Sum of Euclidean Distance Differences and Sum of Absolute Manhattan Distance Differences: multicriteria decision making tools for small data tables

Abstract

Background

Despite its advantages, rank transformation leads inevitably to information loss. This work presents an extension for sum of ranking differences (SRD) algorithm for non-ranking environment. It is expedient to elaborate a new algorithm, which overcomes this difficulty. The procedure has been developed by the analogy of SRD, i.e., pairwise comparisons of (column) vectors, fixing one of them as gold standard and introducing two validation steps (the randomization and Wilcoxon tests after assigning uncertainties by cross-validation).

Results

Two emblematic distance metrics were involved in the development: the most frequently applied Euclidean distance and its robust counterpart the city block (Manhattan) distance. Such a way two new dissimilarity measures have been defined: Sum of Euclidean Distance Differences (DnE) and Sum of Absolute Manhattan Distance Differences (DnM) along with their randomization tests and Variance Analysis (ANOVA). Unfortunately, when leaving the safe rank environment, we also leave the well-known permutations and the theoretical backgrounds (Spearman footrule), as well. This study is limited to a maximum of eight rows in the input matrix, where exact theoretical random distributions are available. Sixteen carefully chosen data sets were selected covering a wide range of scientific disciplines and of numbers for columns and rows in the input matrix: between three to 80 and five to eight, respectively. Three case studies illustrate the advantages and disadvantages of the new dissimilarity measures and statistical tests.

Significance

Superior discrimination ability characterizes DnE and DnM; they provide a more sophisticated ranking (and grouping) patterns than SRD despite their smaller visualization (applicability) domain. The randomization test loses its sensitivity in the order of SRD>DnE>DnM. The latter two realize different clustering patterns from SRD and from each other but (almost) the same ordering. Hence, only one of them is recommended in a ranking environment. Although the random distributions of DnE and DnM is distorted a little, the probability of first kind error (say 5%) can safely be determined from the cumulated frequencies. Comprehensive enumeration of advantages and disadvantages has been completed for SRD, DnE and DnM as dissimilarity measures, clustering tools, multicriteria decision making (MCDM) techniques and their competitors. While preserving great advantages of SRD (simplicity, generality, MCDM character and lack of subjective weights), both new techniques are suitable dissimilarity measures, clustering and MCDM tools in non-ranking environments. DnE and DnM also inflict universal scales for later ANOVA and Wilcoxon tests.