Symmetry of Nonparametric Statistical Tests on Three Samples

Problem statement: Many different nonparametric statistical procedures can be used to analyze ranked data. Inconsistencies among the outcomes of such procedures can occur when analyzing the same ranked data set. Understanding why these peculiarities can occur is imperative to providing an accurate analysis of the ranking data. In this context, this study addressed why inconsistent outcomes can occur and which types of data structures cause the different procedures to yield different outcomes. Approach: Appropriate properties were identified and developed to explain why different methods can define different rankings of three samples with the same data. The approach identifies certain symmetry structures that are implicitly contained within the data and analyzes how the procedures utilize these structures to produce an outcome. Results: We proved that all possible differences among the nonparametric rules are caused because different rules place different levels of emphasis on the specified symmetry configurations of data. Our findings explain and characterize why different procedures can output different results using the same data set. Conclusion: This study therefore served as crucial step in deciding which nonparametric procedure to use when analyzing ranked data. In addition, it serves as the building block to defining new techniques to analyze rankings. Because different procedures use different aspects of the data in different ways, then one may determine the choice of analysis procedure based on what parts of the data one deems important.