Nonparametric Inference on Causal Effects of Continuous Treatments: With and Without the Positivity Condition

In observational studies, causal effects do not always result from standard binary interventions but may rather arise from continuous treatments or exposures. The causal effect of a continuous treatment on an outcome is formally characterized by the dose-response curve (i.e., the mean outcome if all individuals in a population were assigned a given treatment level) and its derivative function. In this talk, we investigate nonparametric inference for the dose-response curve and its derivative, both under the standard positivity assumption and in settings where positivity is violated.

Under the positivity condition, which ensures that every individual has some chance of receiving any treatment level regardless of their covariates, we review and propose doubly robust (DR) inference methods for estimating the dose-response curve and its derivative via kernel smoothing. By leveraging machine learning methods with cross-fitting, our proposed DR estimator achieves asymptotic normality at the standard nonparametric rate of convergence.

Recognizing that the positivity assumption is particularly restrictive for continuous treatments, we develop identification and inference theories for the dose-response curve and its derivative without relying on the positivity condition. Specifically, our approach identifies and estimates the derivative of the treatment effect within regions where positivity holds, then integrates this estimate to the treatment level of interest, thereby mitigating bias from positivity violations. We further demonstrate the inconsistency of conventional inverse probability weighting (IPW) and DR estimators without positivity, and introduce our novel bias-corrected IPW and DR estimators. Notably, our bias-corrected DR estimator not only preserves asymptotic normality at the standard nonparametric rate but also unveils interesting connections to nonparametric support and level set estimation problems. We illustrate our methods with an application to assessing the impact of air pollution exposure (PM2.5) on cardiovascular mortality rates.