Abstract
The relationship between social choice aggregation rules and non-parametric statistical tests has been established for several cases. An outstanding, general question at this intersection is whether there exists a non-parametric test that is consistent upon aggregation of data sets (not subject to Yule-Simpson Aggregation Paradox reversals for any ordinal data). Inconsistency has been shown for several non-parametric tests, where the property bears fundamentally upon robustness (ambiguity) of non-parametric test (social choice) results. Using the binomial(n, p = 0.5) random variable CDF, we prove that aggregation of r(≥2) constituent data sets—each rendering a qualitatively-equivalent sign test for matched pairs result—reinforces and strengthens constituent results (sign test consistency). Further, we prove that magnitude of sign test consistency strengthens in significance-level of constituent results (strong-form consistency). We then find preliminary evidence that sign test consistency is preserved for a generalized form of aggregation. Application data illustrate (in)consistency in non-parametric settings, and links with information aggregation mechanisms (as well as paradoxes thereof) are discussed.
Highlights
The relationship between social choice aggregation rules and non-parametric statistical analysis is well-established [see, e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9]
We prove an additional result as to the nature of sign test consistency
We expect a greater proportional p-value decline given significant sign test results for the constituent, primitive data. This represents something of a strong form consistency result, whereby data aggregation reinforces significant sign test results to a greater degree
Summary
The relationship between social choice aggregation rules and non-parametric statistical analysis is well-established [see, e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9]. While Bargagliotti [34] shows a necessary condition (of the data) to ensure consistent (upon aggregation) results for the KW and Bhapkar’s V tests, respectively, the literature has not determined whether inconsistency upon aggregation is a general paradox among non-parametric statistical tests. Both the title of Haunsperger and Saari (“The Lack of Consistency for Statistical Decision Procedures”) and that of Haunsperger (“Aggregated Statistical Rankings are Arbitrary”) certainly leave open the intriguing possibility of a general result. Whether general consistency under aggregation is an impossibility among non-parametric tests Alternative tests, such as the sign test for matched pairs, apply distinct aggregation rules to non-parametric data in order to assign aggregated statistical rankings of groups.
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