  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.
