  The weighted average treatment effect (WATE) is a causal measure for the
comparison of interventions in a specific target population, which may be
different from the population where data are sampled from. For instance, when
the goal is to introduce a new treatment to a target population, the question
is what efficacy (or effectiveness) can be gained by switching patients from a
standard of care (control) to this new treatment, for which the average
treatment effect for the control (ATC) estimand can be applied. In this paper,
we propose two estimators based on augmented inverse probability weighting to
estimate the WATE for a well defined target population (i.e., there exists a
target function that describes the population of interest), using observational
data. The first proposed estimator is doubly robust if the target function is
known or can be correctly specified. The second proposed estimator is doubly
robust if the target function has a linear dependence on the propensity score,
which can be used to estimate the average treatment effect for the treated
(ATT) and ATC. We demonstrate the properties of the proposed estimators through
theoretical proof and simulation studies. We also apply our proposed methods in
a comparison of glucagon-like peptide-1 receptor agonists therapy and insulin
therapy among patients with type 2 diabetes, using the UK clinical practice
research datalink data.
