# Linear Mixed Model for Genotype Selection of Sorghum Yield
## 4 Discussion  
The paper described the linear mixed model and discussed its importance over GLM analysis, ANOVA, and using the data of the agronomic characteristics of sorghum. It identified sources of variations that lead to invalid conclusions when the classical linear model (ANOVA) was used to analyze the data. The linear mixed model was suggested to be essential for evaluating sorghum agronomic characteristics, primarily grain yield. A descriptive summary result of the sorghum yield, as justified by other investigators, showed  
Figure 3: Marginal and conditional Studentized residual for grain yield (in log scale). that there was an influence from drought on the decline of the production of sorghum, and the variability of the grain yield in drought conditions was less variable compared to the variability of the grain yield production using sufficient moisture. This revealed that effect of drought restricted the genotypes to an extreme yield (too small or too large yield for the genotypes), and the result of the grain yield was consistent for all genotypes [7; 26]. The result agreed with the studies that indicated the genotypes of sorghum having the ability to resist the effect of drought, especially arid and semi-arid areas, as sorghum, in general, has a great drought tolerance relative to other crops, and the preference of framers is different according to their preference criteria [6; 7; 14; 27; 28; 29]. The exploration of the data analysis of the grain yield through a histogram and Q-Q plot indicated the distribution of the data was asymmetrical (not normal), and the remedial measure for the violation of normality assumption, which is the most important assumption for analysis of variance (ANOVA), was the log transformation of the grain yield of sorghum. The transformation reduced the standard deviation from 1.0175 to 0.3305 t/ha, revealing that the transformed data's variability was less than the variability in the original data set. ANOVA considered all factors as a fixed effect, yielding inappropriate modeling and lead to incorrect conclusions drawn due to false positives (Type I errors), powers, and utilities of estimation [30; 31; 32; 33]. This study incorporates both fixed and random effects to identify the effects of genotype and other factors on grain yield, and a mixed model was used for the analysis of grain yield (in log scale) data that compared the results of it with the ANOVA result. The fixed effect parameters in the mixed and ANOVA models were estimated by the BLUE method, and the random effect parameters in the mixed model used the BLUP technique with better accuracy. This study was supported by the investigator who explained the use of both fixed and random effects in the mixed model and also introduced the inappropriate model that led to the incorrect conclusion when considering only the fixed effects instead of incorporating the random effects in the study [34]. The overall performance of the genotypes of this study deviated from the result of [7] the best performing genotypes, as the analysis was performed separately. In addition, the best genotypes under irrigation and stress conditions also deviated, which could be due to the presence of the variability of grain yield among the random effects. This study used the BLUP estimate, which is more precise in comparison to the BLUE estimate, and supported the evidence for the advancement of the use of the linear mixed model for genotype selection in plant breeding. The study conducted by [23] investigated the use of BLUE computing for a linear function of the fixed effects, and BLUP computing for a linear function of the random effect, and, also, showed the selection using BLUP for animal breeding [7; 8; 23; 34]. In this case, treatment was considered a fixed effect [7], whereas the genotypes, interactions of the treatment by the genotype, and the replication within treatments were assumed to be random effects that represented the whole population of sorghum [31].