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Incremental effects for continuous exposures

Kyle Schindl, Shuying Shen, Edward H. Kennedy

TL;DR

This work tackles causal inference with continuous exposures by introducing incremental effects defined via an exponential tilt of the observed treatment density. It derives the efficient influence function and the nonparametric efficiency bound for finite tilts, and proves minimax lower bounds showing that estimation error scales with an effective sample size of $n/\\delta$. A one-step, double-machine-learning estimator is proposed, with explicit $\\delta$-dependent convergence rates and asymptotic normality results. The authors further show that letting $\\delta$ grow yields a novel edge-dose estimator, and introduce a reflected exponential tilt to estimate dose–response at interior points, with an empirical demonstration on political advertisements. The framework thus enables causal effect estimation under weak positivity, quantifies the cost of tilting, and offers practical tools for dose–response analysis in continuous-treatment settings.

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 $δ$ (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 $δ$. In particular, we derive new minimax lower bounds illustrating how the best possible root mean squared error scales with an effective sample size of $n / δ$, 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 $δ$ 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 $δ\to \infty$ yields a new estimator of the dose-response curve across the treatment support.

Incremental effects for continuous exposures

TL;DR

This work tackles causal inference with continuous exposures by introducing incremental effects defined via an exponential tilt of the observed treatment density. It derives the efficient influence function and the nonparametric efficiency bound for finite tilts, and proves minimax lower bounds showing that estimation error scales with an effective sample size of . A one-step, double-machine-learning estimator is proposed, with explicit -dependent convergence rates and asymptotic normality results. The authors further show that letting grow yields a novel edge-dose estimator, and introduce a reflected exponential tilt to estimate dose–response at interior points, with an empirical demonstration on political advertisements. The framework thus enables causal effect estimation under weak positivity, quantifies the cost of tilting, and offers practical tools for dose–response analysis in continuous-treatment settings.

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 (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 . In particular, we derive new minimax lower bounds illustrating how the best possible root mean squared error scales with an effective sample size of , instead of . 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 compared to standard analyses by using mixed supremum and norms. Finally, we define a "reflected" exponential tilt around any interior point and show that taking yields a new estimator of the dose-response curve across the treatment support.
Paper Structure (38 sections, 14 theorems, 280 equations, 5 figures, 1 algorithm)

This paper contains 38 sections, 14 theorems, 280 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup & Notation
  3. Incremental Effects
  4. Some History
  5. Interpretation
  6. Properties
  7. Positivity:
  8. Smoothness:
  9. Static Interventions:
  10. Identification
  11. Fundamental Limits
  12. Efficiency Bound
  13. Minimax Lower Bound
  14. Estimation & Inference
  15. Proposed Estimator
  16. ...and 23 more sections

Key Result

Proposition 1

Suppose $\delta \in [-M,M]$ for some $M<\infty$. Then, the efficient influence function of $\psi(\delta)$ under a nonparametric model is given by $\varphi(Z;\delta) = D_Y + D_{q, \mu} + D_\psi$ for where $\mathbb{E}_Q(\mu(X, A) \mid X) = \int_a \mu(X, a) q_\delta(a \mid X) \, da$ is the conditional mean of $\mu(X, A)$ under the exponentially tilted distribution.

Figures (5)

  • Figure 1: Exponentially tilted densities, where $\delta = 0$ represents no tilt.
  • Figure 2: Histogram of the treatment variable, Total Ads
  • Figure 3: Left: Estimated incremental effect of total political advertisements on individual log campaign contributions. Right: estimated dose-response $a \mapsto \mathbb{E}[Y^a]$ via $\psi_R(\delta, a)$ (blue) and the nonparametric estimator of kennedy2017cte (red).
  • Figure 4: Empirical coverage across $\delta$ for $\alpha \in \{0.25, 0.5\}$ and $n \in \{500, 1000, 2500, 5000\}$
  • Figure 5: Average confidence interval width across $\delta$ for $\alpha \in \{0.25, 0.5\}$ and $n \in \{500, 1000, 2500, 5000\}$

Theorems & Definitions (36)

  • Remark 1
  • Remark 2
  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Lemma 1
  • Theorem 2
  • Remark 3
  • Proposition 3
  • Remark 4
  • ...and 26 more