Measure of Strength of Evidence for Visually Observed Differences between Subpopulations

An increasingly important data analytic challenge is understanding the relationships between subpopulations. Various visualization methods that provide many useful insights into those relationships are popular, especially in bioinformatics. This paper proposes a novel and rigorous approach to quantifying subpopulation relationships called the Population Difference Criterion (PDC). PDC is simultaneously a quantitative and visual approach to showing separation of subpopulations. It uses subpopulation centers, the respective variation about those centers and the relative subpopulation sizes. This is accomplished by drawing motivation for the PDC from classical permutation based hypothesis testing, while taking that type of idea into non-standard conceptual territory. In particular, the domain of very small P values is seen to seem to provide useful comparisons of data sets. Simulated permutation variation is carefully investigated, and we found that a balanced permutation approach is more informative in high signal (i.e large subpopulation difference) contexts, than conventional approaches based on all permutations. This result is quite surprising in view of related work done in low signal contexts, which came to the opposite conclusion. This issue is resolved by the proposal of an appropriate adjustment. Permutation variation is also quantified by a proposed bootstrap confidence interval, and demonstrated to be useful in understanding subpopulation relationships with cancer data.