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In many medical and scientific settings, the choice of treatment or intervention may be de-termined by a covariate threshold. For example, elderly men may receive more thoroughdiagnosis if their prostate-specific antigen (PSA) level is high. In these cases, the causaltreatment effect is often of great interest, especially when there is a lack of evidence fromrandomized clinical trials. From the social science literature, a class of methods known asregression discontinuity (RD) designs can be used to estimate the treatment effect in thissituation. Under certain assumptions, such an estimand enjoys a causal interpretation. Weshow how to estimate causal effects under the regression discontinuity design for censoreddata. The proposed estimation procedure employs a class of censoring unbiased transfor-mations that includes inverse probability censored weighting and doubly robust transfor-mation schemes. Simulation studies are used to evaluate the finite-sample properties of theproposed estimator. We also illustrate the proposed method by evaluating the causal effectof PSA-dependent screening strategies