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Suppose it is of interest to characterize effect heterogeneity of anintervention across levels of a baseline covariate using only pre- and post-intervention outcome measurements from those who received the intervention,i.e. with no control group. For example, a researcher concerned with equity maywish to ascertain whether a minority group benefited less from an interventionthan the majority group. We introduce the `subgroup parallel trends' assumptionthat the counterfactual untreated outcomes in each subgroup of interest followparallel trends pre- and post- intervention. Under the subgroup parallel trendsassumption, it is straightforward to show that a simple `subgroup difference indifferences' (SDiD) expression (i.e., the average pre/post outcome differencein one subgroup subtracted by the average pre/post outcome difference in theother subgroup) identifies the difference between the intervention's effects inthe two subgroups. This difference in effects across subgroups is identifiedeven though the conditional effects in each subgroup are not. The subgroupparallel trends assumption is not stronger than the standard parallel trendsassumption across treatment groups when a control group is available, and thereare circumstances where it is more plausible. Thus, when effect modification bya baseline covariate is of interest, researchers might consider SDiD whether ornot a control group is available.

Subgroup Difference in Differences to Identify Effect Modification Without a Control Group