Estimating treatment effects with a unified semi-parametric difference-in-differences approach

The difference-in-differences (DID) approach is widely used for estimating causal effects with observational data before and after an intervention. DID is traditionally used to assess an average treatment effect among the treated after making a parallel trends assumption on the means of the outcome. With skewed outcomes, a transformation is often needed; however, the transformation may be difficult to choose, results may be sensitive to the choice, and parallel trends assumptions are made on the transformed scale. More recent DID methods estimate alternative treatment effects, such as quantile treatment effects among the treated, that offer a different understanding of the impact of a treatment and may be preferable with skewed outcomes. However, each alternative DID estimator requires a different parallel trends assumption. We introduce a new DID method that is capable of estimating average, quantile, probability, and novel Mann-Whitney treatment effects among the treated with a single unifying parallel trends assumption. The proposed method uses a semi-parametric cumulative probability model (CPM). The CPM is a linear model for a latent variable on covariates, where the latent variable results from an unspecified transformation of the outcome. Our DID approach makes a universal parallel trends assumption on the expectation of the latent variable conditional on covariates. Hence, our method overcomes challenges surrounding outcomes with complicated, difficult-to-model distributions and avoids the need for separate assumptions and/or approaches for each estimand. We introduce the method; describe identification, estimation, and inference; conduct simulations evaluating its performance; and apply it to real-world data to assess the impact of Medicaid expansion on CD4 cell count at enrollment among people living with HIV.