Restricted maximum likelihood / best linear unbiased prediction ( REML / BLUP ) for analyzing the agronomic performance of corn

1 Department of Mathematics and Statistics, Federal University of Pelotas, Capão do Leão, Rio Grande do Sul, Brazil. 2 Plant Genomics and Breeding Center, Federal University of Pelotas, Capão do Leão, Rio Grande do Sul, Brazil. 3 Department of Agronomic and Environmental Sciences, Federal University of Santa Maria Frederico Westphalen, Rio Grande do Sul, Brazil. 4 Agronomy Department, Federal University of Santa Maria, Santa Maria, Rio Grande do Sul, Brazil. 5 Department of Crop Science, Federal University of Pelotas, Capão do Leão, Rio Grande do Sul, Brazil. 6 Breeder Company Plants KSP Seeds Ltda, Pato Branco, Brazil. 7 Federal University of Pampa, Dom Pedrito, Rio Grande do Sul, Brazil.

evaluation in preliminary trials aiming at increasing efficiency and accuracy in selection of lines' combinations are very important for breeding programs achieve their goal.
The need for robust and accurate models in evaluating complex experiments (multi-sites trials, multi-year trials), is making the mixed models-based on restricted maximum likelihood/best linear unbiased prediction (REML/BLUP) more and more popular in plant breeding programs (Resende, 2004) due in many cases, the effect of local/year considered random, where the inferences are general (Piepho et al., 2007).REML/BLUP-based procedures are scarcely penalized by the non-additive nature of traits and by the experimental unbalance as compared to ANOVA-based models (Hu, 2015).In maize, these methods were effective in assessing the performance of hybrids and in accurate prediction of the variance components and breeding values (Nardino et al., 2016;Baretta et al., 2016).
In annual or biennials plants, the use of mixed models is still limited, although growing, as reported in works by Borges et al. (2009).In Brazil, it has been used in several crops such as sugarcane (Bastos et al., 2007;Lopes et al., 2014), soybeans (De Melo Pinheiro et al., 2013), beans (Carbonell et al., 2007) oat (Coimbra et al., 2005) and in potato (Souza et al., 2005).
Adoption of accurate and simplistic biometric models that estimate variance components and predict the genotypic effects are desired in plant breeding, largely due to effects of genotype x environment interaction.In this context, the aim of this research was to estimate the variance components and predict genotypic values by REML/BLUP of maize inbred lines' combinations from two heterotic groups, crossed in partial diallel scheme with F 1 's evaluation in four locations.

MATERIALS AND METHODS
The tests were based on artificial crosses among inbred lines, from company's research station KSP Seeds Ltda., in the city of Clevelândia-PR, Brazil in 2010-2011 growing season.The crosses were carried out in partial diallel design resulting in 25 F1's hybrids, which were evaluated at different locations.The hybrid seeds were hand harvested, dried and prepared for the trials' sowing in four environments.
Sowing of trials was carried out in 2011-2012 growing season in four different locations of Santa Catarina and Paraná States, in southern part of Brazil (Table 1).In previous sowing, each experimental area was analyzed in order to identify potentially disruptive features.A randomized complete block design with three replications was used.Blocks were allocated in order to maintain homogeneity within the block and heterogeneity between the blocks.Experimental units were composed of two 5-m-long cultivar rows, spaced by 0.7 m, adjusting the stand to 42 plants per experimental unit, equivalent to 60,000 plants ha -1 .
The following traits were assessed: plot grain yield (GY, kg per plot), ear height (EH in centimeters) distance from the last node of the stem to the first branch of the tassel (DLN in centimeters), number of tassel branch (NTB, units) and thousand-kernel weight (TKW in grams), measured by counting eight replicates of 100 Nardino et al. 4865 kernels.
Estimates of genetic parameters were carried out based on the restricted maximum likelihood and the best linear unbiased predictor (REML/BLUP).The data set was subjected to analyses in order to estimates genetic parameters based on the procedures of the restricted maximum likelihood and the best linear unbiased predictor (REML/BLUP).The softwares used in the analyses were Selegen (Resende, 2007b) and SigmaPlot v.11.

RESULTS AND DISCUSSION
Components of variance estimated by REML shows that for the DLN trait, genetic variance (σ² G ) represents 9.76% of total phenotypic variance (σ² F ), and the variance of interaction (σ² GxE ) 16.71%, being the great part of total variation coming from environment (σ² E ) 73.53%.Components of variance for the TKW trait show 11.40% for σ² G , 19.70% for σ² GxE and 68.90% for σ² E .Total variance for EH was explained by 12.61% σ² G , 20.43% σ² E and the great contribution was from σ² GxE (66.95%).Total variance for GY was decomposed into 65.52 % from σ² GXE , 10.34% from σ² G and 24.14% from σ² E , indicating that much of the variation is due to effects of genotype × environment interaction.Variance components for NTB trait are great for σ² E (46.44%), for σ² G (39.31%) and for σ² GXE (14.25%).Variance component of GxE interaction is superior to the other variances, except phenotypic, for EH (66.95%) and GY (65.52%).For other traits, the environmental variance represents the largest fraction of the phenotypic variation; however, it can be considered that there is genetic variation among hybrids, especially for NTB trait (Figure 1).
The accuracy values show the precision of inferences of the average of genotype (Table 2).Accuracy values for the traits were classified as moderate too high (Resende and Duarte, 2007).According to Candido et al. (2009), obtaining a successful breeding program depends, among other on parameters for conducting trials with efficient analysis methodologies.
The magnitudes of σ² GXE were lower as compared to σ² G and H 2 G only for NTB trait, resulting in a genotypic correlation between the environments (r gloc ) of 73%.These results indicate the tendency of genotypes' stability in different locations, which is too upper and lower for the trait.For the other traits; however, the magnitude of σ² GXE was over σ² G and high as compared to h² G , resulting in lower values of r gloc for DLN (rg loc = 0.37), TKW (rg loc = 0.37), EH (rg loc = 0.16) and GY (rg loc = 0.15).
Genetic correlations between environments for these last traits show that hybrids do not have the same behavior in different environments, occurring variations in the ordering of genotypes.GxE interaction contains a part related to the difference between the genetic variance of trait in different environments and over the lack of association between genetic treatments in one environment to another.The first is called simple   Table 2. Variance components (VC) estimated by restricted maximum likelihood (REML) for the traits distance from the last node to the first branch of the tassel (DLN), number of tassel branch (NTB, units), thousand-kernel weight (TKW), ear height (EH) and grain yield (GY).  interaction and the second complex or cross interaction (Resende, 2007a).The presence of complex interaction always indicate the existence of specifically adapted cultivars to specific environments, which require the adoption by cautious measures for recommending cultivars (Ramalho et al., 2012).

VC DLN(cm) NTB (units) TKW (g) EH (cm) GY (kg
The predicted estimates of crossings BLUP, resulting in a hybrid with smaller distance from the last node to the first branch of the tassel, are sought, because the greater length of the tassel have negative effects on grain yield (Table 3).This is due to increased demand for assimilates and low performance of hybrids in biotic and abiotic stress conditions (Sangoi et al., 2002b); thus, crosses with average genotypic of lesser magnitude would be desired for selection.The combinations between lines 13x3', 7x4 ', 13x4', 1x1', 15x3', 5x4', 1x4', 15x2', 15x5', 10x5', 9x4', 15x8' and 12x3' have revealed predictions below genotypic overall average (Table 3), standing out among the other combinations.The main contributing factor for selecting smaller size tassel in breeding programs, is because the tassel mass is directly related to the size, negatively influencing the production of grain due to competition with the ear by nutrients.Because of this, larger tassels would be greater-power sink, reducing the availability of assimilates to the ear (Hallauer et al., 2010).
Regarding the predicted components for NTB trait, averages of crossings with lower magnitude are prioritized, because smaller tassels require less assimilate and nutrients.This minimizes protandric nature of the plant, reducing the ontogenic interval between male and female flowering (Sangoi et al., 2002a).The power demand and lower apical dominance of the tassel over the spikes are important characteristics for nutrients to be used for balanced allometric development of inflorescences of the plant.This is reflected in a greater number of ears per plant and best floral synchrony of modern hybrids, especially in high densities.In this context, the predicted genotypic averages of crosses between the hybrids 3x1', 1x1 ', 8x4', 13x4', 9x4', 10x5', 1x4', 13x3', 15x5', 15x8', 5x4' and 15x2' are lesser than other crosses (Table 3).
Regarding the predicted components for EH trait, the crosses combination of 15x2 ', 11x3', 8x4', 4x7', 3x4', 3x1', 9x4', 14x4', 6x4', 5x4' and 4x6' can be highlighted due its average, which is lesser than overall average (Table 3).Such criteria are established due to the maize genetic breeders seeking lines with lower height of ear insertion, being among the major changes sought with the lowest height on the cob, the higher nitrogen use efficiency, allow the center of gravity of plant to stay more balanced, reducing lodging and stem's breakage and favoring the absorption and translocation of nutrients to the grain filling (Sangoi et al., 2002b).
Combination with genotypic averages for GY that can be highlighted among predicted estimates are respectively: 1x3 ', 1x4', 5x4', 15x3', 12x3', 13x3', 15x2', 10x5' and 11x3' (Table 3).These crosses have greater GY than the overall genotypic average of the trials, proving to be promising to produce hybrids with increased GY in the four environments.In this context, the identification of promising crosses for a greater number of environments becomes important, since there are components of genetic, environmental and GxE origin involved in phenotypic expression, which can make difficult and complicate the breeders' work.Ramalho et al. (2012) points out that not always do hybrids with broader adaptation have higher grain yield.This fact can prevent a generalized way recommendation, being the most widely used alternative to solve this problem, to identify hybrids with greater phenotypic stability for different environments and to make a stratification of environments aiming to discover locations' groups within which the variance component associated with the effect of interaction is minimized.
Individual BLUPs in each location allow analysis of each combination in a more thorough form, informing the oscillations of the hybrid and selecting the best combination for a specific micro-region (environment).In this sense, using REML/BLUP-based procedures has numerous advantages and provides more security to the breeder.Resende and Duarte (2007) points that for selection, genotypic averages will represent better the future average of genotypes than phenotypic averages.According to the same authors, even which the assessments are in the same place or region, the effects of blocks and plots are unlikely to be repeated, thus, if the tests are conducted in different regions, the effect on the average will be higher.
Regarding the predicted estimate and their confidence intervals for the 25 combinations for the DLN trait (Figure 2), considering location's average as discriminatory parameter to identify the best combinations, the intersections whose average are lower than the overall average were highlighted, due to the increased demand for energy directed to the growth and development of the tassel, providing competition with the ear (Sangoi et al., 2002a).In Ampére, it was highlighted, the combinations of 1x4', 10x5', 12x3', 13x4' and 15x2' with the smallest magnitudes in relation to overall average (21.73 cm).In Clevelândia, the combinations that can be highlighted are 1x1', 9x4' and 15x8', considering the predicted overall average (22.98 cm).For Itapiranga, the lowest magnitudes for the overall average (21.5) are the crosses, 1x4', 8,4', 9x4', 12x3' and 15x8'.In Pato Branco, the combinations 12x3', 15x2' and 15x8' were lower than overall average (21.44 cm).Based on obtained estimates, it can be considered that none combinations were the same for the four environments.
Regarding MMG trait (Figure 4), the crosses that were higher than the local average in Ampere (302 g) are, respectively, 1x4 ', 5x3' 13x3 '15x3', 15x5' and 15x2' in Ampere environment.The higher predicted estimates in Cleveland are 1x4', 4x7' and 13x3' whose predicted average was 378.79 g, showing that there was a high homogeneity of the crossings for this site.In Itapiranga, the average predicted was 312.63 g, and the superior combinations for this estimate were the combinations 1x4 ',4x7',6x3',10x5',12x3' and 15x5'. The promising combinations in Pato Branco are 1x4',3x1',5x3',10x5',13x3',15x5' and 15x2' above average of 344.97 g.Combinations with superior magnitudes for this trait are desirable in maize breeding, because this trait presents positive association with grain yield.
The GY (Figure 6), target trait of the breeders who seek incessantly cultivars or increasingly productive hybrids, is among the traits more influenced by environmental conditions, due to its complex action in response to different stimuli, resulting in a number of complications to identify more specific hybrids.Under the conditions of Ampere, combinations of superior crosses to average of 7.31 kg plot -1 were from combinations 3x4', 5x3', 9x4', 11x3' and 15x8'.Cleveland revealed between the environments, the highest average (9.96 kg plot -1 ), with the combinations 1x1 ', 1x4', 4x7', 5x3', 7x4', 11x3', 12x3', 13x3' and 15x3' located above this.The crosses that stood out for Itapiranga were 1x4 ', 3x4', 4x3', 4x7', 5x3', 5x4', 13x3', 15x3' and 15x2', with  .For Pato Branco combinations 1x4 ', 5x3', 5x4', 6x3', 10x5', 15x3' and 15x2' were, respectively, they were higher than the average of 8.57 kg plot -1 . The combination of lines 15x3' is higher than the average of the tests in all the environments.The combinations that also showed themselves promising are between 5x4' lines in Itapiranga and Pato Branco environments and 5x3' that reveals superiority in the overall average of all combinations (Table 3), which is promising mainly for Clevelândia and Ampere.The fluctuations revealed by other diallel combinations are in agreement, because of the high effects shown in the variance component GxE.Grain yield is a complex and polygenic trait (Hallauer et al., 2010) controlled by a large number of genes.High effects of GxE interaction component is expected to occur, making it important biometric analysis with accurate models for estimating the components and predicting genotypic values, being REML/BLUP, favorable procedures in estimating these parameters (Resende, 2007a).

Conclusions
Estimates of variance component has revealed that the ear height and grain yield per plot suffer sharp action of GxE interaction.
The GXE interaction has provided oscillations regarding the best combinations in the environments and must be individually analyzed in each environment, the superior crosses for grain yield, except for 15x3' combination which achieved good performance in all locations.
The combination 5x4' is promising to get high grain yields in Itapiranga and Pato Branco.The combination 5x3' has a higher overall average, which is higher than the other combinations in the environments of Clevelândia and Ampere and provides hybrid with fewer branches in the tassel.

+
The climate classification is according toAlvarez et al. (2014).*Average air temperature for the two climate classifications presented in the trial areas.Thot = Average air temperature in the hottest month of the year, Tcold = Average air temperature in the coldest month of the year.

Figure 1 .
Figure 1.Partitioning of phenotypic variance into genetic, environment and interaction effects.

Figure 2 .
Figure 2. Estimates of genotypic averages for the trait distance from the last node to the first branch of the tassel in Ampére, Clevelândia, Itapiranga and Pato Branco regarding 25 combinations of a partial diallel.

Figure 3 .
Figure 3. Estimates of genotypic averages for the trait number of tassel branch in Ampére, Clevelândia, Itapiranga and Pato Branco regarding 25 combinations of a partial diallel.

Figure 4 .
Figure 4. Estimates of genotypic averages for the trait thousand-kernel weight in Ampére, Clevelândia, Itapiranga and Pato Branco regarding 25 combinations of a partial diallel.

Figure 5 .
Figure 5. Estimates of genotypic averages for the trait ear weight in Ampére, Clevelândia, Itapiranga and Pato Branco regarding 25 combinations of a partial diallel.

Figure 6 .
Figure 6.Estimates of genotypic averages for the grain yield per plot in Ampére, Clevelândia, Itapiranga and Pato Branco regarding 25 combinations of a partial diallel.

Table 1 .
Location and climate characteristics of experimental areas.

Table 3 .
Estimates of average components by best linear unbiased predictor (BLUP) for the traits distance from the last node to the first branch of the tassel (DLN), number of tassel branch (NTB, units), thousandkernel weight (TKW), ear height (EH) and grain yield (GY).
*Classification in descending order by the predicted average of the cross's combinations of partial diallel; **Component of genotypic average without GxE interaction for 25 combinations of a partial diallel; ***Combination related to the crosses in partial diallel scheme of 25 hybrids obtained.
their average greater than 8.05 kg plot -1