Stability and genetic parameters for cassava yield attributes in the tropical humid region of Brazil

The performance differences in cassava genotypes arising from genotype vs. environment interactions (G × E) often lead to responses that are significantly lower than expected for selection. The objective of this study was to evaluate different stability methods, both parametric and non-parametric, such as additive main-effects and multiplicative interaction (AMMI), main effect of genotypes plus G × E (GGE), and weighted average of absolute scores (WAASB), in order to quantify the G × E in multi-environmental trials. A total of 12 genotypes were assessed across 12 environments using a completely randomized block design, with three replicates for traits such as fresh root yield (FRY) and dry matter content in the roots (DMC). The data were subjected to analysis of variance and the Scott Knott test (p < 0.05). The sum of squares (SQ) of genotypes, environment, and G × E effects were equally distributed for FRY, whereas for DMC, these effects accounted for 64.1%, 21.9%, and 13.8% of the SQ, respectively, indicating a lower environmental effect on this characteristic. Using the AMMI, GGE, and WAASB methods, genotypes with high agronomic performance and stability for FRY (BR11-34–41 and BR11-34–69) (> 32 t ha−1) and DMC (BRS Novo Horizonte, BR12-107–002, and BR11-24–156) (> 37%) were identified. The broad-sense heritability (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h}^{2}$$\end{document}h2) for FRY and DMC was estimated to be 0.45 and 0.75, respectively. Approximately 72% of the methods identified BRS Novo Horizonte as the genotype with the highest stability and performance for DMC, while 47% identified genotypes BR11-34–41 and BR11-34–69 for FRY and intermediate DMC. Genotype BR11-24–156 exhibited high static stability according to 50% of the methods. Significant correlations were observed between stability and agronomic performance across the different methods, enabling the formation of groups based on stability concepts. Additionally, it was found that two mega-environments existed for FRY, whereas DMC displayed a single mega-environment with similar patterns, indicating an absence of G × E. We identified superior genotypes that could be promoted to national performance trials to develop stable cultivars with better yield attributes in cassava. Supplementary Information The online version contains supplementary material available at 10.1007/s10681-024-03384-5.


Introduction
Cassava (Manihot esculenta Crantz) plays a crucial role as a carbohydrate source for millions of people in various countries, particularly those with a tropical climate.This versatile plant is commercially exploited for all its parts, with the tuberous roots being the main focus due to their starch content.In fact, cassava is the second most significant global source of starch (Stapleton 2012;Ceballos et al. 2020), finding applications in human food, drug production, biopolymers, fuel, and more.
Over the past decade, the worldwide production of cassava has witnessed a remarkable growth of over 27%.This increase can be attributed to several factors, including advancements in cultivation systems and the utilization of genetically improved varieties with enhanced root and starch productivity (Akinwale et al. 2010;Faostat 2020;Oliveira et al. 2020a).Currently, global cassava production stands at approximately 302.66 million tons, cultivated across 28.24 million hectares (Faostat 2020).Among the top four cassava producers worldwide, Nigeria, Thailand, Indonesia, and Brazil, the production of fresh roots reaches 60. 00, 28.99, 18.30, and 18.20 million tons, respectively.
Cassava cultivation spans across multiple countries, primarily in regions located between latitudes of 30° N and 30° S. The cultivation of cassava faces diverse edaphoclimatic conditions, characterized by significant variations in factors such as rainfall, temperature, soil attributes, and sunlight availability (El-Sharkawy 2007).In this context, the varietal component assumes great importance, as the choice of variety is a critical agricultural input.Regardless of the farmer's technological level, the selected variety must effectively utilize these factors to express its genetic potential and ensure high yields.
The process of developing and recommending a new cassava variety for planting is time-consuming, typically taking between 8 to 10 years.Additionally, there are considerable costs associated with phenotyping in various environments during conventional breeding (Oliveira et al. 2014;Ceballos et al. 2015;Wolfe et al. 2017;Andrade et al. 2019).In the initial five years, a large number of genotypes are evaluated with few repetitions and in a limited number of environments.Subsequently, in the last three years, experiments known as value of cultivation and use (VCU) trials are conducted with a smaller set of genotypes.These trials typically involve three or four replications and are conducted across a wide range of environments.This approach allows for a detailed exploration of the genotypes × environments (G × E) interaction, enabling a comprehensive assessment of adaptability and stability parameters.These parameters play a crucial role in recommending the bestperforming clones.
The G × E arises from the inconsistent phenotypic expression of traits when genotypes are exposed to different environmental conditions.This inconsistency leads to a reduction in the correlation between the phenotypic and genotypic values of evaluated genotypes, which is a crucial factor to consider when selecting and recommending cultivars with high genetic potential (Yan and Holland 2010).However, G × E can also be beneficially exploited to identify genotypes with greater stability and adaptability across specific environments of interest (Tumuhimbise et al. 2014).In general, the adaptability of a genotype refers to its ability to respond favorably to a particular environment, while stability relates to the predictability of the genotype's agronomic performance across a range of environments (Finlay and Wilkinson 1963;Eberhart and Russell 1966).According to these authors, the genotype's average performance compared to the overall average of all tests determines the favorable and unfavorable environments.This information is valuable for regionalized recommendations of new clones.
Several studies have demonstrated the presence of G × E in cassava (Maroya et al. 2012;Agyeman et al. 2015;Mtunguja et al. 2016;Adjebeng-Danquah et al. 2017;Chipeta et al. 2017;Nduwumuremyi et al. 2017;Fotso et al. 2018;Masinde et al. 2018;Jiwuba et al. 2020).Therefore, it is important for research studies to identify specific clones that perform well in each environment or exhibit general adaptability.To interpret and utilize G × E, various parametric and non-parametric statistical models have been developed.In general, the additive main effects and multiplicative interaction (AMMI) and genotype main effects plus genotype × environment interaction effects (GGE) methodologies have been widely employed.These methods capture the most significant portion of the sum of squares of G × E interaction for agronomic traits related to the performance of elite genotypes, disease resistance, and root quality in the final evaluation stages (Kvitschal et al. 2009;Tumuhimbise et al. 2014;Esuma et al. 2016;Agustina et al. 2020;Peprah et al. 2020).
The AMMI method combines analysis of variance (ANOVA) for the additive main effects of genotypes and environments with principal component analysis (PCA) to assess the multiplicative interaction effects of G × E (Gauch and Zobel 1996 This approach enhances the model's predictive capacity and is particularly useful for identifying specific adaptations between genotypes and environments.However, the AMMI method lacks discriminative capacity and the ability to represent biplot graphs due to the absence of mathematical properties of vector products (Gauch andZobel 1996, Gauch Jr 2013).On the other hand, the GGE method does not separate the genotype effect from the G × E interaction.By considering the combined value of G + G × E, this method captures the total genetic variation rather than exclusively focusing on the G × E component.It also allows for the identification of the performance of the best genotypes, mega-environments, and heterotic groups (Yan 2016).
Various authors have utilized AMMI and/or GGE methods for analyzing G × E data in cassava, recognizing their unique characteristics and complementary nature (Alwala et al. 2010;Frutos et al. 2013;Noerwijati et al. 2014;Nduwumuremyi et al. 2017).However, some researchers suggest combining multivariate methods like AMMI and GGE with other complementary methodologies such as mixed models, uniparametric approaches, and nonparametric approaches to fully explore the sum of squares of the G × E interaction and enhance the reliability of results.This is particularly important due to the lack of agreement among different methods in identifying genotypes with high agronomic performance and stability or adaptability to specific environments.
One example of a complementary approach is the use of the harmonic mean of predicted genotypic values (HMGV) and the weighted mean of absolute scores (WAASB), which simultaneously select genotypes based on stability and yield parameters (Olivoto et al. 2019).Despite its crucial importance in selecting and recommending new genotypes for cultivation in specific target regions, research on the complementarity of methods for evaluating the G × E interaction in cassava has been limited.The objectives of this study were: i) to estimate the adaptability and stability of cassava genotypes evaluated in multi-environment trials (MET) for subsequent planting recommendations; ii) to explore the correlation between parametric methods (univariate and multivariate), mixed models, and nonparametric models in estimating the G × E interaction; and iii) to identify the methods that provide the best estimation of stability and adaptability in cassava, considering dynamic and static stability as well as the magnitude and significance of G × E effects.

Variety value for cultivation and use tests (VCU)
The VCU trials in multi-environment trials (METs) were conducted by Embrapa Mandioca e Fruticultura over three harvests in the crop's seasons of 2017, 2018, and 2019.Each year, the experiments were set up in four different locations, resulting in a total of 12 environments in the state of Bahia, which is characterized as a tropical hot and humid region.According to the Köppen classification, the predominant climate in the region is Aw and Am.Rainfall is concentrated between the months of April and July, with an average of 1100 mm per year, irregularly distributed, and a prolonged dry period lasting 5-7 months.During the period from planting to harvesting, meteorological data including temperature, precipitation, and humidity were collected by interpolating data from the three closest automatic meteorological stations: Amargosa, Cruz das Almas, and Valença-Bahia (INMET 2020) (Fig. 1).Descriptions of the test sites are provided in Table 1.
The VCU trials involved 12 genotypes selected from the cassava breeding program, including six clones in the final validation phase and six control varieties (Table 2).Soil preparation for cultivation followed conventional practices, including weed desiccation, plowing, and harrowing to incorporate crop residues.A cassava planter was used to open planting furrows and apply fertilizers based on soil analysis.Planting was done manually using standard cuttings measuring 18-20 cm obtained from 12-month-old stems that were free of pests and diseases.The cuttings were placed horizontally along the planting line.
The VCU trials were arranged in a completely randomized block design with three replications.
Vol:. ( 1234567890) Each plot consisted of four rows with 25 plants each, spaced at 0.90 m between rows and 0.80 m between plants.Cultural treatments after planting followed the recommended practices for cassava cultivation (Souza et al. 2006).

Traits evaluated
At 12 months after planting, the following traits were evaluated: 1) root yield (FRY, in t ha −1 ) considering the weight of all roots in the plot, and then adjusted to tons per hectare; and 2) root dry matter content (DMC, in %) obtained from the gravimetric methodology (Kawano et al. 1987).

Individual and joint analysis
The data obtained from the VCU trials were subjected to individual analysis of variance (ANOVA) to assess the significance of different factors.The homogeneity of residual variances was tested using Bartlett's test.The joint analysis of variance was then performed  without restrictions, ensuring that the ratio between the highest and lowest mean square value of the residue was less than 7, as suggested by Pimentel-Gomes (2009).
For the individual ANOVA, the ea1 and ea2 functions of the easyanova package (Arnhold et al. 2013) in the R software version 4.2.0 (R Core Team 2022) were utilized.The statistical model employed for individual analysis is represented by the equation: Here, y ij denotes the vector of phenotypic values associated with genotype i in block j; μ is the overall mean; g i is the effect of genotype i; b j is the effect of block j; and e ij is the experimental error associated with the plot receiving genotype i in block j.
The joint analysis of variance was performed using the AMM" function of the agricolae package (Mendiburu 2021).The statistical model for the joint analysis is given by the equation: Here, y ijk represents the vector of phenotypic values of genotype i associated with block j within environment k; μ s the overall mean; i is the main effect of geno- type; j is the main effect of the environment; k is the singular value for the axis of the principal component of the G × E interaction (IPCA), explaining the interaction of the data; ik is the i-th element of the k-th eigenvector; jk is the j-th element of the k-th eigenvector; ij represents the residue if not all IPCAs are used; and ij is the residual for genotype i in environment j, which is not explained by the model.The phenotypic means for each genotype and environment were then grouped using the Scott-Knott test implemented in R software version 4.2.0 (R Core Team 2022).
The experimental quality of the VCU tests was evaluated using the test accuracy values ( rĝg ), which were calculated using the formula: rĝg = √ 1 − 1 F , where F is the ratio of the mean square of the genotype effects to the mean square of the G × E interaction effects.The test accuracy provides an indication of the reliability and precision of the experimental results.

Variance components and genetic parameters
The variance components, including genotypic, phenotypic, environmental, and G × E interaction, were estimated using the mean squares obtained from the analysis of variance.The heritability for each trait was also estimated.The equations used for these estimations are as follows: ER : where 2 g is the genetic variance; 2 p is the phenotypic variance; 2 gxe is the variance of the G × E interaction; 2 e is the variance of the pooled error; E the number of environments; and R the repetition number.
The genotypic coefficients of variation ( CVG ), phenotypic ( CVP ) and residual (CVr ) coefficients of variation were estimated according to Singh and Chaudhary (1979), where: , where is the overall mean.The expected genetic gain with selection (GA) was estimated using the formula: GA = (K) p h 2 , where GA is the expected genetic gain, K represents the selection differential (15% selection intensity), p is the phenotypic standard deviation.
The genetic advancement as a percentage of the mean (GAM) was estimated by the formula: GAM = GA∕ × 100 , where GAM is the genetic

GGEbiplot
A / E Yan ( 2002) Mixed model Harmonic mean of genetic values advancement as a percentage of the mean, GA is the expected genetic gain, represents the overall mean.The CVratio was obtained by calculating the ratio between CVG CVr , and the P − G was obtained by calculating the difference between CVP and CVG .These calculations and estimations were performed using the "metan" package (Olivoto and Lúcio 2020) in the R software version 4.2.0 (R Core Team 2022).

Adaptability and stability analysis
The adaptability and stability parameters were estimated using various statistical models, including univariate and multivariate parametric models, as well as mixed and non-parametric models (Table 3).The agricolae package (Mendiburu 2021) was used for the (AMMI) and (ASV) analyses, while the GGEBi-plotGUI package (Frutos et al. 2013) was used for the GGE analysis.The metan package (Olivoto and Lúcio 2020) was employed for other methods.
The multivariate parametric analyses GGE and AMMI (ASV) were represented using two-dimensional graphs based on genotypes and environmental scores, represented by PC1 (phenotype of the trait under analysis) and PC2 (stability parameter).The GGE biplots were constructed using the method (centralization = 2) (scale = 0) and (PVS = 2), which represent performance, stability, and adaptability.The AMMI value (ASV) was used to quantify and classify genotypes based on performance stability.The ASV is determined by the distance from the coordinate point to the origin of the biplot, i.e., the ratio between the sum of squares on the PC1 axis and the sum of squares on the PC2 axis.A smaller ASV value indicates a more stable genotype (Purchase et al. 2000).
Simultaneous selection between performance and stability, based on the mixed linear model (LMM), employed two criteria: the harmonic mean of predicted genotypic values ( HMGV ) and the weighted mean of absolute WAASB scores.A higher HMGV indicates a lower standard deviation of genotype behavior in different environments.Thus, the genotype with the highest HMGV in the evaluation envi- ronments received rank 1, up to the gth genotype, and was considered the most stable.In the case of WAASB , the genotype with the lowest mean received rank 1, up to the gth genotype.
Subsequently, the genotypes were classified based on different adaptability and stability parameters using the phenotypic values of each variable under study.For methods based on variance estimates ( DP , S 2 d , Di , Wi , and S 2 x ), lower values indicate more stable genotypes.Conversely, for the R 2 parameter of Eberhart and Russel and Bi of Perkins and Jinks, values close to 1 indicate more adapted genotypes.Therefore, linear regression coefficients ( bi and Bi ) equal to 1 and non-linear regression deviation ( S 2 d and Di ) equal to zero received rank 1 as the best gen- otype, up to the gth genotype.
The Annicchiarico method involves the calculation of the confidence index for the general environment li(g) , with decomposition for favorable environ- ment li(f ) and unfavorable environment li(d) .Positive values of the index indicate favorable environments, while negative values indicate unfavorable environments for cultivation.The average performance of the genotype compared to the overall average of all tests determines whether an environment is favorable or unfavorable.Genotypes with higher parameter values are ranked higher, with rank 1 assigned to the genotype with the highest parameter value.
The nonparametric method of Nassar and Huehn (1987) considers the classification of genotypes in each evaluation environment.A genotype is considered stable if its classification remains relatively constant across multiple evaluation environments, indicating low environmental variation.The stability parameters include Si 1 , which is the average absolute differences in genotype classification across environments, and Si 6 , which is the sum of squares of geno- type classifications relative to the mean classification across environments.Genotypes with lower values of Si 1 and Si 6 are considered stable.Genotypes are ranked based on these stability indices.Rankings of the traits were also calculated using the Scott-Knott test, where the genotype with the highest value receives rank 1, and so on until the gth genotype.
To assess the correlations between stability and adaptability methods, Spearman's correlation coefficients (rs) were estimated using a matrix of ranks obtained from all methodologies for each characteristic.This allows for an exploration of the relationships and groupings of methods based on stability.The correlogram, representing the correlation coefficients, was generated using the metan package (Olivoto and Lúcio 2020).Principal component analysis (PCA) was performed on the ranking correlation matrix using the Factoextra package to further explore the relationships between method groups.All these analyses were implemented in R software version 4.2.0 (R Core Team 2022).

Analysis of variance of the agronomic data
Based on the joint AMMI analysis of variance, the three sources of variation, genotypes, environments, and G × E interaction, were found to be significant (p < 0.001) for both traits (Table 4).The phenotypic expression of FRY and DMC varied among the evaluated genotypes, indicating the presence of selectable variation even with a small number of genotypes in the final evaluation phase.It is likely that the high variation in climatic factors (Fig. 1) contributed to the observed differences in the agronomic performance of cassava genotypes.The significance of the G × E suggests that genotypes responded differently to environmental stimuli based on their variable behavior across different evaluation environments.Therefore, quantifying and exploiting G × E through adaptability and stability parameters becomes necessary for more accurate recommendations of genotypes with greater agronomic interest.
For FRY, the environment effect accounted for 35.54% of the total sum of squares (SQ), followed by the effect of genotypes and the G × E interaction, which contributed 34.99% and 29.46%, respectively (Table 4).The first two principal components (PC1 and PC2) of the AMMI biplot analysis explained 32.83% and 20.70% of the SQ of the G × E interaction.On the other hand, for DMC, the SQ of the effect of genotypes represented 64.18%, while the SQ of the environment and the G × E interaction accounted for 21.98% and 13.82%, respectively.For DMC, the two principal components captured 38.33% and 20.88% of the SQ.
The selective accuracy estimates ( rĝg ) for both agronomic traits exceeded 95% (Table 4), indicating high experimental precision of the VCU tests, enabling reliable inferences about the average p -phenotypic variance, 2 e -residual variance,h 2 -broad sense heritability, CV g -genotypic coefficient of variation (%),CV p -phenotypic coefficient of variation (%), CV r -residual coefficient of variation, CV ratio -relative coefficient of variation, P − G-difference between CV p and CV g , GA-genetic  performance of the genotypes in the assessment environments.Estimates of genetic parameters exhibited wide variation for the two traits, indicating the potential for genetic gains.For FRY, genetic variance ( 2 g ) contributed to 45% of the phenotypic variance ( 2 p ), followed by G × E interaction variance ( 2gxe ) at 40.0% (Table 5).In the case of DMC, 2 g and 2 gxe accounted for 76.0% and 15.0% of the variance 2 p , respectively.Estimates of broad-sense heritability ( h 2 ) were relatively low for FRY (0.45) and high for DMC (0.75).
The high values of CV g , CV p and CV ratio for both traits indicate that a significant portion of the observed variation is attributed to the effect of genotypes, particularly for DMC.The genetic advance estimates as a percentage of the average (GAM) of 24.40% (FRY) and 8.64% (DMC) suggest the potential for significant improvement, especially in fresh root productivity, through the selection of the best genotypes.

Agronomic performance and comparison between the methods for G × E analysis
The range of root yield varied from 16.03 t ha −1 for the genotype IAC-90 to 33.57t ha −1 for the genotype BR11-34-41.Other genotypes such as BR11-34-69 and BRS Formosa also exhibited high root productivity, with yields of 31.49t ha −1 and 29.67 t ha −1 , respectively.In general, the average root yield of the new clones was higher than that of the control varieties, with the six new clones averaging 27.89 t ha −1 compared to 24.38 t ha −1 for the controls.
Regarding DMC, among the five genotypes with high DMC, two new genotypes, BR11-24-156 and BR12-107-002, were among those with the highest average DMC (37.13% and 37.46%, respectively).On average, the six new clones exhibited a DMC of 35.43%, while the controls had an average of 37.01%.However, some genotypes, such as BR11-24-156 and BR12-107-002, surpassed the average DMC of the controls.The highest DMC average was observed in the BRS Novo Horizonte genotype (39.33%), which performed well in the NR-2019 and RA2-2019 environments, indicating a certain adaptability to these specific environments (Figure S1).
There was divergence in the classification of genotypes based on simultaneous selection for stability and agronomic performance.The genotype BR11-34-41 showed greater stability for FRY according to nine methods, ranking 1st based on the HMGV , Si 6 , and three Annicchiarico indices.However, it showed high deviation based on the classification of multivariate methods such as WAASB , ASV , DP , and Wi , and non-parametric Si 1 andASV (Table 6).Genotypes BR11-34-69, BR11-24-156, BR11-34-64, and BRS Novo Horizonte exhibited stability according to most of the methods used.On the other hand, genotypes IAC-90, BRS Mulatinha, and BRS Formosa were ranked as the most unstable for FRY.
For DMC, BRS Novo Horizonte was the most stable genotype according to thirteen methods, except for three univariate parametric methods ( DP,Wi and Bi ) and one non-parametric method ( Si 1 ), as well as the GGE biplot analysis.Three other genotypes, Corrente, BR11-24-156, and BR11-34-69, also ranked high in terms of stability and adaptability.Conversely, genotypes IAC-90, BRS Mulatinha, and BR11-34-41 were the most unstable for DMC across the twelve evaluation environments, occupying positions between 9 and 12th in the ranking (Fig. 2).

Correlations and clustering of the stability and adaptability methods
The Spearman correlations between adaptability and stability methods showed that approximately 23% and 24% of the correlations were significant (p-values < 0.01) and of high magnitude for FRY and DMC, respectively (Figs. 3A and 4A).These correlations were positive and different from zero, indicating associations between the methods.However, the magnitude and direction of the correlations varied, ranging from − 0.83 to 1.0 for FRY and − 0.88 to 1.0 for DMC, indicating that most methods were complementary but with some disagreement in the classification of genotypes in terms of adaptability and stability.
For FRY, high correlations (0.99, 0.98, and 0.84) were observed between the Annicchiarico parameters li(g) , li(d) , li(f ) and the harmonic mean HMGV (Fig. 2).There were also high correlations (r ≥ 0.80) between these three parameters and environmental variance ( Si 6 ) as well as maximum correlations between RMSE and S 2 di and Dji .Approximately 65.0% of the methods showed low correlations (− 0.40 ≤ r ≤ 0.40) for both characteristics, suggesting their use as complementary methods of analysis.Only 1.5% of the methods exhibited a significant negative correlation of high magnitude, such as Bi vs DP (rs = − 0.83***), which can be used as a complementary approach for selecting low-yield genotypes to be discarded in the breeding program.
For DMC, correlations greater than 0.90 were observed between Huehn's parameter ( Si 6 ) and Annicchiarico's parameters ( li(g) , li(f ) and li(d) ) as well as with the ranking based on the mean of the trait (Fig. 3).These correlations indicate that these four parameters are associated with high stability Multivariate statistical methods ( ASV and GGE ) showed positive correlations among themselves, ranging from 0.58 for DMC to 0.78 for FRY.This indicates that these approaches have the same correlation direction but may vary depending on the specific trait being analyzed, suggesting their complementary use.On the other hand, the two linear regression-based methods ( Dji and Bi ) showed low associations for both FRY and DMC, suggesting that the environmental variance can influence the response pattern of these methods due to changes in genotype performance across different environments.
The first two principal components (PCs) explained 71.78% of the phenotypic variation for FRY and 67.47% for DMC (Fig. 2B and 3B), with both characteristics being divided into four groups to allow for classification based on the static (biological) and dynamic (agronomic) concepts of stability.For both FRY and DMC, Group I comprised six methods ( li(g) , li(f ) , li(d) , Si 6 , Pi and HMGV ) with significant, positive correlations (superior to 0.77 for FRY and 0.87 for DMC) among them.The phenotypic data for FRY and DMC were also allocated to this group, indicating a strong association between these characteristics and the stability parameters of Group I, and thus, dynamic stability.
Group II for FRY consisted of nine methods ( R 2 , Dji , Bi , S 2 di , GGE , ASV , RMSEWi and S 2 x ), includ- ing a negative and significant correlation with FRY (r ≥ − 0.40).Although the R 2 and Bi methods were included in this group, their FRY values were of intermediate magnitude, characterizing the group by greater static stability, where genotype performance is relatively constant across most environments regardless of the magnitude of the evaluated characteristic.Group III comprised only the DP and WAASB methods, while Group IV consisted of the Si 1 method, which showed a low correlation with FRY, indicating static stability.
For DMC, Group II can be characterized by the dynamic concept with methods such as R 2 , Dji , WAASB , S 2 di , GGE , ASV , RMSE e S 2 x , as they were associated with the same direction of the response variable.Groups III and IV included three and one stability methods, respectively, with moderate associations with DMC.
In general, methods based on variance estimates ( RMSE , R 2 , S 2 di , Dji and S 2 x ) tended to clas- sify genotypes similarly, as did the Si 6 and HMGV parameters, which simultaneously select for stability and performance.For FRY, there were significant and positive correlations (r ≥ 0.87***) between univariate parametric and non-parametric methods (lig∕Si 6 , lif ∕Si 6 , lid∕Si 6 and Pi∕Si 6 ); four correlations between univariate and multivariate parametric methods (r ≥ 0.69*) ( Wi∕ASV ; Dji∕ASV , S 2 di∕ASV) and RMSE∕ASV ); and four correlations between parametric univariate and mixed models (r ≥ 0.84***) ( lig∕HMGV ; lif ∕HMGV ; lid∕HMGV and Pi∕HMGV ).One correlation between the mixed model and non-parametric method was significant and negative (r = − 0.70*) between ( MHVG and Si 6 .For DMC, the results were similar, except for the correlations between univariate and multivariate parametric methods, which were not significant, indicating random classification similarity among some genotypes.There was also no significant positive correlation between the mixed and multivariate models for both traits. The results obtained for DMC showed similar trends, with the exception of the four correlations observed between univariate and multivariate parametric methods (r ≥ 0.66*) such as ( R 2 ∕ASV ; Dji∕ASV , S 2 di∕ASV) and RMSE∕ASV .On the other hand, the correlations between nonparametric and multivariate methods were not found to be significant, indicating that certain classifications of genotypes were randomly assigned.Furthermore, there was no positive and significant correlation observed between the mixed and multivariate model for the two characteristics.Among the various methods employed, S 2 di∕Dji, RMSE∕Dji and RMSE∕ASV exhibited the highest correlation values (r ~ 1.0) for FRY and DMC.Consequently, some of these methods can be disregarded in the analysis.However, it is worth noting that the complementarity between certain methods can aid in the selection of the most suitable genotypes, particularly those based on multivariate approaches and mixed models, which consider simultaneous selection for stability and yield.

Stability analysis: AMMI x GGE x WAASB x HMGV
Utilizing multivariate models, such as the GGE biplot methodology, enables a more efficient and reliable exploration of the genotype-by-environment (G × E) interaction in multi-environment trials.The GGE analysis explained a greater proportion of the variation in both FRY and DMC compared to the AMMI analysis.The first two principal components of the GGE analysis captured 21.74% more SQG × E variation for FRY and 31.45%more for DMC compared to the AMMI analysis (Figs. 5 and  6).The AMMI analysis had double the residual variation (42.20%) compared to the GGE analysis (21.22%), indicating a greater influence of factors without biological explanation in the AMMI model.The environmental factor represented a higher proportion of the sum of squares for FRY, while the genotypic factor was higher for DMC.
In the AMMI methodology, genotypes close to the origin of the axes are less sensitive to the G × E interaction, while those further away from the origin are more sensitive and influenced by the environment.For FRY, three genotypes (BR12-107-002, BR11-24-156, and BRS Corrente) showed a low contribution to the G × E interaction (Fig. 4).The BR11-24-156 genotype ranked well according to several stability methods, including WAASB, with a median FRY (Table 6).For DMC, the most stable genotypes were BR11-34-69 and BRS Novo Horizonte, with the latter also showing wide adaptation due to its proximity to the center of the biplot and high FRY according to the HMGV and WAASB methods (Fig. 5).
Considering the environmental analysis using the AMMI method, three environments (PP-2018, RA1-2018, and NR-2019) exhibited edaphoclimatic conditions that allowed for greater stability in FRY, and three environments (SA-2017, RA1-2018, and RA2-2019) did so for DMC.Only the RA1-2018 environment was common for both traits.On the other hand, the contribution of the G × E interaction captured by the GGE methodology varied greatly depending on the trait.The UFRB-2018, UFRB-2019, and SV-2017 environments facilitated greater stability for FRY, while four other environments (NR-2019, GA-2019, RA2-2019, and PP-2018) had a low influence on the phenotypic expression of DMC (Figs. 5 and 6).
The WAASB analysis incorporates both the AMMI technique and best linear unbiased predictions (BLUPs) while utilizing all the axes of the principal components (PCs).In the WAASB analysis for FRY, two quadrants are particularly important: i) upper right quadrant that includes the BR11-34-41 and BR11-34-69 genotypes with high yield but some instability.Additionally, the NR-2019 environment demonstrates a high ability to discriminate between genotypes; ii) lower right quadrant where the BR-34-64 and BRS Novo Horizonte genotypes are allocated.These genotypes exhibit above-average productivity and low WAASB values, indicating wide adaptation and stability, albeit with moderate yield (7th and 5th for HMGV, respectively).For DMC, BRS Novo Horizonte, located in the lower right quadrant, demonstrates high performance and stability.This is evident from Table 6 and Figure S2.By conducting a joint analysis of the GGE and AMMI methods, breeders can increase their confidence in recommending new genotypes.This integrated approach provides a comprehensive understanding of genotype performance and stability across multiple environments, enabling better decision-making in genotype selection and recommendation.
In the GGE analysis, the angulation between the eigenvectors of genotypes and environments offers valuable insights into specific adaptabilities.According to Yan (2016), an angle smaller than 90° in the GGE biplot indicates that the performance of the corresponding genotype surpasses the average, exhibiting a high positive correlation.Conversely, angles above 90° suggest below-average performance with a negative correlation, while angles close to 90° indicate performance similar to the mean.Regarding FRY, genotypes 2011-34-41 and 2011-34-69 demonstrated specific adaptations to the NH-2017 and GA-2019 environments, respectively, followed by the control genotype BRS Formosa in the RA1-2018 environment.The control genotype BRS Novo Horizonte exhibited an angle of less than 90° in all environments for both traits, highlighting its high adaptive potential, although it achieved the best FRY and DMC in the UFRB-2018 and GA-2019 environments, respectively.Among the six new genotypes evaluated with potential for cultivation recommendation, three (2011-34-41, 2011-34-64, and 2011-34-69) and the control genotype IAC-90 presented DMC values below the average for all evaluated environments (Figs. 5 and 6).
The interrelationship between environments can also be explored through the biplot.For FRY, the NR-2019 environment exhibited a positive correlation with all other environments, except UFRB-2019.As for the DMC variable, all environments displayed some degree of correlation, with angles smaller than 90° in all analyzed environments.Overall, the environments demonstrated similar ability to discriminate among the evaluated genotypes.In the WAASB method, the first quadrant showcases environments with a high discrimination capacity, indicating the need to explore specific adaptations.For instance, IAC-90 exhibited promising performance in the RA2-2019 and SA-2017 environments for FRY, while IAC-90 and BR11-34-41 displayed potential in the SJ-2017 and NH-2017 environments for DMC (Figure S2).

Mega-environments identification
The GGE analysis provides a valuable tool for exploring the "which-won-where" pattern, which involves forming a polygon connecting genotypes located farthest from the origin of the biplot.This approach emphasizes the correlation between environments and genotype performance, leading to the identification of mega-environments.In the case of the FRY variable, the first two principal components explained 62.42% and 12.85% of the variance, respectively (Fig. 6).The biplot was divided into four sectors by the rays, with only two of them representing mega-environments.Group 1 consisted of nine environments, including UFRB-2018, PP-2018, and NR-2019, with the genotype BR11-34-41 at its apex, indicating the highest FRY.This was followed by genotypes BR11-34-69 and BRS Novo Horizonte.Group 2 comprised two environments, namely UFRB-2019 and RA2-2019, with BRS Mulatinha at the apex.These environments exhibited similar characteristics, leading to their grouping.However, the IAC-90 genotype displayed below-average yield.
The equality line (represented by red arrows) divides the biplot into sectors and represents equal values between two genotypes in a given environment.For example, IAC-90 and BRS Mulatinha showed equal values for FRY in hypothetical columns within this row.Regarding DMC, the genotypes BR11-34-41, BR11-34-69, BR12-107-002, BRS Novo Horizonte, BRS Mulatinha, IAC-90, and Cigana Preta formed the vertices, with BRS Novo Horizonte being the main representative of the vertex, indicating the highest DMC across all environments.
Although the polygon consists of six sectors, only one mega-environment was formed due to smaller environmental influence on this variable.Additionally, the variation within the environment was greater than between groups.The equality line indicated equal values between IAC-90 and the BR11-34-41 genotype in some hypothetical environments among the evaluated assays.
The environments were grouped based on their projection and vector length in the biplot graph, which resulted from the effects of representativeness and discrimination.The discrimination capacity of an environment was determined by the length of its vector in the biplot, while representativeness was measured by the cosine of the angle between the test environment and an ideal environment (Yan and Holland 2010).
The environments in Group 1 exhibited a high representativeness capacity, indicating consistent means with low standard deviation.For instance, the UFRB-2018 and UFRB-2019 environments for FRY, as well as the PP-2018 and NR-2019 environments for DMC, displayed short vectors, indicating stability in performance over the years.While these environments may not effectively discriminate phenotypic variations among genotypes, they can be considered representative and stable.In contrast, the environments in Group 2 (GA-2019 and SV-2017 for FRY, GA-2019 and UFRB-2018 for DMC) had long vectors and small angles with the AEC axis, making them more discriminating and suitable for selecting highly adapted genotypes with high yield.These environments are more informative and can effectively detect differences between genotypes.The positive correlations between the environments in Groups 1 and 2, indicated by the angles smaller than 90°, suggest that genotypes exhibit similar responses in these environments.Utilizing these environments alternately allows for minimal loss of information.
The environments in Group 3 have long vectors and large angles, indicating high discriminative capacity.These environments are informative for detecting differences between genotypes due to their high standard deviation.However, they are strongly influenced by G × E interactions and may not be suitable for selecting superior genotypes.Nonetheless, they can be useful for better classifying genotypes based on their instability or identifying genotypes with specific adaptations to certain environments.Examples include NR-2019and RA2-2019for FRY, and NH-2017, SJ-2017, and RA1-2018 for DMC.These environments are not correlated, as indicated by the angles larger than 90° (Yan and Tinker 2006).

Genetic parameters in cassava VCUs trials
Cassava, being an allogamous species with a high cross-pollination rate, exhibits a significant degree of heterozygosity in its genome (Halsey et al. 2008).This characteristic leads to substantial variability between and within progenies, particularly during the initial and intermediate stages of selection (Ceballos et al. 2012).This genetic variability, combined with environmental factors, contributes to the significant genotypic effects (> 35% for FRY and > 64% for DMC) and G × E interactions (29% for FRY and 13% for DMC) observed in the phenotypic expression.Consequently, conducting MET is crucial to unravel the G × E interactions, explore predictable responses to different environments, and assess stability and adaptability, ultimately advancing breeding efforts.
Previous studies have reported substantial variation in the G × E interaction for FRY and DMC, with variances of 43.80% and 49.08%, respectively (Adjebeng-Danquah et al. 2017;Jiwuba et al. 2020).The genetic control of these traits helps explain these findings.FRY is influenced by numerous genes with a quantitative and polygenic nature, making it highly susceptible to environmental influences (Ssemakula et al. 2007;Wolfe et al. 2016).As a result, FRY exhibits moderate broad-sense heritability ( h 2 > 0.45).On the other hand, DMC is governed by a few genes, leading to low variation between genotypes and high heritability (h 2 > 0.75) due to reduced environmental influence (Ceballos et al. 2015).
In VCU trials, a high genetic variance plays a crucial role in selecting genotypes with diverse response patterns across different growing environments, enabling subsequent planting recommendations.The joint analysis of variance confirmed the significance (p < 0.01) of both environmental effects and the G × E interaction.For DMC, the genetic variation was predominant over the environmental variation, while the opposite situation was observed for FRY.These findings align with the results reported by Adjebeng-Danquah et al. (2017), who analyzed the stability and performance of cassava genotypes in multi-environment trials in Ghana, Africa.In their study, they reported that the genetic, environmental, and G × E interaction components accounted for approximately 32.77%, 37.55%, and 29.68% of the phenotypic variation for FRY, and 54.10%, 9.78%, and 36.10% for DMC, respectively.
Based on these results, different criteria should be applied in the development of new cassava varieties.For FRY, there are indications of a possible dominance of non-additive genetic action with low heritability (h 2 ), suggesting the application of methods based on recurrent phenotypic selection or the exploration of heterotic effects through crossing different genetic pools (Tumuhimbise et al. 2014).On the other hand, the higher heritability estimates for DMC and the potential predominance of additive gene effects suggest the use of simple phenotypic selection methods with increased selection intensity and a smaller number of environments, given their higher predictive capacity (Wolfe et al. 2016;Andrade et al. 2019).Therefore, heritability estimates provide valuable guidance to breeders in determining the most appropriate breeding methods, selection strategies, and experimental arrangements to minimize experimental errors and maximize genetic gains in each selection cycle (Oliveira et al. 2015).

Yield potential of new cassava genotypes
Despite having contrasting genetic controls, our findings suggest that there is potential for improvement in both traits.The genetic advance as a percentage of the mean (GAM) provides a measure of the actual progress, and despite FRY demonstrating moderate heritability, significant gains can still be achieved due to the substantial genetic variation present in the germplasm, despite the adverse environmental effects when compared to DMC.The selection of the best genotypes in relation to the control varieties has resulted in noteworthy advancements in genetic improvement, even in the face of the significant G × E. Out of the six new clones evaluated, five  exhibited FRY values exceeding 24.6 t ha −1 .Additionally, two highly productive clones , along with the BR12-107-002 genotype, displayed high DMC (> 36.69%).While the BRS Novo Horizonte genotype had the highest DMC, the BR11-34-41 and BR11-34-69 genotypes exhibited high FRY but relatively lower DMC values of 33.45% and 33.68% respectively.Consequently, it is crucial to prioritize the introgression of alleles associated with DMC in new cassava clones, aiming to achieve a more balanced combination of different characteristics within a single genotype (Table 6).
In the study area, cassava holds significant economic and social importance, encompassing a cultivation area of approximately 389,000 hectares and a gross production value of around R$1.12 billion.The region ranks as the third-largest cassava producing area in Brazil.However, the root yield is considerably low, standing at 9.80 t ha −1 , even when compared to the national average of 14.9 t ha −1 (IBGE, 2021).On the other hand, current studies in Northeast Brazil indicate the potential for genetic advances of approximately 14% with the new genotypes, which translates to a substantial increase in root yield, around 3.49 t ha −1 , compared to the average of the region's varieties.With an average price of R$567.60 per ton (adjusted for inflation over the past 10 years), cultivating these new genotypes could yield an additional value of approximately R$1,980.00 per hectare.Similarly, although the GAM for DMC was relatively lower (approximately 8.6%), it is possible to achieve additional profitability in cassava starch production, amounting to R$2,660.00 per hectare.This calculation considers the average deflated price of starch over the last 10 years, which stood at R$3,803.67 per ton (CEPEA et al. 2022), and a starch productivity of 8.58 and 7.88 t ha −1 for the improved clones and local varieties analyzed in this study respectively (Table 7).
Cassava exhibits wide phenotypic variation, even through asexual propagation, due to factors such as the low uniformity of plant material and the diverse edaphoclimatic conditions.For instance, the precipitation in this study ranged from 1198.3 mm (UFRB-2018) to 689.43 mm (RA2-2019), demonstrating the significant G × E (Fig. 1).The substantial variation observed between evaluated environments, even within the same region (State of Bahia, Brazil), tends to elicit different responses from the clones during their physiological and productive cycles.Cassava, requiring approximately 400 mm of rain in the initial months of planting (1 to 5 MAP) and a temperature around 25ºC, displays remarkable adaptability and resilience in the face of climate variations, as evidenced by the highly variable meteorological data across years and evaluation periods (El-Sharkawy 2007, 2012;Tironi et al. 2019;Dwamena et al. 2022).
The extent of environmental influence can be quantified by analyzing the amplitude and the ratio between the genotypic variation coefficient ( CV g ) and the phenotypic variation coefficient ( CV p ). Values below 1 indicate an unfavorable situation for selection (Akinwale et al. 2011).Both FRY and DMC displayed higher CV p values compared to CV g , indi- cating a significant environmental effect and demonstrating that the variability is not solely of genetic origin.However, this effect was more pronounced for FRY, with a CV g of 17.18 and CV p of 25.53, while it was less prominent for DMC, with a CV g of 5.14 and CV p of 5.84.Similar findings of lower CV g values for CMD have been reported by other authors.For instance, Peprah et al. (2020) observed significance for the genotypic effect on yield variables of roots and DMC, with CV g and CV p values of 15.58 and 17.63 for FRY, and 8.0 and 8.5 for DMC, respectively.In another study, authors reported a greater magnitude of CV p for FRY and DMC (Kundy et al. 2015;Ewa et al. 2017).
In summary, situations like these make it challenging to directly select the best genotypes when aiming for genetic progress in breeding programs.Therefore, strategies for obtaining genotypes with high performance and stability for each trait must differ.Alternatively, it is suggested to select genotypes for specific environments as a means to achieve more promising results, in terms of performance and stability, using various methods for both FRY and DMC.

Comparison between the stability and adaptability methods
When comparing different methods for studying the G × E interaction in cassava, it was found that the methods of phenotypic stability analysis showed greater concordance in the classification of genotypes when there was a higher explanation of genetic variance over the environmental effect.For example, in the case of FRY, 47% of the methods ranked the best genotype between 1st and 2nd place, while for DMC, this percentage was higher at 72%.This suggests that when the environmental effect is relatively low, stability methods tend to be more consistent in identifying the best-performing genotypes.
Stability methods aim to explore and explain the patterns and biological evidence present in genotypes and environments.In the clustering analysis using principal component analysis (PCA), it was observed that DMC showed positive loadings along PC1 for all models, indicating a more consistent pattern in the ranking order of genotypes.On the other hand, FRY showed loadings of different directions, suggesting more discrepancies in the ranking order of genotypes among the stability methods.This implies that combining multiple stability methods could be a more effective approach to explore the G × E interaction in cassava breeding (Van Eeuwijk et al. 2016).
It is worth noting that recent publications have shown a strong trend towards using mixed models to explore the G × E interaction in breeding programs.However, in the PCA grouping, a clear distinction was observed for group I, where the methods most correlated with FRY and DMC were allocated.These methods are associated with the concept of agronomic stability, wherein a genotype is considered stable if its response is consistent with changes in the environment.Genotypes in group I exhibited a high capacity to respond to environmental stimuli, making them suitable for cultivation in commercial areas with a high technological level.
In the field of agronomy, stability refers to the ability of genotypes to respond differently to environmental conditions, enabling the selection of the best-performing genotype (Becker and Leon 1988).For instance, genotypes BR11-34-41 and BR11-34-69 exemplify this concept for FRY, while BRS Novo Horizonte and Corrente demonstrate agronomic stability for DMC.Therefore, these genotypes are recommended for planting in both favorable and unfavorable environments, as well as unpredictable conditions, as they exhibit positive responses to environmental improvements.This is supported by the li(g) index, which measures genotypic confidence across all environments, li(f ) for favorable environ- ments, and li(d) for unfavorable environments (Annic- chiarico 1992).Stable genotypes with wide adaptation, assigned to group I, exhibit high agronomic performance in both favorable and unfavorable environments, while adaptable genotypes tend to excel in specific environments (Adetoro et al. 2021).
On the other hand, groups II and III, which exhibit weak or no association with the analyzed traits, are classified as statically stable (for FRY) and dynamically stable (although weakly associated) for DMC.The variance of the genotypes significantly influences the stability methods.For instance, the ASV (AMMI), GGE, and WAASB methods are assigned to group II, indicating simultaneous selection for agronomic performance and yield stability.However, for FRY, which shows higher variance of genotype-byenvironment interaction (29.47%) compared to DMC (13.83%),ASV, GGE, and WAASB methods are categorized as static stability, while dynamic stability is associated with DMC.Therefore, the performance of FRY is penalized due to the high heterogeneity among the genotypes (resulting from a strong environmental effect, as indicated by the dispersion of methods in the biplot, and a. CV g /CV r ratio < 1, unfavorable for selection), which is not the case for DMC.In fact, dynamic stability in cassava has been reported by other researchers (Tumuhimbise et al. 2014;Chipeta et al. 2017).
The hypothesis raised by Flores et al. (1998) suggests that simultaneous selection methods tend to correlate in group 2, which serves as an intermediate group between group 1 (yield) and group 3 (stability).This hypothesis has been confirmed by other studies in wheat and rice, showing similar patterns of correlation between simultaneous selection methods and stability methods (Verma et al. 2020;Bornhofen et al. 2017;Sharifi et al. 2021).Genotypes allocated in groups 2 and 3 tend to exhibit a consistent response to environmental variation without being responsive to environmental improvements.This makes them suitable for cultivation in regions where growing conditions may be less favorable for cassava production.However, when the main target of selection is yield, the accuracy of selection based on these stability methods may be lower because they are more related to the static concept of stability (Becker and Leon 1988).
Based on the methods allocated in groups 2 and 3, genotypes BR11-24-156 and BR11-34-64 could be recommended for cultivation in farms with low use of agricultural inputs and technology.These genotypes exhibit competitive yields under such conditions (24.64 and 26.53 t ha −1 of FRY, respectively), but they do not respond significantly to environmental improvements.This recommendation takes into account parameters such as regression deviation ( S 2 d ), root mean square error ( RMSE ), and regres- sion coefficient of determination ( R 2 ) (Eberhart and Russell 1966).However, it is important to note that all these genotypes were associated with the dynamic concept of stability for DMC, despite the formation of distinct groups.
It is important to note that both the concept of stability and the correlation between stability and adaptability methods may vary depending on the specific characteristic being evaluated.However, utilizing methods with diverse approaches and forming groupings provides greater confidence in selection, which should align with the objectives of the breeding program.It is worth mentioning that methods that quantify both stability and yield simultaneously tend to yield better results than those that solely provide information about stability.
Most parametric methods require the assumption of homogeneity of variance for analyzing genotypeby-environment interaction.In contrast, nonparametric methods are not bound by this assumption and therefore tend to reduce bias caused by outliers present in multi-environment trials (METs).Nonparametric methods also have little to no dependence on the distribution of the data, making them easily interpretable and complementary to use (Agustina et al. 2020).However, nonparametric models have limitations in multivariate analyses, and implementing biplot plots for better visualization of G × E interaction can be challenging (Agyeman et al. 2015).
The utilization of stability methods allocated in group I, such as HMGV and Si 6 , can assist cassava breeders in selecting the best genotypes in VCU trials for subsequent cultivation recommendations.On the other hand, methods more closely related to static stability can be employed in the selection of parents for breeding crosses (Kadhem et al. 2010).Considering the hypothesis that variance between genotypes can impact the discriminant power of the methods, both Pi and Si 6 can be used when there is greater heteroge- neity of variance among genotypes and environments.Conversely, parameters such as S 2 di, R 2 andRMSE , are more suitable when the variance is of low magnitude, as they provide a concise and reliable separation between biological and agronomic stability.
Selection via multivariate index AMMI x GGE x WAASB x HMGV for performance, stability, and adaptability An important strategy in breeding programs to address differences in genotype ratings is to explore stability and broad or specific adaptability, considering the significant nature of G × E interaction, through parametric (uni-and multivariate) and nonparametric approaches (Nassar and Huehn 1987).The combined use of these approaches has been widely explored in breeding programs due to their unique characteristics and benefits, as well as the graphical tools that facilitate efficient visualization of G × E.
Among the multivariate models, the AMMI is one of the main statistical tools for understanding G × E interaction.It separates and explores the main effects (additive variance) from the interaction (multiplicative) and identifies specific adaptations based on the magnitude and signs of the scores.Stable genotypes and environments are characterized by scores near zero on the IPCA1 and IPCA2 axes (Gauch andZobel 1996, Gauch Jr 2013;Silva and Benin 2012).On the other hand, the GGE method removes the effect of the environment and expresses the response solely as a function of the genetic effect (G) and G × E interaction.This methodology allows the identification of broad and/or specific adaptations based on the distance of the genotype (or environment) from the origin of the biplot.It also considers the existence of mega-environments, which are sets of environments that share similar soil and climate characteristics.The discrimination and representativeness of the test environments are indicated by the size and angle of the vectors (Yan and Tinker 2006).
Both the AMMI and GGE methods can explore variance and genetic correlations between pairs of genotypes and environments.They aim to distinguish the pattern of G × E interaction from random error and have proven to be efficient in capturing a high proportion of the SQ G×E (> 40%).They have also successfully identified genotypes that are stable and adaptable to different environments.However, in the present study, the GGE method explained a higher proportion of the total variance for FRY (75.27%) and DMC (90.66%) compared to the AMMI method, which captured 53.53% and 59.21% of the variance, respectively.Similar results were reported by Agyeman et al. (2015) when evaluating 10 cassava genotypes in six environments.These differences in explained variance may be attributed to: i) the lack of the inner product property of vectors in AMMI, ii) the fact that GGE does not directly explore the main effects, which may increase the influence of noise in the model due to G × E, resulting in a higher proportion of the SQ, or iii) the higher accuracy of the GGE model in detecting patterns of G × E (Agyeman et al. 2015;Yan 2016).
Previous multi-environment trials, both in the clonal selection phase of evaluation and VCU trials, have also reported higher sum-of-squares uptake in the GGE methodology (85%) compared to AMMI (72%) (Esuma et al. 2016;Rad et al. 2013).Considering simultaneous selection with G × E tending to zero, the AMMI and GGE methods identified the genotypes BR11-34-69 and BRS Novo Horizonte, respectively, as having the highest FRY and stability.Similarly, the BRS Novo Horizonte genotype showed the highest DMC and stability according to both the AMMI and GGE models, with Corrente also exhibiting similar characteristics according to the GGE method.However, it should be noted that a higher explanation of SQ G×E does not always confer superiority to the GGE method, as a greater capture of total variance does not necessarily represent higher accuracy in the selection of superior genotypes due to the presence of noise in the model (Kvitschal et al. 2009;Yan 2016).
In this study, equal weights (50%) were assigned to average performance and stability for both FRY and DMC by using the WAASB index.The WAASB index offers several advantages: it is based on mixed models, allowing flexibility in choosing economic weights based on breeders' interests; it enables the exploration of G × E interaction patterns using more PC; and it exhibits lower sensitivity to outliers inherent in METs trials (Olivoto et al. 2019).In the biplot, genotypes and environments are grouped into four categories: unproductive and unstable genotypes (Group 1), productive and unstable genotypes (Group 2, characterized by high G × E and discriminative environments), low yield (Group 3), and high yield (Group 4, with lower WAASB value, broad adaptation, and low discriminative ability) (Olivoto et al. 2019).In this scenario, BRS Novo Horizonte stood out in Group 4, as did the PP-2018 environment for both traits.
Lastly, the HMGV based on BLUPs offers several advantages, including greater flexibility in dealing with heterogeneity of variances and covariances between environments, elimination of G × E noise, results generated on the scale of the characteristic itself, and taking into account heritability in the estimation of the model with simultaneous selection for performance and genotypic stability.Therefore, when a genotype exhibits small deviations within and between sites, it is considered stable.Thus, HMGV highlighted the genotype BR11-34-41 and BRS Novo Horizonte as the most important genotypes for increasing FRY and DMC, respectively.

Characterization of contrasting environments and mega-environments via GGE
Environments vary in terms of the quality and quantity of resources available to plants, as well as climatic conditions, leading to different phenotypic expressions of genotypes due to unpredictable environmental factors (environmental covariates).Quantifying and exploring environmental variations are crucial for determining the performance profile of a genotype, as the realization of its productive potential depends on meeting ideal soil and climatic conditions (Malosetti et al. 2013).Identifying and classifying genotypes and environments suitable for cultivation is of fundamental importance in any breeding program, as it guides the regionalized recommendation of the best genotypes for target environments where inferences and predictions are valid and accurate (Van Eeuwijk et al. 2016).
In general, optimal environments should effectively discriminate genotypic effects while also being representative of mega-environments.Mega-environments are defined as growing regions with reasonably homogeneous characteristics, but they don't necessarily have to be contiguous areas that produce similar performance in genotypes (Yan and Rajcan 2002;Yan 2016).Within each mega-environment, the effects of G × E interaction are limited or not significant, indicating similar soil and climate conditions that lead to comparable phenotypic responses in groups of genotypes with angles smaller than 90° (within the megaenvironment).Angles greater than 90° indicate uncorrelated environments, and angles greater than 180° indicate negative correlations associated with distinct edaphoclimatic conditions.For example, a difference in performance was observed between UFRB-2019 and NR-2019 for FRY, although more than 70% of the environments showed a high correlation with each other (Yan 2016).
Climatic conditions tend to have a greater influence on the phenotypic expression of genotypes compared to soil conditions.This can be observed in the environments UFRB-2019 and NR-2019, which had distinct responses between traits, with long vectors in the AMMI biplot for FRY (indicating high instability) and short vectors in the GGE biplot for DMC (indicating high stability).Despite this, there was high homogeneity among environmental covariates such as temperature and precipitation, resulting in a high correlation among environments with weak or no G × E patterns (Fig. 1).In sorghum, Oliveira et al. (2020b) reported that historical climate data can guide the recommendation and classification of target genotypes and environments, as agronomic performance is affected by precipitation and temperature, leading to changes in classification.
The perpendicular lines drawn from the origin of the biplot in the mega-environments divide the polygon into sectors.Genotypes located at the vertices of the polygon can be the best performers if they are associated with one or more environments in that sector, or the worst performers if they have no environments in that sector (Yan 2016;Van Eeuwijk et al. 2016).When all evaluated environments fall within the same sector, it indicates that a single genotype performs well in all environments.For example, BRS Novo Horizonte was allocated at the vertex with the highest DMC values in all environments and can be considered a reference genotype for evaluation.Similar findings were reported by Adetoro et al. (2021), where DMC showed five sectors in all environments, and only one megaenvironment was identified, demonstrating the stability and high performance of this genotype and indicating a high correlation between environments.
Indeed, if the environments are located in distinct sectors in the biplot, it suggests the presence of more than one genotype with high yield in different environments.This is exemplified by the FRY trait, where two mega-environments were formed, and the genotype BR11-34-41 exhibited the highest root productivities in a total of nine environments.For instance, NR-2019 and UFRB-2018 showed long vectors (indicating high G × E interaction) and short vectors (indicating low G × E), respectively, demonstrating their discriminative and representative capacities (Yan 2016;Van Eeuwijk et al. 2016).
Recommendation of a genotype that combines high agronomic performance across multiple variables with stability and adaptability is a complex task.However, the present study indicates that the new cassava genotypes outperformed the control genotypes in the face of environmental changes.It is important to further explore the repeatability, reliability, and suitability of different methods to establish alternative approaches that ensure the broad applicability of these methods in multi-environment trials.

Future perspectives
The primary objective of the breeding program is to identify genotypes with superior performance and increased phenotypic stability.However, the METs trials exhibit an inherent characteristic of having an imbalanced number of genotypes evaluated across different years.This imbalance occurs due to the exclusion of genotypes with inferior performance and the inclusion of new clones in subsequent crop years.Consequently, traditional fixed effect methods demonstrate limited efficacy in such situations.To address this, the exploration of G × E interaction becomes more accurate by employing mixed models.Modern approaches, like the MPS and MTMPS indices (Olivoto et al. 2019), offer improved precision by considering both performance and stability across multiple traits.These indices are particularly valuable for identifying superior genotypes that exhibit desirable agronomic characteristics such as root nutritional quality and disease resistance.By incorporating diverse stability indices, these models contribute to the identification of genotypes that excel in various traits of agronomic interest.Such advancements aim to enhance the efficiency of recommending new genotypes while better understanding the influence of environmental factors, especially for complex traits with substantial environmental impact, such as FRY.

Conclusion
The recommendation of genotypes that exhibit high yield, stability, and adaptability across multiple traits is indeed a complex task due to the nature and significance of the G × E interaction.However, even in the final validation phase of genotypes, this study was successful in selecting genotypes with high agronomic performance and potential for improvement through selection.
The genetic parameters indicated that the best genotypes  had the potential to achieve yields greater than 32 t ha −1 and genotypes BR12-107-002 and BR11-24-156 had high dry matter content in the roots (> 37%).These genotypes, along with BRS Novo Horizonte, outperformed the reference varieties.Different methods used in the study showed that 72% identified BRS Novo Horizonte as the genotype with the highest stability and performance for dry matter content, while 47% identified BR11-34-41 and BR11-34-69 as potential genotypes for commercial release with high root yield and intermediate dry matter content.
The positive and highly significant correlations between parametric and non-parametric methods provided a better understanding of their ability to exploit the G × E. Four groups were formed based on the stability concept, with the first group highly correlated with the traits under analysis (agronomic stability).The genotype BR11-24-156 exhibited high static stability according to 50% of the methods (groups II, III, and IV), although it may not be recommended for favorable environments due to its low yield in those conditions.However, it can still be used as a parent in the breeding program.The multivariate approaches (AMMI, GGE, and WAASB) were efficient and complementary in studying the G × E and simultaneous selection.The genotype BR11-24-156 exhibited static stability and BRS Novo Horizonte showed dynamic stability for root yield.For dry matter content, the genotypes BR11-34-69 and BR12-107-002 exhibited static stability, while BRS Novo Horizonte showed dynamic stability.Therefore, these methods can be used in a complementary and simultaneous manner to select genotypes, thereby increasing the reliability of the breeding program in recommending superior genotypes for cassava genetic improvement.
analyzed the dataset and verified the analytical methods.E.J.O. and J.S.S.F.contributed to the interpretation of the results.J.S.S.F.took the lead in writing the manuscript.All authors provided critical feedback and helped shape the research, analysis and manuscript.

Declarations
Conflict of interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Fig. 2 A
Fig. 2 A Spearman's correlation coefficients and B biplot scatter plot of the first two principal components of adaptability and stability parameters for fresh root yield (FRY) evaluated in 12 cassava genotypes in multi-environments

Fig. 4
Fig. 4 Biplot of adaptability and stability values based of additive main effects and multiplicative interaction -AMMI A and genotype main effects plus genotype x environment inter-

Fig. 5
Fig. 5 Biplot of adaptability and stability values based of additive main effects and multiplicative interaction -AMMI A and genotype main effects plus genotype x environment inter-

Table 1
List of the 12-variety value for cultivation and use (VCU) trials carried in the tropical humid region of Brazil

Table 2
Description of the genotypes evaluated in 12-variety value for cultivation and use (VCU) trials evaluated in the crop value and use trials **CNPMF Embrapa Mandioca e Fruticultura, IAC Insituto Agronomico de Campinas, DP Public Domain, BRA Brazil

Table 4
Analysis of variance by the additive main effects and multiplicative interaction (AMMI) method and sum of squares of the genotype × environment interaction for fresh root yield (FRY) and dry matter content (DMC), evaluated in 12 cassava genotypes in 12 environments 1 GL: Degree of freedom ***Significant at p < 0.001% probability, by F test rĝg : Selective accuracy

Table 5
Estimates of variance components and genetic parameters for fresh root yield (FRY) and root dry matter content (DMC), in 12 cassava genotypes evaluated in 12 environments 2 g -genetic variance, 2 gxe -genotype × environments interaction variance, 2 gain (considering 15% selection intensity), GAM-genetic advance as a percentage of the mean

Table 6
Phenotypic adjusted means and ranking (within parentheses) obtained by ordering stability values for fresh root yield (FRY), obtained from the evaluation of 12 cassava genotypes, in 12 environments *Means followed by the same letter in the column belong to the same grouping by Scott-Knott test (p < 0.05); R = ranking; DP

Table 7
Phenotypic adjusted means and ranking (within parentheses) obtained by ordering stability values for dry matter content in roots (DMC), obtained from the evaluation of 12 cassava genotypes, in 12 environments *Means followed by the same letter in the column belong to the same grouping by Scott-Knott test (p < 0.05)