On the relative efficiency of the intent-to-treat Wilcoxon–Mann–Whitney test in the presence of noncompliance

Abstract

Summary A general framework is set up to study the asymptotic properties of the intent-to-treat Wilcoxon–Mann–Whitney test in randomized experiments with nonignorable noncompliance. Under location-shift alternatives, the Pitman efficiencies of the intent-to-treat Wilcoxon–Mann–Whitney and t tests are both derived. It is shown that the former is superior if the compliers are more likely to be found in high-density regions of the outcome distribution, or in other words, if the noncompliers tend to reside in the tails. By logical extension, the relative efficiency of the two tests is sharply bounded by a least and most favourable scenario where the compliers are segregated in regions of lowest or highest density, respectively. Such bounds can be derived analytically as a function of the compliance rate for common location families such as Gaussian, Laplace, logistic, and t distributions. These results help empirical researchers pick the more efficient test for existing data and calculate sample size for future trials in anticipation of noncompliance. Results for nonadditive alternatives and other tests follow along similar lines.