## 2 Materials and Methods
### Statistical Analysis  
**Mahalanobis Distances:** The database compiled for research often contains missing data and outliers, suggesting the need to begin any statistical analysis by implementing a comprehensive data analysis protocol. Among the analytical tools for data processing in multivariate samples, the use of Mahalanobis distances stands out. This technique quantifies the number of standard deviations that a specific observation is from the mean of a distribution. Since outliers do not follow a behavior pattern analogous to regular observations, applying this measure allows for identifying such anomalies. In contrast, from a geometric perspective, the Euclidean distance represents the shortest length between two points; however, it does not consider the correlation between highly inter-related variables. The Mahalanobis distance differs from the Euclidean distance in its ability to incorporate the correlation between variables in its calculation. In this study, the Mahalanobis distance was employed as a scale-invariant metric that allows for calculating the distance between a point \(x\in\mathbb{R}^{p}\) from a \(p\)-variate probability distribution \(f_{X}(. )\), a \(p\)-value, and the mean \(\mu=E(X)\) of that distribution. It was assumed that the distribution \(f_{X}(. )\) has a finite number of second-order moments, facilitating the definition of the covariance matrix \(\Sigma=E(X-\mu)\). Under these conditions, Mahalanobis distances are established through the following mathematical relationship:  
\[D_{M}(x)=\sqrt{(x-\mu)^{T}\Sigma^{-1}(x-\mu)} \tag{1}\]  
**ANOVA Test. ** The ANOVA test is a powerful statistical tool for testing the equality of means across groups. Using [PERSON]'s notation, a one-way ANOVA model can be mathematically represented as follows:  
\[Y_{ij}=\mu+\alpha_{i}+\varepsilon_{ij} \tag{2}\]  
where \(i=1,\cdots,k,j=1,\ldots,n_{i},Y_{ij}\) denotes the outcome of the \(j\)-th observation in the \(i\)-th treatment, \(\mu\) is the overall mean effect expressed by the formula  
\[\mu=\sum_{i=1}^{k}\mu_{i}\alpha_{i}/n, \tag{3}\]  
and \(\alpha_{i}\) represents the fixed or random effect attributable to the \(i\)-th treatment. This implies that, in the absence of differences between treatments and random causes, the performance of each observation would be \(\mu\). The effect \(\alpha_{i}\), corresponding to the \(i\)-th treatment, is defined as  
\[\alpha_{i}=\mu_{i}-\mu \tag{4}\]Therefore, the \(i\)-th treatment increases or decreases the performance by \(\alpha_{i}\). The two fundamental assumptions of this model are (i) that the dataset follows a normal distribution, \(Y\sim\ N\left(\mu,\ \sigma^{2}\right)\) and \(\varepsilon\_i\ \sim\ N\left(0,\ \sigma_{e}^{2}\right)\), and (ii) that the variances in the groups are equal. The test hypothesis is then formulated as  
\[H_{0}:\mu_{1}=\mu_{2}=\cdots=\mu_{k} \tag{5}\]  
for the test statistic \(F\), which is defined by the relationship  
\[F=\frac{MS_{tr}}{MS_{e}} \tag{6}\]  
where \(F\) is the test statistic and \(MS_{tr}\) and \(MS_{e}\) are the mean squares of the treatment and error, respectively. Based on this equation \(H_{0}\ p\) is rejected for a given significance level \(\alpha\) if \(F\) is greater than the critical value \(F>F_{k-1,\ k(n-1)\alpha}\). **Tukey Test. ** The Tukey test can be generalized to the complete family of all contrasts of k means. In the case of the Tukey test, the confidence interval for the contrast, defined in the equation, is calculated using the formula:  
\[\sum_{i=1}^{k}c_{i}\overline{y_{i}}\pm q_{a,\
u,k}\sqrt{\frac{1}{n}}s^{2}\sum _{i=1}^{k}\frac{|c_{i}|}{2} \tag{7}\]  
In this formula, \(c_{i}^{+}=max(c_{i},\ 0)\) and \(c_{i}^{-}=min\left(c_{j},\ 0\right)\). The quantile \(q_{a,\
u,k}\) comes from the distribution of the studentized range with parameters \(\
u\) and \(k\). For both intervals, the margin of error is not influenced by the number of contrasts.