Rule selection invariance as a robustness check in collective choice and nonparametric statistical settings

This study establishes conditional equivalence between Borda rule and rank sum collective choice. We apply the equivalence condition toward a comparison of the Wilcoxon–Mann–Whitney (WMW) rank sum test and sign test in non-parametric statistics, where the sign test is shown to be procedurally-equivalent to pairwise Borda rule aggregation. We further establish a social choice theoretic robustness check on the WMW test by determining whether a significant WMW rank sum winner could be a raw Borda loser (i.e., could have a p-value greater than 0.5 in a corresponding one-sided sign test). We then test whether a significant sign test winner could be a raw WMW winner. The WMW test is robust against significant victory by a raw Borda loser for almost all small sample cases (specifically, for n < 27) at any standard significance level (i.e., for α≤ 0.10). The sign test is robust against significant victory by a raw rank sum loser only for n < 14 at any standard significance level. The results provide caution against using the α = 0.10 significance level in some small sample cases of the WMW test. The robustness checks developed herein can be used generally or specifically to check whether a given set of non-parametric data passes the robustness check or whether significance for one test guarantees raw victory in the alternative test (for a given sample size n) in general. These robustness checks present a methodology to test for a weak form of qualitative equivalence across non-parametric tests. We develop a web application (http://avisss.com/visualization/RankSumHistograms.html) for such robustness checks.