## 4 Data Analysis and Discriminatory Models Used in Breath Biomarker Research
### Data Analysis  
Traditionally, researchers used \(p\)-values as a guide to identify VOCs that show statistically significant variation between the patient and control sample. However, \(p\)-value usage is very limited and if determined using a wrong statistical treatment may lead to false positives and negatives. Particularly, the \(p\)-value \(<0.05\) that is very often used, is not the most appropriate threshold for studies involving multiple hypothesis testing, leading to the detection of false positives [251]. Therefore, a more stringent threshold value with Bonferroni correction has been recommended [251]. Nevertheless, multivariate statistical analysis seems much more suitable to find a set of biomarkers for many different diseases and this assumption is nowadays widely consensual [252; 253; 254]. This can be done in an unsupervised and supervised approach. Normally, unsupervised analysis is the first approach applied in the data generated after sample analysis. It allows data exploration in order to find biological changes between the study groups. The most common method used is the Principal Component Analysis (PCA), which essentially does a reduction in the multidimensional data. However, it does not provide information regarding the variable responsible for the clustering and the outliers are a major concern [252]. Another method used is the clustering, which is a data mining technique to make automatic data groupings according to their degree of similarity. There are different clustering approaches, of which \(k\)_-means_ is very often used [255; 256]. However to deal with many outliers and complex data sets, more robust and sophisticated approaches should be used, as distribution-based (e.g., Gaussian mixture models [252]) or graph-based clustering methods (spectral or transitivity clustering [252; 257]). The unsupervised methods are popular for data visualization but very limited to develop a classification model (do not allow data classification). Therefore, unsupervised methods are usually followed by the supervised methods which allow the characterisation of the relation between a matrix of predictors (in this case VOCs) and vector (or matrix) of responses (e.g., class membership) [252]. To achieve this, linear or nonlinear statistical techniques can be applied, according to the purpose of the analysis. In the first case, the linear discriminate analysis (LDA) is the most often used in breath analysis. Its objective is discovering a linear function based on the original compounds founded, in order to distinguish between the studied groups (e.g., control and disease groups) [258]. LDA is a fast and powerful technique, requiring few optimizing operations, but it can be only applied when there are more samples than the variables [259]. This is, however, the opposite of the most frequent situation in breath analysis and thus PCA is previously used for dimensionality reduction and only then LDA can be applied for classification purpose [252]. Partial least-square (PLS) was originally developed for the quantitative analysis in regression problems [260], but its ability to cope with highly collinear data made it also very suitable for breath analysis [261]. Using this method, it is possible to distinguish the studied classes and to find which compounds are responsible for each of them [252]. PLS-DA is another important method for samples classification. It requires some model optimization, as it frequently causes data overfitting, but it is advantageous in the fact that its model precision improves as the number of relevant variables increases [262]. Metabolomics data frequently show non-linear patterns, but these problems are well handled by using non-linear methods. There are many non-linear methods available, but the most often used are the Kernel-based models (as the support vector machine (SVM) and kernel-PLS-DA [263]), and artificial neural network (ANNs) ([264,265]).