# Linear Mixed Model for Genotype Selection of Sorghum Yield
## 3 Results of the Study  
### Descriptive Statistics  
Grain yield is the most important phenotypic characteristic of sorghum. The average value of the measurements of grain yield was 2.48 t/ha for the sorghum grown under drought conditions with a standard deviation of the grain yield 0.71 t/ha in log scale. For the sorghum genotype grown under sufficient water availability, the average grain yield was distributed at 3.17 t/ha with the corresponding standard deviation grain yield of 1.15. The absence of sufficient water revealed the influence of drought on the growth of the sorghum crop, which lead to the reduction of sorghum production in grain yield. The coefficient of variation in the grain yield of sorghum showed the consistency of grain yield under drought conditions (non-irrigated) compared with grain yield variation under the availability of water (Table 2). The production of grain yield of sorghum in drought was more consistent than the sorghum production with sufficient water availability. Grain yield variability difference under treatment levels shows that the yield of genotypes under non-irrigated conditions were concentrated close to the mean point and generate less variable grain yields for the genotypes. The variation of grain yield in non-irrigated conditions was less than in irrigated conditions since the standard deviation of grain yield of sorghum in the non-irrigated conditions was smaller than that of the irrigated conditions. The distribution of grain yield of sorghum has a longer tail to the right (positively skewed) and shows that the distribution of the measurements of the grain yield of the sorghum is non-normal (histogram of the grain yield). The estimated kernel density of observations of grain yield clearly shows the non-normality of the response variable as the kernel density and normal distribution do not overlap each other. The kernel density of the grain yield shows the skewness of the distribution of the grain yield, specifically skewed to the right, and ensures the non-normality of the grain yield (Figure 1). The scatter plot of the measurements is nonlinear relative to the normal line of the Q-Q plot, in which most of the points deviate from the normal line at the right of the distribution. The patterns of the points of the grain yield scatter plot are curved with the slope increasing from left to right and the result of the theoretical distribution is skewed to the right, which does not provide a better fit for the grain yield of the measurements (Figure 1). The Shapiro-Wilk test is important to test the normality assumption of the grain yield that tests the hypothesis of no significant departure from normality for the observation of a sample size less than 2000. The test statistic for the Shapiro-Wilk test is 0.9548 with the associated \(p\)-value < 0.05, which indicates rejecting the null hypothesis. The normality of the distribution of grain yield is highly significant, which reveals that the distribution of grain yield significantly differs from the normal distribution. One of the remedial measures for violating the normality assumption is transforming the data using the log transformation technique. It is the most familiar technique for transforming non-normal data set into a normal one [25]. Due to the non-normality of \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Non-Irrigated (\(n\) = 392)** & **Irrigated (\(n\) = 392)** \\ \hline Min & 1.27 & 1.29 \\ Mean & 2.48 & 3.17 \\ Std Dev & 0.71 & 1.15 \\ Max & 4.47 & 6.81 \\ CV (\%) & 28.63 & 36.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary statistics for grain yield of sorghum. Figure 1: Histogram and Q—Q plot of grain yield. the observations of grain yield, to maintain the normality assumption, the data points are transformed using the natural logarithmic function. After that, the histogram of the new data set indicates the symmetrical nature of the distribution, which shows the satisfaction of the normality assumption (left hand side of Figure 2).