  We consider the challenges associated with causal inference in settings where
data from a randomized trial is augmented with control data from an external
source to improve efficiency in estimating the average treatment effect (ATE).
Through the development of a formal causal inference framework, we outline
sufficient causal assumptions about the exchangeability between the internal
and external controls to identify the ATE and establish the connection to a
novel graphical criteria. We propose estimators, review efficiency bounds,
develop an approach for efficient doubly-robust estimation even when unknown
nuisance models are estimated with flexible machine learning methods, and
demonstrate finite-sample performance through a simulation study. To illustrate
the ideas and methods, we apply the framework to a trial investigating the
effect of risdisplam on motor function in patients with spinal muscular atrophy
for which there exists an external set of control patients from a previous
trial.
