45653 F3 -3.2249 2.0848 1.95914 F4 -0.1408 4.5982 -3.98555 F5 3.0481 -2.4802 -2.30306 F6 4.6443 -0.9625 4.30467 F7 2.1540 0.2680 -0.29588 F8 -2.7205 2.0172 3.53419 F9 -3.0488 -4.0507 -1.499010 M1 -2.3197 -1.4001 -1.257911 M2 3.0311 2.8749 -0.456512 M3 3.2249 -2.0848 1.959113 M4 0.1408 -4.5982 -3.985514 M5 -3.0481 2.4802 -2.303015 M6 -4.6443 0.9625 4.304616 M7 -2.1540 -0.2680 -0.295817 M8 2.7205 -2.0172 3.534118 M9 3.0488 4.0507 -1.4990Figure 1. Biplot AMMI3 for specific combiningability (SCA) in maize grain yield of a diallel cross;the cubes identify female genotypes (F), and thepyramids, the male genotypes (M).the original diallel table, was shown to be onlyreasonable by the present analysis. This happenedbecause the AMMI method discards part of theinformation contained in the original matrix as ofpredominantly erratic in nature. The option for thebiplot-AMMI3 implied in disregarding 26% of theSS SCA. These results can be confirmed in the Bˆmatrix for ŝ AMMIij prediction.When only one kind of marker is taken, for examplethe F points (or the M points), the divergences andsimilarities among the genotypes which make up theparent group can be assessed. Thus the divergencebetween genotypes 3 and 6 is notorious, as well asthose between 4 and 9. However, it is also evidentthe similarity between the genotypes 3 and 8, andbetween 2 and 9 (Figures 1 and 2). Observe also thatthe first principal axis (IPCA1), capturing 32% ofthe SCA, is strongly determined by the divergenceamong two groups of genotypes (2, 3, 8, 9) and (1, 5,6, 7). The two best hybrid combinations foundresulted exactly from the cross between genotypesbelonging to distinct groups (3x6 and 6x8). The twoother axes, each explaining about 20% of the referredvariability, relate more with intragroup genotypicdivergences. Therefore, the biplot display was shownto be an additional tool to guide the formation ofheterotic groups and the planning of future crosses.CONCLUSIONSThe adoption of biplot-AMMI to represent the SCAallows a quick visualization of the favorable andunfavorable genotypic combinations for hybridproduction. The use of this graphic display also allowsa quick and easy assessment of the genotypicdivergences and similarities among the parents2002, Brazilian Society of Plant Breeding
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