# Linear Mixed Model for Genotype Selection of Sorghum Yield
## 2 Materials and Methods
### Linear Mixed Model  
The linear mixed model provides flexibility in fitting models for various combinations of fixed and random effects. Grain yield is the response variable that represents the phenotypic characteristics of sorghum, with different factors and a linear mixed model assumed for the analysis. Various studies consider the genotypes, replications, blocks, and all interactions as random and fixed effects [14; 15]. The analysis of the phenotypic data for plant breeding and the variety trials depends on a linear mixed model of the form [9; 14; 16; 17]  
\[y_{ijkm}=\mu+\alpha_{i}+\rho_{j(i)}+\beta_{k}+(\alpha\beta)_{ik}+\varepsilon_{ ijkm} \tag{1}\]  
where \(y_{ijkm}\) is the outcome of the grain yield of the \(m^{\text{th}}\) observation of the \(i^{th}\) treatment of the \(j^{th}\) replication and the \(k^{th}\) genotype; \(\mu\) is the overall mean of the grain yield; \(\alpha_{i}\) is the \(i^{th}\) treatment (irrigation) effect, which is a fixed effect; \(\beta_{k}\) is the random effect of the \(k^{th}\) genotype; \(\rho_{j(i)}\) is the random effect of the \(j^{th}\) replication nested within the \(i^{th}\) treatment; \((\alpha\beta)_{ik}\) is the interaction effect of the \(i^{th}\) treatment by \(k^{th}\) genotype. In this case, \(\rho_{j(i)}\), \(\beta_{k}\), \((\alpha\beta)_{ik}\), and \(\varepsilon_{ijkl}\) are the random effects, and they are uncorrelated to each other. The effect of genotypes, the effect of replication within treatment, and the effect of the interactions of the treatment by the genotype are normally distributed with mean zero and variances  
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\(\sigma_{\beta}^{2},\ \sigma_{\rho(a)}^{2}\), and \(\sigma_{\alpha\beta}^{2}\). \(\epsilon_{ijkm}\) is the random term from a normal distribution with mean zero and the variance -covariance matrix \(\sigma_{\varepsilon}^{2}\). The estimates of the fixed effects are estimated based on the BLUE method, which depends on having the minimum variance among all linear unbiased estimators [18]. The estimates of the random effects are obtained by the BLUP method because it has good predictive accuracy compared to the other classical procedure BLUE [9; 10]. When using ANOVA (GLM), all factors in the analysis are considered to be fixed effects, and the estimates of the parameters are obtained from the BLUE technique [19]. However, considering the genotypes as the random effect is important to investigate the impact of genetic gain for the response and its estimate is obtained by BLUP, which is crucial to select the best genotypes for future plant breeding, and BLUP is more accurate than the fixed effect estimates, estimated by BLUE, in ANOVA [9]. There are various estimation techniques for the fixed and random effects parameters in the mixed model. The most common methods of estimation for linear mixed model parameters are maximum likelihood (ML), restricted maximum likelihood (REML), and Minimum Norm Quadratic Unbiased Estimator (MINQUE) [19; 20]. Of these methods, REML is used to estimate the parameters of the random effects of a mixed model, having important properties that account for the degree of freedom lost in the estimation of fixed effects [21]. The estimated values of the random effects assist in estimating the parameters of the mixed model's fixed effects that use the generalized least square method (GLS). BLUP is a method of predicting unobservable random effects depending on the response observations that has better accuracy than other estimators (BLUE) [22; 23]. The estimated values that are estimated by BLUP for each random effect parameter are obtained, and the hypothesis test can be performed using the \(p\)-value to identify the specific genotype being a significantly high/low performer in grain yield. The hypothesis tests of the best/worst performing genotypes, considering the standardized distributed BLUPs as a Student t-distribution with denominating degrees of freedom.