Ordinal Dominance Curve Based Inference for Stochastically Ordered Distributions

Abstract

SummaryIn a variety of applications researchers collect data that are sampled under ordered experimental conditions. In such situations it is reasonable to assume that outcomes are ordered stochastically by the level of the treatment. For example a toxicologist may want to assess the effect of a toxin on the reproductive system by studying the number of offspring produced by an animal as a function of the dose that it received. A variety of scientific questions arise in such a context. The most basic is whether a dose–response relationship is present. If so, determining the minimal toxic dose is of great interest. The strength of the ordering between groups i<j may be assessed by the probabilities ℙXi⩽Xj or monotone functions thereof. Large probabilities indicate a strong ordering. We find that such (and other) questions are naturally addressed by studying the ordinal dominance curve and the area under the curve (AUC). We develop new, non-parametric, order-restricted estimators for the AUCs in K⩾2 samples and investigate them in detail. We show that the restricted estimators are strongly consistent and have lower mean-squared error compared with the unrestricted estimators. In the two-sample case estimators are considerably improved in those situations where the data are highly variable and/or sample sizes are small, and where the AUC is close to its boundary value. Note that these are precisely the conditions under which estimating the AUC is difficult. Simulations indicate that in the multisample case estimators are improved under a much broader set of conditions with substantial reductions in mean-squared error. These estimators provide a basis for a family of distribution-free tests for the usual stochastic order in K⩾2 samples. Limiting distributions are derived. An accurate and simple approximation to the tail area is given for K = 2. Simulation studies show that the new tests perform well compared with existing alternatives. The methodology proposed is illustrated by applying it to an aquatic toxicology study where it is used to determine the minimal toxic dose more accurately.