Adaptability, Stability and Multivariate Selection by Mixed Models

The aim of this work was to estimate the adaptability 
and stability of grain yield per hectare and percentage of crude protein of 
maize grains combined in an index, and to establish a multicharacter selection 
through mixed models based on an objective character and 15 auxiliary traits. 
The trials were conducted in the 2013/2014 agricultural year in four growing 
environments of the Rio Grande do Sul, BR state. The experimental design was 
randomized blocks arranged in a factorial scheme, being four growing sites × 15 
single cross maize hybrids, arranged in three repetitions. The genotypic index, 
composed by the grain yield and the crude protein percentage in the grains, is 
the best selection strategy to achieve maize superior genotypes. The 
multivariate genotypes selection, considering grain yield and crude protein, is 
efficient. The genotypes FORMULA TL®, 
AS1656PRO®, P30F53Hx®, 
LG6304YG® and 30F53 are more adapted and stable 
for grain yield and percentage of crude protein, in the conditions of this 
study. The mixed models were efficient to employ the multicharacter selection 
and to contribute for maize genetic breeding.


Introduction
Maize (Zea mays L.) is one of the most produced cereals in the agribusiness scenario. Its importance is justified because of the wide utilization in animal nutrition, with 70% of the production in the form of silage or feed formulation, and even ethanol production in some countries such as USA [1]. The maize cultivation covers a wide range of growing environments. However, the genotypes may present differential behaviors as function of the environment modifications known as genotype environment interaction (G × E). The G × E interaction causes implications for breeding programs of any species, since the evaluation or recommendation of cultivars. Therefore, the study of this interaction is extremely important in order to find alternatives to minimize its effects, mainly by identifying genotypes highly responsive to environmental improvement, which are characterized by broad adaptability, predictable behavior and good stability [2].
Currently, breeding programs seek to identify high yielding genotypes, and posteriorly target their selection strategies in the quality of the grains, thus, the search for the ideal genotype that gathers productive and qualitative superiority demands elevated financial and labor resources of the breeding program, as well as suitable selecting strategies [3] [4] [5] [6]. An alternative to minimize this obstacle is the multivariate selection, which aims to select a set of simultaneous traits. In this way, the selection index proposed [7] [8] is used, which consists of a linear function of the predicted phenotypic or genotypic values of the characteristics pondered by estimated coefficients in order to maximize the correlation between the index and the true genetic values [9]. Therefore, genetic gain may be maximized when compared to direct selection, or selection individually performed for each trait [10]. The selection indexes have been successfully used in species of agronomic interest such as popcorn [11] [12], baby-corn and green corn [13]. However, there are few studies involving the selection of maize genotypes with high grain yield and protein content simultaneously.
Given the lack of information regarding multivariate selection in maize, this work aimed to estimate the adaptability and stability of grain yield per hectare and percentage of crude protein of maize grains combined in an index, and to establish a multicharacter selection through mixed models based on an objective character and 15 auxiliary traits.

Materials and Methods
The trials were conducted in the 2013/14 agricultural year, in four growing environments of the Rio Grande do Sul-BR state ( Table 1). The climate for all growing environments is classified by Köppen as Cfa subtropical [16]. The experimental design was randomized blocks arranged in a factorial scheme, being four growing environments × 15 single cross maize hybrids, arranged in three repetitions. The genotypes used were: 1) 2A106, 2) 30F53, 3 [14]. **Historical averages of temperature and precipitation [15]. 13) MAXIMUS VIP3®, 14) DEFENDER VIP® and 15) IMPACTP VIP3® ( Table   2). The experimental units were composed by four lines of five meters length, spaced 0.5 meters, totalizing 10 m 2 [17]. No-tillage system was used for all growing environments, with population of 80 thousand plants per hectare. It was used 300 kg•ha −1 of NPK in the formula (10-20-20) as base fertilization, and 135 kg•ha −1 of N in the amidic form as topdressing, applied at V 4 and V 6 vegetative stages. The management of weeds, pest and diseases were carried out preventively, in order to reduce interferences in the experiment's results.
The traits of interest were measured in the useful area of each experimental unit, which was composed by two central lines, discarding 0.5 m of each edge. The measured traits were: spike diameter (SD), results in millimeters (mm); spike length (SL), results in centimeters (cm); spike mass (SM), results in grams (g); cob diameter (CD), results in millimeters (mm); cob mass (CM), results in grams (g); spike insertion height (SH), results in meters; number of rows with grains in the spike (NRG), results in units; plan height (PH), results in meters (m); number of grains per row in the spike (NGR), results in units; prolificity (PRO), results in units; mass of a thousand grains (MTG), results in grams (g); grain yield (GY), results in kg•ha −1 [5] [17]; percentage of crude protein (CP) and mineral material (MM) in the grains [18].
The phenotypic index (PI) was generated by the product of grain yield per hectare and the percentage of crude protein of each genotype's grains [19].
where: PI = phenotypic index combining grain yield per hectare and percentage of crude protein in the grains; GY = grain yield per hectare; CP: percentage of crude protein in the grains; GY S = standard deviation of grain yield; CP S : standard deviation of crude protein. Equal relative economic weights were attributed to both traits (GY and CP), i.e., this phenotypic index was taken as objective character.  where: y, b, g, ge, and e are the data vectors. The model fixed effects are given by the average of the blocks through the sites, aleatory genotypic effects, aleatory G × E interaction effects, respectively. X, Z and W are matrices of incidence for b, g and ge, respectively [20]. The joint selection by PI, and the genotype's stability and adaptability were based on the statistic called harmonic mean of the relative performance of predicted genotypic values (HMRPGV) [20]. In this model, the interaction free predicted genotypic values consider all growing environments, are given by u + g, where u refers to the average of all environments. The predicted values for each trait in the univariate form were used in the genotypic selection index exemplified below. In addition, the genotypic correlation was obtained between the analyzed traits to elaborate the selection index. All the analyses were performed through Selegen software (Reml/Blup) [21]. The predicted genotypic values were used for estimating the pair to pair joint correlation between growing environments. The predicted genetic values for each trait from the univariate analysis may be used to compose the selection indexes considering one objective character and the others as auxiliaries [22], being PI (GY × CP) the objective trait, and the other 15 traits, GY, CP, CD, NRG, MTG, CM, PH, SD, SL, SH, NGR, PRO, SM, SGM and MM, considered auxiliaries, a selection index may be derived using this 16 information simultaneously: where o g is the standardized genotypic value of the objective character, and ai g is the standardized genotypic values of the auxiliary traits. The index's weighting coefficients ( i b ) are given by [22]: a a a a a g g g g ga g r r r r r r r r r r r r r r r r r P r r r r Sim r Vector of genetic covariance between the predicted genetic value of the objec- Thus, the variance of the index is given by: Consequently, the accuracy of GI is given by the root of reliability.

Results and Discussion
The Deviance analysis revealed significance at 5% of probability by the chi-square The genetic parameters estimated for the traits of interest (Table 3)    cates that the G × E interaction for these traits expressed simple effects, in other words, although there was differentiated behavior, the genotypes classification was not substantially altered in function of the different tested environments [30]. The coefficient of genotypic variation (CVgi) ranged from 2.15% to 12.91%, indicating the presence of genetic variation for the evaluated traits. Researches define that the higher magnitude of coefficient of genotypic variation allows genetic gains in the genotypes selection [20]. Regarding the coefficient of experimental variation (CVe), low magnitudes were observed, which reflects the suitable experimental conditions and reliable estimates. The coefficient of relative variation (CVr) ranged from 0.09 (SGM) to 1.96 (GY), with higher contribution of the genotypic value for the trait's total variation, indicating they may be less influenced by environment effects [31].
The genetic correlations for growing environments obtained pair to pair, and referent to the PI objective character, were all low [29], revealing elevated dissimilarity among environments and indicating the absence of breeding zones, therefore, the selection strategies must be exclusively proceeded in each one (Table 4). Studies [32] with maize open pollinated varieties grown in 15 environments in the Goiás state-BR, evidenced formation of two groups of stable environments over the agricultural years studied, and a reduction of 16% of the environments currently used. Research [33] stratified the environments regarding maize lodging and breaking, thus, when considering these traits, the experimental net can be reduced because the genotypes do not present differential responses as function of environmental variations. Besides the best genotypes recommendation through the interaction free genotypic values (u + g), a general recommendation for all environments of the experimental net can be realized by the capitalization of the mean interaction (u + g + gem) among environments (Table 5). This ordering is greatly relevant for plant breeding because it considers the mean genotypes performance in the experimental net environments. The gains with selection through u + g + gem were superior to gains achieved through u + g (Table 5) due to the average performance increment of each genotype in the four environments. Therefore, the use of mixed models methodology and the REML/BLUP procedure allows to access important effects to guide genetic selection by the breeder. Table 5. Ordering of maize hybrids through genotypic values free from genotypes × environments interaction effects (u + g), genotypic values plus one mean effect of interaction (u + g + gem) and predicted gains for the objective character or phenotype index (PI), in the joint analysis among environments. By comparing the ordering for PI through the predicted genotypic value (u + g), genotypic value plus the mean interaction (u + g + gem), stability (HMGV), adaptability (RPGV) and stability, adaptability and grain yield simultaneously (HMRPGV*GY) ( Table 6) The index (GI) with an objective character (PI) and 15 auxiliary traits was elaborated according to methodology of global optimization and multivariate BLUP initially derived by Viana and Resende [22], for utilization with three characters. In this study, the approach was expanded for genotypes selection using 16 characters, being a pioneering work in this sense ( Table 7). The GI is Gain related to the overall mean through HMRPGV*GY (best hybrid): 62% Note: the underlined hybrids are the best five according to the ordering of mean genotypic effects (u + g + gem) in the selection among environments, also present in the selection ordering for stability, adaptability and stability and adaptability (15 in 15, 100%). American Journal of Plant Sciences Coincidence (five best genotypes) between PI and GI: 80% + GY: grain yield per hectare (kg•ha −1 ); CP: percentage of crude protein in the grains (%); CD: cob diameter (mm); NRG: number of rows with grains in the spike (unit); MTG: mass of a thousand grains (g); CM: cob mass (g); PH: plant height (cm); SD: spike diameter (cm); SL: spike length (cm); SH: spike insertion height (cm); NGR: number of grains per row in the spike (unit); PRO: prolificity (unit); SM: spike mass (g); SGM: spike grains mass (g) and MM: mineral material of the grains (%).
composed by the PI objective character which combines grain yield per hectare and percentage of crude protein in the grains, jointly to the 15 auxiliary traits optimally weighted by their accuracies, heritabilities and genetic correlations.
All these factors are adequately considered in the weighting coefficients (Table 7), which will be higher as higher the correlations of auxiliary traits with the objective character are [22]. The GI selective accuracy was 0.63, being 210% higher than the PI objective character individually considered (accuracy of 0.30).
Selective accuracy refers to the correlation between true genotypic value and predicted value through experimental information [29]. This parameter's utilization is considered ideal for choosing the best selection method, mainly because the genetic gain is directly proportional to the accuracy, i.e., as higher the accuracy is, better is the precision of selection [34].
It was verified a change of position between the genotypes selected by PI and A. J. de Pelegrin et al.
GI, with coincidence of 80% among the five best maize hybrids. Therefore, the ordering generated by GI should be used for the final recommendation of the genotypes, since it is a more accurate index than the PI, as it aggregates information of the auxiliary traits, their genotypic correlations with the objective character, genotypic values and selection reliability. In addition to accuracy increment, the GI character provided higher genetic gains than PI, where the use of GI increased genetic gain by 2.33% due to the selection of the best genotype, and 1.72% by the selection of the three best ones. In genetic breeding programs, there is an imminent difficulty for selecting superior genotypes of traits with low genetic control, due to the great effect that the environment exerts on the genotype's phenotypic variation. Therefore, the use of auxiliary traits becomes a viable practice to improve the selecting process efficiency of superior maize genotypes.

Conclusions
1) The genotypic index, composed by the grain yield and the crude protein percentage in the grains, is the best selection strategy to achieve maize superior genotypes.
2) The multivariate genotypes selection, considering grain yield and crude protein, is efficient.
3) The genotypes FORMULA TL®, AS1656PRO®, P30F53Hx®, LG6304YG® and 30F53 are more adapted and stable for grain yield and percentage of crude protein, in the conditions of this study.
4) The mixed models were efficient to employ the multicharacter selection and to contribute for maize genetic breeding.