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The Mann–Whitney effect is a nonparametric measure for comparing the distribution between two groups, which can be estimated by right-censored data. However, the traditional estimator of the Mann–Whitney effect based on the Kaplan–Meier estimators can be inconsistent when the independent censoring assumption fails to hold. Investigation is made on the asymptotic bias of the traditional estimator of the Mann–Whitney effect when the independent censoring assumption is violated due to dependence between survival time and censoring time. A new estimator of the Mann–Whitney effect is proposed by applying the copula-graphic estimator to adjust for the effect of dependent censoring. The proposed estimator and test are consistent when the assumed copulas for the two groups are correct. Some consistency properties under misspecified copulas are also given. Simulations are conducted to verify the proposed method under possible misspecification on copulas. The method is illustrated by a real data set. We provide an R function "MW.test" to implement the proposed estimator and test. • Two-sample tests under dependent censoring. • The traditional Mann–Whitney test yields substantial bias if censoring is dependent. • A new estimator of the Mann–Whitney effect is proposed by applying the copula-graphic estimator. • We provide an R function "MW.test" to implement the proposed estimator and test. • In some cases, even if the copula is misspecified, the new test can be unbiased (consistent). [ABSTRACT FROM AUTHOR]