A sensitivity analysis for the average derivative effect

In observational studies, exposures are often continuous rather than binary or discrete. At the same time, sensitivity analysis is an important tool that can help determine the robustness of a causal conclusion to a certain level of unmeasured confounding, which can never be ruled out in an observational study. Sensitivity analysis approaches for continuous exposures have now been proposed for several causal estimands. In this article, we focus on the average derivative effect (ADE). We obtain closed-form bounds for the ADE under a sensitivity model that constrains the odds ratio (at any two dose levels) between the latent and observed generalized propensity score. We propose flexible, efficient estimators for the bounds, as well as point-wise and simultaneous (over the sensitivity parameter) confidence intervals. We examine the finite sample performance of the methods through simulations and illustrate the methods on a study assessing the effect of parental income on educational attainment and a study assessing the price elasticity of petrol.

💡 Research Summary

This paper addresses the problem of conducting sensitivity analysis for the average derivative effect (ADE) when the exposure of interest is continuous—a setting common in epidemiology, economics, and the social sciences. The ADE, defined as the limit of the expected change in outcome per infinitesimal increase in exposure, provides a scalar summary of causal impact that is especially interpretable in applications such as price elasticity or marginal propensity to consume. Because continuous exposures are almost always observed in non‑experimental studies, the ADE is vulnerable to unmeasured confounding, and a formal sensitivity analysis is required to assess the robustness of any causal claim.

The authors introduce a “Marginal γ sensitivity model” (Assumption 5) that bounds the odds ratio between the latent generalized propensity score (including the unobserved confounder U) and the observed generalized propensity score (marginalized over U) at any two dose levels a and a′. Formally, for all x, u, a, a′, \