## 5 Results and Discussion  
### Effect of Sample Size  
From the theory of the distances under study, we understand that their standard deviation decreases with increasing sample size, tending to zero as the sample size tends to infinity (see Supplement-S1-Moments in [15]). Therefore, when the sample size increases, the variability of the generated simulation distances decreases, resulting in smaller confidence intervals. That is, with great sample sizes, the confidence ellipses of the superimposed MDS solutions tend to become points that show the location of the populations in the MDS plot. Under these conditions, the probability of the most likely dendrogram obtained from the confidence ellipses tends to 1, and the same holds for its cluster probabilities. Thus, the uncertainty in a MDS plot, irrespective of whether it is assessed through the area of the ellipses or the corresponding dendrogram and its cluster probabilities, depends upon the size of the samples and tends to zero as that size tends to infinity. Based on these observations, we examined the dependence of \(STma\) and the probability \(Pr\) of the most likely dendrograms upon the sample size in the simulated datasets. The most representative results are shown in Figures 1-4. We observe that the stability measure \(STma\) calculated using the PCE method does not converge to zero when the sample size increases. Moreover, the value of \(STma\) obtained from PCE depends on the argument _eps_\(=\varepsilon\). However, if we change _eps_, the plot in Figure 1 remains intact, and only the scale of the \(y\)-axis changes, but without a change in the position of the zero value. Similarly, the \(STma\) obtained from the BOOTres technique does not converge to zero when the sample size increases. Note that this property is also observed when using the conventional (not-squared) Euclidean distance. In addition, the ellipses generated from BOOTres using the corMDS technique largely overlap, yielding low \(STvr\) values. For example, the \(STvr\) is close to 1 when using the Euclidean distance and drops to around 0.3 and 0.45 when using the squared and not-squared Euclidean distances, respectively. This indicates that, in addition to nMDS, corMDS is also possibly incompatible with the BOOTres method. In any case, BOOTres and PCA do not calculate confidence ellipses. Figure 1: Dependence of the stability measure \(STma\) upon \(n\) for the ED (\(\bullet\)) and MD1 (o) distance measures when the PCE method for pseudo-confidence ellipses has been used with eps = 0.01. Figure 3 shows the dependence of the stability measure \(STma\) upon sample size calculated from the JACKmds method using ordinal and ratio MDS of the ED distance. A decrease in \(STma\) is generally observed when the sample size tends to zero. However, extreme \(Stma\) values appear many times, almost in all sample sizes studied. Thus, zero \(STma\) values can appear even in small sample sizes, equal to 20 or 50, while particularly large values can also be observed. For example, in Figure 3, we can observe such large values when the sample size is equal to 1000 and 2000. Such extreme values have not been observed in any of the other methods examined and lead to a large amount of uncertainty regarding the accuracy of the method. Due to this uncertainty, its results should be treated with caution, and for this reason, it was not examined further. Finally, in what concerns the MAPsov method, in all datasets examined, \(STma\) tends rapidly to 0 with the increase in the sample size (Figure 4--left). It was found that the dependence of \(STma\) on \(n\) follows the power model:  
\[STma=A\ n^{-B} \tag{8}\]  
with \(B\) > 0 and high R\({}^{2}\) fitting values, most of them around 0.99. As expected, this decrease in \(STma\) with the increase in \(n\) is associated with an increase in the probability of the  
Figure 3: Dependence of the stability measure \(STma\) upon \(n\) calculated using the JACKmds method with an ordinal (o) and ratio (+) MDS of the ED distance.