Early detection of cancer by screening advances the date of diagnosis, but may or may not alter time to death. Screening programme need to assess the true benefit of screening, that is, the length of time by which survival has been extended, beyond merely the time by which the diagnosis is advanced (lead-time). One method is to estimate the distribution of the time survived post-lead-time using total survival time data for screen-detected cancer cases, under the assumption of independence of the lead-time and the post-lead-time survival. However, it seems biologically reasonable that the lead-time and the post-lead-time survival are positively correlated. This paper investigates the consequences of departures from independence of lead-time and post-lead-time survival on estimation of post-lead-time survival. We introduce a new model that involves dependence between the lead-time and the post-lead-time survival. We show that the new model can be converted to the model discussed by Xu and Prorok. We consider the non-parametric maximum likelihood estimator of the post-lead-time survival under the new model. We apply the method to data from the HIP (Health Insurance Plan of Greater New York) breast cancer screening trial. We make comparisons with the survival of cancer cases not detected by screening, such as interval cases, cases among individuals who refused screening, and randomized control cases.