Rank Correlation and Cluster Testing in Multi-Objective Data Analysis of Medical Data

Multi-Objective Data Analysis is a method of exploring conflicting patterns in static data. This method is used to explore complex relationships commonly found in medical data, where human body systems often have overlapping variables, creating a complex web of relationships. Within multi-objective optimization, a relationship is usually described as conflicting if an increase in optimality in one objective leads to a subsequent decrease in the other. This relationship is difficult to extract from static data, where no optimization is taking place. Multi-Objective Data Analysis extracts conflicting relationships using non-dominated sorting to find multiple fronts that represent the gradient of the fitness landscape between the decision and objective spaces, and checks to see if this gradient is related to variables in the decision space that are of interest. In this work, we showcase the use of the Wilcoxon (or Proportions) test and the Pearson Correlation test as objective measures of the usefulness of a found conflicting pair of objectives by comparing the distributions of a chosen response variable relative to the sorting. The shown examples include a clean synthetic dataset, a synthetic dataset with noise in the objective space, and a real-world dataset consisting of breast cancer data. From these examples, the argument is made that the statistical tests are compatible with Multi-Objective Data Analysis.