  We consider how increasingly available observational data can be used to
improve the design of randomized controlled trials (RCTs). We seek to design a
prospective RCT, with the intent of using an Empirical Bayes estimator to
shrink the causal estimates from our trial toward causal estimates obtained
from an observational study. We ask: how might we design the experiment to
better complement the observational study in this setting?
  We propose using an estimator that shrinks each component of the RCT causal
estimator toward its observational counterpart by a factor proportional to its
variance. First, we show that the risk of this estimator can be computed
efficiently via numerical integration. We then propose algorithms for
determining the best allocation of units to strata (the best "design"). We
consider three options: Neyman allocation; a "naive" design assuming no
unmeasured confounding in the observational study; and a "defensive" design
accounting for the imperfect parameter estimates we would obtain from the
observational study with unmeasured confounding.
  We also incorporate results from sensitivity analysis to establish guardrails
on the designs, so that our experiment could be reasonably analyzed with and
without shrinkage. We demonstrate the superiority of these experimental designs
with a simulation study involving causal inference on a rare, binary outcome.
