The Wilcoxon–Mann–Whitney Estimand Versus Differences in Medians or Means

There is a renewed interest in defining the target of estimation when designing randomized trials. Motivated by design work in trials of HIV-1 curative interventions, we compare the Wilcoxon-Mann-Whitney (WMW) estimand to a difference in medians or means in a two-arm study. First, we define each estimand along with an appropriate estimator. Then, we highlight relevant asymptotic relative efficiency (ARE) results for the estimators under normal distributions (ARE: WMW/mean = <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mi>π</mml:mi></mml:mrow> <mml:annotation>$$ 3/\pi $$</mml:annotation></mml:semantics> </mml:math> , median/mean = <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mi>π</mml:mi></mml:mrow> <mml:annotation>$$ 2/\pi $$</mml:annotation></mml:semantics> </mml:math> , median/WMW = <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn></mml:mrow> <mml:annotation>$$ 2/3 $$</mml:annotation></mml:semantics> </mml:math> ), as well as normal mixtures. Measurement of outcomes related to HIV-1 cure involve laboratory assays with lower limits of quantification giving rise to left-censored data. In our simulation study, we compare the estimators in the presence of left-censored observations and at small sample sizes, illustrating that under a censored normal mixture distribution the WMW approach is unbiased, powerful, and has confidence intervals with nominal coverage. We apply our findings to a randomized trial designed to reduce HIV-1 reservoirs. We further expose several extensions of the WMW approach that allows for assessment of interactions between subgroups in a trial, adjustment for covariates, and general ranking methods for clinical outcomes in other disease areas. We end with a discussion summarizing the merits of a WMW based intervention effect estimate versus an estimate summarized on the scale the intervention was originally measured such as the difference in medians or means.