Parameterising the effect of a continuous treatment using average derivative effects

The average treatment effect (ATE) 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 summarise. We focus instead on average derivative effects (ADEs) that are scalar estimands related to infinitesimal shift interventions requiring only local overlap assumptions. ADEs, however, are rarely used in practice because their estimation usually requires estimating conditional density functions. By characterising the Riesz representers of weighted ADEs, we propose a new class of estimands that provides a unified view of weighted ADEs/ATEs when the treatment is continuous/binary. We derive the estimand in our class that minimises 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 ADEs that avoid density estimation and are amenable to modern machine learning methods, which we evaluate in simulations and an applied analysis of Warfarin dosage effects.