How mature are survival data at the time of an interim analysis in a clinical trial with a survival outcome?

In a clinical trial with a survival outcome, an interim analysis is often performed to allow for early stopping for efficacy. If the interim analysis is early in the trial, one might conclude that a new treatment is more effective (compared to e.g.\ a placebo) and stop the trial, whereas the survival curves in the trial arms are not mature for the research question under investigation, for example because the curves are still close to 1 at that time. This means that the decision is based on a small percentage of the events in the long run only; possibly the events of the more frail patients in the trial who may not be representative for the whole group of patients. It may not be sensible to conclude effectiveness based on so little information. Criteria to determine the moment the interim analysis will be performed, should be chosen with care, and include the maturity of the data at the time of the interim analysis. Here, the expected survival rates at the interim analysis play a role. In this paper we will derive the asymptotic distribution of the Kaplan-Meier curves at the (random) moment the interim analysis will be performed for a one and two arm clinical trial. Based on this distribution, an interval in which the Kaplan Meier curves will fall into (with probability 95\%) is derived and could be used to plan the moment of the interim analysis in the design stage of the trial, so before the trial starts.