Individual and combined analyses of variance
Significant genotype × experiment interaction was observed for most of the traits evaluated in one or more combined analysis of variance (Table 1). Therefore, the agronomic performance of the common bean genotypes was not constant in the different years and growing seasons. Similarly, significant genotype × experiment (environment) interaction has been described for most of the plant architecture and grain yield traits evaluated in common bean genotypes (Moura et al. 2013; Boros et al. 2014; Maziero et al. 2015; Soltani et al. 2016; Delfini et al. 2017; Ribeiro et al. 2018; Nadeem et al. 2020). When a significant genotype × experiment interaction is observed for several agronomic traits in common bean, it is not recommended that cluster analysis be performed based on data obtained in a single experiment, because this strategy does not consider the environmental variability between years and growing seasons for the same site of conduction of the experiments (Cargnelutti Filho et al. 2009).
There was a significant genotype effect for most of the evaluated traits in individual and combined experiments. Therefore, there is genetic variability for the plant architecture and grain yield traits in common bean genotypes and this allowed the study of genetic divergence. However, no significant genotype × experiment interaction and genotype effects were observed to the first-internode length, fourth-internode length and fifth-internode length in the experiments combined (I and II; I, II and III; and I, II, III and IV). Therefore, these traits were not included in the cluster analyses.
The diagnosis of multicolinearity revealed CN = 3273.23, which corresponds to the severe collinearity class according to the criteria proposed by Montgomery et al. (2012). Therefore, it was necessary to exclude the traits with high correlation and with greater weight in the last autovectors: third-internode length, number of grains per plant, plant height and epicotyl diameter. The removal of these four traits resulted in weak collinearity (CN = 75.26) and this prevented multicollinear variables from implicitly receiving greater weights in cluster analyses (Cruz and Carneiro 2006), allowing the correct interpretation of the results obtained in the cluster analyses.
Tocher's cluster analysis
The results obtained in the Mahalanobis' generalized distance showed that the order of the three traits which showed greater participation for the differentiation of common bean genotypes was different when considering data obtained in individual (I, II, III and IV) experiments (Table 2). However, the mass of 100 grains was the trait that most contributed to the differentiation between common bean genotypes in the combined experiments I and II (29.36%), I, II and III (29.44%) and I, II, III and IV (31.51%). Similarly, it was found that the mass of 100 grains exhibited the greatest relative contribution to the separation of common bean genotypes, although the order of the other agronomic traits important the recognition of these differences varied in each evaluation year, when the Mahalanobis' generalized distance was used (Coelho et al. 2010). The mass of 100 grains, too, has been described as the most important agronomic trait to assess the genetic dissimilarity of common bean genotypes based on data obtained in one (Correa and Gonçalves 2012) or three (Cabral et al. 2011) experiments.
In the present study, the mass of 100 grains has a greater contribution in the formation of different groups in the cluster analyses, using the results obtained in two, three or four experiments. Therefore, data obtained in two experiments were sufficient to recognize differences between carioca and black bean genotypes, based on plant architecture and grain yield traits, in the Mahalanobis' generalized distance analysis.
When Tocher's cluster analysis was performed from data obtained in individual experiments, it was observed that the number of groups formed and the composition of these groups was different (Table 3). This can be explained by the fact that most of the plant architecture and grain yield traits showed a significant genotype × experiment interaction effect (Table 1). When this happens, the selection of superior parents for agronomic trais will be different in each experiment, and this represents difficulties for the common-bean breeding programs. Previous studies have also shown that the groups formed in the Tocher's cluster analysis were not exactly the same for each of the different environments, since significant genotype × experiment interaction was found for most of the agronomic traits evaluated by Ceolin et al. (2007) and Coelho et al. (2010). These authors observed that the groups formed in the Tocher's cluster analysis were different for each evaluated experiment (environment). Therefore, when a significant genotype × experiment interaction is observed for most agronomic traits, the strategy of presenting the results obtained in the Tocher's analysis for each growing environment is not the best option for the breeding program. This is because the identification of superior or redundant genotypes will vary with the growing environment.
Therefore, the environmental variability between growing seasons and years for the same location where the experiments are conducted must be considered in the cluster analysis. Cargnelutti Filho et al. (2009) recommended that data obtained from six experiments were sufficient to identify divergent cultivars by Tocher's cluster analysis based on grain yield, phenology and morphology traits. However, using data obtained from six experiments to identify superior parents or duplicate accessions in a breeding program implies an increase in the time needed to assess genetic divergence and a lot of delay in making decisions regarding the genotypes that can be used in the crossbreeding blocks.
Data from two experiments (I and II) resulted in the division of common bean genotypes into two groups (Table 3), making it possible to differentiate the line TB 02-19 (group 2) from the other evaluated genotypes (group 1). However, when considering the data obtained in three (I, II and III) or four (I, II, III and IV) experiments, it was possible to differentiate three groups with identical composition. Group 1 included 15 carioca and black bean genotypes, corresponding to 88.23% of the evaluated genotypes; group 2 consisted of the cultivar Pérola (carioca beans); and group 3 was composed of line TB 02-19 (black beans). Tocher's cluster analysis has been efficient in differentiating common bean genotypes for agronomic traits, despite the fact that group 1 normally concentrates the largest number of evaluated genotypes (de Lima et al. 2012; Gonçalves et al. 2016; Pereira et al. 2019; dos Santos et al. 2019). The use from data obtained in three or four experiments allowed greather reliability in the formation of groups in the Tocher's cluster analysis, in the presesent study.
The data obtained in three or four experiments allowed recognition of the differences between the common bean genotypes grouped in each of the three groups. Group 1 included the common bean genotypes of upright plant architecture, characterized by the lowest lodging and general adaptation score values and the largest hypocotyl diameter (Table 4). These genotypes, also, had the highest number of pods per plant, number of grains per pod and grain yield values, among the three groups formed. Groups 2 and 3 were characterized by common bean genotypes of upright plant architecture, that is, higher second-internode length value; however, they showed low grain yield. All evaluated genotypes had medium-sized grains (25 to 40 g), which meet the market demand for carioca and black beans (Carbonell et al. 2010). The results obtained showed that the common bean genotypes in group 1 have a greater number of traits that confer an upright plant architecture and high grain yield, with great potential for use in controlled crosses.
In the present study, the groups formed in the Tocher's cluster analysis were identical when considering data obtained in three or four experiments. Therefore, data obtained in three experiments were sufficient for studies of genetic divergence in the Tocher's cluster analysis, allowing the identification of carioca and black common bean parents with a greater number of taits that confer upright plant architecture and high grain yield.
UPGMA cluster analysis
The lowest CCC was obtained in the UPGMA cluster analysis performed with data obtained in the 2016 rainy (0.5591) and the largest CCC was found with data obtained in the 2017 dry (0.9335) season crops (Figure 1), all of which were significant at 1% probability by the t test. A similar amplitude of variation was observed for CCC values obtained in the UPGMA cluster analysis, considering agronomic and/or morphological traits evaluated in one (Gonçalves et al. 2014, 2016), two (Arteaga et al. 2019) and three (Cabral et al. 2011) experiments with different common bean genotypes. The closer to one the CCC value, the greater the adjustment between the cophenetic matrix and the dissimilarity matrix based on the Mahalanobis' generalized distance, resulting in greater cluster reliability (Cabral et al. 2011). In the present study, the highest CCC values (≥ 0.8746) were obtained in the 2017 dry season (Figure 1) and the combined experiments I and II; I, II and III; and I, II, III and IV (Figure 2), indicating greater reliability in the representation of the groups formed in this UPGMA cluster analysis.
The groups formed in the 2016 rainy, 2017 dry, 2017 rainy and 2018 dry season crops were different in the dendrograms obtained in the UPGMA cluster analysis (Figure 1), confirming the results observed in the Tocher´s cluster analysis (Table 3). This is because when cluster analysis was performed based on data obtained in a single experiment, the environmental variability between years and growing seasons for the same site of conduct of the experiments was not considered. In the present study, it was possible to verify that when the UPGMA cluster analysis was performed with data from one experiment, there was no repeatability in the characterizing the genetic divergence of carioca and black bean genotypes for plant architecture and grain yield traits.
However, the dendrograms generated in the UPGMA cluster analysis, considering data from two, three and four experiments, formed two identical groups adopting 70% similarity as a criterion for defining the groups (Figure 2). Group 1 contained the line TB 02-19 and group II was composed of other carioca and black common bean lines and cultivars. When the UPGMA cluster analysis was applied to morphological traits evaluated in two experiments with common bean genotypes, it was also possible to group the genotypes into just two groups (Guidoti et al. 2018; Arteaga et al. 2019). In the present study, it was not possible to gather in different groups carioca and black common bean genotypes for the plant architecture and grain yield traits. The difficulty of separating carioca and black common bean genotypes into different groups by cluster analysis was also reported for agronomic (Pereira et al. 2019) and molecular (Veloso et al. 2015) traits. As in the process of developing new carioca and black common bean cultivars, crosses between parents with both types of grains were carried out, this resulted in genetic similarity (Veloso et al. 2015). For this reason, carioca and black common bean lines and cultivars have a narrow genetic basis, making it difficult to differentiate common bean genotypes from grain types to agronomic traits (Delfini et al. 2017; Pereira et al. 2019).
The results obtained in the UPGMA cluster analysis showed that the inclusion of data obtained in three or four experiments did not change the clustering pattern of common bean genotypes in relation to the analysis with data from two experiments. Therefore, data from two experiments were sufficient in the UPGMA cluster analysis to obtain a dendrogram with high reliability in the formation of the groups. This allows the identification of promising carioca and black common bean parents for plant architecture and grain yield traits in a more assertive manner.