Suppose it is of interest to characterize effect heterogeneity of an intervention 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 may wish to ascertain whether a minority group benefited less from an intervention than the majority group. We introduce the `subgroup parallel trends' assumption that the counterfactual untreated outcomes in each subgroup of interest follow parallel trends pre- and post- intervention. Under the subgroup parallel trends assumption, it is straightforward to show that a simple `subgroup difference in differences' (SDiD) expression (i.e., the average pre/post outcome difference in one subgroup subtracted by the average pre/post outcome difference in the other subgroup) identifies the difference between the intervention's effects in the two subgroups. This difference in effects across subgroups is identified even though the conditional effects in each subgroup are not. The subgroup parallel trends assumption is not stronger than the standard parallel trends assumption across treatment groups when a control group is available, and there are circumstances where it is more plausible. Thus, when effect modification by a baseline covariate is of interest, researchers might consider SDiD whether or not a control group is available.

翻译：假设研究者仅基于干预接受者的干预前后结局测量值（即无对照组），需描述某一基线协变量不同水平下干预效应的异质性特征。例如，关注公平性的研究者可能希望确定少数群体是否比多数群体从干预中获益更少。我们提出“亚组平行趋势”假设，即各感兴趣亚组中的反事实未治疗结局在干预前后遵循平行趋势。在该假设下，可直观证明简单的“亚组双重差分”（SDiD）表达式（即一个亚组的平均前后结局差减去另一亚组的平均前后结局差）能识别两个亚组间干预效应的差异。尽管各亚组的条件效应无法被识别，但跨亚组的效应差异是可识别的。当存在对照组时，亚组平行趋势假设并不比标准处理组间平行趋势假设更强，在某些情况下甚至更合理。因此，当关注基线协变量的效应修饰时，无论是否存在对照组，研究者均可考虑采用SDiD方法。