Distributions and Effects

Abstract

The nonparametric, rank-based inference methods presented in this book do not rely on means, medians, or other measures of location that have traditionally been used for parametric and semiparametric inference. As a consequence, effect sizes have to be quantified differently. To this end, the nonparametric relative treatment effect is defined. It measures how strong the stochastic tendency of observations from a particular treatment group is to assume greater values than observations from the other groups. For a precise definition of the relative effect, distribution functions are introduced first, including their normalized version. Properties of the nonparametric effect are presented, it is compared to other effect measures and related to the area under the receiver operating characteristic (ROC) curve. Finally, it is demonstrated how the theoretical relative effect can be estimated from data using the ranks of the observations. Thus, a natural link exists between a robust measure of stochastic tendency and rank-based statistical inference.