Evaluation of Efficiency of Weighted Average of Stability and Mean Performance Estimated by Linear Mixed Models for Identifying High‐Yielding Lentil Genotypes Adapted to Rainfed Regions

A combination of the two methods of stability analysis, the additive main effect and multiplicative interaction (AMMI) and the best linear unbiased prediction (BLUP), based on weighted average of stability (WAASB) estimated by the linear mixed models (LMM) index identified the improved genotypes. In this study, 17 advanced genotypes of lentil were studied at two locations, Zanjan and Maragheh, during the two seasons. To examine the genotype × environment interaction, the AMMI and BLUP methods using the WAASB and weighted average of mean performance (WAASBY) index were combined to evaluate the performance stability of genotypes according to different experimental plots. Considering the significant genotype × environment interaction based on likelihood ratio test (LRT), data were analyzed by the BLUP method. The highest grain yield was detected for genotype 13, followed by genotypes 7, 11, 20, 5, 12, and 19 with higher productivity than the grand mean. To select genotypes according to yield and stability, the WAASBY index was defined by combining mean grain yield and stability. Considering 50:50 contributions for the two components, the grain yield and stability of the 13 genotypes were higher than the grand mean. The highest WAASBY was observed for genotypes 7, 20, and 12, which were determined as the best genotypes of lentil under agroecological conditions encountered in the current study regions.

such environmental conditions (Pezeshkpour et al. 2021).The significant interaction of the genotype with the environment for seed yield has loosened the correlation of phenotype to genotype in lentils (Meng et al. 2016).The significant interaction of the genotype with the environment to produce seeds does not reduce the phenotypic correlation and genotype × environment and thus the instability of yield in diverse environments.This is dependent on different agroecological, climatic, agronomic, and genetic variables.Those genotypes with high productivity under different environments are considered as stable genotypes according to the mixed model (Olivoto et al. 2019).
To better interpret the genotype × environment interaction, the additive main effect and multiplicative interaction (AMMI) model is one of the most common methods in the study of multienvironment experiments (Olivoto et al. 2019).This model combines the analysis of variance (ANOVA) for additive or main effects and principal component analysis (PCA) for multiplicative or interactive effects.Thus, the AMMI model provides not only a better understanding of the complex genotype × environment interaction but also an increased accuracy.Due to shortcomings of the AMMI model such as sensitivity to the presence of individual outliers and lack of successful cases of linear mixed-effects model (LMM) analysis, Olivoto et al. (2019) merged the AMMI model and the best linear unbiased prediction (BLUP) method.The WAASB, which combines the AMMI and BLUP methods, determines the weighted average of absolute scores from the singular value decomposition of the matrix of best estimation of the genotype × environment interaction effects generated by LMM (Sharifi 2020).The WAASB method selects the genotypes according to mean performance and stability.A wide range of studies have previously reported the significant interaction of the genotype with the environment (Abbas et al. 2019;Shobeiri et al. 2021;Tadesse et al. 2021).However, the simulation of the genotype × environment interaction in previous studies has been mainly performed using AMMI and bi-plot GGE (the genotype main effect plus the genotype × environment interaction), particularly in chickpea and lentil.Therefore, the objective of this research is to evaluate the efficiency of the commonly used models in stability analyses to identify the genotypes adaptable with climatic conditions in temperate regions using the AMMI model in combination with the BLUP and WAASB methods.

| Materials and Methods
In this research, 17 advanced large-seed lentil genotypes (selected based on yield comparison trials on advanced lentil genotypes registered in 2016-2017 for cold regions) and three check cultivars (Kimia, Bile-savar, Sana) were studied for 2 years (2018-2019 and 2019-2020) at two locations across Zanjan Agricultural and Natural Resource Research Center and Maraghe based on a randomized complete block design with three replications.In Zanjan province, Iran, the Khodabandeh research station (longitude 48°49′ east, latitude 36°9′ north) is located at 1875 m high from sea level.In Maragheh district, Iran, the research station (longitude 46°15′ east, latitude 37°15′ north), with a semiarid cold climate, is located at 1720 m high from sea level.The experimental field had a history of rotation of cereals with legumes.The standard agronomic practices recommended for study regions were conducted to prepare experimental fields.
For fertilization, 100 kg/ha phosphate ammonium and 30 kg/ ha urea were used.To avoid fungal diseases, the seeds of lentil genotypes and cultivars tested were treated with carboxin thiram fungicide at 2 ppm.Seeds were planted on four rows 4 m in length, spaced by 0.25 m.Climatic data and characteristics of the studied genotypes are shown in Tables 1 and 2, respectively.Plant density was set at 200 seeds per meter square.Weeds were controlled by both hand-weeding and herbicide applications.
To measure grain yield (t/ha) at physiological maturity stage, an area of 1.75 m 2 was harvested manually.Two rows in the center of each plot, 3.5 m long, and the grain weight were harvested to measure grain yield.Before subjecting the data to ANOVA, the normality of data and experimental errors and homogeneity of variances were examined using Bartlett's test.Data normality was tested according to the Kolmogorov-Smirnov test using MINITAB16 software.Thus, the stability of 20 lentil genotypes for four environments was analyzed.Considering the genotype and environment as fixed and random terms, a mixed ANOVA was performed.To calculate stability performance, AMMI and BLUP were combined using the R software.In addition, parameters estimated by AMMI, WAASB, and the weighted average of mean performance (WAASBY) were used based on BLUP (Olivoto et al. 2019).The variance components of agronomic traits were determined by the restricted maximum likelihood (REML) method.The stability index of each genotype in multiple environments named WAASB index was determined by the following formula: where WAASB i is the weighted average of absolute scores of the ith genotype and IPCA ik is the score of that genotype in the kth interaction principal component axis.Genotypes with a lower WAASB value are defined as more stable genotypes.Hence, a genotype with a greater WAASB value is commonly regarded as unstable.

| Results and Discussion
The variance estimates were divided into three variance sections: genotypic, genotype × environment, and residual.According to Table 3 and Figure 1, 15.55% of the total variance was justified by the interaction of the genotype with the environment, whereas the genotype effect accounted for 3.66% of the total data variance.A major part of data variance, 80.75%, was explained by residuals.The effect of residuals is attributed to environmental impact, which could be explained according to differences in soil parameters, rainfalls, and distribution of residuals between experimental environments.Earlier findings on other crops showed that a major part of data variance is justified by the environment effect (Akbari, Akbarpour, and Pezeshkpour 2021;Karimizadeh, Pezeshkpour, and Mirzaii 2021).The present research indicated the significant effects of genotype in the first and second years of Zanjan (unshown data).Thus, the genotypes studied in experimental environments presented a reasonable diversity, suggesting that a stability analysis was needed to evaluate the genotype × environment interaction.The mean grain yield in genotypes, which varied from 0.2209 t/ha in genotype 17 to 0.3401 t/ha in genotype 13, was 0.271 t/ha (Table 4).The genotypes studied in the first year and in the first environment (E1) and also in the second year in the third environment (E3) had significant differences; however, in the composite analysis, these genotypes did not have significant differences in the seed yield averaged across environments.
In this study, the sum of squares for the environmental effect indicated the highest contribution, which suggested a wider variability of the main effect of environment compared with the effect of genotype.At first, the ANOVA test was performed for each environment, and Bartlett's test was done to examine the homogeneity of error variances of the experiments.Furthermore, using Bartlett's test to analyses the data of the four environments confirmed the assumption of homogeneity of the variance of experimental lines.The Kolmogorov-Smirnov test (Sa'diyah and Hadi 2016) confirmed the normality of data distribution.In comparisons of long-period datasets on rainfall and average temperature among the experimental locations, it seems that yield reductions highly corresponded with lower rainfalls and higher temperatures, which were considered as a result of random variations in the year.This observation is also evident in the significant genotype × year × location interaction.

Code Genotype
Pedigree Origin The results of the composite analysis showed significant differences in yield levels for the environment, genotype, and the interaction of environment with genotype (Table 5).This suggested variations in yield production among genotypes across different environments.The environment explained the highest variance in the yield dataset, 80.75%, followed by the interaction of environment with genotype (15.55%) and the genotype (3.66%).A major part of variation in yield data can be explained by the environment, which is in turn affected by the year and location.From AMMI results, the three principal components accounted for 100% of variations in environment and genotype (Table 6).
In Figure 1, the mean yield values of the genotypes in each environment have been represented.The third environment showed the highest yield variability, followed by the first environment.This observation could be attributed to climatic fluctuations during the two study years across the two cold regions.Likewise, there are other reports on different reactions of lentil genotypes in diverse environments (Abbas et al. 2019;Karimizadeh, Pezeshkpour, and Mirzaii 2021).According to the scree test, the three principal components accounted for a considerable proportion of variations in the genotype in interaction with the environment determined by the BLUP method (Figure 2).Therefore, these three principal factors explained 72.50%, 14.17%, and 13.33% of the total data variance, respectively.
Figure 3 indicated that 3.7% of the total variance was explained by the genotype effect (darker zones).The lighter zones, which accounted for 96.35% of the total variance, corresponded with the interaction of the genotype with the environment.The first principal component accounted for 72.50% of the total sum of squares.The second principal component also explained 14.17% of the total sum of squares.These two principal components justified a total of 94.60% (37.47% plus 57.10%) of the sum of squares detected for the genotype and 87.27% (70.46% plus 16.81%) of the sum of squares determined for the genotype in interaction with the environment.Considering a greater proportion of the sum of squares explained by the genotype (83.19%) than the genotype interacting with the environment (16.80%), the second principal component was defined as the genotype factor.Thus, this factor could be used to compare the differences in the grain yield between genotypes.
For the first principal component, the genotype interacted with the environment accounted for a greater proportion of the sum of squares (70.46%) when compared to the genotype effects (29.53%).Hence, this principal component was defined as the genotype in interaction with the environment factor.The third principal component explained 13.30% of the sum of squares that could be considered when evaluating the stability of the genotypes in different environments.
The heatmap plot was used to unravel the diversity of grain yield among the genotypes at different environments (Figure 3).A darker color for each environment indicated a higher yield level for the genotype.Moreover, the cluster analyses of the genotypes and environments were represented as vertical and horizontal axes, respectively.This graph showed that a genotype stable in all the environments provided yield stability.Therefore, genotypes 13, 7, and 12 could be considered as stable.With this regard, the yield levels of genotypes at each environment indicated internal variability in that environment.In the second and third environments, the genotypes showed the highest variability.On the other hand, the genotype yield detected for different environments could be also used to determine variations between environments.Thus, genotypes 19, 2, and 17 had greater variations across the environments studied (Figure 3).Such diverse reactions of genotypes across different environments have been reported earlier (Chen et al. 2022;Shobeiri et al. 2021).
According to the results, there was a significant interaction of the genotype with the environment regarding grain yield.Therefore, the BLUP method has been suggested for the analysis of such datasets (Olivoto et al. 2019).The REML results indicated that the variance determined for the three components of the genotype, genotype × environment, and residual phenotypic variance accounted for 3.66%, 15.55%, and 80.75% of the total phenotypic variance, respectively (Table 3).There are reports on the significant interaction of the genotype with the environment in the grain yield using the variance analysis by the lowest squares (Dehghani, Sabaghpour, and Sabaghnia 2008;Karimizadeh, Mohammadi, and Sabaghnia 2013).Because many of the breeding programs are based on multi-environment experiments, the predictive value is important to select genotypes accurately, recommend genotypes, and identify megaenvironments (Olivoto et al. 2019).Thus, some of the parameters estimated in the present research demonstrated a general heritability value of 0.036 for the grain yield.The coefficient of regression for the genotype × environment interaction and the heritability value were 0.1555 and 0.2566, respectively.The genotypic accuracy of selection and correlation of genotypic values in all the environments were 0.5066 and 0.1614, respectively.The coefficients of genotypic and residual variances and the ratio of these two criteria were 5.55%, 26.05%, and 0.213, respectively.The interaction of the genotype with the environment was four  times greater.To improve the predictive value, the use of statistical models such as the BLUP method has been considered.Based on the AMMI method, there were significant effects of the genotype (unshown data), suggesting the important role of three principal components to explain 100% of the total variance in the interaction of the genotype with the environment.Such significant effects of the environment and the genotype × environment interaction have been reported for the grain yield in lentils (Dehghani, Sabaghpour, and Sabaghnia 2008;Namdari et al. 2022;Shobeiri et al. 2021;Tadesse et al. 2021).To the best of our knowledge, this is the first report on the AMMI and BLUP as the two potential tools to gain a better understanding of factors effecting the genotype × environment interaction in order to identify lentil genotypes with high yield stability.
The BLUP method improves the predictions made by the AMMI model to analyses the performance stability of genotypes (Olivoto 2019).In Figure 4, the mean yield values estimated by BLUP indicated the highest yield levels for genotype 13, followed by genotypes 11, 7, 20, 5, 12, 19, and 8.The lowest mean yield value was detected for genotype 17.The predicted values for the check cultivars Bile-savar and Sana were above the grand mean, while the predicted mean for the cultivar Kimia was below the grand mean.Thus, the BLUP method is used to estimate the grain yield accurately (Olivoto et al. 2019).Furthermore, this method optimizes the predictions of random effects when mixed linear effects are present (Piepho et al. 2008).Our findings on accurate genotype performance and predicting genotypic variance components in lentil are in agreement with previous reports on maize (Baretta et al. 2016) and sugar cane (Barbosa et al. 2014).
In this study, the stability of the grain yield components of the studied genotypes was examined according to the WAASBY scores.
Based on the 50:50 weights, genotypes 7, 20, and 12 showing the highest WAASBY scores were determined as stable genotypes with a high grain yield (Figure 5).Many researchers considered the stability value of AMMI (ASV) for selecting and recommending  stable genotypes in lentil (Karimizadeh et al. 2008).However, if the contribution of these components into the genotype × environment interaction is to be low, it is possible that a major contribution of this interaction could be ignored in such bi-plots (Olivoto et al. 2019).In these cases, the WAASB × GY bi-plots in Figure 6 and the WAASBY bi-plots in Figure 4 are used to identify genotypes with high yield and wide-ranging stability.
The third bi-plot, which determined the grain yield versus WAASB values, was divided into four quadrants (Figure 6) to evaluate genotypes in terms of mean grain yield and genotype stability.In the first quadrant, genotypes 10, 3, and 17 were considered to be the most unstable for grain yield with the productivity lower than the grand mean based on the high WAASB scores.In the second quadrant, genotypes 13, 5, and 11 with a grain yield higher than the grand mean were unstable due to the high WAASB scores.In the third quadrant, genotypes 9, 2, 6, 16, 18, 14, and 4 with the productivity lower than the grand mean were stable due to low WAASB scores.Thus, a lower WAASB score has been found to determine stable genotypes in terms of grain yield.In the fourth quadrant, genotypes 7, 12, 20, 15, 14, 19, 1, and 8 with a high grain yield and stable performance due to low WAASB scores were considered as highly productive and stable genotypes.
In Figure 7, which presents the combination of stability and yield interpretation, the WAASB × GY bi-plot could be used to identify stable genotypes for all environments.The most important advantage of this method is to classify genotypes according to all principal components by considering those contributions missed by the first and second principal components (Olivoto et al. 2019).Moreover, this bi-plot uses mixed models that estimate parameters important in quantitative genetic studies such as genotypic variances, genotype × environment interaction, general heritability, mean-based heritability, and correlations of genetic variance (Olivoto et al. 2019).In Figure 7, the two criteria of dependent variable (GY) and stability (WAASB) obtained different weights.Thus, the classification of genotypes based on the WAASBY criterion differs greatly depending on the ratio of WAASB and GY.So, every 5% increase in the grain yield (response variate) corresponded with 5% decrease in the stability weight.In the first column at left, genotypes 15, 8, 7, 18, and 6 were the most stable, while genotypes 14, 9, and 5 showed the lowest stability.Of course, this analysis ignored the grain yield of genotypes; thus, such classification seems unreliable.For the last column, the classification is performed based on the grain yield, and the stability is ignored.Genotypes 13, 11, 7, and 20 were the best in terms of the grain yield, and the lowest yields were determined for genotypes 17 and 3.These findings support the results presented in Table 3.The left-hand clusters of this graph (Figure 7) were used to identify genotypic groups with similar levels of performance stability and mean grain yield.Genotypes 7, 8, 13, and 15 produced high and stable yield (blue color), and genotypes 11, 12, 19, and 20 were productive but unstable (black color).
Genotypes 1, 4, 6, 10, and 18 were stable with low productivity (green color), while genotypes 2, 3, 5, 9, 14, 16, and 17 were unstable with low productivity (red color).According to 50:50 weights determined for the two criteria, these genotypes were also ranked as four groups according to performance stability and grain yield (WAASBY).Considering the equal contributions (50:50) detected for stability and yield, it is observed that genotypes 7, 13, 8, 15, 11, and 12 provided the best rankings.If this 50:50 weight changes to 35:65, for instance, then the selection of genotypes is to be based on heavier weights for the grain yield than the WAASB, suggesting genotypes 7, 11, 13, and 20 to be stable.Therefore, one of the advantages of WAASBY is dependent on the aims of breeding programs in order to estimate the parameters using stability and grain yield components in the ranking of genotypes.Hence, the 7 was ranked as the best, followed by genotypes 20 (cultivar Sana) and 12.

| Conclusion
In the current research, the mosaic plot indicated visually that the first and second principal components obtained high contributions from the genotype × environment and genotypic variance, respectively.Considering the AMMI bi-plots, only the first and second principal components were used for the analysis of genotype × environment.Thus, the performance stability of the WAASBY index was used to estimate the parameters accurately and provide a two-dimensional graph involving the mean grain yield and performance stability for interpreting.According to this index, genotypes 7, 13 and 15 were highly productive and stable.In general, using the WAASBY graph with scoring the variable weight from 0 to 100 for the WAASB index and mean grain yield may provide more reliable and accurate results of the performance stability.Based on the linear mixed model and all the components to calculate the WAASBY index, this index appeared to be the best criterion for such studies.

FIGURE 2 |
FIGURE 2 | Eigenvalues and percentage of variance explained by principal components (PC) of the best linear unbiased prediction (BLUP) and genotype × environment interaction (GEI) matrix for the grain yield of lentil genotypes in multi-environment yield trials.

FIGURE 5 |
FIGURE 5 | Estimates of weighted average of stability (WAASB) and mean performance (WAASBY) for 20 lentil genotypes using 50:50 weights for yield and stability, respectively.

FIGURE 7 |
FIGURE 7 | Heatmap of genotypes ranked depending on the number of principal component axes used to estimate weighted average of stability (WAASB) index; Euclidean distance-based dendrogram used for grouping genotype ranking for both genotypes and principal component axes; rankings of lentil genotypes considering different weights for stability and yield (GY).

TABLE 2 |
Genotype code and parents of 20 lentil genotypes studied during 2 years in four environments at Khodabandeh (Zanjan province, Iran) and Maragheh (Iranian district) research stations.

TABLE 3 |
Evaluation of significance of factors by LRT (χ 2 ) and estimation of variance components by restricted maximum likelihood (REML) in lentil genotypes.
FIGURE 1 | Boxplot showing average grain yield (above) and genotypes yield (below) determined in four environments at Khodabandeh (Zanjan province, Iran) and Maragheh (Iranian district) research stations during two growing seasons; blue and red circles (with error bars) indicate above and below average performance for genotypes, respectively.

TABLE 6 |
Combined analysis of variance of seed yield of experimental lentil genotypes based on AMMI model.