  Randomized experiments can provide unbiased estimates of sample average
treatment effects. However, estimates of population treatment effects can be
biased when the experimental sample and the target population differ. In this
case, the population average treatment effect can be identified by combining
experimental and observational data. A good experiment design trumps all the
analyses that come after. While most of the existing literature centers around
improving analyses after RCTs, we instead focus on the design stage,
fundamentally improving the efficiency of the combined causal estimator through
the selection of experimental samples. We explore how the covariate
distribution of RCT samples influences the estimation efficiency and derive the
optimal covariate allocation that leads to the lowest variance. Our results
show that the optimal allocation does not necessarily follow the exact
distribution of the target cohort, but adjusted for the conditional variability
of potential outcomes. We formulate a metric to compare and choose from
candidate RCT sample compositions. We also develop variations of our main
results to cater for practical scenarios with various cost constraints and
precision requirements. The ultimate goal of this paper is to provide
practitioners with a clear and actionable strategy to select RCT samples that
will lead to efficient causal inference.
