Doubly Robust Inference on Causal Derivative Effects for Continuous Treatments

TL;DR

This work develops nonparametric, doubly robust inference for the derivative of the dose-response curve, $\theta(t)=\frac{d}{dt}m(t)$, under continuous treatments. It first establishes identification and DR estimation for $\theta(t)$ with positivity using kernel smoothing, achieving asymptotic normality and nonparametric efficiency guarantees. The paper then addresses identifiability issues when positivity fails by introducing an additive confounding model and bias-corrected IPW/DR estimators that rely on interior or level-set density constructions, enabling consistent inference without full positivity. A comprehensive asymptotic theory accompanies both the positivity and no-positivity settings, including cross-fitting, variance estimation, and uniform inference via bootstrap. Numerical experiments and a Job Corps case study illustrate the practical advantages of the proposed DR approaches over finite-difference methods, with robust performance under model misspecification and positivity violations. The work also reveals connections to nonparametric set estimation, expanding the methodological toolbox for causal derivative inference in continuous-treatment settings.

Abstract

Statistical methods for causal inference with continuous treatments mainly focus on estimating the mean potential outcome function, commonly known as the dose-response curve. However, it is often not the dose-response curve but its derivative function that signals the treatment effect. In this paper, we investigate nonparametric inference on the derivative of the dose-response curve with and without the positivity condition. Under the positivity and other regularity conditions, we propose a doubly robust (DR) inference method for estimating the derivative of the dose-response curve using kernel smoothing. When the positivity condition is violated, we demonstrate the inconsistency of conventional inverse probability weighting (IPW) and DR estimators, and introduce novel bias-corrected IPW and DR estimators. In all settings, our DR estimator achieves asymptotic normality at the standard nonparametric rate of convergence with nonparametric efficiency guarantees. Additionally, our approach reveals an interesting connection to nonparametric support and level set estimation problems. Finally, we demonstrate the applicability of our proposed estimators through simulations and a case study of evaluating a job training program.

Figures (10)

• Figure 1: Illustrations of a dose-response curve $m(t)$, its corresponding derivative effect curve $\theta(t)$, and average derivative effects $\mathbb{E}\left[\theta(T)\right]$. Here, $m(t)$ is symmetric with respect to $t=3.5$ so that $m(t_1)=m(t_2)$ and $\mathbb{E}\left[\theta(T)\right]=0$ but $\theta(t_1) > 0 > \theta(t_2)$.
• Figure 2: Graphical illustrations of the support discrepancy between $\mathcal{S}(t)$ and $\mathcal{S}(t+\delta)$ for $t\in \mathcal{T}$ as well as Assumption \ref{['assump:cond_support_smooth']}, where $\delta$ can take its value as $uh\in \mathbb{R}$.
• Figure 3: Comparisons between our proposed estimators and the finite-difference approaches by colangelo2020double ("CL20") under positivity and with 5-fold cross-fitting across various sample sizes. Rows present estimation bias, RMSE, and coverage probability for each estimator of $\theta(t)$, while columns correspond to different values for $n$.
• Figure 4: Comparisons between our bias-corrected estimators (NP) in \ref{['subsec:bias_corrected_IPW_DR']} and their counterparts (P) in \ref{['sec:theta_pos']} under the violation of positivity and with 5-fold cross-fitting across different sample sizes. Rows present estimation bias, RMSE, and coverage probability for each estimator of $\theta(t)$, while columns correspond to different values for $n$.
• Figure 5: Estimated derivative effect curves with 95% confidence intervals using our proposed estimators and the finite-difference approaches by colangelo2020double ("CL20") under 5-fold cross-fitting. The vertical red dotted lines mark the original treatment range $[320,1840]$ analyzed in colangelo2020double.

Theorems & Definitions (41)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Theorem 3.1: Asymptotic properties of $\widehat{\theta}_{\mathrm{DR}}(t)$ under positivity
• Remark 3.2: Uniform inference via multiplier bootstrap
• Theorem 3.2: Efficient influence function
• Proposition 4.1: Inconsistency of IPW estimators
• Remark 4.1
• Proposition 5.1
• Remark 5.1