## 3 Results
### Linear and Principal Component Regression Analysis  
Linear regression analysis was undertaken on individual VOCs (37 with \(p\)-value by Student's \(t\)-test \(<\) 0.05 between Cancer Week 1 and Week 3) to look for significant trends with respect to the day on which urine was collected after tumor injection. Determination coefficients (\(R^{2}\) value), regression coefficients and standard errors for all 37 individual VOCs can be observed in Supplementary Table S2. ANOVA determined that 23 out of the 37 VOCs had statistically significant trends by linear regression (\(p\)-value \(<\) 0.05). Of the 23 VOCs, only 5 VOCs had a positive regression coefficient while the other 18 features had negative regression coefficients. Even though statistically significant correlations were observed, none of the VOCs had an adjusted \(R^{2}\) greater than 0.35. Next, PCR was implemented on the same 37 VOCs. After employing PCR on all 37 principal components, a relatively good fit was obtained (\(R^{2}\) = 0.94), but the adjusted \(R^{2}\) value was much lower (0.61), indicating too many variables were being utilized. A scatter plot showing the \(R^{2}\) and adjusted \(R^{2}\) value as a function of the number of principal components utilized for this model can be seen in Figure 5a, which indicates that analyzing more than 19 principal  
Figure 4: (**a–c**) First LDA Model distinguishes all Cancer (Weeks 1–3) from Control with high accuracy using five VOCs showing results in (**a**) one dimension, (**b**) the receiver operator characteristic (ROC), and (**c**) First LDA model in two dimensions. (**d–f**) Second LDA Model distinguishes Cancer Week 1 from Cancer Week 3 with high accuracy showing results using five VOCs in (**d**) one dimension, (**e**) the ROC for Week 1 vs. Week 3, and (**f**) Second LDA model in two dimensions. Week 2 samples shown in (**f**) not included in (**d**) LDI or (**e**) ROC analysis. (**g–i**) Third LDA Model separates Cancer Week 1, Cancer Week 3 and Control with high accuracy. The ROC curves are shown for (**g**) Cancer Weeks 1 and 3 vs. Control and (**h**) the Cancer Week 1 vs. Week 3. (**i**) The third LDA model in two dimensions. ***\(p\) \(<\) 0.001.  
components will result in an overfit model. PCR using the first 19 components resulted in a linear correlation with \(R^{2}\) = 0.82, adjusted \(R^{2}\) = 0.68 and root mean square error (RMSE) = 3.3 (Figure 5b). The 95% confidence interval for the linear regression model is additionally illustrated in Figure 5b. The standardized coefficients for all 37 VOCs can be observed in Table S2. To increase the stability of the model, PCR was implemented using only the 19 VOCs (principal component loadings) statistically significantly contributing toward the first iteration of regression analysis (Figure 5b). Again, when plotting the \(R^{2}\) and adjusted \(R^{2}\) value as a function of the number of principal components (Figure 5c), utilizing more than 10 principal components results in an overfit model. The first 10 principal components resulted in a linear correlation with \(R^{2}\) = 0.71, adjusted \(R^{2}\) = 0.63 and RMSE = 3.6 (Figure 5d). The 95% confidence interval for this PCR model is can also be observed in Figure 5d. The standardized coefficients for all 19 VOCs for this model can also be observed in Table S2.