# Linear Mixed Model for Genotype Selection of Sorghum Yield
## 3 Results of the Study
### The Selection of the Genotypes Using a Mixed Model
The least three genotypes having the least performance on grain yield in the log scale were 41, 78, and 108, with the associated standard error of perdition being 0.0457, 0.0457, and 0.0458, respectively. Genotypes 66, 180, 85, and 137 were genotypes with inaccurate BLUE estimates compared to BLUP performance as their standard deviations were higher than the standard deviations under the arithmetic mean (BLUE). However, the other estimates of the worst ten genotype performances were a more precise estimate of BLUE than that of the BLUP of the worst performing genotypes. The performance of the genotypes on grain yield for the top ten best performing genotypes in BLUP was more accurate than the estimates of BLUE of the best top ten performer genotypes. Table 7 shows the performance of the genotypes under irrigation for the selected genotypes, which was obtained by considering the interactions of the genotypes by treatment. The top three best performing genotypes under irrigation were genotypes 142, 184, and 173 depending on the estimate of BLUP, in contrast, the selection result using BLUE was different, yielding genotypes 190, 149, and 145 in rank order. Genotype 142 was the tenth best performing genotype for the overall comparison of the genotypes and yielded the highest sorghum production under irrigation conditions, while the other genotypes were not among the top ten best performing genotypes, indicating that the top three genotypes provided the highest sorghum production. The least performing genotypes were presented, with the top three being 55, 49, and 41, in that order. In terms of estimating the random effect, genotype 41 was the worst performer. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Solution for Fixed Effects**} \\ \hline
**Effect** & **Estimate** & **Standard Error** & \(t\) **Value** & **Pr \textgreater{} \(\mid\) \(t\) \(\mid\)** \\ \hline Overall mean; \(\mu\) & 0.8698 & 0.02791 & 31.16 & \textless{}0.0001 \\ Treatment (Irrigated); & 0.2226 & 0.02591 & 8.59 & \textless{}0.0001 \\ Ref (non-irrigated) & 0 & & & \\ \hline \multicolumn{5}{c}{**Test of Fixed Effect**} \\ \hline
**Fixed Effect** & **Numerator** & **Denominator** & **F value** & **Pr \textgreater{} F** \\ \hline Treatment & 1 & 782 & 73.77 & \textless{}0.0001 \\ \hline \multicolumn{5}{c}{**Random Effects Variance Parameter Estimates**} \\ \hline
**Random Component** & **Estimate** & **Standard Error** & **Z Value** & **Pr \textgreater{} Z** \\ \hline Replication; \(\sigma_{\rho(\text{a})}^{2}\) & 0.000579 & 0.000448 & 1.29 & 0.0978 \\ Genotype; \(\sigma_{\rho}^{2}\) & 0.08688 & 0.009246 & 9.4 & \textless{}0.0001 \\ Genotype * treatment; \(\sigma_{\alpha\beta}^{2}\) & 0.008596 & 0.000918 & 9.37 & \textless{}0.0001 \\ Residual; \(\sigma_{\varepsilon}^{2}\) & 0.00093 & 0.000067 & 13.96 & \textless{}0.0001 \\ \hline \hline \end{tabular}  
* show interaction.  
\end{table} 
Table 5: Tests on fixed and random effects significance for grain yield. Table 8 shows the performance of the genotypes under stress (drought), including the top best performing genotypes and the worst performing genotypes. According to the results, the top three genotypes with a high grain yield were 55, 49, and 8, which were among the worst performers under irrigation conditions. The genotypes with the lowest grain yield production under stress were 137, 142, and 184, which were expected to provide insufficient production for community food. Genotypes 142 and 124 were the genotypes that indicated high grain yield under irrigation conditions; however, these were incapable of resisting drought (stress) conditions. This shows that the genotype yielding the high yield under irrigation may be inappropriate for the drought condition. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**The Worst Performing Genotype (DF = 782)**} & \multicolumn{6}{c}{**The Best Performing Genotype (DF = 782)**} \\ \hline
**Genotype** & **Estimate** & **Std.Err.