Journal of Data Science

Abstract: When comparing two independent groups, the shift function compares all of the quantiles in a manner that controls the probability of at least one Type I error, assuming random sampling only. Moreover, it provides a much more detailed sense of how groups compare, versus using a single measure of location, and the associated plot of the data can yield valuable insights. This note examines the small-sample properties of an ex tension of the shift function where the goal is to compare the distributions of two specified linear sums of the random variables under study, with an emphasis on a two-by-two design. A very simple method controls the proba bility of a Type I error. Moreover, very little power is lost versus comparing means when sampling is from normal distributions with equal variances.