yijk = µ+ b(E)jk + Gi + Ej + (G*E)ij + eijk
Here yijk is the observation of the kth replication of the ith genotype of the jth environment, m is the population overall mean, Gi is the fixed effect of ith treatment, Ej is the randomized effect of jth environment, (g*a)ij is the genotype x environment interaction effect, b(E)jk is the replications nested in environments effect and eijk is the experimental error randomized variable.
Stability analysis was made using bilinear regression models (Verma et al., 1978, Silva et al., 1985, Cruz et al, 1989). Each genotype is described by three parameters: two regression coefficients (b1 and b2) and the variance of regression deviation S2di. Coefficient b1 indicates genotype response in unfavorable environments and b1 + b2 measures response in the favorable ones. Environmental indexes are the independent variables of this multiple regression method and the zero value is the intercept of each one. The advantage of using this method is the ability to evaluate genotypes also under unfavorable environmental conditions. The model is:
E(Yij) = B0 + B1 Ij + B2Jj
Expected observation of the ith genotype in the jth environment, B0 is the mean for each genotype, B1 is the unfavorable environments regression coefficient, B1 + B2 is the favorable environments regression coefficient for each genotype, Ij = the environmental index (Eberhart and Russel, 1966), Jj = Ij(+) � Îj(+) the environment index of favorable ones minus their average.
The following matricial equation represents Yij values:
Xb + E = Y
The X matrix has three columns, the unity, the environmental effects, and thirdly, the favorable environmental effects minus its average. Unfavorable effects have zero value. b is the vector of the unknown regression coefficients and E is the vector of the experimental error for each genotype. Y is the vector of the observations. Applying the minimum square method we obtain the following system:
X'Xb = X'Y
( X' is the trans X matrix and b the regression coefficients estimations vector)
Table 1 shows genetic x environment significant effect results; effects for genotypes and environments also were significant.
Table 1. Analysis of variance for (YIELD)
and (EXVOL) .(**p<0.01, *p<0.05).
| S V / G L
Variables |
Environments
9 |
Rep(Env.)
20 |
Treatments
13 |
Treat x Env
117 |
Error
260 |
VC(%) |
| (YIELD) | 93.43** | 0.631 | 0.822** | 0.188** | 0.097 | 12.34 |
| (EXVOL) | 22052** | 1923.4 | 5707.4** | 1461.6** | 1154.2 | 9.09 |
Tables 2a and 2b show the environment average for each hybrid in two conditions: unfavorable (E(-) ) and favorable (E(+)). B0 is the overall mean including both environment conditions. The slopes (B) feature responses for genotypes on each environment (B1 y B1 + B2). R2 is the determination coefficient, which measures fitness for the model and Sdi is the regression device mean squares, measuring stability response of the hybrids. For each genotype, yield shows high R2 values; expansion volume (exvol) shows high R2 values except for H3, H9, H11 genotypes.
For yield, all genotype slopes are 1 despite the environment (Table 2a) However, for H1, H6 , H9, H10, H14, Sdi was significant. The hybrids differ not in their responses but in their stability. For the mentioned hybrids, behavior is not predictable on the environment range of this experiment.
Table 2a. Stability parameters for (YIELD).
Test t student for B1=1 y B1+ B2 =1
.(*p<0.05). Test F.(*p<0.05).
| HYBB | E(-) | E(+) | B0 | B1 | B2 | B1+ B2 | R2 | Sdi |
| 1 | 0.993 | 3.796 | 2.675 | 1.056 | 0.213 | 1.269 | 96.37 | 0.383* |
| 2 | 1.006 | 3.757 | 2.657 | 1.035 | -0.06 | 0.972 | 98.61 | 0.126 |
| 3 | 1.094 | 3.752 | 2.689 | 1.000 | 0.177 | 1.177 | 99.45 | 0.049 |
| 4 | 0.877 | 3.488 | 2.442 | 0.981 | 0.034 | 1.015 | 98.25 | 0.148 |
| 5 | 0.748 | 3.260 | 2.255 | 0.935 | -0.275 | 0.660 | 97.67 | 0.166 |
| 6 | 0.981 | 3.491 | 2.487 | 0.947 | 0.021 | 0.968 | 94.51 | 0.450* |
| 7 | 0.999 | 3.898 | 2.739 | 1.089 | -0.664 | 1.023 | 99.16 | 0.084 |
| 8 | 0.732 | 3.583 | 2.442 | 1.066 | -0.192 | 0.874 | 98.86 | 0.107 |
| 9 | 0.861 | 3.715 | 2.574 | 1.063 | -0.117 | 0.945 | 97.58 | 0.233* |
| 10 | 1.254 | 3.792 | 2.777 | 0.949 | 0.214 | 1.163 | 97.51 | 0.211* |
| 11 | 0.901 | 3.506 | 2.464 | 0.979 | 0.175 | 1.154 | 98.74 | 0.11 |
| 12 | 0.850 | 3.200 | 2.260 | 0.885 | 0.078 | 0.963 | 98.95 | 0.073 |
| 13 | 0.893 | 3.659 | 2.553 | 1.040 | -0.151 | 0.890 | 99.33 | 0.059 |
| 14 | 0.849 | 3.435 | 2.407 | 0.972 | -0.048 | 0.924 | 97.34 | 0.218 |
Table 2b. Stability parameters for the variable (EXVOL).
Test t student for B1=1 y B1+ B2 =1
.(*p<0.05). Test F.(*p<0.05).
| HYB | E(-) | E(+) | B0 | B1 | B2 | B1+ B2 | R2 | Sdi |
| 1 | 351.87 | 389.14 | 366.78 | 1.034 | -0.482 | 0.552 | 68.02 | 917.4 |
| 2 | 379.76 | 427.11 | 398.70 | 1.162 | -0.448 | 0.713 | 53.18 | 2198* |
| 3 | 351.39 | 370.47 | 359.02 | 0.455* | -0.812 | -0.357* | 28.11 | 1013 |
| 4 | 325.11 | 386.80 | 349.79 | 1.397 | -1.365 | 0.032 | 69.18 | 1517 |
| 5 | 362.61 | 405.17 | 379.63 | 1.021 | 0.042 | 1.062 | 58.18 | 1533 |
| 6 | 340.24 | 394.30 | 361.87 | 1.306 | -0.018 | 1.288 | 57.27 | 2566* |
| 7 | 347.59 | 394.94 | 366.53 | 1.261 | -0.608 | 0.653 | 72.06 | 1122 |
| 8 | 352.17 | 395.50 | 369.50 | 1.036 | 0.051 | 1.087 | 72.63 | 830 |
| 9 | 388.16 | 397.56 | 391.92 | 0.221* | 0.929 | 1.151 | 27.21 | 1229 |
| 10 | 372.86 | 425.49 | 393.91 | 1.354 | 1.101 | 2.455* | 96.24 | 191 |
| 11 | 371.93 | 385.39 | 377.31 | 0.435* | 0.263 | 0.697 | 24.94 | 1404 |
| 12 | 347.76 | 411.81 | 373.38 | 1.730* | 0.710 | 2.440* | 82.37 | 1476 |
| 13 | 361.87 | 390.29 | 373.24 | 0.651 | 0.201 | 0.852 | 45.98 | 1109 |
| 14 | 356.25 | 390.79 | 370.07 | 0.935- | 0.435 | 1.370 | 66.25 | 1046 |
H10 and H7 had yield upper values in both environments, but H10 was unstable while H7 was stable and predictable in this evaluation (Fig 1). For exvol, genotype responses differ under unfavorable environments. For H3, H9 and H11 B1 <1 and for H2 B1>1. In favorable environments, for H3, B1+B2 < 1 while for H10 and H12 >1. Sdi was significant only for H2 and H6. Even when H2 showed the best popping expansion, H9 retained its capability in both environment situations, becoming better for this character (Fig. 2).
Table 3 shows means and the environment index for both variables. For yield, environments 1, 4, 5, 6, 3 and 2 were profitable, for exvol environments 7, 8, 1, 3 were. Instead of profitability of environment 1 for both characters, we found that a favorable environment for one is unfavorable for the other. This agrees with other classical studies describing a negative correlation between yield and expansion volume. These results contribute to aiding farmers in selecting better genotypes with yield stability, quality properties and adaptation for this no traditional zone.
Table 3. Means and environments. (+)
favorable. (-) unfavorable.
| Env | (YIELD) | (EXVOL) | ||||
| Mean | Ij | Ij(+) | Mean | Ij | Ij(+) | |
| 1 | 4.41 (+) | 1.88 | 0.81 | 390.4 (+) | 16.71 | -7.08 |
| 2 | 3.56 (+) | 1.03 | -0.032 | 364.5 ( -) | -9.19 | - |
| 3 | 2.66 (+) | 0.13 | -0.94 | 380.2 (+) | 6.50 | -17.29 |
| 4 | 4.26 (+) | 1.73 | 0.67 | 362.7 ( -) | -10.99 | - |
| 5 | 3.97 (+) | 1.44 | 0.37 | 360.5 ( -) | -13.21 | - |
| 6 | 2.71 (+) | 0.18 | -0.89 | 362.8 ( -) | -10.88 | - |
| 7 | 1.13 ( -) | -1.41 | - | 412.0 (+) | 38.31 | 14.51 |
| 8 | 0.79 ( -) | -1.73 | - | 407.3 (+) | 33.65 | 9.85 |
| 9 | 0.81 ( -) | -1.72 | - | 346.1 ( -) | -27.56 | - |
| 10 | 0.98 ( -) | -1.54 | - | 350.4 ( -) | -23.34 | - |
Figure 1. Genotype H7
Figure
2. Genotype H9
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