When effect modifiers influence the decision to participate in randomized trials, generalizing causal effect estimates to an external target population requires the knowledge of two scores - the propensity score for receiving treatment and the sampling score for trial participation. While the former score is known due to randomization, the latter score is usually unknown and estimated from data. Under unconfounded trial participation, we characterize the asymptotic efficiency bounds for estimating two causal estimands - the population average treatment effect and the average treatment effect among the non-participants - and examine the role of the scores. We also study semiparametric efficient estimators that directly balance the weighted trial sample toward the target population, and illustrate their operating characteristics via simulations.
Keywords: Causal inference; empirical likelihood; external validity; propensity score; sampling score; semiparametric efficiency bound.