We apply covariate adjustment to the Wincoxon two sample statistic and Wincoxon-Mann-Whitney test in comparing two treatments. The covariate adjustment through calibration not only improves efficiency in estimation/inference but also widens the application scope of the Wilcoxon two sample statistic and Wincoxon-Mann-Whitney test to situations where covariate-adaptive randomization is used. We motivate how to adjust covariates to reduce variance, establish the asymptotic distribution of adjusted Wincoxon two sample statistic, and provide explicitly the guaranteed efficiency gain. The asymptotic distribution of adjusted Wincoxon two sample statistic is invariant to all commonly used covariate-adaptive randomization schemes so that a unified formula can be used in inference regardless of which covariate-adaptive randomization is applied.
Consider a random sample of n units, each of which is assigned to one and only one treatment j and results in an outcome Y j distributed with an unknown continuous distribution F j , j = 1, ..., J, where J ≥ 2 is the number of treatments. An example is a clinical trial to study effects of J medical products, in which units are typically patients. The simplest way of assigning treatments is simple randomization that assigns n units completely at random with a pre-determined probability π j > 0 to treatment j, J j=1 π j = 1.
Let A ∈ {1, …, J} be the treatment assignment and A i be the treatment assignment for unit i in the random sample, i = 1, …, n. Details of generating A i ’s are given in Section 2. If A i = j, then unit i is assigned to treatment j and the observed outcome from unit i is denoted by Y iAi = Y ij ∼ (distributed as) Y j . Estimation and inference on unknown characteristics in F 1 , …, F J can be carried out based on outcomes Y iAi , i = 1, …, n.
For comparing two fixed treatments j and k, the well-known Wilcoxon-Mann-Whitney rank-sum test (Lehmann, 1975, pages 5-9) is a valuable nonparametric alternative to the two sample t-test (based on sample means of outcomes from two treatment groups) that is criticized when the outcome Y j or Y k is not normally distributed and/or has large variance. The Wilcoxon-Mann-Whitney test statistic is given by
focusing on the number of outcome pairs Y ij and
, where I(B) is the indicator of event B, n t is the number of i’s with A i = t, and t = j, k. The U jk in (1) is called the Wilcoxon two sample statistic (Serfling, 1980, page 175), a special case of the two sample U-statistic for estimating the treatment effect θ jk = E(U jk ) = P (Y j ≤ Y k ). Under the null hypothesis H 0 : F j = F k (i.e., there is no difference in populations under treatments j and k), θ jk = 1/2 and, thus, the Wilcoxon-Mann-Whitney test rejects H 0 when U jk is far way from 1/2. More details are given in Section 3.1. In many studies there exists covariate information related with the outcomes, useful for gaining estimation efficiency. In clinical trials, for example, there are baseline covariates not affected by treatments, such as patient’s age, sex, geographical location, occupation, education level, disease stage, etc. Utilizing covariates to improve efficiency of estimation and inference is referred to as covariate adjustment.
The Wilcoxon two sample statistic U jk in (1) does not make use of any covariate and, thus, it may be improved by covariate adjustment. If a correct model between outcomes and covariates can be specified, then U jk can be improved through model fitting with covariates. However, such a modelbased approach relies heavily on the model correctness. Because a correct model may not be easily specified in applications, model-free approaches for covariate adjustment have caught on recently. In the regulatory agencies of clinical trials, for example, it is particularly recommended to utilize covariates “under approximately the same minimal statistical assumptions that would be needed for unadjusted estimation” (ICH E9, 1998;EMA, 2015;FDA, 2021). Note that the consistency and asymptotic normality of Wilcoxon two sample U jk in (1) is established under no assumption other than simple randomization and n → ∞ (Jiang, 2010).
The purpose of this paper is to apply covariate adjustment to U jk in (1) and the related Wilcoxon-Mann-Whitney test, through model-free covariate calibration considered as early as in Cassel et al. (1976) for survey problems and well summarized in Särndal et al. (2003). The covariate calibration has been shown to be effective in gaining efficiency for functions of sample means or estimators from generalized estimation equations (Yang and Tsiatis, 2001;Freedman, 2008;Zhang et al., 2008;Moore and van der Laan, 2009;Lin, 2013;Vermeulen et al., 2015;Wang et al., 2019;Liu and Yang, 2020;Benkeser et al., 2021;Zhang and Zhang, 2021;Cohen and Fogarty, 2023;Wang et al., 2023;Ye et al., 2023;Bannick et al., 2025, among others). But our covariate adjustment for U jk in (1) is created by directly using the covariance between U jk and adjusted covariates without any assumption. It guaran-tees an asymptotic efficiency gain over the unadjusted U jk and provides an invariant inference formula for all commonly used covariate-adaptive randomization schemes (Section 2.1), including simple randomization. It also gains additional efficiency in comparing two treatments when there are more than two treatments (J > 2).
After introducing notation and randomization for treatment assignments, in Section 2.2 we derive covariate adjustment/calibration for the Wilcoxon two sample statistic U jk in (1), motivated by why this adjustment guarantees efficiency gain. In Section 2.3 we establish that the proposed covariate adjusted statistic is asymptotically normal under all commonly used covariate-adaptive randomization, without any assumption. The covariate adjusted
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