Performing a Difference-in-Difference Analysis where the Control Group is Already Treated

Question

I'm involved in a project where the outcome is the proportion of cancer patients who have received surgery. The treatment event is a state-level policy change that mandated moving all these cancer patients from FFS insurance plans to MMC insurance plans (abbreviation meanings not relevant to the question at hand).

Around two-thirds of the patients live in counties that already mandate MMC plans for cancer patients (already-treated control group) while one-third live in counties that mandated MMC plans after the treatment event (treatment group).

Our hypothesis is that the proportion of people who have received surgery have increased more in the treatment group relative to the control group after the treatment event.

My question is how to conceptualize this scenario using DiD, if possible?

I got conflicting answers from others where I work about the feasibility of a model like this with some saying it wouldn't work because it would be impossible for an already-treated and newly-treated group to have parallel trends in the pre-period while others have said it wouldn't really matter because the already-treated control group can be thought of as a never-treated control group in the model, with the methodology being the same otherwise.

Any additional insight or help here is much appreciated.

Answer 1

Technically, the mechanics of difference-in-differences (DiD) can be reversed.

"[S]ome sa[id] it wouldn't work because it would be impossible for an already-treated and newly-treated group to have parallel trends in the pre-period..."

This is not necessarily true.

The control units constitute the group of states that have always imposed MMC insurance plans, while the treated units represent the subset of states that adopt MMC insurance plans later. Thus, the FFS states start in the treatment condition, then move back into the control condition in some future time period. Note that the identification condition is now tested by future parallel paths across the two groups after the event, rather than past parallel "untreated" paths. Try assessing the proportions over time in the later periods once MMC plans become state policy everywhere. In this sense, once treated states switch their policies, we need more and more time periods after this change to test for any differential trend trajectories across the groups. Whereas in classical DiD settings we assess for a stability of the group trends before some event, here we're doing the same thing but now after the event.

[W]hile others have said it wouldn't really matter because the already-treated control group can be thought of as a never-treated control group in the model, with the methodology being the same otherwise.

This is correct, so long as we understand how to reverse the process.

Traditionally, we think of DiD as the interaction between a treatment dummy (treatment versus control) and a time indicator (before versus after). With respect to the treatment dummy, the control group equals 0 for those state-county pairs where MMC insurance plans where always the law of the land, while the treatment group (i.e., "later MMCs") equals 1 for those state-county pairs that eventually change their policy from FFS to MMC insurance plans. That seems fairly straightforward. However, the big twist with respect to the time indicator is that it's reverse-coded. So, states start with a value of 1 pre-event (FFS-era), then change to 0 post-event (MMS era). Note how DiD in this reverse-engineered setting identifies "pre-treatment" period effects (i.e., "FFS policy era"), whereas in traditional settings we usually say "post-treatment" period effects. The mechanics are the same, but the interpretation is different. Review work by Kim and Lee (2019) for a neat empirical application.