Parameterising the effect of a continuous exposure using average derivative effects

Abstract

The (weighted) average treatment effect is commonly used to quantify the main effect of a binary exposure on an outcome. Extensions to continuous exposures, however, either quantify the effects of interventions that are rarely relevant (e.g., applying the same exposure level uniformly in the population), or consider shift interventions that are rarely intended, raising the question how large a shift to consider. Average derivative effects (ADEs) instead express the effect of an infinitesimal shift in each subject's exposure level, making inference less prone to extrapolation. ADEs, however, are rarely considered in practice because their estimation usually requires estimation of (a) the conditional density of exposure given covariates, and (b) the derivative of (a) w.r.t. exposure. Here, we introduce a class of estimands which can be inferred without requiring estimates of (a) and (b), but which reduce to ADEs when the exposure obeys a specific distribution determined by the choice of estimand in the class. We moreover show that when the exposure does not obey this distribution, our estimand represents an ADE w.r.t. an 'intervention' exposure distribution. We identify the 'optimal' estimand in our class and propose debiased machine learning estimators, by deriving influence functions under the nonparametric model.

Niwen Zhou and Xu Guo’s Contribution to the Discussion of ‘Assumption-Lean Inference for Generalised Linear Model Parameters’ by Vansteelandt and Dukes

“…By connecting to regression models, we believe that we may often connect better to researchers' a priori understanding of the causal data-generating mechanism, while merely inferring specific features of it. Though the postulated model could be misspecified, our estimands retain close connections to (and sometimes equal) average derivative effects (Hines et al, 2021), which-by virtue of focussing on the effect of a small change in in everyone's observed exposure-tend to be quite 'safe' for general use. An alternative would be to focus on shift interventions that express the effect of increasing the exposure uniformly by, say, 1 unit in the population.…”

Authors' reply to the Discussion of ‘Assumption-Lean Inference for Generalised Linear Model Parameters’ by Vansteelandt and Dukes

“…Even so, the estimation of curves adds complications in view of their high dimensionality, both when it comes to inference and reporting. Our focus on low-dimensional parameters thus remains of interest, even more so as linear approximations are often relevant, for example they sometimes express how much the average outcome would change if each subject's observed exposure were slightly increased (Hines et al, 2021). Sensitivity analyses are useful, but the truth is that subject-matter researchers will often want to present results for a single selected model.…”