Longitudinal Generalizations of the Average Treatment Effect on the Treated for Multi-valued and Continuous Treatments

Abstract: The Average Treatment Effect on the Treated (ATT) is a common causal parameter defined as the average effect of a binary treatment among the subset of the population receiving treatment. We propose a novel family of parameters, Generalized ATTs (GATTs), that generalize the concept of the ATT to longitudinal data structures, multi-valued or continuous treatments, and conditioning on arbitrary treatment subsets. We provide a formal causal identification result that expresses the GATT in terms of sequential regressions, and derive the efficient influence function of the parameter, which defines its semi-parametric efficiency bound. Efficient semi-parametric inference of the GATT requires estimating the ratios of functions of conditional probabilities (or densities); we propose directly estimating these ratios via empirical loss minimization, drawing on the theory of Riesz representers. Simulations suggest that estimation of the density ratios using Riesz representation have better stability in finite samples. Lastly, we illustrate the use of our methods to evaluate the effect of chronic pain management strategies on the development of opioid use disorder among Medicare patients with chronic pain.