An exploration of the effects of mathematical data structures on nonparametric statistical techniques based on ranks

In the case where repeated observations on a set of alternatives are made available, nonparametric statistical tests based on ranks can be used to determine whether or not the alternatives are significantly different. This class of tests includes the Kruskal-Wallis test, the V test, the Mann-Whitney test, and Wilcoxon test. When trying to determine if the alternatives are significantly different or not, inconsistencies amongst the test results can occur. A first step in understanding why inconsistent results occur between the test outcomes, is to study the overall rankings of the alternatives the tests intrinsically provide. The test statistic of each test is based on the overall ranking formed by analyzing the data. Data structures are characterized that cause inconsistencies among the test's overall rankings. To do this, a composition mapping is formed from the space of nx3 data to the space of possible rankings of three alternatives.;Methods of data aggregation using the class of test are discussed. In some cases, the addition of new data can occur at the raw data stage or at the ranked data stage while in others, the only option is to combine ranked data. Information is lost about the alternatives due to the many options in which the aggregation can occur.;The rankings are susceptible to a Simpson-like paradox as well. If two component data matrices give rise to the same overall rankings while using a rank-based test, it would seem that the aggregation of the two should output the same ranking. However, this is not the case. A condition is given that the data must satisfy in order for the paradox not to occur.