## 4 Results
### Comparison of the Classification Results  
In this section, we conduct a comparison assessment between the proposed permutation-based group feature ranking method and an existing method called the group lasso. Initially, the original lasso method was employed to remove irrelevance features on an individual basis. We then examined the performance of group feature selection approaches by employing the selected group features in classification tasks. The tasks were executed through machine learning algorithms, aiming at identifying which one performed better in terms of the classification tasks. The comparison involved the utilization of machine learning techniques, specifically three algorithms: logistic regression (LR), support vector machine (SVM), and random forest (RF). Leave-one-out cross-validation was employed to evaluate the performance of the methods, providing a robust measure of the classification performance. Important performance metrics, accuracy, and \(F1\) score (\(F1\)) were used to evaluate the performance of the classification methods. To compute these performance metrics, a confusion matrix was determined as outlined in Table 3. In Table 3, _tp_ represents true positive, _tn_ denotes true negative, _fp_ denotes false positive, and _fn_ denotes false negative. Accuracy helped us understand the number of correct predictions made by our method compared to the total number of predictions. The formula for accuracy can be expressed as follows:  
\[\text{accuracy}=\frac{tp+tn}{tp+tn+fp+fn} \tag{4}\]  
The F1 score balances precision and recall as follows:  
\[F1=2\times\frac{precision\times recall}{precision+recall} \tag{5}\]  
where precision and recall denote \(\frac{tp}{tp+fp}\) and \(\frac{tp}{tp+fn}\), respectively. In the context of a multi-class classification, we extended the confusion matrix by calculating an \(F1\) score for each class individually. These scores were then averaged to arrive at a macro-averaged \(F1\) score. This macro-averaging technique ensured an equitable contribution from each class to the final score. Consequently, this approach fairly evaluated the performance of the classification model, regardless of class imbalance. The following three gene expression datasets were examined: GLA-BRA-180, CLL-SUB-111, and TOX-171. The proposed and group lasso methods were applied to these datasets, and each method selected group features separately. The selected group features were used to train the machine learning algorithms and calculate their respective accuracies and F1 scores on the testing dataset. Machine learning algorithms were also trained and evaluated using the original datasets, without employing any feature selection technique (No-FS). The results are summarized in Table 4. In comparison to No-FS, both feature selection methods (group lasso and the proposed method) performed better. Notably, using the CLL-SUB-111 dataset, the proposed method outperformed the group lasso using LR, SVM, and RF. For the GLA-BRA-180 dataset, the performances of both methods were almost the same for the SVM and RF classifiers in terms of accuracy. However, the proposed method performed slightly better with LR. Finally, concerning the TOX-171 dataset, the proposed method demonstrated a slight advantage over LR and RF and nearly the same performance as the SVM. This comparative analysis helped us see how well \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & \multicolumn{2}{c}{**Predicted Class**} \\ \cline{3-4}  & & Positive & Negative \\ \hline \multirow{3}{*}{**TOX-171**} & \multirow{3}{*}{Positive} & _tp_ & _fn_ \\ \cline{2-4}  & & Negative & _fp_ & _tn_ \\ \hline \hline \end{tabular}
\end{table} 
Table 3: An example of a confusion matrix. our approach stands up against an established method and which method works best on which algorithms. For group feature selection, both the proposed method and group lasso were effective, which is why the results of these two methods in terms of accuracy and F1 score did not show a significant difference.