3.1. 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 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). The scatter plots of the new data points fitted on the normal line show that the new data sets have a normal distribution (0.9811 ± 0.33) with a lower standard deviation compared to the untransformed data values of the grain yield (2.82 ± 1.017) (right hand side of
Figure 2).
3.2. The Selection of Best Genotypes Using the Mean Performance of the Genotypes
Table 3 presents the analysis of variance of grain yield (in log scale), considering treatment, genotype, replication within treatment, and interactions of the genotype by treatment as sources of variations for grain yield (log scale). According to the result, there were highly significant differences in grain yield (in log scale) among the genotypes and the interaction between the genotypes. The result also indicates that there was a significant difference between the levels of the treatment on the grain yield (in log scale), which explained the presence of sufficient water and absence of sufficient water yield different sorghum production. There was also a significant difference on grain yield characteristics among the effect of replication within treatment (
p-value < 0.0001). The effect of the interactions of the genotypes by treatment on grain yield was highly significant (
p-value < 0.0001).
The mean performance of the genotypes was obtained using the arithmetic mean technique, which is one of the best linear unbiased estimators of the genotypes at large, the grain yield over replication, and the estimates ranked to select the best and or worst performing genotypes of sorghum. The worst and the best performing of the genotypes are presented in
Table 4, which were obtained via the arithmetic mean performance. The top three best performing genotypes were 149, 190, and 145, with the arithmetic mean of the yield of the genotypes provided being 1.686, 1.677, and 1.6635 t/ha in the log scale, and their corresponding standard deviations were 0.245, 0.251, and 0.260, respectively. The least three performing genotypes were 41, 78, and 108, with corresponding standard deviations of 0.0445, 0.0606, and 0.0409, respectively (
Table 4).
3.3. 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.
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 perdition 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. 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.
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.
The model was diagnosed using the Studentized residual for grain yield (log scale), which indicated the linearity of the predicted value on residuals, as well as the conditional Studentized residual for grain yield (in log scale). The linearity of the fixed effects, including the intercept and treatment effect, was represented by the top left plot of the marginal mean versus the residual. The marginal mean demonstrated the fixed effects’ linearity to the residual. The residual statistics show that the mean was close to zero and the standard deviation was one. The descriptive statistics of the residual were shown in the lower right corner of the log scale marginal residual for grain yield. The Q-Q plot checked the residual’s normality assumption, whether the residuals have a normal distribution or not. According to the Q-Q plot, the normality of the studentized residual for grain yield was satisfied as the scatter plot was linear to the normal curve (
Figure 3). The histogram and scatter plot showed the normality and randomness of the conditional studentized residual of grain yield, and the assumptions are satisfied (top left and right of conditional studentized residual for gran yield (in log scale)).
Given the random effects, the conditional the Studentized residual for grain yield represents the difference between the observed and predicted values. The scatter plot depicts the conditional residuals’ homoscedasticity. The residual statistics are shown in the lower right, with the mean of the residual being zero and the standard deviation being one. The residual versus predicted value of the conditional is represented by the left top plot. The Studentized residual indicates the homoscedasticity of the conditional error, and the histogram and Q-Q plot of the residual show that the conditional residual meets the normality assumption (
Figure 3).