Many research questions in public health and medicine concern sustained
interventions in populations defined by substantive priorities. Existing
methods to answer such questions typically require a measured covariate set
sufficient to control confounding, which can be questionable in observational
studies. Differences-in-differences relies instead on the parallel trends
assumption, allowing for some types of time-invariant unmeasured confounding.
However, most existing difference-in-differences implementations are limited to
point treatments in restricted subpopulations. We derive identification results
for population effects of sustained treatments under parallel trends
assumptions. In particular, in settings where all individuals begin follow-up
with exposure status consistent with the treatment plan of interest but may
deviate at later times, a version of Robins' g-formula identifies the
intervention-specific mean under SUTVA, positivity, and parallel trends. We
develop consistent asymptotically normal estimators based on
inverse-probability weighting, outcome regression, and a double robust
estimator based on targeted maximum likelihood. Simulation studies confirm
theoretical results and support the use of the proposed estimators at realistic
sample sizes. As an example, the methods are used to estimate the effect of a
hypothetical federal stay-at-home order on all-cause mortality during the
COVID-19 pandemic in spring 2020 in the United States.Comment: 15 pages, 2 figure