A note on the amount of information borrowed from external data in hybrid controlled trials with time-to-event outcomes

In situations where it is difficult to enroll patients in randomized controlled trials, external data can improve efficiency and feasibility. In such cases, adaptive trial designs could be used to decrease enrollment in the control arm of the trial by updating the randomization ratio at the interim analysis. Updating the randomization ratio requires an estimate of the amount of information effectively borrowed from external data, which is typically done with a linear approximation. However, this linear approximation is not always a reliable estimate, which could potentially lead to sub-optimal randomization ratio updates. In this note, we highlight this issue through simulations for exponential time-to-event outcomes, because in this simple setting there is an exact solution available for comparison. We also propose a potential generalization that could complement the linear approximation in more complex settings, discuss challenges for this generalization, and recommend best practices for computing and interpreting estimates of the effective number of events borrowed.