Theoretical basis for multi-environment trial data analysis using the ProbStab framework

In MET, the primary goal is to select genotypes that exhibit high yield and stable performance across diverse environments, while also accounting for uncertainty and risk in the selection process. Choosing and recommending genotypes for target environments always comes with hidden risks that plant breeders face [29].

In the ProbStab software package, uncertainty is quantified using bootstrap resampling. This method creates thousands of resampled datasets (equal to the number of bootstrap repetitions) to estimate the probability distributions of performance and stability indices for each genotype. Using this approach, breeders can quantify both the probability of genotype superiority and the associated selection risk, enabling decisions to be based on probabilistic evidence and stability, rather than relying on point estimates.

The non-parametric nature of QRF in ProbStab enhances the accuracy and reliability of genotype recommendations. To address the complexity inherent in GEI within MET, ProbStab integrates ML with stability analysis. The workflow encompasses the following steps: (1) Random Forest modeling of complex GEIs to predict genotype performance, and (2) Bootstrap resampling to estimate empirical probabilities of superior performance and/or stability of genotypes in the resampled datasets. This integrated approach enables the simultaneous evaluation of genotype performance and stability, providing a strong selection criterion for breeding programs (Fig 1).

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Fig 1. Overview of the ProbStab framework for probabilistic assessment of genotype performance and stability.

https://doi.org/10.1371/journal.pone.0352098.g001

Random forest modeling and performance prediction

To predict genotype performance in different environments, an RF model was used. This model builds multiple decision trees on bootstrap samples and random subsets of predictors and averages the predictions from all trees, which increases accuracy and reduces overfitting.

The predicted performance of genotype i in environment j is defined as [28]:

where f(⋅) is the non-parametric function learned by the RF, and is the residual error.

This algorithm represents an advanced extension of the RF, estimating the complete distribution of responses (in this context, performance) rather than solely the mean. Consequently, it facilitates the calculation of quantiles and the evaluation of predictive uncertainty, rendering QRF particularly suitable for analyzing variable or complex datasets.

Hyperparameter optimization and model calibration

Hyperparameter optimization was conducted using an exhaustive grid search combined with k-fold cross-validation (CV) to ensure robust parameter selection and to mitigate overfitting. Specifically, the training dataset was partitioned into k folds (for example, k = 5), where in each iteration, four folds were used to train the model and the remaining fold was used for validation. This process was repeated across all folds, such that each fold served as a validation set once.

For each candidate hyperparameter configuration in the search grid, comprising the number of randomly sampled predictors at each split (mtry) and the minimum node size (min.node.size), the model’s predictive performance was assessed via cross-validation. The configuration yielding the lowest prediction error and highest coefficient of determination (R²) was selected as the optimal set of hyperparameters. The split rule was fixed to “variance,” and bootstrap resampling was employed with a sampling fraction of 0.632, consistent with standard RF methodology [30].

It is generally recommended to employ a sufficiently large number of trees in the forest to stabilize predictions and reduce variance. The default value in ProbStab is 1000 trees, which balances predictive accuracy and computational efficiency. Users may adjust this parameter according to their data size and available computational resources [31–33].

Following hyperparameter selection, the final RF model was refitted on the full training dataset using the optimal parameter configuration. Predictive performance was then evaluated on the independent analysis dataset.

Evaluation of random forest model prediction performance

To evaluate the prediction performance of the RF model for estimating genotype performance, three common statistical metrics were used: Mean Error (ME), Root Mean Square Error (RMSE), and Coefficient of Determination (R²).

In addition to these numerical metrics, a density plot of predicted versus observed values was constructed to visually assess the agreement between predicted and real observations in the test data. These metrics were calculated for each fold and reported as an average to indicate the model’s generalization accuracy.

Quantifying prediction uncertainty of the model

To evaluate the uncertainty associated with RF predictions, QRFs were employed [34]. QRF estimates the conditional distribution of the response variable and allows the construction of prediction intervals (PIs) based on specific quantiles of this distribution.

To quantify prediction uncertainty, the 0.05 and 0.95 quantiles of the conditional distribution were used. The prediction interval width (PIW) is defined as: PIW = q0.95 − q0.05, where q0.95 and q0.05 represent the 95th and 5th percentiles, respectively. The PIW indicates the range of uncertainty around each prediction. These quantiles correspond to a 90% prediction interval, a standard level in ML uncertainty quantification that balances interval width and coverage probability [35].

To assess the accuracy of these uncertainty estimates, the Prediction Interval Coverage Probability (PICP) was calculated for different interval levels. PICP is the proportion of observed values that fall within the corresponding prediction interval. For example, for PI90, a well-calibrated model should have a PICP close to 0.90, meaning that approximately 90% of observed values lie within the interval [36].

Significant deviations between PICP and the nominal PI level indicate miscalibration: if PICP is lower than nominal, the model underestimates uncertainty, while if PICP is higher than nominal, the model overestimates uncertainty [16].

In addition to PICP, the Mean Prediction Interval Width (MPIW) is commonly utilized as a performance metric in statistical and ML models. MPIW reflects the model’s capacity to minimize prediction uncertainty and is calculated as follows [35]:

where n is the number of observations, and q0.95(x_i) and q0.55(x_i) are the 95th and 5th percentiles of the conditional distribution of the response trait for observation i.

The acceptable range for MPIW depends on the context and type of prediction. Generally, a smaller MPIW indicates more precise predictions, while a larger MPIW reflects greater uncertainty. As a guideline, if the MPIW is less than 10% of the observed range of the trait, it is considered good; values between 20–30% are acceptable, and values above 40% indicate that prediction intervals are excessively wide [37].

Probability of superior performance

Let B denote the total number of bootstrap iterations and G the complete set of genotypes evaluated. In each iteration, b ∈ {1, 2,  ..., B}, a dataset is generated by resampling the original data, and a prediction model is fitted. Based on model predictions, a subset T_b of top-performing genotypes is identified according to the specified selection intensity (e.g., 15%) based on predicted performance.

To determine whether a specific genotype g belongs to T_b, the following indicator function is used:

The probability of superior performance for genotype g is calculated as:

This metric represents the relative frequency with which genotype g appears among the top genotypes across all bootstrap iterations, thereby providing a non-parametric, robust, and reliable estimate of performance that accounts for sampling uncertainty and model variability. Higher P_g(p) values indicate superior predicted performance across diverse environments.

Probability of superior stability

A stable genotype is traditionally defined as one exhibiting lower variance in GE, which can be estimated using Shukla’s stability variance for each genotype. Shukla’s stability variance (σ²ᵢ), also known as the stability variance statistic, estimates the contribution of each genotype to the overall genotype × environment interaction. It is calculated as the squared deviation of each genotype’s performance across environments, adjusted for the interaction mean square. This parameter was chosen because it is computationally straightforward, widely used in plant breeding literature, and provides a genotype-specific stability measure that is independent of the overall mean [25]. In each bootstrap iteration, stable genotypes are those with the lowest stability variance values, forming a subset S_b of the genotypes selected at the specified selection intensity (e.g., 15%).

The indicator function for determining whether genotype g is among the most stable genotypes (S_b) is:

The probability of superior stability for genotype g is then calculated as:

This probability reflects the frequency with which genotype g appears among the most stable genotypes across bootstrap samples, thereby providing a robust estimate of stability that accounts for both sampling and model prediction uncertainty. It is particularly valuable for identifying genotypes exhibiting consistent performance under variable or unpredictable environmental conditions.

Joint probability of performance and stability

To estimate the probability that a genotype is simultaneously among the highest-performing and most stable genotypes, in each iteration, the subsets T_b and S_b represent the top-performing and most stable genotypes at the defined selection intensity. Their intersection is defined as:

The joint indicator function is

The joint probability of superior performance and stability for genotype g is:

This metric captures the simultaneous achievement of high performance and stability for genotype g across all bootstrap iterations, serving as a comprehensive index for selecting candidate genotypes best suited for breeding programs that target stable performance in variable environments.

Risk and potential of genotypes

Let B_CI denote the number of bootstrap iterations for evaluating genotype risk and potential. In each iteration, b ∈ {1, 2,  ..., B_CI}, a resampled dataset is generated, and the prediction model is fitted. Predicted genotype performance across environments is then computed.

Genotype “risk” is defined as an indicator of low performance. For each genotype g, in each bootstrap iteration, the lower 5th percentile of predicted performance is calculated:

where represents predicted performance in all environments for iteration b, and Q0.05 denotes the 5th percentile. Genotype risk is calculated as the average across all iterations:

A lower indicates greater sensitivity to environmental conditions and a higher likelihood of poor performance.

Genotype “potential” is defined as an indicator of high-performance capacity. For each genotype g, in each bootstrap iteration, the upper 80th percentile of predicted performance is computed:

The genotype potential is the mean across all bootstrap iterations:

Higher values indicate greater performance capacity.

The 5th percentile was chosen to represent the ‘worst-case scenario’ or lower bound of expected performance, which is consistent with risk-averse selection strategies in plant breeding [22]. The 80th percentile was chosen to represent high-performance potential while avoiding the instability of extreme upper quantiles (e.g., 95th percentile), which may be overly sensitive to outliers.

For each genotype, the probability of exceeding the 80th percentile threshold -based on the mean 80th percentile across all genotypes- is calculated:

The indicator function I (⋅) is:

This metric reflects the frequency with which genotype g surpasses the defined threshold, thereby assisting in the selection of genotypes with high, stable performance while accounting for sampling uncertainty and model variability.

Confidence interval estimation

The 95% confidence intervals (CIs) for the predicted performance of each genotype were estimated using a bootstrap approach. Let B_CI denote the number of bootstrap iterations. In each iteration, a resampled dataset is generated from the data with replacement, and a RF model is refitted using the optimized hyperparameters. The predicted performance for each genotype across all environments is calculated, and the mean predicted yield is recorded. After all iterations, the 2.5th and 97.5th percentiles of the predicted yields across bootstrap samples are used as the lower and upper bounds of the 95% CI, respectively, while the mean provides the average predicted performance. Genotypes are classified as “Above Average” or “Below Average” depending on whether their mean predicted yield exceeds or falls below the grand mean across genotypes.

This approach accounts for sampling variability and model uncertainty, thereby providing a robust estimate of prediction accuracy. The results can be visualized in a plot depicting the mean predicted yield, confidence intervals, and the grand mean reference, thereby facilitating the comparison of genotypes across multiple environments.

For comparing genotypes with each other or with a check, the confidence intervals offer a straightforward visual criterion: if the confidence intervals of two genotypes (or a genotype and a check) overlap, the difference is not statistically significant at the 95% confidence level.

Software implementation

The stable version of the ProbStab package will be available from CRAN (https://CRAN.R-project.org/package=ProbStab) and can be installed directly via the R console using the following command:

install.packages(“ProbStab”)

The development version of the package will also be available on GitHub (https://github.com/Mohammadreza-shiri/ProbStab) and can be installed with the devtools package:

# install.packages(“devtools”) # Run this line if devtools is not installed

devtools::install_github(“Mohammadreza-shiri/ProbStab”)

After installation, the package may be loaded using:

library(ProbStab)

Motivational example

The following example demonstrates the application of the ProbStab package. The dataset used in this example comes from a maize breeding program conducted in Iran. The final/VCU trial included 11 maize test hybrids evaluated across 20 environments (location-year combinations) using a Randomized Complete Block Design (RCBD) with four replications. The preliminary trials comprised 99 hybrids evaluated using an alpha lattice design with two replications at three locations, and the intermediate trials comprised 19 hybrids evaluated using an RCBD with three replications at four locations.

To load and prepare the data, two datasets are required. The first, the training dataset, contains all experimental data, including PYT, AYT, and final yield/VCU trials. The second, the analysis dataset, comprises only the data from the final/VCU trials. The training dataset, in addition to the analysis dataset, contains data from the preliminary trials and intermediate trials. Notably, the hybrids tested in the final/VCU trial are also included among those evaluated in the PYT and AYT trials.

Columns in the datasets are organized by environment, replication (rep), genotype, and yield, with genotype and environment defined as factors. Raw phenotypic data (plot-level yield) from each environment are used directly for model training and hyperparameter tuning. No prior adjustment (e.g., spatial correction or mixed-model adjustment) is applied to the data before RF modeling. The QRF algorithm learns environmental effects implicitly through the environment predictor variable, which is treated as a categorical factor. This approach maintains transparency and avoids additional parametric assumptions.

The QRF algorithm does not rely on any statistical design assumptions. It learns purely from the predictor variables (environment and genotype) and does not require information about experimental design, blocking, or replication structure. The replication column is included in the dataset because any experimental design (RCBD, lattice, alpha design, etc.) includes replication. Therefore, ProbStab can analyze data from any experimental design without modification. If a user has unreplicated data, they can simply place the constant value 1 in the replication column. The replication column is ignored by the QRF model and is only used for the optional classical ANOVA function (perform_anova), which is provided separately for users who wish to compare ML results with traditional analysis.

To execute the program, the function load_data is called to read and format the training and analysis datasets from the specified paths. This function prepares the data and returns it as a list.

In addition to the probabilistic ML pipeline, ProbStab also provides a classical ANOVA function (perform_anova) for users who wish to compare the results with traditional stability analysis.

Comparison ProbStab with ProbBreed

To benchmark ProbStab against an established probabilistic approach, a direct empirical comparison was conducted using the same maize dataset. ProbStab was run with 1000 bootstrap iterations (default setting in prob_stab_pipe, ci_perf_pipe, and risk_potential_pipe). To match this sampling effort, the ProbBreed package [25] was applied to the final/VCU trial data using a Bayesian multi‑environment model (genotype and environment as fixed effects, including genotype × environment interaction). Four MCMC chains were run, each with 500 iterations; the first 50% of iterations (250 samples per chain) were discarded as burn‑in, resulting in a total of 1000 posterior samples for inference. To ensure comparability, a selection intensity of 15% was used in both methods, matching the default setting in ProbStab (top_geno_frac = 0.15). The results obtained from ProbBreed are presented as highest posterior density (HPD) intervals for genotypic main effects (Supplementary S2 Fig), along with the probability of superior performance (Supplementary S3 Fig), the probability of superior stability (Supplementary S4 Fig), and the joint probability of superior performance and stability (Supplementary S5 Fig).

Hyperparameter tuning and model evaluation

In the first step of this analysis, after loading and preparing the data using the load_data function and introducing the training and analysis datasets, the QRF model was trained. Hyperparameter tuning via cross validation prevented data leakage, and the default number of trees was set to 1000 (user modifiable) for stable predictions. All these steps were carried out using the ml_tcp_pipe function (Box 1). Depending on the dataset and the breeder’s strategy, the user can change the default values and the input hyperparameter grid as needed (Box 2).

Optimal hyperparameters of RF for yield prediction

By running the ml_tcp_pipe function, the best model hyperparameters were obtained. These can be viewed with the command: print(analysis_results$hyperparameters); and are summarized in Table 1.

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Table 1. Final Hyperparameters Used for the Model.

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Another output of the ml_tcp_pipe function is the evaluation of the model performance for both the training and analysis datasets. These outputs can be obtained with the following commands (see Supplementary S1 Table for training data, and Table 2 for analysis data):

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Table 2. Model performance metrics on training and analysis data.

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print(analysis_results$training_performance)

print(analysis_results$analysis_summary_stats)

Table 2 shows the model performance metrics and the summary statistics of predictions on the analysis dataset. The Root Mean Square Error (RMSE) was 1.5927 t/ha, and the Mean Absolute Error (MAE) was 1.1796 t/ha. Both values were much smaller than the standard deviation of yield (3.8067 t/ha). The coefficient of determination (R² = 0.8525) indicated that the model explained approximately 85% of the observed yield variance, reflecting its high predictive power. Thus, model predictions were consistent with observations, and systematic errors in yield prediction were very small.

For evaluating prediction uncertainty, the PICP was 0.8920, indicating that over 89.2% of the observed data fell within the predicted intervals. This indicates accurate and reliable intervals. The MPIW was 4.7837, nearly 20% of the yield range. This indicates that the prediction intervals were relatively narrow and precise, making them suitable for practical applications in genotype selection.

A comparison of means showed that the observed yield mean (12.6736 t/ha) and the predicted mean (12.6718 t/ha) were almost identical, indicating unbiased predictions. Standard deviations (Observed = 4.1485 t/ha; Predicted = 3.8067 t/ha) showed that the model slightly underestimated the variability among genotypes and environments. Medians and quartiles of observed and predicted data also showed strong agreement, confirming that the model well captured the central tendency and distribution of yield. Overall, these results demonstrate that the model produced accurate and reliable predictions, effectively representing uncertainty and variability within the dataset.

For visualization, the density plot of observed versus predicted data can be generated using: print(analysis_results$analysis_density_plot) (S1 Fig). Note that for predicted data, only one prediction per genotype–environment combination was used (constant across replicates), whereas observed values vary across replicates. This provides a more conservative evaluation than using replicate means.

Finally, if the user approves the hyperparameters, another output of the model is a comparison table including observed data, predicted data, and prediction intervals (lower and upper bounds). This can be viewed with:

print(analysis_results$analysis_comparison_sample)(S2 Table).

Probability analysis of superior performance, superior stability, and joint probability

After modeling the MET data using ML algorithms and tuning the optimal hyperparameters, the predicted yield of each genotype was calculated. Based on these predictions, the mean yield, the probability of superior performance, the probability of superior stability, and the joint probability of both performance and stability were estimated for each genotype using the bootstrap method. These calculations were implemented through the function prob_stab_pipe (Box 3).

In this function, the argument model_results contains the final RF model, the optimized hyperparameters, and the outputs of the data analysis, which are automatically taken from analysis_results. The argument boot_reps indicates the number of bootstrap repetitions (default = 1000), and it can be adjusted by the user. The argument top_geno_frac defines the selection intensity, or the proportion of superior genotypes, with a default of 15%, which is also user-adjustable. After execution, the function calculates the probabilities of superior performance and superior stability, as well as their joint probability, for each genotype. The results are presented in a formatted summary table using the command:

print(stability_results$final_summary_table_kable) (S3 Table).

The following plots can be generated using the respective print commands:

• Mean predicted yield of genotypes, obtained with print(stability_results$plot_yield) (Fig 1);
• Probability of superior performance, obtained with print(stability_results$plot_prob_performance) (Fig 2);
• Probability of superior stability, obtained with print(stability_results$plot_prob_stability) (Fig 3);
• Joint probability of superior performance and stability, obtained with print(stability_results$plot_prob_joint) (Fig 4);
• A biplot of the probability of superior performance (x-axis) versus the probability of superior stability (y-axis), obtained with print(stability_results$plot_Superior_performance_prob_vs_Superior_stability_prob) (Fig 5).

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Fig 2. Probability of superior performance across environments.

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Fig 3. Probability of superior stability across locations.

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Fig 4. Joint probability of superior performance and stability.

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Fig 5. Biplot of superior performance probability versus superior stability probability.

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These visualizations provide a clear and user-friendly display, helping plant breeders to select superior genotypes more effectively by combining performance and stability information.

Based on the selected model, the probabilities of superior performance and stability were calculated with a selection intensity of 15% to increase the mean yield of the selected genotypes. The results, presented in Fig 2, show that hybrids H03 and H02 had the highest probability of superior performance (1.00 and 0.98, respectively). Hybrids H04 and H07 showed probabilities of 0.02 and 0.01, while hybrids H01, H05, H11, H08, H06, H09, and H10 were not selected as superior in any bootstrap sample. The 100% probability of H03 indicates no detected risk of low performance relative to others, while H02 shows a 2% risk under the chosen selection intensity.

The stability analysis (Fig 3) revealed a different pattern compared to performance. Hybrid H06 showed the highest probability of superior stability (0.69), indicating the most consistent performance across environments. Hybrids H02 and H03 followed with probabilities of 0.24 and 0.23, respectively. Other hybrids had very low or zero stability probabilities.

The joint probability analysis (Fig 4) integrated both performance and stability measures. The results revealed that H02 and H03 were the only hybrids with meaningful joint probabilities (0.25).

The biplot of superior performance probability versus superior stability probability (Fig 5) provided a visual representation of genotype classification. Hybrids H03 and H02 were clearly positioned in the upper-right quadrant, reflecting their high probability of superior performance along with moderate stability. In contrast, H06 was located higher on the stability axis but close to zero on the performance axis, confirming its strength in stability but lack of yield advantage. The remaining hybrids clustered near the origin, indicating negligible probabilities of achieving either criterion.

Confidence Intervals (95%) for predicted genotype performance

The function ci_perf_pipe was designed to calculate the 95% confidence interval (CI) of predicted genotype performance using the bootstrap method (Box 4). Its arguments include model_results, which contains the final RF model, optimized hyperparameters, and analysis data (automatically taken from analysis_results), and stability_results, which includes the results of genotype stability analysis. The argument boot_reps_ci specifies the number of bootstrap repetitions for estimating the confidence intervals (default = 1000).

After running this function, the mean predicted performance of each genotype and its 95% confidence interval are estimated using bootstrap. The genotypes are then compared with the overall mean and classified into two groups: “above mean” and “below mean.” The results are shown graphically as mean points with error bars for the confidence intervals, along with a vertical line indicating the overall mean. This plot can be viewed by:

print(ci_results$plot_ci) (Fig 6).

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Fig 6. Confidence Interval of Predicted Performance for Each Genotype (95%, 2.5th and 97.5th percentiles).

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This visualization provides a clear tool for comparing genotype performance and checking the reliability of predictions within confidence intervals. It is useful for identifying superior genotypes across environments.

Fig 6 shows a caterpillar plot of the predicted mean performance and 95% bootstrap confidence intervals for candidate genotypes. The intervals demonstrate different levels of overlap among genotypes. Hybrids H03 and H02 had the highest predicted mean yields, while H10 and H09 showed the lowest. The CI bars of H02 and H03 did not overlap with most other genotypes (except H04), suggesting significant differences in yield performance. The ranking of hybrids based on probability of superior performance (Fig 6) was consistent with the ranking based on confidence intervals (Fig 6).

Risk and potential analysis of genotype performance in METs

The function risk_potential_pipe was developed to evaluate the risk and potential yield of genotypes across multiple environments using the bootstrap method (Box 5). Its arguments include model_results (the final RF model, optimized hyperparameters, and analysis data, automatically taken from analysis_results) and boot_reps_ci, which specifies the number of bootstrap repetitions for estimating the risk and potential indices (default = 1000).

After execution, the function provides quantitative indicators that assist in identifying genotypes combining low risk with high yield potential, which is particularly useful for breeding decisions under diverse environmental conditions. The analysis produces two main outputs. First, the risk analysis, expressed as the 5th percentile yield, estimates the lower bound of predicted yield and reflects the worst-case scenario for each genotype (Table 3). Second, the potential analysis, expressed as the 80th percentile yield along with the probability of achieving high yield, represents the expected performance under favorable conditions and highlights genotypes with the greatest potential for high yield (Table 4). Additionally, the analysis calculates the probability that a genotype surpasses the mean yield of the 80th percentile, providing further evidence for distinguishing superior candidates. The results are summarized in two output tables, risk_summary_table (Table 3) and potential_summary_table (Table 4), which can be displayed using the respective print commands in R.

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Table 3. Genotype risk summary (5th percentile yield).

https://doi.org/10.1371/journal.pone.0352098.t003

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Table 4. Genotype potential summary (80th percentile yield).

https://doi.org/10.1371/journal.pone.0352098.t004

The results (Table 3) showed that hybrids H03 and H02 had the highest lower-bound predicted yields (13.89 and 13.51 t/ha, respectively), indicating a low risk of yield reduction. By contrast, H10 and H09 showed the lowest lower-bound yields (11.05 and 11.59 t/ha), indicating a higher risk of poor performance. Genotypes with high 5th percentile yields are therefore considered safer options for stable production.

The results (Table 4) indicated that hybrids H02 and H03 had the highest potential, with 100% probability of achieving high yield. Hybrids H04 and H07 also showed high potential (0.99 and 0.94, respectively). In contrast, H06, H09, H10, and H11 showed almost zero probability of achieving high yield.

Classical ANOVA

The perform_anova function is designed to conduct a classical analysis of variance (ANOVA) on observed data from MET arranged in a randomized complete block design (RCBD). This function utilizes the columns yield, genotype, environment, and replication to estimate the effects of environment, replication within environment, genotype, and the genotype × environment interaction. In addition, it provides the percentage contribution of each source of variance (S4 Table).

To compare genotype means, the function automatically performs Fisher’s LSD test, and the results are presented in a table (S5 Table). This implementation does not introduce new methodology; it is included to provide a standard, convenient tool for performing ANOVA directly within the package, without requiring additional software (Box 6):

The outputs, including the ANOVA table and genotype mean comparisons, can be displayed with the following commands:

print(classical_anova_results$anova_table_kable)

if (!is.null(classical_anova_results$lsd_groups_kable)) {

print(classical_anova_results$lsd_groups_kable)

}

Comparison ProbStab with ProbBreed

The direct comparison between ProbStab and ProbBreed (Supplementary S2–S5 Fig) showed strong agreement for the top hybrids. Both methods identified H03 and H02 as the highest‑performing genotypes (ProbStab: 1.00 and 0.98; ProbBreed: 0.99 and 0.98). Both also gave low probabilities to H07 and H04. For stability, both ranked H06 first (ProbStab: 0.69; ProbBreed: 0.57). The joint probability of performance and stability confirmed H02 and H03 as the best compromise (ProbStab: 0.25 each; ProbBreed: 0.16 and 0.15). Overall, ProbStab provides a valid alternative to ProbBreed, with the advantage of integrating multi‑stage training data (PYT, AYT) for more precise estimates.