Covariate Adjustment for Wilcoxon Two Sample Statistic and Test

TL;DR

The paper addresses inefficiency of the Wilcoxon two-sample statistic under covariate-adaptive randomization by introducing a model-free covariate calibration that yields a covariate-adjusted statistic $U_{jk}^{\rm C}$. It derives the calibration $U_{jk}^{\rm C}=U_{jk}+(\bar{\mathbf X}_j-\bar{\mathbf X})^T\boldsymbol\beta_j-(\bar{\mathbf X}_k-\bar{\mathbf X})^T\boldsymbol\beta_k$ with $\boldsymbol\beta_j=\boldsymbol\Sigma^{-1}\boldsymbol C_{jk}$ and establishes its asymptotic normality under a wide class of covariate-adaptive randomizations, yielding a unified inference formula that is invariant to the randomization scheme when relevant covariates are included. The paper quantifies the efficiency gain via explicit Pitman ARE results and provides testing and confidence interval procedures based on plug-in estimators $\hat{\boldsymbol\Sigma}, \hat{\boldsymbol\beta}_j, \hat{\boldsymbol\beta}_k$, supported by simulations and a real clinical trial example. These contributions enable robust, more powerful nonparametric comparisons across covariate-adaptive designs and diverse outcome distributions, with interpretable results on the original outcome scale.

Abstract

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.