## 2 Materials and Methods
### Theory and Implementation of ANOVA-PDR  
The calculation procedure of the ANOVA in a multivariate scenario is demonstrated in Figure 1. Briefly, before the ANOVA, the obtained spectral data matrix was mean-centered for each measured variable to acquire the grand means matrix. Afterwards, the original matrix was subtracted by the grand means matrix to obtain the grand residuals matrix [12]. The grand residuals matrix was then used to construct multiple sub-matrices, i.e., the means and residuals matrix of each factor. Specifically, the factors of this EVOO adulteration study included origin, adulteration level, and adulteration type. The sub-matrices allowed calculation of the percentage of total variance for each factor, the significance level of the variance, and the variance associated with factor interactions.  
The PDR is a straightforward multivariate metric for rapidly quantifying the degree of separation from multivariate data objects for a pair of classes [17]. The PDR performs multivariate resolution by generating a set of projections onto the difference vectors of two class averages between pairs of means divided by 2 times the summed standard deviations, given by  
\[R_{s}(a,b)=\frac{\left|\overline{P_{a}}-\overline{P_{b}}\right|}{2(S_{a}+S_{b })} \tag{1}\]  
from which \(R_{s}(a,b)\) is the PDR of class \(a\) and \(b\), \(P_{a}\) and \(P_{b}\) are obtained by projecting the objects in the corresponding category onto the difference vector between the mean of two classes, given by  
\[P_{i}=(\overline{X_{a}}-\overline{X_{b}})X_{i}^{T} \tag{2}\]  
for which \(X\) is the two-way matrix for each target class of the UV-Vis dataset; thus, \(X_{i}^{T}\) is the transposed two-way matrix for each target class, where \(i\) stands for class \(a\) or \(b\). Theprojection of an object \(i\) on the difference vector is calculated by \(\overline{X_{a}}-\overline{X_{b}}\). From these projections, the averages \(\overline{P_{a}}\) and \(\overline{P_{b}}\), and their corresponding standard deviations \(S_{a}\) and \(S_{b}\) are calculated, and, finally, the resolution between class a and class b is obtained by Equation (1). A PDR greater than 1.5 indicates that the two classes are well-resolved. The larger the PDR, the greater the resolution between the two classes. The PDRs of a dataset that contains three or more classes can be sorted in the order of an upper or lower triangular matrix with a size equal to the number of classes, where the PDR of classes \(a\) and \(b\) was given in the columns \(a\) and rows \(b\), respectively. The geometric mean of all PDRs was used as the average PDR of all classes in the dataset.  
By combing the ANOVA and PDR, a straightforward metric for class separation under the influence of multiple factors can be achieved. For each individual factor, the same mean submatrix constructed for ANOVA-PCA were also resolved through PDR according to their respective internal classes. For the main influencing factors, ANOVA-PDR analyzes each class in the corresponding effect matrix in pairs, and expresses it as a triangular matrix to measure the class separation of the data object [19]. To better visualize the separations of various factors between classes, these PDR matrices are plotted in grayscale in this work. The PDRs from small to large corresponds to the color in the color bar from dark to light, so the lighter background color indicates larger PDR that represents a better separation between different classes, and vice versa. The average PDR of each effect matrix was also given to evaluate the factor matrix resolution. The ANOVA-PDR calculation was performed with an in-house script written in MATLAB R2021b (The MathWorks, Natick, MA, USA).