  Quasi-experimental methods have proliferated over the last two decades, as
researchers develop causal inference tools for settings in which randomization
is infeasible. Two popular such methods, difference-in-differences (DID) and
comparative interrupted time series (CITS), compare observations before and
after an intervention in a treated group to an untreated comparison group
observed over the same period. Both methods rely on strong, untestable
counterfactual assumptions. Despite their similarities, the methodological
literature on CITS lacks the mathematical formality of DID. In this paper, we
use the potential outcomes framework to formalize two versions of CITS - a
general version described by Bloom (2005) and a linear version often used in
health services research. We then compare these to two corresponding DID
formulations - one with time fixed effects and one with time fixed effects and
group trends. We also re-analyze three previously published studies using these
methods. We demonstrate that the most general versions of CITS and DID impute
the same counterfactuals and estimate the same treatment effects. The only
difference between these two designs is the language used to describe them and
their popularity in distinct disciplines. We also show that these designs
diverge when one constrains them using linearity (CITS) or parallel trends
(DID). We recommend defaulting to the more flexible versions and provide advice
to practitioners on choosing between the more constrained versions by
considering the data-generating mechanism. We also recommend greater attention
to specifying the outcome model and counterfactuals in papers, allowing for
transparent evaluation of the plausibility of causal assumptions.
