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Shapley values as metrics for studying genotype-by-environment interaction

This study demonstrates that Shapley values (SHAP) derived from machine learning serve as a promising complementary tool for plant breeding by effectively quantifying genotype adaptability and stability while providing integrated insights into environmental contributions to genotype-by-environment interactions.

Original authors: Luan Tiago Dos Santos Carbonari, Carlos Zacarias Joaquim Junior, Mauro Bitencourt Souza, Altamir Frederico Guidolin, Sydney Antonio Frehner Kavalco, Jefferson Luís Meirelles Coimbra

Published 2026-07-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Luan Tiago Dos Santos Carbonari, Carlos Zacarias Joaquim Junior, Mauro Bitencourt Souza, Altamir Frederico Guidolin, Sydney Antonio Frehner Kavalco, Jefferson Luís Meirelles Coimbra

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a coach trying to pick the best athletes for a sports team. You have a group of players (the genotypes) and you want to know who will perform best. But there's a catch: these players have to compete in different stadiums with different weather, grass types, and crowd noises (the environments).

Sometimes, a player who is a superstar on a sunny day crumbles in the rain. This is what scientists call Genotype-by-Environment Interaction (G×E). It's the reason why a crop that grows huge in one field might be tiny in the next.

For decades, plant breeders have used old-school math (like linear regression) to figure out who is "stable" (consistent) and who is "adaptable" (does well everywhere). But the authors of this paper argue that these old tools are like trying to measure a complex, swirling storm with a ruler. They miss the messy, non-linear details.

So, the researchers tried something new: Shapley Values.

The New Tool: The "Fair Share" Calculator

To understand Shapley values, imagine a group of friends playing a video game together to win a big prize. The total score they get depends on who is playing, what character they chose, and how well they work together.

The Shapley value is a math concept from game theory that answers one question: "How much did each specific person contribute to the final win?"

It doesn't just look at the final score; it calculates the "fair share" of the victory for every single player, considering every possible combination of teammates. If a player always helps the team win, they get a high "fair share." If they drag the team down, their share is negative.

How the Researchers Used It

In this study, the scientists used a computer model (an Artificial Neural Network) to predict how much common beans would grow in 12 different locations in Brazil. Then, they applied the Shapley "Fair Share" calculator to the results.

Here is what they found, translated into everyday terms:

1. Adaptability = The Average Score

The Concept: If a bean plant has a high positive Shapley value on average, it means it consistently adds points to the team's score, no matter where it's planted.
The Finding: Some beans, like SCS 204 Predileto, had high positive values. They were the "reliable superstars" that boosted the yield everywhere. Others, like BRS Campeiro, had negative values, meaning they tended to lower the team's score.
Simple Takeaway: The average Shapley value tells you if a plant is generally good at adapting to new places.

2. Stability = The Consistency of the Score

The Concept: If a player's contribution swings wildly from game to game (sometimes +100, sometimes -50), they are unstable. If they always give +50, they are stable.
The Finding: The researchers looked at how much the Shapley values varied for each plant. A plant with a small variation (low "swing") is stable.
Simple Takeaway: The "wobble" in the Shapley values tells you how reliable a plant is. If the wobble is small, the plant is a safe bet.

3. The Stadiums (Environments)

The Concept: Just as players have fair shares, the stadiums (environments) also have a fair share.
The Finding: Some locations, like Ponte Serrada, had positive Shapley values. This means the soil and weather there naturally helped the beans grow better. Other locations had negative values, acting like a "bad stadium" that made it hard for the plants to succeed.
Simple Takeaway: Shapley values let breeders see exactly which fields are naturally good and which are naturally tough.

Did the New Tool Work?

The researchers compared their new "Shapley" method against the old "Ruler" methods (like Eberhart & Russell and AMMI).

For Adaptability: The new method agreed perfectly with the old methods. If the old math said a plant was adaptable, the Shapley value said the same thing.
For Stability: The agreement was "moderate." They didn't always rank the plants in the exact same order. The authors explain this isn't a failure; it's because the new method sees stability through a different lens (looking at complex, non-linear patterns) while the old methods look at simple, straight-line patterns.
The Verdict: The Shapley method didn't replace the old tools; it acted like a complementary pair of glasses. It gave the breeders a clearer, more detailed picture of why a plant performed the way it did, especially when the relationship between the plant and the environment was complicated.

The Bottom Line

This paper shows that using "Fair Share" math (Shapley values) helps plant breeders understand the complex dance between a plant's genes and its environment. It confirms which plants are reliable all-stars, which ones are risky, and which fields are naturally fertile. It's a new, smarter way to use Artificial Intelligence to help us grow better food.

Problem Statement
Genotype-by-environment interaction (G×E) is a fundamental challenge in plant breeding, where the differential performance of genotypes across varying environmental conditions complicates the selection and recommendation of superior cultivars. Traditional approaches to analyzing G×E rely on classical statistical methods, such as linear regression (Finlay and Wilkinson, 1963; Eberhart and Russell, 1966), variance decomposition (Wricke, 1962; Shukla, 1972), and multivariate models like AMMI (Gauch, 1988) and GGE Biplot (Yan et al., 2000). While these methods are established, they often depend on strict statistical assumptions, including effect additivity, normality, and homogeneity of variances. Furthermore, they may struggle to capture complex, nonlinear relationships inherent in biological systems. Conversely, machine learning models (e.g., Random Forest, neural networks) can model these complex interactions but are often criticized as "black boxes," lacking the interpretability required for biological insight and decision-making in breeding programs.

Methodology
This study proposes the application of Shapley values (SHAP), a concept derived from cooperative game theory (Shapley, 1953) and adapted for machine learning interpretability (Lundberg and Lee, 2017), as a metric for studying G×E interaction. The research utilized grain yield data from Value for Cultivation and Use (VCU) trials of eight common bean (Phaseolus vulgaris L.) genotypes evaluated across 12 distinct environments in Santa Catarina, Brazil.

The methodology involved the following steps:

Modeling: An Artificial Neural Network (ANN) using the MLPRegressor from the scikit-learn library was trained on the yield data. The model utilized one-hot encoding for genotypes, environments, and their explicit interactions to capture nonlinear patterns.
SHAP Calculation: SHAP values were computed to quantify the contribution of each feature to the model's predictions. The features were grouped into three sets: Genotype ( $G$ ), Environment ( $E$ ), and Genotype $\times$ Environment interaction ( $G \times E$ ).
Metric Derivation:
- Adaptability (A-SHAP): Defined as the mean SHAP value for a genotype across environments, representing its average contribution to yield.
- Stability (S-SHAP): Defined as the standard deviation of SHAP values for a genotype, reflecting the consistency of its contribution across environments.
- Environmental Contribution (E-SHAP): The mean SHAP value for each environment, indicating whether conditions were favorable or unfavorable.
Comparative Analysis: The SHAP-derived metrics were compared against traditional methods, specifically the Eberhart and Russell (1966) regression approach and the AMMI model. Consistency was evaluated using Spearman correlation, Kendall's concordance coefficient, and the Jaccard similarity index for genotype rankings.

Key Results

Adaptability and Yield: The mean SHAP value (A-SHAP) showed a strong association with mean grain yield (Pearson's $r = 0.96$ ) and the regression coefficient ( $\beta_i$ ) from the Eberhart and Russell method ( $r = 0.65$ ). Genotypes such as SCS 204 Predileto and IPR Campos Gerais exhibited positive A-SHAP values, indicating broad adaptability and superior performance.
Stability: The variability of SHAP values (S-SHAP) served as a stability metric. While S-SHAP showed moderate positive correlation with traditional stability indices (S²di and ASV), the agreement was not perfect (Kendall's $\tau$ ranged from 0.57 to 0.71). This suggests that SHAP captures a distinct aspect of stability, likely due to its foundation in a nonlinear predictive model rather than linear variance decomposition.
Environmental Characterization: The method successfully identified favorable environments (e.g., Ponte Serrada 16-17) with positive E-SHAP values and low variability, and unfavorable environments (e.g., Campos Novos 15-16) with negative values and high variability.
Integrated View: Unlike traditional methods that often separate main effects from interaction effects, the SHAP approach provided an integrated assessment, simultaneously quantifying the contributions of genotypes, environments, and their interactions to the final prediction.

Significance and Claims
The authors claim that Shapley values represent a promising, complementary tool for plant breeding programs, particularly for interpreting G×E interaction in the context of machine learning models. The study posits that SHAP metrics offer several advantages:

Interpretability: They transform "black box" neural network predictions into interpretable components (genotype, environment, interaction) expressed in original response units (yield), facilitating biological understanding.
Flexibility: The method is applicable to any predictive model and does not rely on the strict statistical assumptions (linearity, normality) required by classical methods.
Complementarity: While SHAP-based adaptability metrics align closely with traditional measures, the stability metrics offer a different perspective. The authors argue that this divergence is not a limitation but a form of complementarity, providing multiple viewpoints on the same phenomenon.
Decision Support: By enabling the simultaneous evaluation of genotype performance, stability, and environmental quality, SHAP values can support more informed decisions regarding genotype recommendation, whether for broad adaptation or specific environmental targeting.

The paper concludes that Shapley values do not replace traditional methods but extend the analytical repertoire of plant breeders, enhancing the accuracy and efficiency of cultivar recommendation by integrating artificial intelligence with interpretable statistical metrics.

The New Tool: The "Fair Share" Calculator

How the Researchers Used It

Did the New Tool Work?

The Bottom Line

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