## 3 Results and Discussion
### Chemometric Analysis of Urine Samples
The top-ranking features in terms of importance were used to build the classification models. Figure 6a shows the ROC curves of a set of six volatiles based on the average cross-validation performance. The obtained values for the area under the curves (AUC) between 0.867 and 0.968, with a 95% confidence interval, are excellent and represent a good accuracy in discriminating both groups. Figure 6b shows the plot of the predictive accuracy of biomarker models with an increasing number of features. \begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Significant VOMs** & **\(t\)-Stat** & **\(p\)-Value** & **=\(-\)LOG10(p)** & **FDR** \\ \hline
1 & 3,5-Dimethylbenzaldehyde & \(-\)7.479 & 6.87 \(\times\) 10\({}^{-10}\) & 9.1628 & 3.92 \(\times\) 10\({}^{-8}\) \\
2 & TDN & \(-\)5.7798 & 3.84 \(\times\) 10\({}^{-7}\) & 6.4162 & 1.09 \(\times\) 10\({}^{-5}\) \\
3 & D-Carvone & 4.363 & 5.82 \(\times\) 10\({}^{-5}\) & 4.235 & 0.001106 \\
4 & 6-Methylphenanthridine & 3.8847 & 0.000282 & 3.55 & 0.004016 \\
5 & \(\alpha\)-Methylcinnamaldehyde & 3.6128 & 0.000665 & 3.1772 & 0.007581 \\
6 & 2-Bromophenol & 3.486 & 0.000982 & 3.0079 & 0.009328 \\
7 & TONEA & 3.3169 & 0.001633 & 2.7871 & 0.013294 \\ \hline \end{tabular} Abbreviations: TDN: 1,1,6-Trimethyl-1,2-dihydrononphthalene; TONEA: 2,5,5,8a-tetramethyl-1,2,3,5,6,7,8-octahydro-1-naphthalenyl ester acetate; FDR: false discovery rate.  
\end{table} 
Table 3: Important features identified using the \(t\)-tests. Figure 4: **(a)** Partial least-squares discriminant analysis (PLS-DA). **(b)** Variables of importance in projection (VIP) scores plot, representing the important features identified by the PLS-DA. The coloured boxes on the right indicate the relative concentrations of the corresponding metabolites in each group under study. **(c)** 10-fold CV performance of the PLS-DA classification using a different number of components (* means best Q2 value, the best classifier). **(d)** PLS-DA model validation by permutation tests based on 1000 permutations of the VOMs obtained by GC-MS of the urine samples from the groups under study. Figure 5: Hierarchical cluster analysis of CTRL (control) and Pca (prostate cancer) groups **(a)** Dendrogram analysis of the volatomic data, using Euclidean distance measure and Ward’s linkage. **(b)** Clustering result shown as heatmap illustrates the concentration of the urinary volatile organic metabolites identified in each sample. Columns correspond to Pca and CTRL sample groups, respectively, whereas rows correspond to the most relevant VOMs detected. The colour of the cells corresponds to the normalised peak areas of the compounds (minimum \(-1\), dark blue; maximum \(+1\), dark red). The performance of the classification model was assessed through a confusion matrix was performed based on the classification method: PLS-DA. The columns represent the actual classes the outcomes should have been, while the rows represent the predictions we have made. The number of correct and incorrect predictions is summarized in Figure 7.  
Our model predicted that 10/12 were from control groups when there were 12/12. The accuracy corresponds to the proportion of predictions that the model classified correctly. In this case, the accuracy of the model was 91.3% as it predicted that two healthy individuals belong to the PCa group (two false positives). The precision of the model, related to the proportion of positive identifications that were correct, was 85%. The sensitivity which expresses the proportion of actual positives identified correctly was 100%, whereas the specificity associated with the proportion of actual negatives that are correctly identified was 83.3%. Figure 6: (**a**) ROC curves for the most important features with the highest ability to discriminate both groups. (**b**) Plot of the predictive accuracy of biomarker models with an increasing number of features. The most accurate biomarker model is highlighted with a red dot. Figure 7: Structure of a 2 \(\times\) 2 confusion matrix to assess the performance of a classification model.