Отбор соев в Кении подкрепляется моделями многотраля и генотипного взаимодействия между окружающей средой.

Introduction

Soybean ( Glycine max (L.) Merr.) is one of the most important global crops, widely cultivated for vegetable oil, protein production, and animal feed 1 – 3 . In the recent years, soybean expansion has reached tropical and subtropical regions of Africa 4 , driven by initiatives such as the Pan-African Soybean Trials Network, which aims to evaluate and promote varieties adapted to local conditions 5 . In Kenya, soybean has emerged as an important alternative for agricultural diversification and a food security crop, highlighting the need to develop genotypes adapted to local edapho-climatic conditions. Conducting multi-environment trials (METs) is essential to identify genotypes with high performance and stability, as these can capitalize on genotype-by-environment (G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E) interaction across different scenarios 6 , 7 .

One of the main challenges breeders face in soybean improvement is the G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction, especially when selecting and recommending cultivars. This interaction can significantly alter the phenotypic expression of soybean traits, such as yield, disease resistance, and tolerance to abiotic stress. When G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction is at crossover, genotype rankings vary considerably across environments, complicating the precise identification of superior materials 8 , 9 . Therefore, quantifying and comprehensively understanding G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interactions facilitates the development of more efficient strategies, either through defining mega-environments for targeted selection or broad selection aiming at cultivars with consistently good performance across diverse environmental conditions 8 .

An effective way to investigate G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction is through reaction norm analysis, which models the phenotypic response of genotypes across an environmental gradient and allows identification of both plasticity and stability patterns 10 . For proper modeling of this interaction, approaches such as mixed linear models (MLM) are widely used to efficiently address these imbalances, enabling improved genetic evaluations and accurate predictions. MLMs effectively handle unbalanced data, optimize genetic value predictions, and capture different types of G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E expression, such as parallel responses (no interaction) or changes in scale and rank across environments 11 , 12 . They are also effective for multi-trait analyses, supporting more comprehensive genotype evaluations, especially when analyzing large datasets 11 .

Traditionally, soybean breeding programs have prioritized single-trait selection strategies, which often overlook the complexity of improving multiple traits simultaneously. This simplification can limit overall genetic gains, especially when traits differ in scale, heritability, and economic relevance 13 . To overcome these limitations, several multi-trait selection approaches have been developed, including the classical Smith-Hazel index 14 , the Pesek-Baker index 15 , and the multi-trait genotype-ideotype distance index (MGIDI) 16 . These methods allow for the integration of phenotypic data across environments and facilitate the identification of superior genotypes based on multiple selection criteria. However, reducing the subjectivity involved in assigning economic weights remains a key challenge. In this context, approaches such as the Desired Gain Index are essential, as they enable breeders to explicitly define the expected genetic gains for each trait, aligning selection more closely with breeding objectives.

With the support of Pan-African Soybean Trials Network, several African countries have either begun cultivating soybean or significantly expanded their evaluation areas to identify superior genotypes adapted to local conditions 5 , 17 . Given Kenya’s recent expansion and limited history of soybean cultivation, it is necessary to conduct targeted studies to identify high-yielding and stable genotypes. Therefore, this study aimed to: (i) quantify the magnitude of the genotype-by-environment (G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E) interaction for soybean genotypes cultivated under Kenya’s edaphoclimatic conditions; and (ii) perform simultaneous selection for multiple traits using data from multiple environments and cropping seasons.

Methods

Experimental site characterization and description

Figure 1 A sets the geographical context by mapping Africa’s biomes and pinpointing Kenya in the eastern part of the continent. The experimental sites in this study span distinct ecological zones with diverse environmental conditions. Busia and Siaya, in western Kenya, lie within the Tropical and Subtropical Moist Broadleaf Forest biome, which is characterized by high humidity, fertile soils, and evenly distributed rainfall favorable for agricultural practices. Nakuru and Njoro, situated centrally, fall within the Tropical and Subtropical Grasslands and Savannas biome, presenting fertile volcanic soils and seasonal climatic conditions conducive to agricultural experiments. Murang’a, Kirinyaga, and Kiambu, located in high-altitude regions, belong to the Montane Grasslands and Shrublands biome, which is characterized by moderate temperatures and soils rich in organic matter. Kibwezi and Taita Taveta, in the southeast, are positioned in transitional zones between humid biomes and savannas, exhibiting climatic variability that affects agricultural performance. Environmental characterization is supplemented by soil data (Fig. 1 B). Figure 1 C shows the delimitation of Kenya and its biomes, while Fig. 1 D presents the altitude map. All maps were generated in R software (R Core Team, 2025; version 4.5.1) 18 , using the packages sf 19 , terra 20 , ggplot2 21 , and rnaturalearth 22 . Fig. 1 ( A ) Shows the map of Africa with biome distributions across its entire continent, highlighting Kenya in East Africa. ( B ) Presents the soil classifications in Kenya, emphasizing soil variability in the studied regions. ( C ) Illustrates the diversity of biomes within Kenya. ( D ) Depicts Kenya’s altitudinal ranges, distinguishing lowland areas, transition zones, and mountainous regions, which directly influence climatic conditions and agricultural suitability. The geometric shapes represent the locations evaluated in Kenya, and the information in parentheses indicates the number of trials conducted at each location. All maps were generated in R Software.

Plant materials

The soybean genotype trials used in this study are part of the Pan-African Trials Network, which aims to accelerate the introduction of test varieties across Africa, the USA, Australia, and Latin America, thereby providing farmers with a broader seed selection. These genotypes, managed under the coordination of SIL 17 , consisted of cultivars and experimental lines developed by both public and private breeding programs from Africa, Brazil, and the United States. Field evaluations were conducted in Kenya during the cropping seasons from 2019 to 2023/2024, through international collaborations focused on identifying genotypes adapted to the diverse target population of environments (TPE) across the African continent. A total of 65 unique soybean genotypes (G001 to G065) were tested across nine multi-environment trial sites located mainly in western Kenya (Supplementary Table 1 ). It is important to note that the number 65 refers to the cumulative set of genotypes evaluated throughout the entire study period; each individual trial typically included a subset of approximately 30 genotypes, depending on the year and location. Some genotypes were discontinued over time due to poor adaptation to specific environments, making them unsuitable to remain in the Pan-African network. The trials were designed to capture agroecological variability representative of key soybean-producing regions in the country.

Experimental design and data collection

Each environment was defined by a combination of location and year, resulting in 19 trials (Supplementary Figure 1 ). The trials were conducted using a randomized complete block design (RCBD) with three replications and the experimental plots comprised of four (4) rows, each 5 m long, spaced at 0.5 m apart, with 25 plants per meter, resulting in a plot area of 10 m 2 . Data were collected in the two central rows, corresponding to a net plot area of 5 m 2 , for the agronomic traits; plant height at maturity (PH, cm), measured from the base of the stem to the apex of the main stem; number of days to maturity (NDM, days), recorded as the number of days from sowing to physiological maturity; and grain yield (GY, kg ha −1 ), calculated from the total grain mass harvested from the net plot area, adjusted to 13% moisture content. Pest and disease management followed standard agronomic recommendations for soybean production.

Statistical analysis

Estimates of variance components for each evaluated trait were obtained by the Restricted Maximum Likelihood (REML) method 23 . Prediction of genotypic values was carried out using Best Linear Unbiased Prediction (BLUP) 24 , accounting for the variance-covariance structure of the trials.

Single-environment trial analysis

The analysis of single-environment trials was performed using the following linear mixed model: 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{y} = \mu \textbf{1} + \textbf{X} \textbf{b} + \textbf{Z} \textbf{g} + \varvec{\epsilon } \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{y}$$\end{document} is the vector of phenotypic observations ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times 1$$\end{document} ); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} is the intercept; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{1}$$\end{document} is a vector of ones with dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times 1$$\end{document} ); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}$$\end{document} is the vector of fixed replication effects, with incidence matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{X}$$\end{document} of dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times p$$\end{document} ), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} is the number of levels of the fixed effect; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{g}$$\end{document} is the vector of random genotype effects, assumed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{g} \sim N(\textbf{0}, \textbf{I}\sigma _g^2)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _g^2$$\end{document} is the genotypic variance and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{I}$$\end{document} is an identity matrix of appropriate dimension; the incidence matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} has dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times v$$\end{document} ), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} being the number of genotypes. The residual errors are represented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\epsilon } \sim N(\textbf{0}, \textbf{I}\sigma _{\epsilon }^2)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\epsilon }^2$$\end{document} is the residual variance, with dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times 1$$\end{document} ). This model was fitted independently for each environment and trait to estimate the genetic effects within specific trial conditions.

We applied the likelihood ratio test (LRT) 25 to assess the significance of genotype effects according to Eq. ( 2 ): 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} LRT = -2(Log{\textit{L}} - Log{\textit{L}}_{f}) \end{aligned} \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document} represents the log-likelihood of the reduced model, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{f}$$\end{document} corresponds to the log-likelihood of the full model. The LRT value was compared against the chi-square distribution.

Multi-environment trial analysis

Analysis of multi-environment trials (MET) was performed using the following model (Eq. ( 3 )): 3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textbf{y} = \mu \textbf{1}_n + \textbf{X}_1 \textbf{a} + \textbf{X}_2 \textbf{b} + \textbf{Z} \textbf{g} + \varvec{\epsilon } \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{y}$$\end{document} is the vector of phenotypic observations ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times 1$$\end{document} ); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} is the intercept; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{1}_n$$\end{document} is a vector of ones with dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times 1$$\end{document} ); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{a}$$\end{document} is the vector of fixed environmental effects ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \times 1$$\end{document} ), with incidence matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{X}_1$$\end{document} of dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times a$$\end{document} ); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{b}$$\end{document} is the vector of fixed replication effects nested within environments ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$pa \times 1$$\end{document} ), with incidence matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{X}_2$$\end{document} of dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times p$$\end{document} ); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{g}$$\end{document} is the vector of random genotypic effects within environments, assumed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{g} \sim MVN(\textbf{0}, \Sigma _g \otimes \textbf{I}_v)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _g$$\end{document} is the genetic covariance matrix across environments with dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \times g$$\end{document} ), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{I}_v$$\end{document} is an identity matrix of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} representing the number of genotypes. The corresponding incidence matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}$$\end{document} has dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times v$$\end{document} ); \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\epsilon }$$\end{document} is the vector of residual errors, assumed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\epsilon } \sim MVN(\textbf{0}, \Sigma _{\epsilon } \otimes \textbf{I}_n)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _{\epsilon }$$\end{document} is the residual covariance matrix with dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document} ), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{I}_n$$\end{document} is the identity matrix of dimension ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document} ). n is the number of observations, p the number of replications, v the number of genotypes, and a the number of environments. This model was fitted separately for each evaluated phenotypic trait (PH, NDM, and GY), enabling the estimation of genotype effects and G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction across the conducted trials.

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}$$H_a^2$$\end{document} ) was estimated using Eq. ( 4 ). 4 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathop H\nolimits _a^2 = \frac{\mathop {\hat{\sigma }} \nolimits _g^2}{\mathop {\hat{\sigma }} \nolimits _g^2 + \mathop {\hat{\sigma }} \nolimits _{ge}^2 + \mathop {\hat{\sigma }^2_{\epsilon }}} \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma}^2_{g}$$\end{document} represents the genetic variance, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma}^2_{ge}$$\end{document} corresponds to the G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction variance, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma}^2_{\epsilon} $$\end{document} denotes the residual variance, all estimated from Eq. ( 3 ).

The coefficient of determination for G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction effects ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^2_{ge}$$\end{document} ) was calculated using: 5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} r^2_{ge} = \frac{\hat{\sigma }^2_{ge}}{\hat{\sigma }^2_g + \hat{\sigma }^2_{ge} + \mathop {\hat{\sigma }^2_{\epsilon }}} \end{aligned}$$\end{document} The G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E correlation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{ge}$$\end{document} ) was estimated using: 6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathop r\nolimits _{ge} = \frac{\mathop {\hat{\sigma }} \nolimits _g^2}{\mathop {\hat{\sigma }} \nolimits _g^2 + \mathop {\hat{\sigma }} \nolimits _{ge}^2} \end{aligned}$$\end{document} The genotypic coefficient of variation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_g$$\end{document} ) and the residual coefficient of variation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_r$$\end{document} ) were estimated using: 7 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} CV_g = \frac{\sqrt{\hat{\sigma }^2_g}}{\hat{\mu }} \times 100, \quad CV_r = \frac{\sqrt{\hat{\sigma }^2_{\epsilon }}}{\hat{\mu }} \times 100 \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mu }$$\end{document} represents the overall mean of the trials.

The CV ratio (coefficient of variation ratio) was used to assess the magnitude of genetic variation relative to residual variation. This ratio is defined as: 8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} CV_{ratio} = \frac{CV_g}{CV_r} \end{aligned}$$\end{document} The genetic correlation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{r}_g$$\end{document} ) between traits was estimated using the BLUPs of plant height (PH), number of days to maturity (NDM), and grain yield (GY). The correlation was calculated as: 9 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{r}_g = \frac{\hat{\sigma }_{gXY}}{\sqrt{\hat{\sigma }_{gX}^2 \hat{\sigma }_{gY}^2}} \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{r}_g$$\end{document} = estimated genetic correlation between traits \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y$$\end{document} ; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma }_{gXY}$$\end{document} = genetic covariance between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y$$\end{document} ; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma }_{gX}^2$$\end{document} = genetic variance of trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} ; \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma }_{gY}^2$$\end{document} = genetic variance of trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y$$\end{document} .

We estimated the stability of each genotype using the weighted average of absolute scores. These scores were derived from the singular value decomposition of the matrix containing the BLUPs for the G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction effects 26 . The WAASB \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_i$$\end{document} was computed as: 10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {WAASB}_i = \frac{\sum _{k=1}^{p} \left| \text {IPCA}_{ik} \times \text {EP}_k \right| }{\sum _{k=1}^{p} \text {EP}_k} \end{aligned}$$\end{document} where WAASB \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_i$$\end{document} represents the weighted average for the i th genotype. In this equation, IPCA \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_{ik}$$\end{document} corresponds to the score of the i th genotype on the k th interaction principal component axis (IPCA), while EP \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_k$$\end{document} denotes the proportion of variance explained by the k th IPCA. Lower WAASB \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_i$$\end{document} values indicate higher stability.

Olivoto et al. 26 proposed the WAASBY index, which enables simultaneous selection by weighting mean performance (Y) and stability (WAASB \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_i$$\end{document} ) based on Eq. ( 11 ): 11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {WAASBY}_i = \frac{(r Y_i \times \theta _Y) + (r W_i \times \theta _S)}{\theta _Y + \theta _S} \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$WAASBY_i$$\end{document} is the simultaneous selection index for the i th variety, balancing mean performance and stability. The parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _Y$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _S$$\end{document} represent the weights assigned to the response variable and WAASB \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_i$$\end{document} , respectively. The variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$rY_i$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$rW_i$$\end{document} correspond to the rescaled values of the response variable and WAASB \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_i$$\end{document} , respectively. The analysis was conducted using R Software 18 , employing the Metan package 26 .

Desired gain selection index

The Desired Gain Index was proposed by Pesek and Baker 27 to achieve predetermined gains for a set of traits while optimizing the overall selection response. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} be the total number of traits included in the index, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j$$\end{document} be the number of traits for which a targeted gain is desired. The phenotypic variance matrix of the traits in the index is represented as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {var}({X}) = P_{n \times n}$$\end{document} , while the covariance between X and the genetic values of the target traits is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {cov}(X, A) = G_{n \times j}$$\end{document} . Thus, the fundamental relationship of the index can be expressed as: 12 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{i}{\sigma _I} {\textbf {G}}^\top {\textbf {b}} = {\textbf {d}}, \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {d}}$$\end{document} represents the vector of desired genetic gains, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i$$\end{document} corresponds to the selection intensity, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _I$$\end{document} is the standard deviation of the selection index, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {G}}$$\end{document} represents the genetic covariances among traits, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {b}}$$\end{document} is the vector of selection index weights, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {G}}^\top$$\end{document} denotes the transpose of matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {G}}$$\end{document} . This methodology was later refined and formalized by Yamada et al. 28 and Itoh and Yamada 29 . The calculation of index coefficients follows the equation: 13 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textbf {b}} ={\textbf { P}}^{-1} {\textbf {G}} ({\textbf {G}} {\textbf {P}}^{-1} {\textbf {G}})^{-1} {\textbf {d}}, \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {P}}$$\end{document} represents the phenotypic variance matrix among traits, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\textbf {G P}}^{-1} {\textbf {G}})^{-1}$$\end{document} is the generalized inverse ensuring that the coefficients are adjusted to optimize the correlated response among traits. We employed the NLopt package 30 using the COBYLA method (Constrained Optimization by Linear Approximations).

We selected the top 20% of genotypes to calculate the genetic gain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\Delta G\%)$$\end{document} based on the Desired Gain Index, following Eq. ( 14 ). 14 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta G\% = \left( \frac{\hat{\mu _s } - \hat{\mu _o}}{\mu } \right) \times 100 \end{aligned}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mu _s}$$\end{document} represents the mean BLUP of the selected genotypes, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\mu _0}$$\end{document} is the mean BLUP of the evaluated population, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} is the phenotypic mean. We used the ASReml-R 31 package for the analysis.

Results

The trials were conducted in 19 distinct environments in Kenya. Among them, Bungoma had the highest concentration, covering 31.6% of the total, followed by Murang’a (21.1%) and Njoro (15.8%). The remaining 31.5% was distributed across the other six locations (Table 1 ). Approximately two-third (68.4%) of the trials were sown in the first growing season, which typically begins between April and August, while 31.6% corresponded to the second growing season, with a sowing window between September and January. Overall, the mean values for GY, NDM, and PH were similar between the two seasons, with respective averages of 1899 and 1794 kg ha −1 , 125 and 111 days, and 67.4 and 47.8 cm. Three trials were conducted at altitudes above 2000 m, exclusively in Njoro, while three others were established below 1200 m, in the locations of Kibwezi (1178 m), Kirinyaga (1159 m), and Busia (1010 m), representing an altitudinal variation of approximately 1150 m. Table 1 Summary of the overall means of grain yield (GY), number of days to maturity (NDM), and plant height at maturity (PH) in soybean trials across different environments in Kenya, Africa. Location Environment Year Season Trait PH (cm) NDM (days) GY (kg ha −1 ) Bungoma E01 2019/20 1 71.84 115 2634.09 Njoro E02 2019/20 1 107.18 182 2200.53 Kiambu E03 2019/20 1 62.23 135 1773.83 Kibwezi E04 2019/20 1 71.10 111 2331.35 Njoro E05 2020/21 1 105.15 189 1108.54 Kirinyaga E06 2020/21 1 61.16 113 1135.24 Muranga E07 2020/21 1 60.52 117 2622.62 Bungoma E08 2020/21 2 34.61 119 1212.83 Muranga E09 2020/21 2 45.96 107 2641.98 Nakuru E10 2020/21 2 53.03 122 1092.41 Siaya E11 2020/21 2 56.21 103 1186.63 Bungoma E12 2021/22 1 54.44 95 675.72 Njoro E13 2021/22 1 86.64 176 2004.27 Muranga E14 2021/22 1 57.29 104 1809.31 Bungoma E15 2022/23 1 41.44 90 2044.71 Bungoma E16 2022/23 2 44.61 101 1673.33 Muranga E17 2022/23 2 52.47 113 2954.25 Bungoma E18 2023/24 1 50.83 105 2710.91 Busia E19 2023/24 1 46.22 97 1630.31 Overall mean 62.35 120 1783.66

Based on the boxplot, the traits PH, NDM, and GY exhibited median values of 59, 111, and 1630, respectively (Fig. 2 ). For GY, the overall mean was 1783.66 kg ha −1 , with the highest yield observed in E17 (2954.25 kg ha −1 ) and the lowest in E12 (675.72 kg ha −1 ), demonstrating high variability among environments (Table 1 ). For NDM, the scenario is similar. At E15, the genotypes reached maturity early, around 90 days, while at E05 in Njoro, this stage was achieved 98 days later. Despite this fluctuation, the overall mean across all environments was approximately 120 days (Table 1 ). Finally, a lower mean PH was observed at E08, Bungoma, with 34.61 cm. In contrast, taller plants were depicted in trials at Njoro (E02 and E05), the region with the highest altitude in the experiment network, with 107.18 cm and 105.15 cm respectively. However, in this study, the overall mean recorded for soybean monogenic trait (PH) was 62.35 cm (Table 1 ). Table 2 Summary of the likelihood ratio test for plant height (PH, cm), number of days to maturity (NDM, days), and grain yield (GY, kg ha −1 ) in 65 soybean varieties evaluated across 19 yield trials in Kenya from 2019 to 2023/24 crop seasons. Source LogL LRT \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^2$$\end{document} PH NDM GY PH NDM GY PH NDM GY Gen -6372.90 -4984.30 − 12359.00 333.44 235.38 42.94 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2e^{-16}$$\end{document} *** \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2e^{-16}$$\end{document} *** \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.655e^{-11}$$\end{document} *** Env -6438.10 − 4974.40 − 12382.00 130.49 215.70 89.71 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2e^{-16}$$\end{document} *** \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2e^{-16}$$\end{document} *** \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.2e^{-16}$$\end{document} *** Gen \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} Env -6552.60 − 5136.80 − 12573.00 359.47 540.55 471.77 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2e^{-16}$$\end{document} *** \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2e^{-16}$$\end{document} *** \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.2e^{-16}$$\end{document} *** LRT (Likelihood Ratio Test), LogL (Log-Likelihood), ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^2$$\end{document} ) (Chi-Square Test), Gen (Genotypes), and Env (Environment)

Fig. 2 Boxplot for plant height (PH, cm), number of days to maturity (NDM, days), and grain yield (GY, kg ha −1 ) in soybean varieties evaluated in Kenya from the 2019 to 2023/24 crop seasons.

The LRT revealed significant effects ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p < 0.001$$\end{document} ) of genotype, environment, and G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction for all the three traits evaluated (Table 2 ). The estimated variance components indicated differences in the magnitude of genetic variability among the traits (Table 3 ). Phenotypic variance was higher for grain yield compared to PH and NDM. 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}$$H_a^2$$\end{document} ) revealed that most of the phenotypic variance is explained by genetic variation. The genotype-environment correlation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^2_{ge}$$\end{document} ) was high for all traits, indicating changes in the ranking of varieties across environments. For GY, the G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction was large, indicating meaningful changes in genotype performance ranking across environments, which reflects a strong crossover interaction pattern. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_g$$\end{document} indicated greater genetic variability for PH. Furthermore, high experimental precision was confirmed by the low values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_r$$\end{document} . The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_{ratio}$$\end{document} indicated that genetic selection will be efficient for PH and NDM ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_{ratio} > 1$$\end{document} ), whereas for GY, the residual variance exceeded the genotypic variance, making selection more challenging. Table 3 Parameters for plant height (PH), number of days to maturity (NDM), and grain yield (GY, kg ha −1 ) were estimated. Parameters Traits PH (cm) NDM (days) GY (kg ha −1 ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma ^2_g}$$\end{document} 4,281.30 70.42 75,785 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma ^2_p}$$\end{document} 778.1 991.00 928,856 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^2_{ge}$$\end{document} 0.211 0.275 0.486 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2_a$$\end{document} 0.976 0.972 0.842 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{ge}$$\end{document} 0.518 0.657 0.584 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_g$$\end{document} 26.70 7.010 15.500 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_r$$\end{document} 15.40 3.480 22.200 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_{ratio}$$\end{document} 1.740 2.010 0.699 These include genetic variance ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma ^2_g}$$\end{document} ), phenotypic variance ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\sigma ^2_p}$$\end{document} ), 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}$$H_a^2$$\end{document} ), the proportion of variance due to G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^2_{ge}$$\end{document} ), G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E correlation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{ge}$$\end{document} ), genetic coefficient of variation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_g$$\end{document} ), residual coefficient of variation ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_r$$\end{document} ), and the coefficient of variation ratio ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$CV_{ratio}$$\end{document} ).

Genetic correlation analysis indicated a moderate positive association between PH and NDM (r = 0.28), suggesting that taller plants tended to reach maturity later (Fig. 3 ). In contrast, GY showed a strong negative correlation with NDM (r = − 0.71), implying that late-maturing genotypes were generally less productive. Additionally, PH and GY exhibited a moderate negative correlation (r = − 0.42), indicating that increased plant height did not translate into higher yields (Fig. 3 ). These findings highlight the importance of balancing plant architecture and productivity in soybean breeding for Kenyan conditions. Fig. 3 Genetic correlations among soybean traits evaluated in Kenya during the cropping seasons from 2019 to 2023/24. The chord diagram illustrates the relationships between Plant Height (PH), Number of Days to Maturity (NDM), and Grain Yield (GY). The color gradient represents the strength and direction of the correlations: red indicates a strong negative correlation (−1), transitioning through orange (moderate negative), white (neutral, 0), cyan (moderate positive), and green for a strong positive correlation (+1). The width of the connections reflects the magnitude of the correlation. Only unique correlations are displayed to avoid redundancy.

Based on the BLUP values for the evaluated traits, specific genotype performance was observed (Fig. 4 ). For PH, G29, G11, G48, G07, and G01 exhibited the shorter heights. Regarding NDM, genotypes G10, G32, G09, G36, and G34 had a shorter cycle, maturing earlier than the others. For GY, the highest-yielding genotypes were G42, G26, G46, G35, and G53 respectively. In contrast, G21, G22, G23, and G24 were characterized by taller plant heights, longer growing cycles, and lower grain yield, making them less desirable for selection. Fig. 4 Visualization of BLUP values for soybean genotypes related to plant height (PH), days to maturity (NDM), and grain yield (GY). Darker shades of green indicate genotypes with more favorable BLUP values, while stronger shades of red represent genotypes with less desirable trait performance.

The Biplot analysis (Fig. 5 ) categorized genotypes and environments based on overall performance and stability. Quadrant I comprised of unstable genotypes in highly discriminative environments but with trait values below the overall mean. Quadrant II included unstable genotypes (high WAASB) with performance above the mean in environments with strong discriminative ability. In Quadrant III, genotypes exhibited high stability but low overall performance, while the environments had limited discriminative capacity. Quadrant IV encompassed high-performing and stable genotypes (low WAASB), making them the most suitable for traits targeted for improvement, such as GY. For PH and NDM, selection prioritized stable genotypes with lower trait values, highlighting G37, G58, G29, and G17 for PH and G10, G36, G32, and G13 for NDM. Regarding GY, the focus was on identifying genotypes within Quadrant IV to maximize productivity, with G42 as the highest-yielding genotype, exceeding the overall mean.

Concerning the environments, although E17 (Muranga, 1355 m, 2022/23) stood out for having the highest average yield among all experiments (Table 1 ), its position in the graph (Fig. 5 C) indicates a low ability to discriminate genotypes. On the other hand, E12 (Bungoma, 1517 m, 2021/22), which recorded the lowest average yield, was positioned in the first quadrant of the WAASB graphs for PH (Fig. 5 A) and GY (Fig. 5 C), suggesting that this region has a high discriminative ability for genotypes (except for NDM), a pattern similar to that observed in E05 (Njoro, 2162 m, 2020/21). A comparable performance was observed in E02 (Njoro, 2162 m, 2019/20), where the materials exhibited greater height compared to the others (Table 1 ). This environment also demonstrated a high discriminative ability due to its location in quadrant II. However, it may be less desirable, as the PH and NDM values in this environment were above the average (Fig. 5 ). Fig. 5 The figure presents scatter plots illustrating the relationship between different phenotypic variables and the weighted average of absolute scores in a plant breeding experiment. In panel 1( A ) the relationship between plant height (PH) and the weighted average of absolute scores is shown, while panel 1( B ) presents the relationship between days to maturity (NDM) and the weighted average of absolute scores. In panel 1( C ) grain yield (GY) is correlated with the weighted average of absolute scores. Green points represent the evaluated genotypes, whereas red points correspond to different experimental environments. The vertical and horizontal lines indicate the mean limits of the variables, allowing the identification of genotypes and environments with contrasting performances.

Twelve (12) genotypes; G42, G26, G46, G35, G53, G41, G12, G54, G39, G52, G02, and G37, were selected based on the Desired Gain Index. Among them, two genotypes, G42 and G26, exhibited the highest index values, being considered the most desirable ideotypes for soybean under Kenya’s diverse soybean growing conditions (Fig. 6 ). Fig. 6 The circular bar chart illustrates the selected (green) and non selected (red) genotypes. The bar length represents the genotype’s superiority across the three traits based on the desired genetic index.

The density distribution of selected and non-selected genotypes for the variables PH, NDM, and GY is presented in Fig. 7 . It is observed that the selected genotypes (in green) tend to shift compared to the non-selected ones (in red), reflecting the impact of selection based on the Desired Gain Index (Fig. 7 ). Specifically, the selected genotypes exhibited a genetic gain of 6.5% for plant height (PH, Fig. 7 A), 1.3% for days to maturity (NDM, Fig. 7 B), and 15.6% for grain yield (GY, Fig. 7 C). These results indicate that applying the index enabled the identification of superior genotypes based on the established selection criteria. Fig. 7 Density plot illustrating the distribution of phenotypic traits: ( A ) plant height (PH), ( B ) number of days to maturity (NDM), and ( C ) grain yield (GY). The red normal curve represents the mean of the original population ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _0$$\end{document} ), while the green curve indicates the mean of the selected population ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _s$$\end{document} ).

Discussion

This study evaluated 65 soybean genotypes across 19 environments in Kenya and revealed substantial variability for PH, NDM, and GY. The LRT demonstrated significant effects of genotype, environment, and G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction for all traits. High-performing and stable genotypes were identified using a multi-trait selection index, leading to genetic gains of 6.5% for PH, 1.3% for NDM, and 15.6% for GY. Twelve genotypes stood out for their superior performance and adaptability under Kenyan growing conditions. As part of the Pan-African Trials (PATs), our study applied multi-environment trials (METs) across a broad range of environments, enabling the simultaneous assessment of multiple traits and genotypes under contrasting conditions of climate, soil, altitude, and management. Such trials are essential for evaluating the reliability and performance of candidate varieties, as emphasized by Gesteira et al. 32 . The key objective in this context is to expand METs across different target populations of environments (TPEs), seasons, and years, while broadening the diversity of assessed genotypes and facilitating the identification and recommendation of those best suited to new agroecological zones 33 – 35 . However, a significant challenge in this process is the strong influence of crossover G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interactions, which alter the ranking of varieties across sites and complicate the selection of candidates that combine high performance and stability in agronomic traits, as highlighted in Table 2 .

Throughout the selection process, genotypes with poor adaptation to the TPEs, lower GY, or undesirable agronomic characteristics are systematically discarded, while new experimental genotypes are continually introduced into the evaluation pipeline. This dynamic reprocessing commonly leads to genetic and statistical imbalance in multi-environment trials, as highlighted by Araújo et al. 36 . Such imbalance becomes more critical in networks that involve highly contrasting environments, as is the case in Kenya. The strong environmental heterogeneity typically intensifies G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction, which poses a major challenge for breeders when selecting genotypes that combine high performance with stability. In particular, previous studies have demonstrated that key environmental factors–such as altitude, maximum temperature, and soil fertility–exert significant and distinct impacts on G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction patterns 37 . In this context, altitude has been consistently reported as a major driver of variation in soybean development, affecting phenological traits such as time to maturity and plant height. This is largely attributed to genotype \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} altitude interactions, which tend to have a stronger effect on phenotypic expression than genotype \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} latitude interactions 38 . Such environmental influences not only modify the absolute performance of genotypes but also contribute to crossover interactions, leading to changes in genotype ranking across environments. Similar patterns of significant G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction for soybean traits, especially grain yield, have been widely documented in studies from various countries and continents, including Africa, Asia, and the Americas 39 – 43 . In Africa, research has been more concentrated in Southern Africa 33 , 44 , as well as in other regions such as Ethiopia 45 , 46 and Nigeria 47 . These findings reinforce the importance of using METs combined with robust statistical models to properly account for environmental variation, improve genotype evaluation, and support more accurate selection decisions in breeding programs targeting diverse and challenging production environments.

The results indicate that the phenotypic variance was substantially higher for GY compared to PH and NDM (Table 3 ), highlighting the complex nature of this trait and its strong dependence on environmental conditions 48 , 49 . Moreover, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^2_{ge}$$\end{document} values were moderate to high across all traits (Table 3 ), confirming a strong G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction and considerable changes in genotype rankings across environments. Furthermore, the genotypic correlation among environments \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^2_{ge}$$\end{document} ranged from 0.518 to 0.657 (Table 3 ) for all traits investigated. A high \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{ge}$$\end{document} facilitates the identification of stable and superior genotypes, suggesting a consistent pattern across environments. Conversely, a low \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{ge}$$\end{document} presents a challenge for selecting such genotypes, as it indicates greater variability in their performance across different conditions 50 . Therefore, integrating strategies that explicitly account for G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E–such as METs–and incorporating multi-trait selection is essential for identifying genotypes not only with superior agronomic performance but also with broad adaptability and stability across diverse environments 51 . The high estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2_a$$\end{document} for all traits suggest that a large portion of the phenotypic variance is attributable to genetic factors, reinforcing the potential for genetic gain through selection and the feasibility of maintaining progress via recurrent selection 52 . In this regard, METs provide an optimal framework for assessing heritability–a key parameter reflecting the reliability of observed phenotypes–across varying environmental conditions, particularly through the application of eBLUP, which enhances the precision of its estimates and enables the accurate selection of genetically superior individuals 53 – 55 .

One of the major challenges in plant breeding is managing unfavorable correlations between traits, which can hinder progress simultaneous genetic improvement. This complexity was manifest in our study through the analysis of genetic correlations among the evaluated traits, where plant height (PH) showed a positive correlation with NDM (r = 0.28), suggesting that taller genotypes tend to exhibit longer growth cycles, which may be an undesirable combination in production systems that require early-maturing cultivars with low lodging risk. More notably, GY was negatively correlated with both PH (r = − 0.42) and NDM (r = − 0.71), signaling that taller or late-maturing genotypes generally yielded less, possibly due to prolonged exposure to adverse conditions during the reproductive phase 56 , similar findings were reported in soybean by Bhartiya et al. 57 , Nataraj et al. 58 , and Wang et al. 59 . This behavior was markedly evident in genotype G05, which exhibited the lowest overall yield (669.47 kg ha −1 ) while presenting the longest growth cycle (148 days to R8). These findings align with previous studies conducted in China, where architectural traits such as plant height, number of nodes, and branches showed significant correlation with maturity stages, supporting that these genetic relationships, often driven by pleiotropy or linkage, impose constraints on the simultaneous optimization of desirable traits 60 .

Further supporting these results, the breeder’s equation, expressing genetic gain as the product of selection accuracy, genetic variation, and selection intensity over time 61 , highlights how such unfavorable correlations can diminish long-term breeding efficiency 53 , 61 . In your database, PH showed the highest CV \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_g$$\end{document} , indicating a greater genetic diversity and higher proportional gain potential, while NDM had the lowest CV \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_r$$\end{document} , reflecting a higher experimental precision in its measurement. The CV \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phantom{0}_{ratio}$$\end{document} values greater than 1.0 for PH and NDM points to a predominance of genetic over residual variance, whereas a value below 1.0 for GY suggests stronger environmental influence, potentially limiting the accuracy of selection for this trait. To neutralize these negative correlations, breeders may imposed other selection methods such as independent culling, tandem selection and index selection, enabling the simultaneous improvement of multiples features.

In this context, given the high G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E interaction detected in our work (Tables 2 and 3 ), the use of traditional frequentist models, such as ANOVA based on the ordinary least squares method, may present restrictions, particularly under the genetic imbalance and environmental heterogeneity of METs. To address these bottlenecks and improve the accuracy of genotype assessment, we employed the WAASB model, which uses a linear mixed-effects model that effectively captures the G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} E patterns. Unlike other stability statistics such as AMMI Stability Value (ASV), its main advantages include the robust estimates using the weighted average of absolute deviations of the interaction principal components (IPCA) scores,instead of squared deviations, making it less sensitive to outliers 26 , 62 . This was crucial in our dataset, where significant crossover interactions were present and genotype varied across environments. The WAASB method enabled the identification of varieties such as G42 that merge high performance and broad adaptability (Fig. 5 ), confirming its suitability for stability analysis in the context of METs 63 – 65 .

The integration of biplots combining WAASB and GY (or another trait) facilitates the visualization of genotypes that achieve a balance between high mean performance and stability (Supplementary Figure 2 ), as observed in the present study with genotype G42 (Fig. 5 ). In addition to GY, other agronomic traits such as PH and NDM may also contribute to genotype selection. In this regard, genotypes showing relatively lower PH and shorter NDM–such as G37, G58, G29, and G17 (for PH), and G10, G36, G32, and G13 (for NDM)–could be of interest, as these traits are often associated with improved agronomic performance, better harvestability, and potential adaptability to environments with shorter growing seasons or stress-prone conditions 66 . With respect to the environments, E15 (Bungoma, 1526 m, 2022/2023) stands out not only for its greater earliness compared to the others but also for its strong discriminative ability for GY and NDM, like E12 for PH (Fig. 5 ). This facilitates genotype selection in these regions, showcasing their necessity in soybean METs 45 , 58 . Thus, WAASBY proved to be an effective tool for selecting genotypes with desirable trait values and broad adaptability, as reported in previous studies on soybean 33 , 47 , 58 , 67 , 68 .

To address this complexity and enhance selection efficiency, breeders commonly adopt selection indices, which consolidate multiple traits into a single value per genotype. This strategy enables more efficient and integrated selection decisions 16 , 69 . The choice of selection index depends on specific breeding objectives, trait heritability, population structure, and developmental stage of the program. In this context, selection indices have proven to be effective in various crops such as soybean 32 , 70 , 71 , common bean 72 , and wheat 73 , with their suitability shaped by genetic correlations and breeding strategies 74 . Among the various methods, we adopted the Desired Gain Index (DGI) proposed by Pesek and Baker 15 , which is particularly valuable due to its flexibility in specifying expected genetic gains for each trait. Empirical studies confirm the superiority of index-based approaches over direct or single-trait selection, especially when dealing for complex traits like grain yield 70 , 73 . In our study, the selection intensity of 20% applied to the calculation of the Desire Index enabled the identification of 12 desirable genotypes, from which G42 and G26 standing out in decreasing order of relevance. Unlike the Smith-Hazel index, which focuses on profit maximization, our approach aimed to enhance the expected genetic gains 75 by selecting varieties with high performance and stability across the 19 environments evaluated in Kenya, Africa.

As a result, we observed genetic gains of 1.3% for NDM, 6.5% for PH, and 15.6% for GY, ensuring continuous progress in population improvement over time 75 and allowing for the recommendation of specific genotypes for different specific regions. Taken together, we demonstrated that the WAASB model efficiently selects soybean genotypes with high stability and broad adaptation 26 to Kenyan agro-climatic conditions. Moreover, we showed that the Desired Gain Index is a robust and effective tool for identifying superior ideotypes, particularly in multi-trait selection scenarios. This approach reduces the inherent subjectivity associated with assigning economic weights to each trait 27 – 29 . Our primary goal was to simultaneously evaluate grain yield, plant height, and physiological maturity, to facilitate an objective and accurate selection of ideotypes best suited to the diverse local agroecological conditions in Kenya. This strategy has the potential to directly benefit small and medium-sized Kenyan farmers by enabling them to incorporate genetically superior and locally adapted materials into their production systems. Thus, prior knowledge of genotype performance and stability empowers farmers to make strategic and informed decisions in improving their agricultural sustainability and profitability while mitigating the risks associated with climate change, which is increasingly becoming a global agricultural crisis.

Conclusion

This study demonstrated a strong genotype-by-environment interaction for soybean under Kenya’s diverse edaphoclimatic conditions. Multi-trait selection enabled the identification of superior genotypes, with genetic gains of 6.5% in plant height, 1.3% in days to maturity, and 15.6% in grain yield. Genotypes G42 and G26 stood out as the most desirable ideotypes for Kenya, offering promising options for advancing soybean breeding in the region.

Supplementary Information

Supplementary Information 1. Supplementary Information 2. Supplementary Information 3.

The original online version of this Article was revised: In the original version of this Article José B. Pinheiro was omitted as a corresponding author. Correspondence and requests for materials should also be addressed to jbaldin@usp.br.

Publisher’s note

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André A. Stella and João P. S. Pavan contributed equally to this work.

Change history

10/22/2025

A Correction to this paper has been published: 10.1038/s41598-025-24771-5

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-10654-2.

Acknowledgements

This research was supported by the Coordination for the Improvement of Higher Education Personnel (CAPES) and São Paulo Research Foundation (FAPESP, Grant 2024/01868), National Council for Scientific and Technological Development (CNPq), Soybean Innovation Lab (SIL), and Luiz de Queiroz College of Agriculture of the University of São Paulo.

Author contributions

Conceived the experiment(s): [A.A.A., J.P.S.P., E.P.L.]; Conducted the experiment(s): [P.G., G.C., M.F.S., T.G., J.N]. Provided insights into the methodology: [M.S.A., B.F.F., B.W.D.]; Analyzed the results: [A.A.A., J.P.S.P., M.S.A.]; Writing—review and editing: [A.A.A., J.P.S.P., M.S.A., I.V.U.]; Provided critical revisions of the paper drafts [I.V.U., J.B.P.]. All authors reviewed and approved the final version of the manuscript.

Data availability

The phenotypic data analyzed during the current study are available from the supplementary material.

Declarations

Competing interests

The authors declare that they have no conflicts of interest.

Ethical approval

The plant material comprises soybean varieties cultivated in Africa by the Soybean Innovation Lab (SIL). We complied with all necessary regulations and guidelines required to conduct this research.

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