Assessing Algorithms Used for Constructing Confidence Ellipses in Multidimensional Scaling Solutions

Abstract

This paper assesses algorithms proposed for constructing confidence ellipses in multidimensional scaling (MDS) solutions and proposes a new approach to interpreting these confidence ellipses via hierarchical cluster analysis (HCA). It is shown that the most effective algorithm for constructing confidence ellipses involves the generation of simulated distances based on the original multivariate dataset and then the creation of MDS maps that are scaled, reflected, rotated, translated, and finally superimposed. For this algorithm, the stability measure of the average areas tends to zero with increasing sample size n following the power model, An−B, with positive B values ranging from 0.7 to 2 and high R-squared fitting values around 0.99. This algorithm was applied to create confidence ellipses in the MDS plots of squared Euclidean and Mahalanobis distances for continuous and binary data. It was found that plotting confidence ellipses in MDS plots offers a better visualization of the distance map of the populations under study compared to plotting single points. However, the confidence ellipses cannot eliminate the subjective selection of clusters in the MDS plot based simply on the proximity of the MDS points. To overcome this subjective selection, we should quantify the formation of clusters of proximal samples. Thus, in addition to the algorithm assessment, we propose a new approach that estimates all possible cluster probabilities associated with the confidence ellipses by applying HCA using distance matrices derived from these ellipses.