Genotype x environment interaction in cowpea by mixed models

Several methods have been proposed to measure effects of genotype × environment interaction (G×E) on various traits of interest of plant species, such as grain yield. Among these methods, mixed models using the Restricted Maximum Likelihood (REML)-Best Linear Unbiased Prediction (BLUP) procedure with random genotype effects have been reported as advantageous, since they allow the obtaining of actual genotypic values for cultivation and use. The objective of this work was to evaluate the response of grain yield to different locations and years, and the effects of G×E on the performance of cowpea genotypes by the methodology of mixed models. Twenty genotypes were evaluated under rainfed conditions in 47 locations in 2010, 2011 and 2012 using randomized block design. After joint analysis, the genotypes adaptability and stability patterns within and between years were tested by the Harmonic Mean of Relative Performance of Genetic Values (HMRPGV) statistics. The analysis within the years showed highly significant effects of the genotype × location interaction in all the years evaluated. The results of the joint analysis presented highly significant effects (p≤0.01) of the genotype, and triple interaction (genotype × location × year) (p≤0.001), denoting a strong effect of the G×E on the genotype performances. The HMRPGV analysis was adequate to identify superior genotypes, highlighting the MNC02676F-3, MNC03-737F-5-1, MNC03-737F-5-9, BRS-Tumucumaque, and BRS-Guariba as the genotypes with best stability and highest grain yield. The selection of these genotypes resulted in a new average yield (1,402 kg ha -1) which is higher than that obtained by selection based only on the phenotype (1,230 kg ha-1).


INTRODUCTION
Cowpea (Vigna unguiculata (L.) Walp.) is a legume that has great nutritional value.This species is grown mainly in Africa, and Nigeria is the world's largest producer.It is widely cultivated in the Northeast region of Brazil, and nowadays also in the North and Center-West regions; Brazil is the world's third largest producer (SINGH et al., 2010;VIJAYKUMAR;SAINI;JAWALI, 2010).Researches on cowpea breeding in Brazil have been developed by the Brazilian Agricultural Research Corporation (Embrapa) which maintains an Active Germplasm Bank for this species and has been conducting a Breeding Program led by Embrapa Mid-North since 1991.
One of the main challenges of plant breeding is the expansion of cultivation to other regions, since it requires taking into account the adaptability of the species to different environmental conditions and the maintenance of its production stability.However, no studies on cowpea using methodologies with mixed models are found in the literature.These methodologies make it possible to predict the true Values for Cultivation and Use (VCU), and in this case, the effects of the genotype need to be assumed as the random effects in the statistical model (PIEPHO, 1997;RESENDE, 2007;ROSADO et al., 2012;VERARDI et al., 2009).
The assumption that the effects of the genotype are random makes it possible to obtain the Best Linear Unbiased Predictions (BLUP) of effects of the genotypes and G×E, which eliminate their noise by weighing such effects through a regressing factor, which is usually referred to as repeatability, which in practice refers to the heritability of the trait, thus leading to shrinkage estimates of such effects and to prediction of genetic values (PIEPHO, 1997;RESENDE, 2007;SEARLE;CASELLA;McCULLOCH, 1992).This concept has been used together with the Restricted Maximum Likelihood (REML) method, developed by Patterson and Thompson (1971) as the optimal procedure for the estimation of components of variance, maximizing the likelihood function of the residuals, instead of the observed data, giving an incidence matrix of the fixed effects.Therefore, the REML-BLUP analysis has been the most recommended for tests using mixed model approaches (BORGES et al., 2010;CARBONELL et al., 2007;MAIA et al., 2009;PIMENTA et al., 2016;RESENDE, 2007;SCHAEFFER, 2004;SILVA et al., 2011;ZENI-NETO et al., 2008).
Using this type of approach, Borges et al. (2010) found responses varying from 1.06-to 1.08-fold the overall average for rice genotypes with better performance, in tests conducted in 11 locations and 11 years.Bastos et al. (2007) evaluated the adaptability and stability of sugarcane genotypes and also found values for the selected clones with better performance than the overall average.
In this context, the objective of this work was to evaluate the response of grain yield of 16 lines and 4 cultivars to different locations and years, and the effects of G×E on the performance of cowpea genotypes by the methodology of mixed models.
An environment was initially considered as each location × year combination, with adjustments made for each year, considering the genotype effects as random and the location effects as fixed, according to the following model: y = Xb + Zg + Ti + e.The joint analysis was performed within each evaluation year according to the model: y = Xb + Zg + Ti + e, wherein y is the vector of observations; b is the vector of effects of the blocks × location combinations added to the overall average (fixed effects); g is the vector of genotype effects (assumed to be random); and i is the vector of effects of the genotype × location interaction (random).Subsequently, the analysis was performed considering the entire dataset, according to the model: y = Xb + Zg + Qm + Ti + Uq + e, wherein y is the vector of observations; b is the vector of effects of the combinations block × location × year added to the overall average (fixed effects); g is the vector of genotype effects (assumed to be random); m is the vector of effects of the genotype × year interaction (random); i is the vector of effects of the genotype × location interaction (random); q is the vector of effects of the genotype × location × year triple interaction (random); e is the vector of errors (random).X, Z, Q, T and U are the incidence matrices for these effects, respectively, assuming g ∼ N(0,Iσ g 2 ) and e ∼ N(0,Iσ e 2 ).
The Restricted Maximum Likelihood (REML)-Best Linear Unbiased Prediction (BLUP) procedure was used to compute the components of variance and subsequent prediction of random effects.The significance test for these effects was given by the Likelihood Ratio Test (LRT), considering the χ 2 .The experimental quality of all tests was measured by the Selective Accuracy statistics (RESENDE; DUARTE, 2007), defined as: SA = [1 / 1 + (σ e 2 / r) / σ g 2 ] 0, 5 , wherein r is the number of blocks used in the tests, σ e 2 is the residue variance, and σ g 2 is the genotypic variance.
Based on the random effects obtained through the joint analysis, the predicted genotypic values were obtained by μ + g i , wherein μ is the mean of all locations and g i is the genotypic effect free from the G×E.The criterion for the joint selection of genotypes, considering simultaneously the grain yield, stability and adaptability was given by the statistic of Mean Harmonic of Relative Performance of Genotypic Values (RESENDE, 2007): wherein n is the number of locations and Vg ij is the genotype value for the genotype i expressed as the proportion of the mean of the location j.All analyzes in the present study were performed using the statistical program R (R 3.1.2).

RESULTS AND DISCUSSION
The experimental precision measured by the accuracy had values ranging from 0.24 to 0.98, and overall mean (considering all locations evaluated) of 0.79.According to Resende and Duarte ( 2007), accuracy values below 0.5 are low, from 0.5 to 0.7 are average, from 0.7 to 0.9 are high, and above 0.9 are very high, thus, 42% of the experiments presented high experimental accuracy and 38% presented very high accuracy.
The components of variance of the random effects and mean squares of the fixed effects obtained by the analysis within each year by mixed models are presented in Table 2.All sources of variation had highly significant effects within the three years of evaluation, except the genotype effect within 2011.The mean squares for the genotype effect presented different variation patterns over the years, highlighting the variation between genotypes in 2010, which was approximately 15-fold higher than the variation within 2011.On the other hand, the mean square for the effect of locations was greater within 2011.
According to the results of the joint analysis considering the whole dataset, the effects of genotypes were significant (p<0.01) by the LRT test; and the genotype × location interaction (G×L) and genotype × year interaction (G×Y) presented no significant differences.However, the triple interaction was significant (p≤0.001),which confirms the strong effect of the G×E on the performance of the genotypes evaluated.The accuracy value was higher than 90%, indicating a satisfactory experimental accuracy for the joint analysis (RESENDE;DUARTE, 2007).Although the effects of genotypes were not significant (p>0.05) in 2011, they were significant in 2010 and 2012.The G×L was significant within the three years evaluated, which justifies the use of all years in the overall joint analysis.According to Silva et al. (2011), the absence of significant effects of genotypes is expected in cases of tests with very contrasting environments.These results indicate that the different performances of the genotypes were caused by a combination of factors that can be represented by the G×L×E triple interaction to the detriment of the other interaction effects and the main effects of genotypes.
According to Yan and Hunt (2002), in cases where the proportion of variation explained by the interaction component is greater than that explained by the differences between genotypes, the locations are grouped within mega-environments; this information is essential for the recommendation of cultivars.The effects of years, and locations were significant (p < 0.001), i.e., both contributed significantly to the triple interaction.
The significance of the effects of locations, and years in the joint analysis (Table 2) provides arguments for these factors (years and locations) to be taken into account with equal importance in the estimation of the G×E in cowpea.Therefore, since much of this interaction is caused by unpredictable factors (such as precipitation, humidity, etc.), the use of mixed models should be prioritized to study the adaptability and stability in this case, because confirms the advantage of using the HMRPGV method in the recommendation of varieties, as well as in the formation of populations for breeding programs.

CONCLUSIONS
1.The genotypes MNC02-676F-3, MNC03-737F-5-1, MNC03-737F-5-9, BRS-Tumucumaque and BRS-Guariba are suitable for planting in the environments evaluated, since they have good stability combined to a superior performance compared to the other genotypes evaluated; 2. The use of HMRPGV statistics in the evaluation of grain yield variations in cowpea is advantageous compared to the selection by phenotypic mean; 3. The present study highlights the use of this methodology (HMRPGV) when there is a need for evaluations of G×E interaction, while considering a wide spatial and temporal variation, since in these cases, an efficient analysis with simplified interpretation of the results is necessary.

Figure 1 -
Figure 1 -Geographic distribution of the experimental stations used for evaluations of Value for Cultivation and Use of cowpea, between the years 2010 to 2012

Table 1 -
Locations and geographic descriptions of the experimental stations used in the Value for Cultivation and Use (VCU) tests during the period 2010-2012