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In this paper we perform inference on the effect of a treatment on survival times in studies where the treatment assignment is not randomized and the assignment time is not known in advance. Two such studies are discussed: a heart transplant program and a study of Swedish unemployed eligible for employment subsidy. We estimate survival functions on a treated and a control group which are made comparable through matching on observed covariates. The inference is performed by conditioning on waiting time to treatment, that is, time between the entrance in the study and treatment. This can be done only when sufficient data are available. In other cases, averaging over waiting times is a possibility, although the classical interpretation of the estimated survival functions is lost unless hazards are not functions of waiting time. To show unbiasedness and to obtain an estimator of the variance, we build on the …