To evaluate the robustness and adaptability of the proposed OEDE model, we used four widely used public medical datasets from the UCI Machine Learning Repository, with different class Imbalance Ratios (IR), as shown in Table 2. These datasets were carefully chosen to cover a broad range of real-life situations where minority class data are clinically significant. The Pima Indiana Diabetes Dataset (IR = 1.89) includes the inception of diabetes in female patients based on diagnostic measurements. The Haberman’s Cancer Survival Dataset (IR = 2.78) has patient records for those who had undergone breast cancer surgery, aimed at predicting post-operative survival. The Thoracic Surgery Dataset (IR = 7.27) is based on predicting survival following major lung surgery for patients, considering clinical and surgical factors. Finally, the Cervical Cancer Risk Dataset (IR = 14.6) uses personal health information and screening results to assess the risk of cervical cancer. These datasets provide a range of imbalance ratios where the minority class proportion is less than 40%, offering a robust testbed for validating the effectiveness of the OEDE.

The proposed ensemble model leverages three different base learners, namely Logistic Regression (LR), Random Forest (RF), and XGBoost (XGB), with the aim of improving predictive performance based on model diversity. The LR is a linear model that predicts the probability of a binary outcome using a sigmoid function. ³¹ Its interpretability and probabilistic output make it a valuable baseline, particularly when modelling linear relationships. ^{32, 33} RF is an ensemble of decision trees that introduces non-linearity and robustness by aggregating predictions from a group of trees trained on bootstrapped datasets and random feature subsets, thus removing variance and overfitting. ³⁴ It performed well in capturing complex feature interactions. ³⁵ XGBoost builds a sequence of trees, where each tree corrects the errors made by its predecessors, optimizing a normalized objective function L = ∑ l l ( y i , y i ̂ ) + ∑ k Ω ( f k ) , where Ω ( f ) = γT + 1 2 λ ‖ w ‖ 2 . ³⁶ Its ability to model non-linearities with good precision, ³⁷ regularization, and the ability to handle missing values ²⁰ makes it a strong learner in the ensemble. Together, these models provide distinct perspectives, such as linear separability, variance minimization, and gradient-based optimization, making the ensemble more robust and less prone to overfitting than any single learner.

We used oversampling techniques, the Synthetic Minority Oversampling Technique (SMOTE), and Adaptive Synthetic Sampling (ADASYN) to address the class imbalance of the original dataset. We assessed the performance of our proposed ensemble model on three different datasets: the original imbalanced dataset, SMOTE-balanced dataset, and ADASYN-balanced dataset. SMOTE creates synthetic instances for the minority class through interpolation between minority samples and their k-closest minority neighbour, thereby expanding the decision boundary and reducing overfitting to specific samples. ³⁸ While SMOTE assumes the same significance to all instances of the minority class, ³⁹ ADASYN focuses on varying the importance of individual minority instances according to their level of difficulty in learning. ⁴⁰ It individually generates a synthetic minority sample, which is harder to classify owing to its lower density in the feature space, thereby promoting the generalization of the challenging part of the data. ⁴¹ Through this cross-dataset comparison of model performance, we aim to evaluate not only the generalizability of the proposed ensemble, but also how different balancing strategies influence its performance.

The Optimized Ensemble by Differential Evolution (OEDE) was designed as a novel ensemble model to alleviate the classification difficulties of imbalanced medical datasets. This method incorporates class-balanced base learners and a Differential Evolution (DE)-driven algorithm for AUC maximization. Three fundamentally different classifiers, LR, RF, and XGB, were used as base models. All base models were configured with a class-weighting mechanism during training to address the effects of class imbalance. LE employs class_weight= ‘balanced’, RF uses class_weight=‘balanced_subsample’, while XGB is fine-tuned for logloss with consideration taken for label imbalance. These learners were independently trained on the same dataset and produced probability estimates for the positive class that were combined using a learned ensemble set of weights. OEDE does not combine the outputs of the base learners using fixed or heuristic-based weights; instead, it uses Differential Evolution (DE) for adaptive weight optimization. DE is a stochastic population-based global optimization method that is well-suited for non-differentiable and non-convex functions. ^{42– 45}

Let the prediction probabilities from M base classifiers for a given instance x be denoted by [p ₁(x), p ₂(x), …, p _M(x)]. Assign weight w _i to each base classifier i, where the weights satisfy:

For the proposed ensemble, assign a weight w _i to each base model i such that: ∑ i = 1 M w i = 1 and w i ≥ 0 for all i (1)

The ensemble’s predicted probability is given by the weighted average: p ens ( x ) = ∑ i = 1 M w i p i ( x ) (2)

The goal was to choose the weights w = [w ₁, w ₂, …, w _M] to maximize the AUC on a validation set. Recall that the AUC represents the probability that a randomly chosen positive instance has a higher score than a randomly chosen negative instance. AUC = 1 | S + ‖ S − | ∑ i ∈ S + ∑ j ∈ S − I ( p ens ( i ) > p ens ( j ) ) (3) where S ⁺: index of positive instances, S ^-: index of negative instances, I: indicator function.

Because many optimization algorithms (such as Differential Evolution) are formulated as minimization problems, the loss function is defined as the negative AUC: L ( w ) = − AUC ( y val , p ens ( val ) ) (4)

Thus, the optimization problem becomes: min w L ( w ) subject to ∑ i = 1 M w i = 1 , w i ≥ 0 ∀ i (5)

DE is selected for this task because of its robustness in optimizing non-differentiable and non-convex functions such as AUC. DE evolves a population of weight vectors over generations using mutation, crossover, and selection to converge to a globally optimal solution. The proposed process is described by Algorithm 1.

We used five common yet important classification metrics such as Accuracy, Precision, Recall, F1-Score, and Area Under the ROC curve (AUC), to evaluate the performance of the proposed OEDE model. While working with the imbalanced dataset, these measures offer a comprehensive insight into the efficiency of the model. Let P _C, N _C, P _E, and N _E represent the numbers of correctly classified positives, correctly classified negatives, false positives, and false negatives, respectively. •

The model’s overall correctness is gauged by accuracy.

When it comes to an imbalanced dataset, the AUC is especially significant, which represents the area under the ROC curve by plotting the true positive rate ( P C P c + N E ) and false positive rate ( P E P E + N C ) across different classification thresholds. A higher AUC represents better separability. Finally, we visualized the ROC curve of each model, which offered an intuitive view of the trade-off between false alarms and sensitivity. The closer the curve is to the upper-left corner, the better is the classifier.

On the original dataset, OEDE achieved high accuracy and AUC score, as shown in Table 3, effectively discriminating between diabetic and non-diabetic samples without artificial rebalancing. Although ensemble models such as RF and XGBoost demonstrated competitive accuracy, their AUC and recall values were low, indicating a bias towards the majority classes. The OEDE tended to have a high F1 score, while maintaining a better balance between precision and recall, as shown in Figure 2. The overall performance of all models improved when the dataset was balanced using SMOTE, as indicated by the increased recall and F1-score, but OEDE still outperformed the others in terms of accuracy, AUC, and F1-score. Notably, the change in the AUC value was marginal, indicating that merely balancing the data may not be sufficient. In the ADASYN-balanced scenario, it tends to echo those of SMOTE but with minor instability for some models because ADASYN introduces noise in the minority samples. OEDE again outperformed other baseline models with a minor drop in precision while maintaining the AUC and showed a balanced performance throughout the datasets.

OEDE achieved a higher accuracy, AUC, and F1-score than traditional ensemble models such as RF, AB, and CB in an imbalanced dataset, as shown in Table 4. While some models show comparable accuracy, OEDE shows stability in precision and recall, leading to a better F1-score as shown in Figure 3, demonstrating its ability to detect the minority class without compromising the overall correctness. After the dataset is balanced with SMOTE, an improvement is observed in recall for most of the models, which shows that the models benefit from the synthetic sample. The OEDE maintained its AUC lead, highlighting its ability to influence informative patterns more effectively, even in a balanced dataset. Its high F1-score reflects food robustness against overfeeding for synthetic data. When ADASYN is used for data balancing, some models show instability in precision and recall, as ADASYN tends to produce harder-to-learn synthetic samples, while OEDE has a high-performance score without sacrificing stability and sensitivity.

The thoracic surgery dataset showed a significant imbalance in class distribution, challenging most of the classifiers that favour the majority classes. OEDE outperformed traditional ensemble models such as RF, LGBM, and BRF in terms of AUC and F1-score, as shown in Table 5 and Figure 4, while the original imbalanced dataset was used. Although many baseline models achieve relatively high accuracy, OEDE maintains its balanced performance, achieving higher recall for minority classes and offering better overall performance. With the SMOTE-balanced dataset, most models improve recall values owing to synthetic samples. OEDE continues to display a superior performance matrix, especially the AUC value, which indicates the robustness of the model even when the dataset is balanced with synthetic data without dropping precision or overfitting. Similar to the previous datasets, the performance of the baseline models fluctuated in the ADASYN-balanced dataset, but OEDE maintained a stable performance and showed high F1 and AUC scores. This generalizability and consistent performance are possible owing to the differential evaluation-based weight optimization.

The Cervical Cancer Risk dataset shows a significant difficulty owing to high-class imbalance, where high-risk cases are less than 7% of the dataset, which tends to affect the performance of the traditional model. On the imbalanced dataset, most baseline models, such as RF, ET, and CB, struggle with the minority class and show a low recall and F1-score, despite considerable accuracy, as shown in Table 6 and Figure 5. In contrast, OEDE is able to distinguish the minority class effectively owing to adaptive weighting, as reflected in the AUC and F1-score. Again, SMOTE improves the recall for all models, including OEDE, while maintaining competitive precision and achieving the highest AUC. For the ADASYN-balanced dataset, OEDE maintained high performance with the highest AUC and F1-score, supporting its resilience and generalizability.

The box plot overlaid with swarm points ( Figure 6) demonstrates the diversity of the different datasets. The median and IQR of AUC on the Cervical dataset presented the highest AUC scores (~0.95 median), demonstrating excellent and stable classification performance. Even for the Pima dataset, a moderate spread was observed, indicating reasonable generalization for the model. The Haberman and Thoracic datasets, on the other hand, demonstrated lower and more variable AUCs, around a median of 0.66–0.68, suggesting struggles possibly with class imbalance or limited separability.

The performance of the OEDE across the four benchmark datasets demonstrates its robustness and adaptability across balanced and imbalanced datasets. Despite varying levels of class imbalance, OEDE achieved superior performance, outperforming a range of traditional baseline models. To support the numerical findings, ROC curves were plotted ( Figure 7) for all the datasets to visually represent the discrimination ability of the model. Consistently across all datasets, the ROC curve of OEDE was favourable, aligning with the AUC values achieved.