Nonparametric Inference on Dose-Response Curves Without the Positivity Condition

TL;DR

The paper develops a nonparametric framework to identify and infer dose–response curves $m(t)$ and their derivatives $\theta(t)$ without relying on the positivity condition for continuous treatments. It introduces an integral estimator $\widehat{m}_{\theta}(t)$ built from a localized derivative estimator $\widehat{\theta}_C(t)$, supported by a fast computation scheme and bootstrap-based inference. Under mild regularity assumptions, it proves uniform consistency and bootstrap validity, with Gaussian-process approximations enabling uniform confidence bands. Empirical validation through simulations and a PM$_{2.5}$-cardiovascular mortality case demonstrates robustness to positivity violations and provides nuanced insights into exposure–outcome relationships beyond standard RA methods.

Abstract

Existing statistical methods in causal inference often assume the positivity condition, where every individual has some chance of receiving any treatment level regardless of covariates. This assumption could be violated in observational studies with continuous treatments. In this paper, we develop identification and estimation theories for causal effects with continuous treatments (i.e., dose-response curves) without relying on the positivity condition. Our approach identifies and estimates the derivative of the treatment effect for each observed sample, integrating it to the treatment level of interest to mitigate bias from the lack of positivity. The method is grounded in a weaker assumption, satisfied by additive confounding models. We propose a fast and reliable numerical recipe for computing our integral estimator in practice and derive its asymptotic properties. To enable valid inference on the dose-response curve and its derivative, we use the nonparametric bootstrap and establish its consistency. The performances of our proposed estimators are validated through simulation studies and an analysis of the effect of air pollution exposure (PM$_{2.5}$) on cardiovascular mortality rates.

Figures (5)

• Figure 1: Simulation results under the single confounder model \ref{['single_conf']} with $n=2000$. Left: The support of the joint distribution of $(T,S)$. Middle: Estimated dose-response curves using naive regression adjustment (RA) and the proposed integral estimators, overlaid with the true one $m(t)$. Right: Estimated derivatives of the dose-response curve by naive RA and our proposed localized estimators, overlaid with the true derivative $\theta(t)$. The middle and right panels also present the 95% confidence intervals (CIs) and/or uniform confidence bands from our proposed estimators, shown as shaded regions and dashed lines, respectively.
• Figure 2: Comparisons between our proposed estimators and the naive RA estimators across various sample sizes under the single confounder model \ref{['single_conf']}. Rows present results for estimating $m(t)$ and $\theta(t)$ respectively, while columns correspond to different values for $n$.
• Figure 3: Comparisons between our proposed estimators and the naive RA estimators across various sample sizes under the linear confounding model \ref{['linear_conf']}. Rows present results for estimating $m(t)$ and $\theta(t)$ respectively, while columns correspond to different values for $n$.
• Figure 4: Comparisons between our proposed estimators and the naive RA estimators across various sample sizes under the linear confounding model \ref{['nonlinear_conf']}. Rows present results for estimating $m(t)$ and $\theta(t)$ respectively, while columns correspond to different values for $n$.
• Figure 5: Estimated relationships between the PM$_{2.5}$ concentration and CMR or its changing rate at the county level. Left: The estimated CMR with respect to the PM$_{2.5}$ concentration. Right: The estimated changing rates of CMR with respect to the PM$_{2.5}$ concentration. We also present the 95% confidence intervals and uniform confidence bands as shaded regions and dashed lines respectively for each regression scenario.

Theorems & Definitions (38)

• Proposition 2.1: Identification for $\theta(t)$
• proof : Proof of Proposition \ref{['prop:iden_theta']}
• Proposition 2.2: Properties of the additive confounding model
• Remark 3.1: RA estimator of $\theta(t)$
• Remark 3.2: Linear smoother
• Lemma 4.1: Uniform convergence of $\widehat{\beta}_2(t,\bm{s})$
• Theorem 4.2: Convergence of $\widehat{\theta}_C(t)$ and $\widehat{m}_{\theta}(t)$
• Lemma 4.3: Asymptotic linearity
• Remark 4.1: Non-degeneracy and validity of pointwise confidence intervals
• Remark 4.2: Bandwidth selection and the curse of dimensionality