The evaluation of cultivars using multi-environment trials (MET) is an important step in plant breeding programs. One of the objectives of these evaluations is to understand the genotype by environment interaction (GEI). A method of determining the effect of GEI on the performance of cultivars is based on studies of adaptability and stability. Initial studies were based on linear regression; however, these methodologies have limitations, mainly in trials with genetic or statistical unbalanced, heterogeneity of residual variances, and genetic covariance. An alternative would be the use of random regression models (RRM), in which the behavior of the genotypes is characterized as a reaction norm using longitudinal data or repeated measurements and information regarding a covariance function. The objective of this work was the application of RRM in the study of the behavior of common bean cultivars using a MET, based on Legendre polynomials and genotype-ideotype distances. We used a set of 13 trials, which were classified as unfavorable or favorable environments. The results revealed that RRM enables the prediction of the genotypic values of cultivars in environments where they were not evaluated with high accuracy values, thereby circumventing the unbalanced of the experiments. From these values, it was possible to measure the genotypic adaptability according to ideotypes, according to their reaction norms. In addition, the stability of the cultivars can be interpreted as variation in the behavior of the ideotype. The use of ideotypes based on real data allowed a better comparison of the performance of cultivars across environments. The use of RRM in plant breeding is a good alternative to understand the behavior of cultivars in a MET, especially when we want to quantify the adaptability and stability of genotypes.
The performance of inbred lines in advanced endogamous generation is commonly evaluated in successive generations of testing and selection, which we defined as "multistage field trials" (MSFT). MSFT data routinely exhibit heterogeneity of (co)variances at several levels due to genetic and/or statistical imbalance. Nowadays, mixed models have been widely used to deal with unbalanced data. However, few studies on common bean have addressed the use of a mixed model approach with modeling of (co)variance structures for random effects in MSFT. Furthermore, factor analysis and genotype-ideotype distance (FAI-BLUP) selection index was originally proposed using best linear unbiased predictions from individual analysis. In this regard, we aimed to study the implications of modeling (co)variance structures for random effects in the estimation of genetic parameters and evaluate the accuracy and efficiency of inbred line selection by the modeled FAI-BLUP approach. A total of five trials were evaluated from 2018 to 2020. The results revealed that the unstructured covariance matrix fitted better for grain yield, whereas the matrix with uniform correlation and heterogeneity of variances fitted better for grain aspect and plant architecture. The modeled FAI-BLUP approach increased the values of selection accuracy and selection efficiency. Our results suggest that modeling the different structures of (co)variances and selecting the best-performing genotypes by modeled FAI-BLUP approach should be used in common bean assays involving unbalanced data.
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