## 2 Materials and Methods
### Statistical Analysis  
A multi-statistical approach (standard and CoDa analysis) was used to analyse the data set. The standard evaluation of the VOC profiles (average, standard deviation, ANOVA and classification tree) was performed using XLSTAT v. 2017 for Excel (Addinsoft, Paris, France). For exploratory tools in the simplex (biplot, variation array, and _clr_ transformations), the package CodaPack v 1.03 (Department of Computer Sciences and Applied Mathematics, Universidad de Girona, Girona, Spain) ([20] was used. Clustering of samples, complete log-ratio analysis, and selecting the most influential VOC log-ratios was performed using WARD, LRA, and STEP from easyCODA [21]. The detection of potential biomarkers [22] was achieved in robCompositions [23]. The last two packages were run in R (v 4.0.3) (R Foundation for Statistical Computing: Vienna, Austria [24], under RStudio (v 1.4.1103) (RStudio, Inc.: Boston, MA, USA) [25]. Because of the number of cases, PLS-R analysis was not intended.  
For CoDa exploratory tools, the original data (in the simplex) were transformed into values in the real (Euclidean) space. The centred log-ratio transformation (_clr_), which preserves distances, is one of the most straightforward procedures [12]. It transforms the composition X into coefficients Y by dividing each cell's value by the geometric mean of the VOCs in the corresponding row and taking then the logarithm of the resulting ratio. Its formula is:  
\[\mathrm{Y}=clr(\mathrm{X}=[\mathrm{y1},\mathrm{y2},\mathrm{\ }\ldots\mathrm{yD}]=ln \frac{x_{1}}{\sqrt[D]{\sum_{i=1}^{D}x_{i}}}\ \ldots ln\frac{x_{D}}{\sqrt[D]{\sum_{i=1}^{D}x_{i}}}, \tag{1}\]  
When Y's variance is estimated column-wise, each VOCs _clr_ variance is obtained following the standard procedure. Its magnitude represents the variability of the components over observations. It may be used to measure the effect of factors without omitting the possible influence of parts with low contents and high estimation errors. The _clr_ transformation is used to build the compositional biplot [26] and allows a straightforward interpretation of the result when applying the standard multivariate statistics.  
For biomarker identification, the statistic Vj developed by [PERSON] et al. [22] was used:  
\[V_{j}=\sum_{k=1}^{D}\frac{(n_{i}+n_{2})\sqrt{\frac{t}{k}}}{n_{1}\cdot\sqrt[ \mathrm{j}_{k}^{[1]}+n_{2}\cdot\sqrt[\mathrm{j}_{k}^{[2]}]}\ \mathrm{for\ j,\ k=1,\ 2,\ \ldots D}, \tag{2}\]  
where n1 and n2 are the numbers of observations corresponding to the two groups. If the jth part (compound) is a marker, the jth column and row of all the three sources of information will not have a similar structure, and Vj will be markedly higher than 1. Otherwise, the value will be \(\sim\)1. Vj average and the empirical standard deviations are respectively:  
\[\overline{V}=\frac{1}{D}\sum_{k=1}^{D}V_{k}\ ;\ s_{V}=\sqrt{\frac{1}{D}\sum_{k=1}^{D} \big{(}V_{k}-\overline{V}\big{)}}, \tag{3}\]  
Then, the Vj value can be standardised as \(V_{j}^{*}=\frac{V_{j}-\overline{V}}{s_{V}}\) and a standard normal percentile for \(p\geq 0.975\) (\(\approx 1.96\)) cut off value established. Parts with higher \(V_{j}^{*}\) could be considered as potential biomarkers.