## 2 Theoretical Background  
### Methods for Creating Confidence Ellipses in MDS Plots  
In the relevant literature, MDS confidence ellipses are closely related to the stability of a MDS solution [4]. The stability methods for a MDS solution examined were the following:  
1. Pseudo-confidence ellipses (PCE). This method has been proposed by [PERSON] [5] and creates pseudo-confidence ellipses around MDS points based on an implementation of the Hessian of the stress loss function. However, the area of the pseudo-confidence ellipses depends upon the value of a parameter \(\varepsilon\), which shows that the stress value at the perturbation region should be at most 100 \(\varepsilon\%\) larger than the local minimum of the stress loss function. Thus, in fact, the calculation of confidence ellipses assumes an arbitrary choice of the value of parameter \(\varepsilon\).
2. Jackknife method (JACmds). This has been developed by [PERSON] and [PERSON] [6], who proposed a leave-one-out method that can be used both in metric and nonmetric MDS. The jackknife MDS plot is a graph with stars, where the centers of the stars are the jackknife centroids and the rays are the jackknife solutions. In the present study, we have added a convex hull to the points of each star, and we have calculated their areas.
3. Bootstrapping of the MDS residuals (BOOTres). This method is described in the manual of the _MultBiplotR_ package of R in the details of the _BootstrapDistance_ function [7]. It is based on the study carried out by [PERSON] [8], [PERSON] and [PERSON] [9], and [PERSON] and [PERSON] [10] and uses random sampling or permutations of MDS residuals to obtain the bootstrap replications that are used to create bootstrap confidence ellipses.
4. Overlapping MDS maps (MAPsov). This is one of the most interesting methods whose origin can be found in the study carried out by [PERSON] and [PERSON] [11]; [PERSON] and [PERSON] [12]; and [PERSON], [PERSON], and [PERSON] [13], whereas its current version is due to [PERSON] and [PERSON] [14]. The steps used in MAPsov to construct confidence ellipses in a MDS plot are the following: Based on the original dataset, a great number of datasets of artificial distances are generated using simulations or resampling techniques. For each artificial dataset of pairwise distances, the MDS map, that is, the first two MDS dimensions, is computed. The MDS maps are scaled, reflected, rotated, translated, and finally superimposed. From the superimposed data, confidence ellipses are constructed.  
Note that in all methods presented above, the original dataset consists of two or more samples of continuous or binary random variables. That is, it is a multivariate dataset of two or more samples. For each stability method, the proposed stability measure is estimated (_STvr_) (see Equations (5) and (9) in [4]). In addition, we calculate the percentage of overlapping areas (_POij_), a stability measure based on the overlapping regions (_STov_), and a stability measure based on the average areas (_STma_) defined by:  
\[PO_{ij}=\frac{A_{i}\cap A_{j}}{A_{i}+A_{j}-A_{i}\cap A_{j}} \tag{1}\]  
\[STov=1-\frac{1}{N}\underset{i=1}{\overset{N}{\sum}}\frac{A_{i}\cap A_{j}}{\min (A_{i},A_{j})} \tag{2}\]  
\[STma=\underset{j=1}{\overset{n}{\sum}}\frac{A_{j}}{n} \tag{3}\]  
where \(A_{i}\) and \(A_{j}\) are the areas of confidence ellipses that correspond to samples \(i\) and \(j\), respectively; \(n\) is the number of samples; and \(N=n(n-1)/2\) is the total number of all pairs of samples \(i\) and \(j\) when \(i<j\).