## 5 Results and Discussion
### Confidence Regions and Cluster Probabilities  
The _MAPsov()_ function, using one or more pre-selected MDS methods and a certain distance measure, provides P% confidence ellipses along with the most likely dendrogram with its cluster probabilities obtained from these ellipses, the probability of its appearance, and the probabilities of the most frequent clusters. In addition, for comparison, the function provides similar information obtained from the direct application of HCA to the original dataset, that is, the basic dendrogram with its cluster probabilities, the probability of its detection, and various cluster probabilities, like the probabilities of the most frequent clusters. It can also provide the first \(m\) most likely dendrograms and their probabilities. Representative results are shown in Figures 7 and 8 and in Supplement-S2-Tables and Dendrograms. Figures 7 and 8 show 95% confidence ellipses based on the corMDS or nMDS technique using the cMD1 or cBMD1 distance measures applied to the datasets EG1, Sim-100, WHI, and NO1. Note that the Sim-100 dataset is an artificial dataset that simulates the EG1 dataset using larger sample sizes, equal to 100. The figures also depict the corresponding most likely dendrograms with their probability, that is, the probability of their detection, and the Monte-Carlo cluster probabilities. More details related to Figures 7 and 8 are provided in Supplement-S2-Tables and Dendrograms. The general observation that comes from all MDS plots with 95% confidence ellipses is that the addition of these ellipses gives a much clearer picture of the biodistance map of the populations under study than the single points of the MDS plots. However, although they help to visualize the effect of uncertainty in the calculated distances on the distribution of populations on the MDS map, they do not eliminate subjective cluster selection based simply on the proximity of various MDS points. This step can be achieved only if we combine 95% confidence ellipses with HCA and, in particular, if we estimate cluster probabilities based on these ellipses. Thus, if we consider Figure 7 and the corresponding results presented in the spreadsheets _EG1_ and _Sim-100_ in Supplement-S2-Tables and Dendrograms, we observe that for the dataset EG1, due to the rather small sample size, which is equal to 30, the 95% ellipses overlap, especially the ellipses of the sample pairs 1-2, 3-4, and 4-5, creating the impression that these pairs form pairwise clusters. This impression is enhanced by the high correlation between the percent of pairwise overlapping areas of 95% confidence ellipses and the corresponding cluster probabilities calculated from these ellipses. The Pearson and Spearman coefficients are 0.994 and 0.899, respectively, and, as expected, they are both statistically significant. Note that, similarly, the Pearson and Spearman coefficients between overlapping areas of 95% confidence ellipses and the corresponding cluster probabilities of the MDS plots of Figure 8 are high (>0.72) and statistically significant (spreadsheet Correlations in Supplement-S2-Tables and Dendrograms). However, this correlation is related to the  
Figure 8: 95% confidence ellipses based on the corMDS and nMDS techniques using the cMD1 and cBMD1 distance measures on the WHI and NO1 datasets and the corresponding most likely dendrograms with cluster probabilities. uncertainty in the MDS plots and not to the presence of clusters. This can be seen if we look at low-uncertainty MDS plots, such as the plot in Figure 7 for the Sim-100 dataset. In such plots, the 95% confidence ellipses do not overlap, and, therefore, no correlation exists between overlapping areas and cluster probabilities. Moreover, at low-uncertainty MDS plots, the confidence ellipses may not help to determine clusters of samples. For example, we observe in Figure 7 that for the dataset Sim-100, the 95% confidence ellipses do not overlap and are approximately equidistant from each other. Therefore, we cannot confidently infer possible clusters among the samples.