AUTHOR=Sanders Shane , Ehrlich Justin , Boudreau James 

TITLE=Simpson's aggregation paradox in nonparametric statistical analysis: Theory, computation, and susceptibility in public health data

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=9

YEAR=2023

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2023.1169164

DOI=10.3389/fams.2023.1169164

ISSN=2297-4687

ABSTRACT=<p>This study establishes sufficient conditions for observing instances of <italic>Simpson's</italic> (data aggregation) <italic>Paradox</italic> under rank sum scoring (<italic>RSS</italic>), as used, e.g., in the <italic>Wilcoxon-Mann-Whitney</italic> (<italic>WMW</italic>) rank sum test. The <italic>WMW</italic> test is a primary nonparametric statistical test in <italic>FDA</italic> drug product evaluation and other prominent medical settings. Using computational nonparametric statistical methods, we also establish the relative frequency with which paradox-generating <italic>Simpson Reversals</italic> occur under <italic>RSS</italic> when an initial data sequence is pooled with its ordinal replicate. For each 2-sample, n-element per sample or 2 x <italic>n</italic> case of <italic>RSS</italic> considered, strict <italic>Reversals</italic> occurred for between 0% and 1.74% of data poolings across the whole sample space, roughly similar to that observed for 2 x 2 x 2 <italic>contingency tables</italic> and considerably less than that observed for <italic>path models</italic>. The <italic>Reversal</italic> rate conditional on observed initial sequence is highly variable. Despite a mode at 0%, this rate exceeds 20% for some initial sequences. Our empirical application identifies clusters of <italic>Simpson Reversal</italic> susceptibility for publicly-released mobile phone radiofrequency exposure data. <italic>Simpson Reversals</italic> under <italic>RSS</italic> are not simply a theoretical concern but can reverse nonparametric or parametric biostatistical results even in vitally important public health settings. Conceptually, <italic>Paradox</italic> incidence can be viewed as a robustness check on a given <italic>WMW</italic> statistical test result. When an instance of <italic>Paradox</italic> occurs, results constituting this instance are found to be data-scale dependent. Given that the rate of <italic>Reversal</italic> can vary substantially by initial sequence, the practice of calculating this rate conditional on observed initial sequence represents a potentially important robustness check upon a result.</p>