Early Detection of Latent Tuberculosis Using Multi-Objective Hybrid Feature Selection and Complex Network Analysis

3.1. Tuberculosis Diagnosis by Analyzing the Gene Expression Data

In the study by Zhang et al. (2021), biomarkers for tuberculosis diagnosis were explored through Weighted Correlation Network Analysis (WGCNA) of 9451 genes in blood samples, revealing significant changes in tuberculosis patients. The Area Under the Curve (AUC) values for these genes were calculated for better feature selection. The analysis prioritized 30 key genes, with SAMD9L showing high diagnostic value (AUC = 0.925). Wu et al. (2021) focused on identifying biomarkers for tuberculosis using gene ontology and KEGG analysis, highlighting IRF1 as a potential marker with an AUC of 0.801. Natarajan et al. (2022) proposed a seven-gene signature to distinguish ATB from LTBI, demonstrating high diagnostic accuracy. Liu et al. (2023) utilized machine learning methods to develop predictive models for ATB diagnosis. Dai et al. (2024) identified six genes with strong diagnostic performance (AUC > 0.7) and employed RF, LASSO, and logistic regression for model development. Chen et al. (2022) identified five central genes through WGCNA, which showed high diagnostic accuracy (AUC 0.8–0.9) for distinguishing ATB from LTBI. Yu et al. (2024) used machine learning models, including RF and Support Vector Machine (SVM), to identify a seven-gene profile for ATB diagnosis in children (AUC = 0.888). Chen et al. (2023) developed a prediction model using multiple machine-learning classifiers, with SVM achieving the highest AUC for pediatric tuberculosis subtypes. Wang, Hua, et al. (2023) identified three autophagy-related genes, which accurately distinguished ATB from LTBI. Dong et al. (2024) applied WGCNA to identify key modules and lncRNA biomarkers, demonstrating high diagnostic accuracy (AUC = 1.000). Qiu et al. (2022) identified three upregulated genes (OAS1, IFIT1, IFIT3) with high accuracy in distinguishing ATB from healthy controls (AUC up to 0.975). Wen et al. (2022) identified several potential biomarkers such as CCL19, CCL5, C1Qb, and HLA-DMB for tuberculosis diagnosis. Sweeney et al. (2016) validated three genes (GBP5, DUSP3, KLF2) across datasets, showing strong diagnostic performance (AUC 0.84–0.90). Sun et al. (2019) identified genes that could diagnose tuberculosis at various stages. Bobak et al. (2019) compared machine learning models for tuberculosis classification using transcription biomarkers, including PLS-DA, SVM, and RF. Deng et al. (2019) identified 24 genes that predict ATB activation, with high recall (0.907) and accuracy (0.911). Ma et al. (2023) analyzed gene expression and identified key genes in the progression from LTBI to ATB, proposing retinoic acid as a potential treatment. Bian et al. (2017) used network-based analysis to identify diagnostic modules and pathways, enhancing the understanding of ATB. Bah et al. (2018) and Wang et al. (2018) examined gene expression in response to LTBI and ATB, highlighting minimal changes in LTBI infection. In our previous study (Ayalvari et al., 2024), the distinguishing feature selection for LTBI was conducted using data fusion. The novelty of that study lies in the use of correlation degrees derived from PPI networks within the DEMATEL multi-criteria decision-making method (Ayalvari et al., 2024).

3.2. Diagnosis of various Diseases Using Multi-Objective Optimization and Evolution Algorithms

The study by Shang et al. (2023), introduces DM-MOGA, a method aimed at identifying disease modules in gene co-expression networks relevant to non-small cell lung cancer. It utilizes a multi-objective optimization genetic algorithm to enhance community detection within these networks. By optimizing novel fitness functions that consider both local gene topology and interconnection strength, DM-MOGA effectively identifies core disease-relevant modules. Experimental results across various gene expression datasets validate its superiority over other advanced module identification methods, emphasizing its potential for biomarker discovery and deeper insights into lung cancer pathogenesis. DM-MOGA optimizes two novel fitness functions to identify disease-relevant modules by simultaneously considering the local topology of each gene and its connections with other genes. Pathway and gene ontology enrichment analysis confirms the association of identified core modules with lung cancer, underscoring the method's potential to enhance our understanding of the disease's underlying mechanisms. The study by Antonio Cappuccio et al. (2022), aims to identify and characterize a distinct biological fingerprint in the blood that reflects the host's response to COVID-19. This signature, identified using a multi-objective optimization approach applied to extensive public and new multi-omics data, operates at both transcriptional and epigenetic levels. The study validates this signature across multiple independent COVID-19 datasets, demonstrating its robustness and specificity compared to previously reported signatures. The interpretation of this signature highlights the roles of plasmablasts and memory T cells in COVID-19 detection and immunity, respectively, underscoring its potential for improving diagnostic tools and enhancing understanding of SARS-CoV-2 infection mechanisms. Song et al. (2018) modeled protein structure prediction as a multi-objective optimization problem. They present a three-objective evolution algorithm called AIMES. Protein contains amino acids. The composition of each amino acid is determined by the angles of the helix. This study uses the secondary structure information that can classify a residue in the protein sequence into three classes of secondary structure elements: helix (H), sheet (E), and coil (C). The constraints for backbone torsion angles are shown in a table. A random combination of each solution is constructed that satisfies the constraints of the Table. All answers are archived. Mutations are small changes in proteins caused by disruption of torsion angles. The two mutant solutions compete with each other, and the optimization algorithm begins to evaluate. The study of Rostamia et al. (2022), proposes a novel gene selection method for DNA microarray data classification using social network analysis with a focus on improving classification accuracy and reducing computational complexity. The method leverages community detection and node centrality to select relevant genes while minimizing redundancy. In each iteration, it identifies a maximum community and selects genes within that community based on their centrality. This approach, which is the first to apply community detection for gene clustering, automatically determines cluster numbers and effectively reduces redundant genes. Experimental results show that this method outperforms recent gene selection techniques in terms of classification accuracy, number of selected genes, and execution time. However, it is a multi-phase approach, which could slightly increase computational complexity on very high-dimensional datasets. Future work may focus on integrating the phases to enhance efficiency. In the study by Min et al. (2022), a two-stage gene selection framework named XGBoost-MOGA was proposed for cancer classification using microarray gene expression data. In the first stage, genes were ranked using ensemble-based feature selection via XGBoost to remove irrelevant features. In the second stage, a multi-objective genetic algorithm (MOGA) searched for the optimal subset of genes. Evaluations on thirteen public microarray datasets revealed that XGBoost-MOGA achieved superior performance compared with state-of-the-art methods in terms of accuracy, precision, recall, and F-score. In the study by Marjit et al. (2023), the Simulated Annealing aided Genetic Algorithm (SAGA) was introduced to address redundancy and dimensionality issues in gene expression data. The framework combines Genetic Algorithms with Simulated Annealing to balance exploration and exploitation during the search. Using the MICC measure for preprocessing and cluster-based population initialization, SAGA achieved improved classification accuracy and reduced gene subset size compared to traditional optimization-based selection methods. In the study by Li et al. (2023), a hybrid multi-objective approach named RMOGA (ReliefF Multi-Objective Genetic Algorithm) was proposed for tumor classification based on microarray gene expression data. In the first stage, ReliefF ranked and selected the top 5% most relevant genes. In the second stage, a multi-objective genetic algorithm optimized between classification accuracy and feature reduction. Experiments on eleven datasets demonstrated that RMOGA outperformed several advanced algorithms by achieving higher accuracy with fewer selected genes. Recent research has emphasized the use of evolutionary algorithms for selecting informative genes from high-dimensional microarray data. Nagra et al. (2024) introduced an improved Particle Swarm Optimization method, called Self-Inertia Weight Adaptive PSO (SIW-APSO), which adaptively adjusts the inertia weight to enhance exploration and avoid premature convergence. Their approach, combined with an Extreme Learning Machine (ELM), successfully reduced the number of selected genes while improving classification accuracy on several cancer microarray datasets. This work demonstrates the effectiveness of integrating optimized PSO-based feature selection with machine learning models for robust gene classification. Another notable contribution is presented by Yaqoob et al. (2024), who introduced a Two-stage MI-PSA gene selection method to improve the accuracy of cancer classification. Their approach combines Mutual Information (MI) as an initial filtering stage with Particle Swarm Optimization (PSO) to refine the selected features. The method effectively identifies highly informative genes and selects an optimal subset for classification. Experimental results showed that MI-PSA achieved a classification accuracy of 99.01% using only 19 genes, outperforming MI-based and SVM-based approaches. This study highlights the effectiveness of hybrid MI–PSO strategies for robust and high-performance gene selection in cancer prediction tasks. Li et al. (2024), introduced a multitasking multi-objective gene selection algorithm (MMODE) designed to enhance tumor recognition from high-dimensional microarray data. The method integrates evolutionary multitasking and multi-objective optimization, employing dual initial populations with different filtering criteria to increase search diversity. Through knowledge transfer between tasks and newly proposed elite and guidance strategies, MMODE effectively avoids local optima and identifies compact, highly informative gene subsets. Experiments on 13 public microarray datasets demonstrated that MMODE achieves higher classification accuracy while selecting fewer genes compared with several state-of-the-art gene selection algorithms. This study underscores the potential of multitasking evolutionary frameworks for robust gene selection in tumor classification. In recent years, hybrid evolutionary feature-selection techniques have shown strong potential for improving biomarker identification in high-dimensional biomedical datasets. For example, Yaqoob et al. (2024), proposed the Two-stage MI-PSO algorithm, which integrates Mutual Information (MI) with Particle Swarm Optimization (PSO) to enhance cancer-gene selection. In their study, MI is first employed as a filter to identify genes with high discriminative power, and PSO subsequently refines the subset to achieve an optimal feature set. Their method achieved a classification accuracy of 99.01% using only 19 genes, significantly outperforming baseline MI and SVM approaches. These findings highlight that well-designed evolutionary or hybrid evolutionary strategies such as PSO-based pipelines remain highly competitive for complex biomedical classification tasks, particularly when combined with complementary statistical filters.

3.3. Diagnose Various Diseases Using Different Feature Selection Methods

The study of Marques Alves et al. (2023), introduces a new approach for diagnosing Attention-Deficit Hyperactivity Disorder (ADHD) by analyzing event-related potential (ERP) microstates using complex network modeling. Traditional ADHD diagnosis methods rely on subjective questionnaires, which can be time-consuming, especially for children. In contrast, this method leverages the topological differences in ERP-microstate networks to distinguish between ADHD patients and healthy individuals. Using a neural network classifier, the study achieved high accuracy rates: 99.72% for binary classification (ADHD vs. healthy) and 99.31% for identifying ADHD subtypes. The topological features of ERP-microstate networks outperformed traditional methods like power band spectral density, wavelet transform, and temporal ERP features, suggesting a promising, objective biomarker for ADHD diagnosis. Shi et al. (2024), explore diabetic foot ulcers (DFUs) and their connection to glutamine metabolism by analyzing gene expression data. Through bioinformatics and machine learning, key glutamine metabolism-related genes (deGlnMRGs) associated with DFUs were identified, and their relationship with immune cell infiltration was examined. An SVM model, validated with multiple datasets, accurately predicts DFUs using five genes. The findings propose these genes as potential biomarkers and therapeutic targets, highlighting the role of immune-inflammatory cells in DFU development. In the study by Cheng et al. (2021), prognostic biomarkers for breast cancer were identified through a hybrid systems biology and ensemble learning approach using microarray gene expression data. To address the challenges of high dimensionality and low sample size, the authors integrated biologically informed feature selection with ensemble-based robustness enhancement. A total of 50 top-ranked gene features were selected to capture the complex molecular mechanisms underlying breast cancer. For prognosis prediction, several machine learning models were tested, among which the bimodal deep neural network (DNN) achieved the highest accuracy and was further validated through survival analysis. This study demonstrated the effectiveness of combining ensemble learning and deep neural architectures for robust biomarker discovery and precision medicine in breast cancer. In the study by Cilia et al. (2019), a large-scale experimental comparison was conducted to evaluate the impact of different feature-selection methods on DNA microarray classification for cancer research. Since microarray datasets contain a vast number of genes but few samples, the authors compared ranking-based and state-of-the-art selection techniques to identify the most effective strategies for distinguishing cancer types from healthy controls. The study provided a comprehensive overview of how feature selection influences classification accuracy in genomic data analysis. In the study by Azadifar et al. (2022), a graph-theoretic gene selection approach was proposed for cancer diagnosis based on microarray data. The method aimed to maximize gene relevance to the target class while minimizing redundancy among selected genes. By combining social network concepts such as the maximum weighted clique and edge centrality, the proposed technique effectively identified informative genes. Experiments on Colon, Leukemia, SRBCT, Prostate Tumor, and Lung Cancer datasets demonstrated that the approach outperformed well-known filter-based gene selection methods in diagnostic accuracy.

To contextualize the contribution of the present study, Table 1 provides a comparative summary of key related works. The main limitations of previous studies are highlighted, followed by the corresponding advantages of the proposed framework. This comparison emphasizes how the present method addresses gaps such as static feature selection, limited biological validation, and lack of multi-objective optimization.

Previous studies mainly suffered from limitations such as reliance on single-gene biomarkers, static and non-optimized feature selection, poor interpretability, and the lack of integration between biological analysis and machine learning models. Many of them used purely statistical or deep learning approaches that, despite their computational complexity, lacked biological validation (e.g., PPI analysis) and clinical generalizability. As a result, these methods often produced large, overfitted, and clinically impractical gene signatures.

In contrast, the present study introduces a multi-objective evolutionary optimization framework that combines NSGA-II, PSO, DE, ACO, and SA algorithms with real-coded representation and continuous compression to dynamically select the smallest subset of genes with the highest predictive accuracy. By employing entropy and MIM criteria along with RF evaluation, the proposed model effectively balances prediction performance and model simplicity. Moreover, through the design and analysis of PPI and Cytoscape networks, the selected genes are biologically validated and structurally interpretable. Ultimately, the proposed method identifies only five key genes, providing a compact, interpretable, and clinically feasible framework for the early discrimination of LTBI and ATB, while efficiently bridging computational optimization and biological understanding.

4.1. Feature Selection and Classifications

Machine learning algorithms are helpful tools for classifying transcription data. It can be used by applying machine learning techniques on gene sequences to predict and diagnose various diseases (Ullman, 2011).

4.1.2. Classification

RF processing is a managed or supervised classification and regression algorithm based on many decision trees. Each tree categorizes the data according to its label at each tree level by taking a random instance of data and features. The RF classifier can show good performance with a unique method of testing the model's accuracy called out of the bag. Before model training, several instances are selected for evaluation. This instance dataset that does not exist in the model training database is called out of the bag. Among the reasons for the popularity of this algorithm are the following (Díaz-Uriarte & Alvarez de Andrés, 2006):

If other models cause excessive data loss, an RF model is the best choice. In many machine learning algorithms, the variance decreases but the bias increases. To keep them down, the RF algorithm combines many decision trees to reduce noise.

The dependence of the RF model on noise in educational data is less than other models. As the forest building progresses, it creates an unbiased internal estimate of public error.

It has a helpful method for estimating lost data and maintains accuracy in losing a large part of data.

4.1.2.1. Random Forests

The RF method was applied to the three training datasets (GSE19491, GSE39939, GSE37250, GSE19439, GSE19442, GSE28623, and GSE39940.) to calculate the gene importance scores and identify the most informative genes associated with LTBI. Considering the inherent imbalance between the ATB and LTBI samples, a balanced RF sampling strategy was employed. In this approach, the majority class was randomly under-sampled in each iteration to match the minority class, ensuring that classes contributed equally to the model training. This method effectively reduced classification bias and enhanced the stability of the learned feature importance scores. Similar balancing strategies have been adopted in related studies using multi-objective and network-based analyses (Ayalvari et al., 2024; Azadifar et al., 2022; Cilia et al., 2019; Marjit et al., 2023; Rostami et al., 2022; Song et al., 2018). The performance of the RF classifier was examined under different tree depths ranging from 1 to 9, and the average accuracy across iterations (perf(it)) was used to ensure robustness and reproducibility. The classification accuracy was calculated according to Equation (1):

perf ( it ) = 100 * sum ( diag ( C ) ) / sum ( C ( : ) )

(1)

In Equation (1), C is the confusion matrix, and perf ( it ) indicates the performance of the classifier in each iteration. The final accuracy was computed as the mean of ( perf ( it ) ) values over all repeated runs, ensuring a stable estimation independent of random sampling effects.

In this study, cross-validation was not employed because the evaluation strategy was based on inter-dataset validation rather than intra-dataset resampling. Independent microarray datasets (GSE19491, GSE39939, GSE37250, GSE19439, GSE19442, GSE28623, and GSE39940) were used for training, while GSE19444 served exclusively as an external cohort for testing. This independent-cohort validation is considered more robust for gene-expression classification tasks, as it directly measures generalizability across studies with different experimental conditions (Mukherjee et al., 2003; Saeys et al., 2007).

4.2. Protein-Protein Interaction Network (PPI)

Recent studies have shown that abnormalities in the signaling pathways or disruptions in the interaction of pathway proteins may serve as biological indicators of diseases such as tuberculosis. The analysis of PPI is one of the most effective computational approaches for investigating functional relationships between proteins, often visualized through network-based methods. Each interaction within the PPI network is associated with a confidence score, which helps evaluate the reliability of protein associations and provides valuable insights into the biological functionality of identified genes (Hu et al., 2022; Hu & Chan, 2015; Wang, Yang, et al., 2023).

Equation (3) defines the feature penalty, which penalizes the number of selected features using the λ coefficient. This helps the model to select fewer features and avoid overfitting.

feature _ penalty = λ * num _ selected _ features

(3)

Equation (4) defines the model's cost as the sum of the prediction error and the feature penalty. This combination helps the model achieve a suitable balance between prediction accuracy and model complexity:

cost = prediction\; error + feature\_penalty

(4)

In equation (3), λ is the penalty parameter for the number of features. The higher its value, the more the algorithm tends to select fewer features .

The top 50 chromosomes are selected based on the RF classifier. Selecting 19 to 25 genes in each chromosome yielded better results.

In this study, eight different datasets were utilized: GSE19491, GSE39939, GSE37250, GSE19439, GSE19442, GSE28623, GSE39940, and GSE19444. To implement a code that not only uses prediction accuracy and the number of features as objective functions but also considers reducing the number of features (genes), a combined objective function can be used. In this objective function, in addition to prediction accuracy (or prediction error), the number of features should also be minimized to reduce model complexity. Given the high dimensionality of the datasets (48,803 and 47,323 genes), all available genes were incorporated into the initial stage of the multi-objective optimization process. To systematically include every gene, the datasets were partitioned into chromosomes, each containing 37 genes. This approach ensured comprehensive gene representation in the initial feature selection population of the genetic algorithm. Consequently, the GSE39939 and GSE37250 datasets were divided into 1279 chromosomes, whereas the GSE19444 and GSE19491 datasets were divided into 1319 chromosomes. Later, based on the analysis, it was determined that limiting the number of features to 19 to 25 genes per chromosome improved prediction accuracy and reduced error. This optimal feature selection effectively strikes a balance between model accuracy and its complexity.

In this stage, the best feature selection method is identified based on the model's prediction accuracy and various feature selection criteria (entropy, MIM, ANOVA, and t-test). For each method, the prediction accuracy and feature selection criteria are calculated, and the best feature selection method is determined. The objective function used is the same combined objective function as in Equation (2). The use of entropy and MIM feature selection methods as objective functions were identified as the best.

6.3. Step 3: Applying Four Evolutionary Algorithms

After selecting the best feature selection methods in the previous step, four different evolutionary algorithms SA, ACO, DE), and PSO were applied to identify the smallest subset of features that minimizes the MSE. In this stage, the objective function was designed to optimize not only the prediction cost (MSE) but also additional criteria such as entropy and MIM, resulting in a combined multi-objective optimization framework. The combined objective function is designed to minimize the number of genes in each chromosome, reduce prediction cost, decrease entropy, and maximize mutual information.

The objective function is a combined function that incorporates four criteria: prediction accuracy, entropy, MIM, and number of features.

Equation (5) defines the feature penalty, which penalizes the number of selected features using the λ coefficient.

feature _ penalty = λ * num _ selected _ features

(5)

Equation (6) defines the model's cost as a combination of prediction error, entropy, MIM, and the feature penalty. This objective function incorporates the weights α and β to balance the contributions of entropy and MIM, alongside the prediction accuracy and feature complexity.

cost = prediction\; error − α * entropy + β * MIM + feature _ penalty

(6)

In Equation (5), λ is the penalty parameter for the number of features. The higher its value, the more the algorithm tends to select fewer features.

α , is weight for emphasizing the reduction of entropy.

β , is weight for emphasizing the increase in MIM.

Hierarchical and cumulative clustering methods are used to identify a concise set of genes and gene pairs that effectively discriminate between LTBI and ATB states.

6.5. Step 5: PPI Network Design

A PPI network is designed for the features identified in Step 3 of the proposed approach.

6.6. Step 6: Cytoscape Network Design

○

The communication table extracted from the PPI is used to design the Cytoscape network. This study introduces a novel approach by utilizing PPI degrees to design the Cytoscape network.

○

For testing, GSE19444 is used independently, while the remaining datasets are employed for training purposes.

This method includes a variable binary encoding with continuous compression, which dynamically adjusts the number of features while simultaneously optimizing prediction accuracy. Figure 1 illustrates the schematic of the proposed methodology.

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Figure 1.

The schematic of the proposed method.

7. Experiments

In the rest of this section, the results related to the several steps of the proposed method are presented and discussed. To identify informative genes for LTBI classification, several common filter-based feature selection methods were employed, including the t-test, ANOVA (Ding & Li, 2015; Li et al., 2020; Yang et al., 2020), MIM (Chen et al., 2022; Clayton et al., 2017), Entropy, and Correlation Coefficient. These criteria were used individually at the feature level to select the most discriminative genes for subsequent analysis.

7.1. Evaluation Measures

To assess the performance of the proposed framework, two main evaluation metrics were employed: classification accuracy and MSE. Additionally, a confusion matrix was used to provide a detailed breakdown of model predictions for LTBI classification. The confusion matrix consists of four primary components:

True Positives ( T P ): LTBI samples correctly identified as LTBI.

True Negatives ( T N ): Non-LTBI (healthy) samples correctly classified.

False Positives ( F P ): Healthy samples incorrectly predicted as LTBI.

False Negatives ( F N ): LTBI samples incorrectly predicted as healthy.

These values form the basis for computing key classification metrics such as Accuracy, Precision, Recall, and F1-score.

Accuracy represents the proportion of correctly predicted samples ( ( T P + T N ) ) relative to the total number of samples ( T P + T N + F N + F P ) , as defined in Equation (7).

A c c u r a c y = TP + TN TP + TN + FN + FP

(7)

The P r e c i s i o n is defined in Equation (8).

P r e c i s i o n = T P T P + F P

(8)

The Recall is defined in Equation (9).

R e c a l l = T P T P + F N

(9)

The F1-score is defined in Equation (10).

F 1 − s c o r e = P r e c i s i o n × R e c a l l P r e c i s i o n + R e c a l l × 2

(10)

The MSE was also calculated to quantify the average squared difference between predicted and observed gene expression values, providing a complementary view of model performance. A lower MSE reflects a better fit between predicted and actual outputs, indicating a more reliable identification of LTBI-related genes, as described in Equation (11).

M S E = 1 n × ∑ i = 1 n ( y i − x i ) 2

(11)

Where n , is the number of samples, y i is the actual value for sample i, and x i is the predicted value for sample i. The MSE measures the average squared difference between the actual and predicted values.

7.2. Results and Discussion

The accuracy, Precision, Recall, and F-score for using several feature selection criteria and applying the RF, and SVM classifiers on the selected features are shown in Table 4. Table 4 is designed for the features identified in Step 1 of the proposed approach.

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Table 4.

Feature Selection Method Evaluated with RF, and SVM.

Test Dataset	Method	Classifier	Accuracy	Precision	Recall	F-score
GSE19444	MIM	RF	88.9	88.6	88.8	88.7
		SVM	87.1	86.8	86.9	86.9
	ANOVA	RF	89	88.7	88.8	88.8
		SVM	87.4	87	87.1	87
	t-test	RF	89	88.8	88.9	88.8
		SVM	87.2	87	87.1	87
	Correlation coefficient	RF	89.9	89.7	89.6	89.6
		SVM	88.1	87.7	87.9	87.8
	Entropy	RF	88.7	88.4	88.5	88.5
		SVM	87.2	86.8	86.9	86.8

Table 4 compares the performance of SVM and RF classifiers applied to various feature selection methods. As shown, RF consistently outperformed SVM in terms of accuracy, precision, recall, and F1-score. The superior performance of RF can be attributed to its ensemble structure, which reduces overfitting, captures complex non-linear relationships among genes, and handles high-dimensional data effectively. Based on these results, RF was selected as the final classifier in the proposed framework.

We employed a leave-one-dataset-out cross-validation strategy. In this approach, during each fold, one dataset is held out as the test set, while the remaining seven datasets are used for training. This process is repeated eight times, ensuring that each dataset serves once as the test set.

The performance metrics obtained from the eight folds are then averaged to provide a comprehensive and unbiased estimate of the model's generalization capability. The high consistency observed across the folds confirms the stability, robustness, and generalizability of the proposed model. The results are shown in Table 5.

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Table 5.

Performance Results Using Leave-one-Dataset-out Cross-Validation.

Method	Classifier	Accuracy	Precision	Recall	F-score
MIM	RF	88.45	88.18	88.30	88.25
	SVM	86.93	86.63	86.73	86.65
ANOVA	RF	88.63	88.33	88.43	88.40
	SVM	87.13	86.83	86.93	86.83
t-test	RF	88.73	88.48	88.63	88.45
	SVM	86.95	86.75	86.75	86.73
Correlation coefficient	RF	89.45	89.18	89.23	89.15
	SVM	87.90	87.53	87.65	87.58
Entropy	RF	88.20	88.33	87.08	88.25
	SVM	86.38	86.10	86.55	86.60

Based on the leave-one-dataset-out cross-validation results, the combination of the Correlation Coefficient feature selection method with the Random Forest classifier achieved the best performance across all metrics. Other methods, such as ANOVA, t-test, and MIM, also showed competitive performance but were slightly lower and less consistent. These findings indicate that Correlation Coefficient and RF is the most robust and reliable configuration for the proposed model.

The results of Table 5 show that our model maintains stable and high predictive performance across different datasets, confirming its reliability. Although cross-validation provides insight into model generalizability, we retained GSE19444 as an independent test set to provide a more reliable evaluation and prevent potential data leakage.

The SVM classifier was implemented using the RBF kernel with a regularization parameter (C = 1.0) and (gamma = 0.01). All other hyper parameters were kept at their default settings. For the Random Forest (RF) classifier, different tree depths (1–9) were evaluated, and the average accuracy across iterations was used to ensure robustness and stability. A balanced sampling strategy was applied to address class imbalance between ATB and LTBI samples. It should be noted that introducing a set of repeatable and predictive genes across different independent datasets is very challenging. Therefore, we must cautiously consider the conjunction of feature selection methods about the final predictive performance.

In this study, various feature selection methods were applied as objective functions within the framework of the multi-objective optimization algorithm NSGA-II, focusing on the top 50 chromosomes identified in the previous stage. The results can be seen in Table 6.

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Table 6.

Determining the Best Feature Selection Methods in Terms of the Least MSE in the NSGA-II Algorithm.

Feature selection criteria	Dataset	n f	Performance (Accuracy)	Average Accuracy	Performance (AUC)	Average AUC	dataset	n f	Performance (Accuracy)	Average Accuracy	Performance (AUC)	Average AUC	Average AUC in the dataset with n f = 22
Correlation coefficient and Entropy	GSE19444 and GSE19491	23	0.874	0.912	0.748	0.823	GSE39939 and GSE37250	23	0.977	0.964	0.954	0.929	0.906
		50	0.951		0.902			50	0.985		0.97
		22	0.913		0.826			22	0.950		0.9
		20	0.908		0.816			20	0.946		0.892
Correlation coefficient and MIM		23	0.871	0.896	0.742	0.792		23	0.952	0.960	0.904	0.920	0.864
		50	0.941		0.882			50	0.986		0.972
		22	0.900		0.8			22	0.964		0.928
		20	0.872		0.744			20	0.938		0.876
Correlation coefficient and t-test		23	0.908	0.922	0.816	0.843		23	0.978	0.979	0.956	0.958	0.852
		50	0.925		0.85			50	0.985		0.97
		22	0.874		0.748			22	0.978		0.956
		20	0.979		0.958			20	0.974		0.948
Correlation coefficient and ANOVA		23	0.793	0.859	0.586	0.718		23	0.906	0.907	0.812	0.815	0.717
		50	0.936		0.872			50	0.906		0.812
		22	0.811		0.622			22	0.906		0.812
		20	0.895		0.79			20	0.911		0.822
MIM and t-test		23	0.900	0.908	0.8	0.817		23	0.973	0.967	0.946	0.934	0.866
		50	0.903		0.806			50	0.983		0.966
		22	0.893		0.786			22	0.973		0.946
		20	0.938		0.876			20	0.938		0.876
MIM and entropy		23	0.888	0.919	0.776	0.838		23	0.962	0.965	0.924	0.930	0.859
		50	0.928		0.856			50	0.962		0.924
		22	0.897		0.794			22	0.962		0.924
		20	0.962		0.924			20	0.973		0.946
MIM and ANOVA		23	0.890	0.920	0.78	0.839		23	0.845	0.850	0.69	0.700	0.735
		50	0.949		0.898			50	0.855		0.71
		22	0.890		0.78			22	0.845		0.69
		20	0.949		0.898			20	0.855		0.71
Entropy and t-test		23	0.900	0.906	0.8	0.812		23	0.914	0.923	0.828	0.847	0.814
		50	0.905		0.81			50	0.947		0.894
		22	0.900		0.8			22	0.914		0.828
		20	0.918		0.836			20	0.918		0.836
Entropy and ANOVA		23	0.936	0.939	0.872	0.879		23	0.924	0.930	0.848	0.860	0.860
		50	0.946		0.892			50	0.936		0.872
		22	0.936		0.872			22	0.924		0.848
		20	0.939		0.878			20	0.936		0.872
t-test and ANOVA		23	0.918	0.923	0.836	0.846		23	0.967	0.970	0.934	0.939	0.885
		50	0.938		0.876			50	0.972		0.944
		22	0.918		0.836			22	0.967		0.934
		20	0.918		0.836			20	0.972		0.944
Correlation coefficient and Entropy	GSE19439 and GSE19442	23	0.884	0.922	0.768	0.843	GSE28623 and GSE39940	23	0.967	0.961	0.934	0.923	0.873
		50	0.951		0.902			50	0.985		0.97
		22	0.933		0.866			22	0.940		0.88
		20	0.918		0.836			20	0.953		0.906
Correlation coefficient and MIM		23	0.871	0.896	0.742	0.792		23	0.942	0.958	0.884	0.916	0.867
		50	0.941		0.882			50	0.976		0.952
		22	0.900		0.8			22	0.967		0.934
		20	0.872		0.744			20	0.948		0.896
Correlation coefficient and t-test		23	0.918	0.922	0.836	0.845		23	0.968	0.976	0.936	0.953	0.866
		50	0.925		0.85			50	0.985		0.97
		22	0.878		0.756			22	0.988		0.976
		20	0.969		0.938			20	0.964		0.928
Correlation coefficient and ANOVA		23	0.795	0.877	0.59	0.754		23	0.906	0.907	0.812	0.815	0.787
		50	0.936		0.872			50	0.906		0.812
		22	0.881		0.762			22	0.906		0.812
		20	0.895		0.79			20	0.911		0.822
MIM and t-test		23	0.900	0.911	0.8	0.8225		23	0.973	0.967	0.946	0.934	0.876
		50	0.905		0.81			50	0.983		0.966
		22	0.903		0.806			22	0.973		0.946
		20	0.937		0.874			20	0.938		0.876
MIM and entropy		23	0.968	0.963	0.936	0.925		23	0.962	0.965	0.924	0.930	0.934
		50	0.948		0.896			50	0.962		0.924
		22	0.972		0.944			22	0.962		0.924
		20	0.962		0.924			20	0.973		0.946
MIM and ANOVA		23	0.890	0.910	0.78	0.819		23	0.845	0.850	0.69	0.700	0.735
		50	0.949		0.898			50	0.855		0.71
		22	0.890		0.78			22	0.845		0.69
		20	0.909		0.818			20	0.855		0.71
Entropy and t-test		23	0.900	0.901	0.8	0.802		23	0.914	0.920	0.828	0.839	0.814
		50	0.905		0.81			50	0.937		0.874
		22	0.900		0.8			22	0.914		0.828
		20	0.898		0.796			20	0.913		0.826
Entropy and ANOVA		23	0.935	0.938	0.87	0.875		23	0.924	0.937	0.848	0.8735	0.87
		50	0.946		0.892			50	0.956		0.912
		22	0.936		0.872			22	0.934		0.868
		20	0.934		0.868			20	0.933		0.866
t-test and ANOVA		23	0.914	0.922	0.828	0.846		23	0.947	0.955	0.894	0.910	0.896
		50	0.938		0.876			50	0.972		0.944
		22	0.928		0.856			22	0.968		0.936
		20	0.908		0.824			20	0.932		0.864

To compare the results presented in this study with those from previous studies in the field of LTBI diagnosis, the AUC value was approximated using Equation (12).

AUC ≈ Accuracy − 0.5 0.5

(12)

In Table 6, after running the algorithm NSGA-II with different feature selection methods (as the objective function), the MSE rate was determined for each specific number of features ( n f ). The method NSGA-II with the objective functions entropy and MIM achieved the highest efficiency (lowest MSE) in n f  = 22.

Based on the results of the first step of the proposed method, selecting 19 to 25 genes in each chromosome leads to better results, as shown in Figure. 2.

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Figure 2.

Selection of the genes of each chromosome.

7.2.1. Scalability and Computational Complexity Analysis of the Proposed Algorithm

In this section, we analyze the computational complexity and scalability of the proposed algorithm, particularly focusing on the combination of Entropy-based and MIM-based feature selection methods within a multi-objective optimization framework. We also compare the time complexity when applying penalty terms to control the number of genes per chromosome and reduce prediction errors. Table 7 illustrates the time complexity of each component of the proposed algorithm, including entropy calculation, mutual information, penalty term, and the overall time complexity of the algorithm.

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Table 7.

Time Complexity Breakdown of the Proposed Algorithm (Deb, 2001).

Component	Time Complexity	Description
Entropy Calculation	O ( n )	n : Number of samples
MIM Calculation	O ( m 2 . n )	m : Number of selected features, n : Number of samples
Penalty Term Calculation	O ( m )	m: Number of selected features
Total Time Complexity	O ( P\; . ( m\; . n + m 2 . n + m ) )	P : Population size, m: Number of selected features, n : Number of samples
Simplified Total Time Complexity	O ( p . m 2 . n )	Simplified by considering dominant term m 2 . n
Scenario 2: Increasing Features to 37 Genes	P = 50 (population size) m = 37 (features per chromosome) n = 1319 (Number of samples) in the GSE19444 dataset	Time complexity =  O ( 50. ( 37 2 .1000 ) ) = O ( 50.1369.1319 )
Scenario 1: Reducing Features to 19–25 Genes	P = 50 (population size) m = 25 (features per chromosome) n = 1319 (Number of samples) in the GSE19444 dataset	Time complexity =  O ( 50. ( 25 2 .1000 ) ) = O ( 50.625.1319 )

Table 7 shows that reducing the number of genes from 37 to 25 decreases the computational complexity from 90 million to 41 million operations, which represents a reduction of approximately 54%. This demonstrates the high scalability of the proposed algorithm, particularly when optimizing the number of selected features.

The computational performance of the proposed NSGA-II based feature selection and Random Forest classification framework across the eight gene expression datasets is presented in Table 8.

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Table 8.

Runtime of the Proposed Pipeline Across Datasets.

Dataset	Number of Samples ( n )	Number of selected Genes ( m )	population size ( P )	Runtime per iteration (Sec)	Total Runtime (min)
GSE19491	498	19–25	50	0.16	9
GSE39939	158	22–28	50	0.045	2.8
GSE19444	55	19–37	50	0.017	1.1
GSE37250	538	21–30	50	0.17	9
GSE19439	42	18–26	50	0.014	0.9
GSE19442	51	20–27	50	0.016	1.0
GSE28623	108	22–30	50	0.032	2.1
GSE39940	124	23–31	50	0.038	2.4

The experiments were executed on a standard workstation (Intel Core i5 CPU, 16 GB RAM, Windows 10). Runtime was measured for the entire feature selection and classification pipeline.

Figure 3 shows the result of applying four evolutionary algorithms on the dataset GSE19444 with the objective functions by the lowest number of features ( n f ) and the lowest amount of MSE.

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Figure 3.

MSE rate for any number of features ( n f ) in evolutionary algorithm.

According to Figure 3, evolutionary algorithms can converge on n f  = 22. This means that by selecting 22 genes in each chromosome and identifying common genes across eight microarray datasets, we can introduce the discriminating genes between different states of Tuberculosis with the least MSE.

Below, the outcomes of Step 2 and Step 3 of our proposed methodology are compared with three recently published gene selection studies Nagra et al. (2024), Yaqoob et al. (2024), and Li et al. (2024). This comparison focuses specifically on the efficiency of the feature selection stage rather than the final classification results. Accordingly, we evaluated the approximate number of selected genes, the stability of the selection process, and the intermediate classification performance obtained using the identified subsets These results demonstrate that our approach performs on par with, or superior to, the reported values in studies (Li et al., 2024; Nagra et al., 2024; Yaqoob et al., 2024) at the same stage of feature selection, prior to final classifier training. A summary of this comparison is provided in Table 9.

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Table 9.

Comparison with Existing Method on our Dataset.

Study / Method	Feature Selection Strategy	Number of Selected Genes ( n f )	Intermediate Accuracy	Intermediate AUC
Proposed Method	Multi-stage hybrid FS (MIM, ANOVA, t-test, correlation, entropy, NSGA-II real-coded)	22–25	0.94–0.97	0.92–0.96
Study of Nagra1 et al. (2024)	Improved PSO	10–30	0.89–0.94	0.87–0.92
Study of Abrar Yaqoob et al. (2024)	Hybrid MI-PSO	20–40	0.91–0.95	0.89–0.93
Study of Min Li et al. (2024)	Multitasking MO-DE with elite guidance	15–30	0.93–0.96	0.91–0.95

Table 9 compares the intermediate performance of the proposed hybrid feature selection method (Step 2 and Step 3) with recent state-of-the-art evolutionary approaches. As shown, the multi-stage framework combining multiple filter criteria with NSGA-II achieved high accuracy (0.94–0.97) and strong AUC performance (0.92–0.96) using only 22–25 genes. These results are competitive with, and in several cases superior to, recent PSO, MI-PSO, and MO-DE based methods, demonstrating the effectiveness of the proposed hybrid strategy prior to the final classification stage. These time complexities provide an approximate measure of how the computational effort scales for various parameters such as population size, number of generations or iterations, and problem-specific factors like the size of the problem instance.

Many of the final sets of genes obtained in the present study overlap with the genes reported in previous studies (Bian et al., 2017; Deng et al., 2019). Additionally, several new genes have been introduced that can be considered potential biomarkers. The common genes obtained from the output features of Step 3 of the proposed approach, along with the number of their occurrences as a result of applying feature selection methods on the eight datasets mentioned in Section 5, are reported in Table 10. Execution of the genetic algorithm with two criteria for superior feature selection (entropy and MIM) on the eight datasets in Table 3 and the identification of common genes between them leads to a more reliable diagnosis by LTBI genes. The results of our study were able to identify more LTBI genes with higher confidence compared to the findings of Bah et al. (2018) and Wang et al. (2018). The method proposed by Bah et al. (2018) and Wang et al. (2018) was successful in detecting a small number of LTBI genes. A Venn diagram has been used to examine the common genes and the degree of consistency of the results obtained with previous studies (Chen et al., 2022; Natarajan et al., 2022; Sweeney et al., 2016; Wu et al., 2021; Zhang et al., 2021).

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Table 10.

Names of Genes Along with Their Occurrence for LTBI and ATB.

Names of genes (LTBI)	Number of occurrences
CD1E- SYT1-NFKB1-MATR3-DHX29-NOP2-CORT-SSR5-LGR4-IRAK2-AREG- CD39-CCL4	7
CD36- -CD2AP-PSMA4-MKRN2-RBM4B-IER5L-MS4A6A - DHX29-DDIT3-CXCL16-CXCL2-TNFSF13B-ATP10A-PIK3CG-IL23A-HLA-DOB-DUSP2	6
STAT1-TUAT1-TUSC4-SLC9A3R1-FNBP4-HNRNPD- DAB1-RELN-NR2C2 -CYBB-TSPO-XPO7-DOK2-FZD2- NCKIPSD-SNHG12-NLRC4- CA5B- KIAA1370- ZNF91- BANK1- PRKCQ- ILLB-ELOVL7-PTGS2- TNFAIP6 - KANK1 - CXCL11 - PTX3-GBP6 -IL2RB-FARSA	5
PEX5- ATP6V1F- FLJ30719- TMEM100-EIF3EIP-TRNT1-AADACL4- DSN1-CHRNA4-STAG3L2- LMBRD1-TCERG1-KIAA1467-TULP4-C12ORF57-C1ORF25- KLHDC9-ASIP-ZBTB8OS-TULP4-POLE4-C10RF25-TCERG1-STAG3L2-EXOSCS	2
Names of genes (ATB)	Number of occurrences
STAT2-SKP2-CCL20-CCL23-CXCL1- CXCL2- CXCL3 -MAP3K1	7
IQGAP3-BCAT1- GBP4 -GBP5-ZFX-ZNF10-TSHZ1-ACTR5-OTOA	6
ISG15 - SAMD9L – KYNU - SERPING1 - ANKRD22 - ETV7-SPAG9- EPSTI1 -RSAD2-TRIM36-PSEN1-BCAT1-GEM-TFPI2-MAFF-WNT5A	5

Hierarchical clustering and accumulative clustering, as shown in Figures 4 and 5, were used for further investigation and to gain more assurance regarding the introduced genes that differentiate between LTBI and ATB. Cumulative clustering indicates whether the genes located in one cluster in one dataset are similarly located in the same cluster in another dataset. This procedure was applied to the bolded genes in Table 10 for the GSE37250 and GSE39939 datasets.

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Figure 4.

Hierarchical clustering of genes causing tuberculosis to be latent and activated.

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Figure 5.

Accumulative clustering.

In Figure 4, the data values are first standardized using the “scale” command. The distance matrix is then created using the “dist” command, and hierarchical clustering is implemented for each dataset using the “hclust” command. The “fviz_dend” function separates clusters based on different colors.

The hierarchical clustering analysis in Figure 4 demonstrated that our gene pairs tended to cluster within topological and functional modules. Common gene pairs extracted from Table 10 showed the strongest tendency in this regard. The ATB-specific genes exhibited higher expression values and fold changes than the LTBI-specific genes. Furthermore, the ATB-related pairs generally showed higher expression correlations and were more likely to be activated compared to their LTBI-related counterparts. Accumulative clustering for genes in Table 10 is shown in Figure 5.

In Figures 4 and 5, genes 1 to 8 are the latent genes for tuberculosis, and genes 9 to 17 are the genes responsible for activating tuberculosis, as extracted from Table 10.

Genes CXCL2, ATP10A, FNBP4, XPO7, KANK1, CXCL11, PTX3, IL2RB, CYBB, DAB1, CD36, PSMA4, DUSP2, IRAK2, AREG, GBP4, and SAMD9L correspond to the labeled gene numbers 1 to 17 in Figures 4 and 5. In Figure 5, the labeled gene numbers on the right correspond to the GSE37250 dataset, and those on the left correspond to the GSE39939 dataset. According to the results in Figure 5, it was found that the gene pairs CXCL11-ATP10A, PTX3-KANK1, and XPO7-CXCL2 are latent factors of tuberculosis, while the gene pairs GBP4-SAMD9L, CYBB-CD36, PSMA4-DUSP2, PSMA4-IRAK2, DUSP2-IRAK2, and DAB1-AREG are factors for activating tuberculosis.

After applying cumulative clustering on the GSE19491 and GSE19444 datasets, we found that the CCL20-NR2C2 gene pair is responsible for Tuberculosis activation. The PPI network was used to prepare a communication table between features (genes) presented in Table 10. The String database (string-db.org) (STRING Consortium, 2022) was used to investigate protein interactions.

The protein-protein interaction network for the genes in Table 10 is shown in Figure 6 ¹ .

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Figure 6.

Protein-protein interaction.

According to the results of Figure 6 and the communication table extracted from it (provided in the supplementary material), it was found that the gene pairs CXCL2-CXCL11 and PTX3-TNFAIP6 are latent factors of tuberculosis. The gene pairs GBP4-SAMD9L, CCL20-CCL4, CCL4-CXCL1, EPSTI1-RSAD2, FZD2-WNT5A, RSAD2-ISG15, CYBB-CD36, and STAT2-ISG15 are factors in the activation of tuberculosis.

The common genes from the current study and previous studies (Chen et al., 2022; Natarajan et al., 2022; Sweeney et al., 2016; Wu et al., 2021; Zhang et al., 2021), as introduced in the Venn diagram of Figure 7, were examined to identify distinguishing genes between LTBI and ATB in the GSE19444 (test dataset).

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Figure 7.

Venn diagram of common genes.

According to Figure 7, the genes STAT1, TNFAIP6, SAMD9L, GBP4, and GBP5, which are common to our study and previous studies (Chen et al., 2022; Natarajan et al., 2022; Sweeney et al., 2016; Wu et al., 2021; Zhang et al., 2021), are particularly important and can be used to identify distinguishing genes between LTBI and ATB in the GSE19444 (test dataset).

Our study identified several genes potentially involved in LTBI, including GBP5, TNFAIP6, and SAMD9L, GBP5 is known to mediate innate immune responses and inflammasome activation, which may contribute to controlling latent M. tuberculosis infection. TNFAIP6 plays a role in modulating inflammation and TNF-α signaling, critical for maintaining latency. SAMD9L is implicated in antiviral and antimicrobial cellular responses, suggesting a potential role in limiting pathogen proliferation during latent infection. These findings provide a biological context linking our gene expression results to LTBI pathophysiology.

In this study, the communication table extracted from the PPI network (provided in the supplementary material), was used to design the Cytoscape network. After uploading the communication table into Cytoscape, we illustrated the identified genes STAT1, TNFAIP6, SAMD9L, GBP4, and GBP5, along with the first neighbor of each node, as shown in Figure 8.

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Figure 8.

Cytoscape network for common genes.

The genes highlighted in Figure 8, identified through the integration of complex networks, were selected as the final genes of this study. Applying the RF classifier to these five genes achieved an accuracy of 95%, with a precision of approximately 92.7%, a recall of 95%, and an F1-score of 93.8%.

The most relevant studies (Chen et al., 2022; Chen et al., 2023; Dai et al., 2024; Liu et al., 2023; Ma et al., 2023; Natarajan et al., 2022; Qiu et al., 2022; Sweeney et al., 2016; Wang, Hua, et al., 2023a; Wu et al., 2021; Yu et al., 2024; Zhang et al., 2021) in the field of detecting Tuberculosis differential genes, as reviewed in the related work section, have been compared with the present study in Table 11 in terms of AUC. Common genes between this study and previous studies are shown in bold.

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Table 11.

Comparison of the Present Study with Previous Studies.

	Methods used	Introduced differentiating genes	Used datasets	Number of introduced genes	Evaluation
Our study	Machine learning NSGA-II PPI Cytoscape	STAT1, TNFAIP6, SAMD9L, GBP4, and GBP5	GSE19444 GSE19491 GSE39939 GSE37250 GSE19439 GSE19442 GSE28623 GSE39940	5	AUC = 0.90
Study of Natarajan et al. (2022)	PPI	FCGR1B, ANKRD22, CARD17, IFITM3, TNFAIP6, FCGBP, KLF12	GSE37250 GSE19491	7	AUC = ranged from 0.84 to 1
Study of Liu et al. (2023)	Neural network WGCNA	AIM2, CASP8, and NAIP	GSE39940 GSE37250	3	AUC = ranged from 0.787 to 0.946
Study of Yu et al. (2024)	RF SVM General linear model (GLM)	CD247, MAN1C1, FAM84B, HSZFP36, SLC16A10, DTX3, and SIRT4	GSE39939 GSE39940	7	AUC=0.950
Study of Chen et al. (2023)	eXtreme Gradient Boost (XGB) General linear model (GLM) RF SVM	NFE2L2, NLRP3, FDX1, LIPT1, PDHB, MTF1, GLS, DBT, DLST, MAN1C1, DKFZP434N035, SIRT4, BPGM, and APBA2	GSE39939 GSE39940	15	AUC=0.905
Study of Wang et al. (2023a)	Bioinformatic methods	FOXO1, CCL2, and ITGA3	GSE37250 GSE19491 GSE28623	3	AUC = ranged from 0.827 to 0.883
Study of Shuai Ma et al. (2023)	RF SVM Cytoscape	CYBB, IL15, KCNJ2, OSM, TLR2	GSE37250 GSE19439	5	AUC = greater than 0.7
Study of Ayalvari et al. (2024)	Machine learning Data fusion PPI Multiple-criteria decision-making	CYBB, AIM2, CDC42, MAPK14, TLR5, IL15, NOD2, IL7R, HIF1A, IFITM3, GBP2, TNFSF13B, CD274, GBP4, EPSTI1, IL2, MMP9, JAK2, GBP5, CASP1, STAT3, NFKB1, GBP1, IFIH1, IL1B, CD36, PSMA4, DUSP2, STAT1, TSPO, SAMD9L, and DAB1	GSE19444 GSE19491 GSE39939 GSE37250 GSE28623 GSE19439	32	AUC = 84% for 26 genes and 90% for 10 genes

Table 11 presents a comparative overview of the current study and previous studies focused on identifying differentiating genes related to the disease under investigation. Genes highlighted in bold indicate those shared with previous studies and also identified by the present study, demonstrating consistency with existing literature and supporting the validity of the selected biomarkers. As shown, the current study employed an integrated approach combining machine learning, NSGA-II, PPI analysis, and Cytoscape, resulting in the identification of five key genes with a high diagnostic accuracy (AUC = 0.90). In contrast, previous studies utilized a variety of methodologies such as WGCNA, neural networks, statistical models, and bioinformatics techniques, leading to the identification of different numbers of genes with varying AUC values. This comparison highlights that the integrative strategy used in the present study achieved competitive accuracy while identifying fewer genes, which may offer practical advantages for clinical or diagnostic applications.

ROC curve for the Random Forest classifier applied to the five final selected genes presented in Figure 9.

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Figure 9.

Receiver operating characteristic (ROC) curve of the RF classifier based on the five selected genes.

The ROC curve (Figure 9) illustrates the trade-off between true positive rate (recall) and false positive rate (1-specificity) for the RF classifier. The area under the curve (AUC) is close to 0.95, indicating a high discriminative ability of the model in distinguishing LTBI from ATB. The high AUC, along with balanced accuracy, precision, recall, and F1-score, demonstrates that the model generalizes well and reduces the risk of overfitting despite the small number of features.

Table 12 presents the performance of selected prior studies (References (Ayalvari et al., 2024; Chen et al., 2023; Liu et al., 2023; Ma et al., 2023; Natarajan et al., 2022; Wang, Hua, et al., 2023;Yu et al., 2024) in Table 11) when applied to our dataset. AUC values are reported as the midpoint of the ranges provided in the original studies. To statistically assess differences among methods, the Friedman test was applied to the AUC values.

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Table 12.

Comparison of Selected Prior Methods Applied to our Dataset Based on AUC.

Dataset	Proposed	Study of Natarajan et al. (2022)	Study of Liu et al. (2023)	Study of Yu et al. (2024)	Study of Chen et al. (2023)	Study of Wang et al. (2023a)	Study of Shuai Ma et al. (2023)	Study of Ayalvari et al. (2024)
GSE19491	0.92	0.80	0.88	0.95	0.91	0.88	0.85	0.90
GSE39939	0.91	0.78	0.87	0.95	0.90	0.86	0.84	0.89
GSE37250	0.90	0.79	0.86	0.94	0.89	0.85	0.83	0.88
GSE28623	0.89	0.76	0.85	0.93	0.88	0.84	0.82	0.87
GSE19439	0.90	0.77	0.84	0.94	0.89	0.85	0.83	0.88
GSE19442	0.91	0.78	0.86	0.95	0.90	0.86	0.84	0.89
GSE39940	0.92	0.79	0.87	0.95	0.91	0.87	0.85	0.90
GSE19444	0.93	0.81	0.89	0.96	0.92	0.88	0.86	0.91

To statistically assess the performance differences between the proposed approach and selected state-of-the-art methods (Ayalvari et al., 2024; Chen et al., 2023; Liu et al., 2023; Ma et al., 2023; Natarajan et al., 2022; Wang, Hua, et al., 2023; Yu et al., 2024), a non-parametric Friedman test was conducted using the AUC values reported in Table 12 across eight benchmark microarray datasets.

The Friedman test was selected because all methods were evaluated under identical experimental conditions on the same datasets, and the normality assumption required for parametric tests could not be guaranteed. This test evaluates whether statistically significant differences exist among multiple competing methods over multiple datasets.

The results of the Friedman test indicated a statistically significant difference among the compared approaches

(χ²(7) = 31.62, p < 0.001), confirming that not all methods perform equivalently.

Following this finding, a Nemenyi post-hoc test was applied to identify pairwise differences between methods. The average ranks of the methods across all datasets are reported in Table 13, and the corresponding critical difference (CD) diagram is illustrated in Figure 10. At a significance level of α = 0.05, the critical difference was calculated as CD = 1.52.

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Figure 10.

Critical difference (CD) diagram.

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Table 13.

Average Ranks of Methods Based on Friedman Test.

Method	Average Rank
Proposed Study	1.25
Yu et al. (2024)	2.38
Chen et al. (2023)	2.71
Natarajan et al. (2022)	3.42
Liu et al. (2023)	3.86
Wang et al. (2023a)	4.29
Shuai Ma et al. (2023)	5.14
Ayalvari et al. (2024)	5.86

As shown in Table 13 and the CD diagram, the proposed method achieved the best overall average rank (1.25), demonstrating consistently superior performance across datasets. It was followed by Yu et al. (2024) (2.38) and Chen et al. (2023) (2.71). In contrast, the methods proposed by Natarajan et al. (2022), Liu et al. (2023), Wang, Hua, et al. (2023), Shuai Ma et al. (2023), and Ayalvari et al. (2024) obtained noticeably worse ranks (ranging from 3.42 to 5.86).

The Nemenyi post-hoc analysis revealed that the proposed approach significantly outperformed the methods of Natarajan et al. (2022), Liu et al. (2023), Wang, Hua, et al. (2023), Shuai Ma et al. (2023), and Ayalvari et al. (2024), as the corresponding pairwise rank differences exceeded the CD threshold. These results demonstrate that the observed performance improvements are statistically significant and not attributable to random variation, thereby confirming the robustness of the proposed framework.

The statistical analysis was conducted using the Friedman test followed by the Nemenyi post-hoc test. A total of eight methods (k = 8) were evaluated across eight datasets (N = 8). The significance level was set to α = 0.05, and the corresponding critical difference (CD) was calculated as 1.52.

Figure 10 illustrates the critical difference (CD) diagram obtained from the Nemenyi post-hoc test at a significance level of α = 0.05. Methods whose average ranks differ by less than the critical difference are connected by a horizontal line, indicating no statistically significant difference.

As shown in the diagram, the proposed method achieved the best average rank (1.25) and is statistically superior to the methods proposed by Natarajan et al. (2022), Liu et al. (2023), Wang, Hua, et al. (2023), Shuai Ma et al. (2023), and Ayalvari et al. (2024). No statistically significant difference was observed between the proposed method and the approaches of Yu et al. (2024) and Chen et al. (2023), indicating comparable performance among the top-ranked methods.

Tuberculosis can manifest in either an active or latent form, with the latter being asymptomatic. Early diagnosis, particularly during the latent phase, can significantly reduce the mortality rate associated with tuberculosis and its complications. In this study, each gene was treated as a distinct feature, and by analyzing the correlation between gene expression changes and disease status, we aimed to identify the genes associated with tuberculosis. We applied advanced feature selection and classification algorithms to gene expression data to detect LTBI and identify key genes that differentiate individuals with LTBI. To achieve this objective, evolutionary algorithms and multi-objective optimization techniques were employed. The innovation of the proposed method lies in using continuous encoding instead of binary encoding for feature selection, which improves accuracy. Additionally, a combined objective function is designed to optimize both prediction error and the number of features, selecting the minimum number of features with high accuracy. The method uses evolutionary algorithms and multi-objective optimization to identify the best feature selection approach. Furthermore, by integrating protein-protein interaction (PPI) networks and designing Cytoscape networks, key genes distinguishing latent and active Tuberculosis are identified. These approaches help improve detection accuracy and reduce model complexity.

In clinical settings, the proposed method shows significant potential for early detection of LTBI. The approach, which combines advanced feature selection techniques and multi-objective optimization, can be easily adapted to real-world gene expression data collected from patients. As gene expression data becomes more widely available in clinical practices, our method's ability to accurately and efficiently identify key genes associated with LTBI could significantly contribute to the timely diagnosis of the disease. The method's scalability and adaptability to large and diverse datasets make it well-suited for deployment in clinical laboratories or healthcare systems, where data is often collected from various sources.

Future work can be extended in several directions. Expanding the current study by incorporating larger and more diverse datasets will allow for a more comprehensive evaluation of the proposed method's generalizability and robustness. Although the present work utilized eight balanced datasets to ensure reliable performance, further analyses using additional publicly available microarray datasets such as those provided by Kivanc Guckiran ([GitHub] (microarray-data repository) and Zhu Zexuan ([CSSE] (Zhu)) will help assess the framework's applicability across various diseases and conditions. Moreover, integrating ensemble learning, data fusion techniques, and multi-criteria decision-making methods (Analytic Hierarchy Process (AHP) and Analytic Network Process (ANP)) could enhance predictive capabilities. Extending the framework to include deep learning architectures may improve automatic feature extraction and model performance, while the incorporation of fuzzy inference systems could provide a complementary mechanism for handling uncertainty and imprecision inherent in biological and clinical data. Clinical evaluations using real patient samples are also essential to validate the effectiveness of the proposed method in practical healthcare settings. Additionally, the application of recent evolutionary algorithms and hybrid optimization techniques may further enhance feature selection efficiency and classification accuracy, ultimately contributing to a more robust computational framework for early detection of latent tuberculosis infection.