A Multivariate Rank-Based Two-Sample Test Statistic

Abstract

The problem of testing whether two samples come from the same or different populations is a classical one in statistics. A new distribution-free test based on standardized ranks for the univariate two-sample problem is studied. Providing a distribution-free (nonparametric) method offers a valuable technique for analyzing data that consists of ranks or relative preferences of data, and of data that are small samples from unknown distributions. The proposed test statistic examines the difference between the average of between-group rank distances and the average of within-group rank distances. This test statistic is closely related to the classical two-sample Cramér–von Mises criterion; however, they are different empirical versions of the same quantity for testing the equality of two population distributions. The advantage of the proposed rank-based test over the classical one is its ease to generalize to the multivariate case. In addition, the motivation of the proposed test is its application to microarray analyses in identifying sets of genes that are differentially expressed in various biological states, such as diseased versus non-diseased. A numerical study is conducted to compare the power performance of the rank formulation test with other commonly used nonparametric tests.