Introduction

he bread wheat (Triticum aestivum L.) crops for drought conditions are an alternative for reducing the present deficit of Mexico, given that there are approximately one million non-irrigated hectares in which wheat cultivation seems to be an adequate option (^{Villaseñor y Espitia, 2000}). However, wheat producers of rainfed conditions face problems for marketing their harvests, principally because the milling industry prefers to import wheat from international markets, with the argument that the Mexican varieties do not have an adequate quality for industry (^{Hortelano et al., 2013}).

Knowledge of the genetic and environmental influence on the variation of industrial quality in bread wheat is fundamental for the production of high quality and stable harvests which favor its commercialization. The evaluation of quality in a high number of different environments is complicated, thus breeders have limited information of the variation of quality attributable to both genotype and environmental conditions (^{Espitia et al., 2003}).

Genotype has an effect on the industrialization quality of wheat (^{Sánchez-García et al., 2015}), but environmental conditions have a higher influence on quality (^{Vázquez et al., 2012}; ^{Mohan et al., 2014}; ^{Rozbicki et al., 2015}). ^{Espitia et al. (2003)} indicated that in the wheat production conditions in Mexico and with local varieties bread volume is controlled by environmental effects in 74 %; although this will be determined by the variability among genotypes and among environments. The environmental conditions under which wheat is cultivated are the most important factor for the expression of its quality (^{Espitia et al., 2003}).

The availability of N affects the protein content, which in turn affects bread volume. Temperature during grain fill (^{Li et al., 2013}), the amount and distribution of rainfall (^{Aslani et al., 2013}) and the presence of frosts (^{Martínez et al., 2015}) are also environmental factors which significantly affect industrial quality (^{Mohan et al., 2014}). The stability of quality, specifically bread volume, is a character of great importance for wheat breeding, but there are few studies (^{Hristov et al., 2010}).

In the evaluation of genotypes in different environmental conditions there are frequently unbalanced sets of data due to diverse causes, such as lost data, variable number of replicates among environments, some genotype-environment combinations are not tested, and according to the type of experiments, there may or may not be replicates per environment. Therefore, the analysis of these experiments is complicated and demands robust computational resources. The methods for the analysis of stability depend on the availability of computational programs to manipulate unbalanced data. Many of the analyses of stability fit in the mixed model, where the environments are random effects and the genotypes are fixed. The parameters of the model can be estimated with the option of restricted maximum likelihood (REML), given that it permits the analysis of unbalanced data; besides, most of the measurements of stability can be estimated with the MIXED procedure of SAS (^{SAS Institute Inc., 2005}).

There is a considerable amount of information about methodologies for the estimation of stability in an ample amount of random values (^{Lin et al., 1986}; ^{Becker and Leon, 1988}; ^{Denis et al., 1997}; ^{Piepho, 1998}). The most utilized models are stability of variance (^{Shukla, 1972}), regression models (^{Finlay and Wilkinson, 1963}; ^{Eberhart and Russell, 1966}) and environmental variance (^{Lin et al., 1986}).

The study of stability was limited by the availability of computational resources. But there are recent programs in statistical packages, such as SAS, for fast analysis of stability in unbalanced and balanced data.

The objective of the this study was to determine the stability of bread volume of wheat varieties for rainfed conditions from a set of unbalanced data, and to determine whether the MIXED procedure of SAS can be a tool in the selection of stable genotypes.

Materials and Methods

Results and Discussion

Bread volume of the varieties and environments in all of the models tested showed significant differences (p ≤0.01), according to the goodness of fit tests of the tested models (Table 1). The highest value of the criterion of information of Akaike corresponded to the model of the variance of stability of ^{Shukla (1972)}; thus according to ^{Wolfinger (1996)}, the stability of the varieties should be determined with this model. Under the hypothesis that the reduced model is different from the complete model, the simplification of these models in relation to the complete model of the environmental variance is supported by the data of the present study. Following this criterion the model of the variance of stability is favored; therefore, it should be used to determine the stability of the varieties of the study.

Table 1 Goodness of fit test for bread volume of nine varieties of bread wheat in 22 rainfed environments in Mexico 

p: number of parameters in the variance-covariance structure, AIC: information criterion of Akaike, LL_R: parameter of restricted maximum likelihood.

The relevant parameters for the evaluated models (Table 2) indicated that the variety Juchi F2000 was the most stable according to the model of variance of stability, corresponding the lowest value of σ²_(i) , the lowest value of environmental variance (σ _ii), the lowest absolute value of the regression coefficient (σ_i) according to the analyses of ^{Finlay and Wilkinson (1963)} and ^{Eberhart and Russell (1966)} and the lowest variance of the deviations) σ² _(ii)) according to the model of ^{Eberhart y Russell (1966)}. Thus, all of the models indicated that Juchi F2000 was the most stable variety, followed by Gálvez M87 and Náhuatl F2000. Bread volume is closely related to the amount and quality of proteins. These varieties are characterized by their high protein content (127 to 139 g kg^-1) (^{Espitia et al., 2003}); besides they present alleles of glutenins of high molecular weight 17+18 and 5+10 in Glu-B1 and Glu-D1, related to good quality (^{Kaya and Akcura, 2014}) and will be able to better cushion the environment effect (^{Espitia et al., 1999}). Therefore, these are stable varieties for bread volume. The least stable varieties were Pavón F76, Temporalera M87 and Batán F96, as they showed the variances with highest values (Table 2), although they also showed the allele 5+10 in Glu-D1. This could be due to differences in alleles of glutenins with low molecular weight (^{Martinez-Cruz et al., 2010}), or to the influence of other factors that will reduce quality, such as drought and diseases (^{Hortelano et al., 2013}). The standard errors of the estimated parameters seem high in all of the varieties, which could indicate that 22 environments is an inadequate number to reliably estimate stability; this agrees with that reported by ^{Piepho (1999)}.

Table 2 Estimated parameters (restricted maximum likelihood, REML) with standard errors (SE) for the variance-covariance structure of stability models for the bread volume of bread wheat 

A variety should be stable and produce bread with high mean of bread volume, thus this variable is key in selection. The means of bread volume with standard errors (Table 3) estimated by the models showed that the estimated means and the standard errors are stable in all of the models. According to the model of the variance of stability, Náhuatl F2000, Juchi F2000 and Gálvez M87 showed the highest averages of bread volume, followed by Pavón F76, Tlaxcala F2000 and Temporalera M87. This is related to the high protein content (^{Espitia et al., 2003}) and its quality (Table 4); as Náhuatl F2000, Juchi F2000 and Gálvez M87 present the allele 5+10 of glutenins of high molecular weight in Glu-D1, known to confer strength to the dough and higher bread volume (^{Espitia et al., 2008}; ^{Malik et al., 2013}; ^{Martínez et al., 2015}).

Table 3 Estimated measurements of bread volume and standard errors for wheat varieties. Restricted maximum likelihood REML with standard errors for different models. 

Table 4 Quality and content of protein of nine varieties of bread wheat in 22 rainfed environments in Mexico 

DMSH, honest minimum significant difference.

In addition to Juchi F2000, the varieties Náhuatl F2000 and Gálvez M87 showed good stability and high averages of bread volume; Tlaxcala F2000 presented high stability and intermediate bread volume. Furthermore, Rebeca F2000 and Temporalera M7 were identified with intermediate values of stability and bread volume. Rebeca F2000, Romoga F96 and Batán F96 presented low stability and bread volumes. The most adequate combination in a variety would be its stable bread-making quality, without response to the variations of the environment, and that would produce bread with high volume. According to the estimations of environmental variances and the average values estimated with the variance model of stability (^{Shukla, 1972}), the varieties that best adhere to what was previously indicated are Náhuatl F2000, Juchi F2000 and Gálvez M87; followed by Pavón F76 and Tlaxcala F2000. This indicates that the varieties of the series 2000 have good bread-making quality; furthermore, three of them have stability and generate bread with volume comparable to Gálvez M87, which is considered in the industry as a reference variety of bread-making quality (^{Espitia et al., 2003}).

The combination of stability and bread volume, through comparisons of the estimated average values by the model of the variance of stability (Table 5), showed that the varieties Náhuatl F2000, Juchi F2000 and Gálvez M87 are the most desirable for their higher stability and bread volume. The order of stability concurred in all of the varieties; that is, the higher the bread volume, the higher the stability. Thus, to cushion the effects of environment restrictive for bread volume, it would be convenient to use the genotypes of very good quality, which when cultivated in the restrictive environments maintain acceptable bread volume as indicated by ^{Espitia et al. (2003)}.

Table 5 Desirability of genotypes combining stability and bread volume 

Average values with different letters are statistically different.

†The order of the genotypes indicates desirability.

The test environments formed three groups (Figure 1). Most of them (12) in quadrant II, with higher stability and bread volume. Quadrant III contains six environments that generated higher stability and low bread volume. Quadrant IV contains four environments, which showed low stability and bread volume. None of the environments was located in quadrant I; this represents an area of low stability and high bread volume. For the evaluation of lines some of the locations in quadrant II could be dispensed with, since they represent good stability and high bread volume. This would reduce costs, as this condition would be represented with half of the environments.

Figure 1 Dispersal of the environments dependent on the environmental variance and bread volume. 

With this program, stability is analyzed easily and rapidly, and allows the analysis with the averages genotype-environment (experiments without repli-cates per environment) and the computational resources are reduced. In addition, with the mixed models the analysis of unbalanced experiments is possible. This type is presented frequently for diverse reasons, such as lost data of plots, different number of replicates among experiments or lack of proof of some genotype-environment combinations. By placing the parameters of stability in terms of variances it is possible to compare between methods; furthermore, by assigning a numerical value of stability to each genotype, they can be compared, categorized and selected. In the breeding programs this methodology can be used to select lines by their stability, and by having a measurement of stability, the genotypes can be ordered from high to low.

Conclusions

The varieties Náhuatl F2000, Juchi F2000 and Gálvez M87 generate bread with higher and more stable volume when they are cultivated in most of the test environments; thus, they are the most desirable for bread-making.

The stability model of the stability variance shows the best fit to bread volume. The mixed models, because of their ease of handling, speed of analysis and lower demand of computational resources, are a tool of great usefulness in the determination of stability in diverse environments and in the selection of stable genotypes