Wilcoxon-Mann-Whitney statistics in randomized trials with non-compliance

Abstract

The Mann–Whitney-type stochastic shift $\mathbb{P}(Y>X)$ has long been used as a scale-free alternative to the mean difference in measuring the distance between two populations. It has recently been recast as a causal estimand, but only in standard settings where confounders are fully captured. We study the Mann–Whitney treatment effect (MWTE) in randomized trials with non-ignorable non-compliance, where the treatment received is confounded by unknown factors. First, we define and estimate a local MWTE on the compliers via the standard principal-stratification approach with randomization status as an instrumental variable (IV). Then, we derive sensitivity bounds for the local effect estimand when key IV assumptions such as exclusion restriction and monotonicity are violated. Finally, we study the asymptotic operating characteristics of the local MWTE estimator in testing the treatment effect. Analytic bounds on the asymptotic relative efficiencies show that this IV-based test is likely superior to standard intent-to-treat tests under location-shift alternatives. The proposed methodology is applied to the famous National Job Training Partnership Act Study as an illustration.