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This study establishes sufficient conditions for observing instances of Simpson's (data aggregation) Paradox under rank sum scoring (RSS), as used, e.g., in the Wilcoxon-Mann-Whitney (WMW) rank sum test. The WMW test is a primary nonparametric statistical test in FDA drug product evaluation and other prominent medical settings. Using computational nonparametric statistical methods, we also establish the relative frequency with which paradox-generating Simpson Reversals occur under RSS when an initial data sequence is pooled with its ordinal replicate. For each 2-sample, n-element per sample or 2 x n case of RSS considered, strict Reversals 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 contingency tables and considerably less than that observed for path models. The Reversal 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 Simpson Reversal susceptibility for publicly-released mobile phone radiofrequency exposure data. Simpson Reversals under RSS are not simply a theoretical concern but can reverse nonparametric or parametric biostatistical results even in vitally important public health settings. Conceptually, Paradox incidence can be viewed as a robustness check on a given WMW statistical test result. When an instance of Paradox occurs, results constituting this instance are found to be data-scale dependent. Given that the rate of Reversal 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. |
