  We present a general framework for using existing data to estimate the
efficiency gain from using a covariate-adjusted estimator of a marginal
treatment effect in a future randomized trial. We describe conditions under
which it is possible to define a mapping from the distribution that generated
the existing external data to the relative efficiency of a covariate-adjusted
estimator compared to an unadjusted estimator. Under conditions, these relative
efficiencies approximate the ratio of sample size needed to achieve a desired
power. We consider two situations where the outcome is either fully or
partially observed and several treatment effect estimands that are of
particular interest in most trials. For each such estimand, we develop a
semiparametrically efficient estimator of the relative efficiency that allows
for the application of flexible statistical learning tools to estimate the
nuisance functions and an analytic form of a corresponding Wald-type confidence
interval. We also propose a double bootstrap scheme for constructing confidence
intervals. We demonstrate the performance of the proposed methods through
simulation studies and apply these methods to data to estimate the relative
efficiency of using covariate adjustment in Covid-19 therapeutic trials.
