Incremental effects for continuous exposures

TL;DR

This paper develops a new framework for estimating incremental effects of continuous treatments in causal inference, avoiding positivity assumptions, and provides theoretical bounds and estimators that depend on the tilt parameter.

Contribution

It introduces stochastic interventions based on exponential tilting for continuous exposures, deriving efficiency bounds, minimax lower bounds, and new convergence rates for estimators.

Findings

Derived the efficient influence function and efficiency bound for incremental effects.

Established how estimation error scales with the tilt parameter .

Proposed a new estimator for dose-response curves using reflected exponential tilting.

Abstract

Causal inference problems often involve continuous treatments, such as dose, duration, or frequency. However, identifying and estimating standard dose-response estimands requires that everyone has some chance of receiving any level of the exposure (i.e., positivity). To avoid this assumption, we consider stochastic interventions based on exponentially tilting the treatment distribution by some parameter $\delta$ (an incremental effect); this increases or decreases the likelihood a unit receives a given treatment level. We derive the efficient influence function and semiparametric efficiency bound for these incremental effects under continuous exposures. We then show estimation depends on the size of the tilt, as measured by $\delta$. In particular, we derive new minimax lower bounds illustrating how the best possible root mean squared error scales with an effective sample size of $n / \delta$, instead of $n$. Further, we establish new convergence rates and bounds on the bias of double machine learning-style estimators. Our novel analysis gives a better dependence on $\delta$ compared to standard analyses by using mixed supremum and $L_2$ norms. Finally, we define a "reflected" exponential tilt around any interior point and show that taking $\delta \to \infty$ yields a new estimator of the dose-response curve across the treatment support.