Aggregated statistical rankings are arbitrary

In many areas of mathematics, statistics, and the social sciences, the intriguing, and somewhat unsettling, paradox occurs where the “parts” may give rise to a common decision, but the aggregate of those parts, the “whole”, gives rise to a different decision. The Kruskal-Wallis nonparametric statistical test on n samples which can be used to rank-order a list of alternatives is subject to such a Simpson-like paradox of aggregation. That is, two or more data sets each may individually support a certain ordering of the samples under Kruskal-Wallis, yet their union, or aggregate, yields a different outcome. An analysis of this phenomenon yields a computable criterion which characterizes which matrices of ranked data, when aggregated, can give rise to such a paradox. Copyright Springer-Verlag Berlin Heidelberg 2003