  Estimating externally valid causal effects is a foundational problem in the
social and biomedical sciences. Generalizing or transporting causal estimates
from an experimental sample to a target population of interest relies on an
overlap assumption between the experimental sample and the target
population--i.e., all units in the target population must have a non-zero
probability of being included in the experiment. In practice, having full
overlap between an experimental sample and a target population can be
implausible. In the following paper, we introduce a framework for considering
external validity in the presence of overlap violations. We introduce a novel
bias decomposition that parameterizes the bias from an overlap violation into
two components: (1) the proportion of units omitted, and (2) the degree to
which omitting the units moderates the treatment effect. The bias decomposition
offers an intuitive and straightforward approach to conducting sensitivity
analysis to assess robustness to overlap violations. Furthermore, we introduce
a suite of sensitivity tools in the form of summary measures and benchmarking,
which help researchers consider the plausibility of the overlap violations. We
apply the proposed framework on an experiment evaluating the impact of a cash
transfer program in Northern Uganda.
