Parameterizing the effect of a continuous treatment using average derivative effects

Summary The average treatment effect is commonly used to quantify the main effect of a binary treatment on an outcome. Extensions to continuous treatments are usually based on the dose-response curve or shift interventions, but both require strong overlap conditions, and the resulting curves may be difficult to summarize. This article focuses instead on average derivative effects, which are scalar estimands related to infinitesimal shift interventions requiring only local overlap assumptions. Average derivative effects, however, are rarely used in practice because their estimation usually requires estimating conditional density functions. By characterizing the Riesz representers of weighted average derivative effects, we propose a new class of estimands that provides a unified view of weighted average derivative (respectively, treatment) effects when the treatment is continuous (respectively, binary). We derive the estimand in our class that minimizes the nonparametric efficiency bound, thereby extending optimal weighting results from the binary treatment literature to the continuous setting. We develop efficient estimators for two weighted average derivative effects that avoid density estimation and are amenable to modern machine learning methods, and we evaluate their performance in simulations and an applied analysis of Warfarin dosage effects.