# Linear Mixed Model for Genotype Selection of Sorghum Yield
## 3 Results of the Study
### The Selection of the Genotypes Using a Mixed Model  
The result in Table 5 indicates that to the test result of the effect of treatment on grain yield, indicating whether the treatment effect parameter estimates are significant or not and if the test of the random effects had a factor on the grain yield. The hypothesis that no mean grain yield difference across treatment levels was rejected as the \(p\)-value was small (\(p\)-value \(<\) 0.0001). This showed that the presence of treatment affected the mean difference in grain yield in the log scale compared to the absence of treatment (non-irrigated) (\(p\)-value \(<\) 0.0001). The estimated value of the parameter of treatment also indicated the mean difference in the level of treatment that the mean grain yield under irrigation was estimated at, which was 1.0924. The estimated value of grain yield under the non-irrigated level of the treatment (in log scale) was 0.8698 (in log scale). The result showed that the presence of variability in grain yield associated with the effect of genotypes, in which 89.58% of the total variance of grain yield (log scale) was related to the effect of the genotype that explained the grain yield variability. The association between the interactions of the genotypes by treatment and the grain yield was indicated by the result. It was shown that 8.86% of the total variance of the grain yield was associated with the interactions of the treatment by the genotype. The variability of grain yield due to the random effect of the genotype and the interactions of the genotypes by treatment was highly significant (\(p\)-value \(<\) 0.5). The random effect replication within the treatment explained 0.597% of the total variance of the grain yield variability, which was an insignificant effect in the variability of the grain yield (\(p\)-value \(>\) 0.5) (Table 5). The variability due to the random block effect was very small, thus the analysis excludes the block effect. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**The Worst Performing Genotypes**} & \multicolumn{4}{c}{**The Best Performing Genotypes**} \\ \hline
**Genotype** & **Mean** & **Std.Dev** & **Genotype** & **Mean** & **Std.Dev** \\ \hline
41 & 0.3031 & 0.0445 & 149 & 1.6860 & 0.2448 \\
78 & 0.3684 & 0.0606 & 190 & 1.6771 & 0.25147 \\
108 & 0.4098 & 0.0409 & 145 & 1.6635 & 0.2603 \\
136 & 0.4117 & 0.0746 & 105 & 1.6482 & 0.1886 \\
39 & 0.4419 & 0.0555 & 174 & 1.6105 & 0.2122 \\
66 & 0.4471 & 0.1903 & 164 & 1.5700 & 0.2243 \\
180 & 0.4659 & 0.0818 & 141 & 1.5576 & 0.2272 \\
56 & 0.4861 & 0.0507 & 68 & 1.5563 & 0.1609 \\
85 & 0.5117 & 0.083 & 99 & 1.5247 & 0.1494 \\
137 & 0.5125 & 0.3017 & 142 & 1.5177 & 0.3171 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Least and top performing genotypes using arithmetic mean performance. Table 6 indicates that the performance of the genotypes and important to select the best genotypes for future breeding, the solution of the random effects helps to estimate the correlation between the predicted value and true estimated value of the genotypes that is important to make a rank for the genotypes of sorghum. For selecting the best genotypes, the ranking of the BLUP solution of the genotypes and the finding of the better genotype with a minimum mean square error are used. Table 6 presents the worst and best performing genotypes estimated through BLUP. The BLUP result is a precise estimate of the random effect of the genotype having the least standard error of prediction compared to the performance of the estimated BLUE genotypes. The top three best genotypes with high performing grain yields were 149, 190, and 145, and the predicted values of the genotypes were 0.668, 0.6589, and 0.6452, respectively, with their corresponding standard error of sorghum partition being 0.0689 for each genotype. The BLUP estimate is more accurate than the estimates of the arithmetic mean (BLUE) performance of the genotypes with standard errors of 0.245, 0251, and 0.260, respectively.