Difference-in-differences (DID) approaches estimate causal effects with observational data before and after an intervention. DID traditionally estimates the average treatment effect among the treated and assumes parallel trends in outcome means. With skewed outcomes, a transformation is often needed to compare means; however, transformations may be difficult to choose and results may be sensitive to the choice and must be interpreted on the transformed scale. Recent DID methods estimate alternative treatment effects that offer additional insights. However, each estimator requires a different parallel trends assumption. We introduce a new DID method that estimates average, quantile, probability, and novel Mann-Whitney treatment effects among the treated with a single unifying parallel trends assumption. The method uses a semi-parametric cumulative probability model (CPM): a linear model for a latent variable on covariates, where the latent variable results from an unspecified transformation of the outcome. Our approach makes a single parallel trends assumption on the expectation of the latent variable conditional on covariates. Hence, we avoid specifying outcome transformations and separate estimand-specific assumptions. We introduce the method; describe identification, estimation, and inference; conduct simulations evaluating its performance; and apply it to assess the impact of Medicaid expansion on CD4 count among people with HIV.

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