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Doubly Robust Inference on Causal Derivative Effects for Continuous Treatments
This paper investigates the construction of a doubly robust estimator for the derivative of the continuous treatment effect using kernel smoothing. The analysis considers scenarios both with and without the positivity condition. Specifically, positivity (Assumption (ref)) requires that every individual has a nonzero chance (measured by a conditional density function) of being exposed to any treatment level $T=t$ across all possible values of the covariate vector $\bm{S}\in \mathcal{S}\subset \mathbb{R}^d$. Let $Y(t)$ be the potential outcome rubin1974estimating that would have been observed under treatment level $T=t$. The focus of this work is the (causal) derivative effect curve $t\mapsto \theta(t) := \frac{d}{dt}\mathbb{E}\left[Y(t)\right]$, where $t\mapsto m(t):= \mathbb{E}\left[Y(t)\right]$ represents the (causal) dose-response curve.
Valid inference on $\theta(t)$ is essential for understanding how the outcome of interest $Y\in \mathcal{Y}$ changes with treatment $t$, offering insights beyond the expected value $\mbox{$\mathbb{E}$}\left[Y(t)\right]=m(t)$ of the potential outcome across the population. Indeed, the derivative effect curve $\theta(t)$ serves as the most natural continuous-treatment counterpart to the average treatment effect $\mbox{$\mathbb{E}$}\left[Y(1)\right] - \mbox{$\mathbb{E}$}\left[Y(0)\right]$ in binary treatment settings. Despite the importance of estimating $\theta(t)$, the current research on continuous treatments has primarily focused on inferring $m(t)$ diaz2013targeted,kennedy2017non,bonvini2022fast,takatsu2022debiased or the average derivative effect $\mbox{$\mathbb{E}$}\left[\theta(T)\right] = \mbox{$\mathbb{E}$}\left[\frac{\partial}{\partial t}\mbox{$\mathbb{E}$}\left(Y|T,\bm{S}\right)\right]$ under some regularity conditions hardle1989investigating,powell1989semiparametric,newey1993efficiency,cattaneo2010robust,hirshberg2020debiased,hines2023optimally, with limited attention to $\theta(t)$ itself. As demonstrated by (ref), a dose-response curve $m(t)$ attains the same value at two distinct treatment levels $t_1,t_2$, and the associated average derivative effect $\mathbb{E}\left[\theta(T)\right]$ is identical to zero. Nevertheless, the actual causal effects differ and remain nonzero at $t_1,t_2$, which can be effectively captured by our target estimand $\theta(t)$. In reality, $\mathbb{E}\left[\theta(T)\right]$ only quantifies the overall causal effects, while $\theta(t)$ provides more precise treatment effects at a personalized level of interest.
To achieve precise inference on $\theta(t)$ without numerical approximation, a straightforward approach is to impose structural assumptions on the conditional mean outcome function $\mbox{$\mathbb{E}$}\left(Y|T=t,\bm{S}=\bm{s}\right)$ or directly on the dose-response curve $m(t)$, known as the marginal structural modeling robins2000marginal,neugebauer2007nonparametric. Although this approach can easily construct an estimator of $\theta(t)$ via a standard differentiation on the estimated dose-response curve, those structural assumptions are difficult to verify in practice. Alternatively, existing methods for derivative estimation gasser1984estimating,mack1989derivative,zhou2000derivative, combined with the inverse probability weighting (IPW) technique hirano2004propensity,imai2004causal, can define an estimator of $\theta(t)$; see, e.g., our proposed IPW estimator in (ref). Yet, this approach requires correct specification and accurate estimation of the conditional density of $T$ given $\bm{S}$. The sensitivity of these approaches to model misspecification and estimation motivates us to propose a doubly robust (DR) inference procedures for $\theta(t)$ that accommodates misspecification in either the outcome regression or the conditional density models robins1986new,van2003unified,bang2005doubly while imposing less stringent requirements on the estimation rates of convergence.
The existing inference methods for $m(t)$ and the above discussion on $\theta(t)$ relies on the positivity condition (Assumption (ref)), which may be violated in observational studies with continuous treatments cole2008constructing,westreich2010invited. When positivity fails, the identifications of both $m(t)$ and $\theta(t)$ become infeasible without structural assumptions; see (ref) for details. zhang2024nonparametric address this problem without positivity by imposing an assumption on the potential outcome model that can be satisfied by additive confounding models and proposing a regression adjustment (RA) estimator of $\theta(t)$. We extend their identification and estimation strategies to propose IPW and DR estimators of $\theta(t)$ under additive confounding models. This extension not only advances the field but also reveals novel connections between the derivative effect curve inference and classical support estimation problems cuevas1997plug,cuevas2009set.
{\bf 1. Identification and Estimation:} Under the positivity and other regularity conditions that are stated in (ref), we propose our IPW and DR estimators of $\theta(t)$ using kernel smoothing in (ref). In particular, our proposed DR estimator leverages a local polynomial approximation to the outcome variable and is robust to the misspecification of either the outcome regression or the conditional density models.
{\bf 2. Challenges and Remedies Under Violations of Positivity:} When the positivity condition is violated, we demonstrate that the conventional approaches exhibit two types of bias due to lack of identification and support discrepancy in (ref). To resolve these issues, we adopt the additive structural assumption to maintain identification and utilize nonparametric set estimation techniques to develop our bias-corrected IPW and DR estimators of $\theta(t)$ in (ref).
{\bf 3. Asymptotic Theory:} We establish the consistency and asymptotic properties of RA, IPW, and DR estimators of $\theta(t)$ when the nuisance functions are nonparametrically estimated under cross-fitting; see (ref) with positivity and (ref) without positivity. Specifically, our proposed DR estimators are asymptotically normal and can be used to conduct valid (uniform) inference on $\theta(t)$ with nonparametric efficiency guarantees.
{\bf 4. Numerical Experiments:} We showcase the finite-sample performances of our proposed estimators of $\theta(t)$ with and without the positivity condition through simulations and a case study of the Job Corps program in the United States in (ref) and (ref). All the codes for our experiments are available at \url{https://github.com/zhangyk8/npDRDeriv}, and we provide some practical considerations for implementing our proposed estimators in (ref).
The dose-response curve $m(t)$ and its derivative $\theta(t)$ are non-regular target parameters, as they lack unique G\^ateaux derivatives and Riesz representers, depending on how the treatment distribution is localized at $t$ van1991differentiable,carone2019toward,ichimura2022influence. As one of the key ingredients in this paper, kernel-based localization is a common approach in the literature, which has been used to construct IPW or DR estimators of $m(t)$ kallus2018policy,su2019non,huber2020direct,colangelo2020double,klosin2021automatic. An alternative localization method is through the basis approach or series estimator chen2014sieve,chen2014sieveM,luedtke2024one. Additionally, a general form of the IPW estimator of $m(t)$ was studied by galvao2015uniformly. Under the positivity condition, the RA or G-computation robins1986new estimators of $\theta(t)$ have been explored by gill2001causal,flores2007estimation,lee2018partial. Recently, colangelo2020double and bong2023local also considered approximating $\theta(t)$ via finite differences of the estimated dose-response curve or a closely related matching method. Shortly after the first version of this paper, zeng2025nonparametric proposed their DR estimator of $\theta(t)$ by regressing the pseudo-outcome in kennedy2017non via local polynomial regression, but they did not address the positivity violation.
Growing interest in relaxing the positivity condition has led to new developments in causal inference. For continuous treatments, branson2023causal studied a smoothed causal effect with trimmed conditional densities, while schindl2024incremental examined stochastic interventions via exponentially tilted treatment distributions. Notably, dynamic stochastic interventions with continuous treatments can be robust to the violation of positivity diaz2020causal,bonvini2023incremental,mcclean2024fair. To our knowledge, no existing works directly consider nonparametric inference on $\theta(t)$ without positivity, and our work takes an initial step to fill in this gap.
Throughout this paper, we consider an outcome variable $Y\in \mathcal{Y}\subset \mathbb{R}$, an univariate continuous treatment $T\in \mathcal{T}\subset \mathbb{R}$, and a vector of continuous confounding variables or covariates $\bm{S}=(S_1,...,S_d)\in \mathcal{S}\subset \mathbb{R}^d$ with a fixed dimension $d$. We write $Y {\perp\!\!\!\perp} \bm{X}$ when the random variables $Y,\bm{X}$ are independent. The common distribution and expectation of $\bm{U}=(Y,T,\bm{S})$ are denoted by $\mbox{$\mathrm{P}$}$ and $\mbox{$\mathbb{E}$}$ respectively, whose Lebesgue density is $p(y,t,\bm{s}) = p_{Y|T,\bm{S}}(y|t,\bm{s})\cdot p_{T|\bm{S}}(t|\bm{s}) \cdot p_S(\bm{s})$. Here, $p_T(t)$ and $p_S(\bm{s})$ are the marginal densities of $T$ and $\bm{S}$, respectively, and $p_{T|\bm{S}}(t|\bm{s}) = \frac{\partial}{\partial t} \mbox{$\mathrm{P}$}\left(T\leq t|\bm{S} = \bm{s}\right)$ is the conditional density of $T$ given covariates $\bm{S}=\bm{s}$. We also denote the joint density of $(T,\bm{S})$ by $p(t,\bm{s}) = p_{T|\bm{S}}(t|\bm{s})\cdot p_S(\bm{s}) = p_{\bm{S}|T}(\bm{s}|t)\cdot p_T(t)$ and the support of $p_{\bm{S}|T}(\bm{s}|t)$ by $\mathcal{S}(t)$ for $t\in \mathcal{T}$. Our methodology and theory in this paper also apply when $\bm{S}$ consists of discrete covariates, where it suffices to, e.g., replace the density $p_{\bm{S}}(\bm{s})$, with the probability mass function $\mbox{$\mathrm{P}$}(\bm{S}=\bm{s})$ and substitute integration with summation. For any real-valued $\mbox{$\mathrm{P}$}$-integrable function $f$, we write $\mbox{$\mathrm{P}$} f = \int f(\bm{u})\, d\mbox{$\mathrm{P}$}(\bm{u})$ and denote the $L_p(\mbox{$\mathrm{P}$})$-norm of $f$ by $\left|\left| f \right|\right|_{L_p} := \left(\int |f(\bm{u})|^p d\mbox{$\mathrm{P}$}(\bm{u})\right)^{\frac{1}{p}}$. If $\widehat{f}$ is estimated on an independent data sample, then $\left|\left| \widehat{f} \right|\right|_{L_p} := \left(\int \left|\widehat{f}(\bm{u})\right|^p d\mbox{$\mathrm{P}$}(\bm{u})\right)^{\frac{1}{p}}$. Additionally, we let $\mathbb{P}_n$ denote the empirical measure so that $\mathbb{P}_n f = \frac{1}{n} \sum_{i=1}^n f(\bm{U}_i) = \int f(\bm{u}) d\mathbb{P}_n(\bm{u})$ and $\mathbb{G}_n(f) = \sqrt{n}\left(\mathbb{P}_n-\mbox{$\mathrm{P}$}\right)f$. Finally, we use $\mathbbm{1}_A$ to denote the indicator function of a set $A$. The big-$O$ notation $h_n=O(g_n)$ means that $|h_n|$ is upper bounded by a positive constant multiple of $g_n >0$ when $n$ is sufficiently large. In contrast, $h_n=o(g_n)$ when $\lim_{n\to\infty} \frac{|h_n|}{g_n}=0$. For random variables, $o_P(1)$ is short for a sequence of random variables converging to zero in probability, while $O_P(1)$ denotes the sequence that is bounded in probability.
Suppose that the data sample consists of independent and identically distributed (i.i.d.) observations $\left\{(Y_i,T_i,\bm{S}_i)\right\}_{i=1}^n \subset \mathcal{Y}\times \mathcal{T} \times \mathcal{S}$. Since the main estimands of interest $\theta(t)=\frac{d}{dt}\mbox{$\mathbb{E}$}\left[Y(t)\right]$ and $m(t)=\mbox{$\mathbb{E}$}\left[Y(t)\right]$ are defined by potential outcomes that are not directly observable, we introduce some identification conditions for identifying $m(t)$ and $\theta(t)$ with observed data $\left\{(Y_i,T_i,\bm{S}_i)\right\}_{i=1}^n$.
Assumption (ref)(a,b) are standard identification conditions for causal dose-response curves gill2001causal,kennedy2017non, while Example 1 in zhang2024nonparametric demonstrates the necessity of imposing Assumption (ref)(c) for identifiability. In particular, Assumption (ref)(c) ensures that the distribution of $(T,\bm{S})$ has a nontrivial support in $\mathcal{T}\times \mathcal{S}$.
Finally, Assumption (ref)(d) only requires the interchangeability of the expectation and (partial) differentiation when $\mu(t,\bm{s})$ is well-defined on $\mathcal{T}\times \mathcal{S}$. It is a mild condition and can be satisfied when $\left|\frac{\partial}{\partial t} \mu(t,\bm{S})\right|$ is upper bounded by an integrable function with respect to the distribution of $\bm{S}$; see Theorem 1.1 and Example 1.8 in shao2003mathematical. We also emphasize that the requirement of $\mu(t,\bm{s})$ being well-defined on $\mathcal{T}\times \mathcal{S}$ in Assumption (ref)(d) is entailed by the following positivity condition. In general, when the positivity condition is violated, the conditional mean outcome (or regression) function $\mu(t,\bm{s})$ is no longer well-defined outside the support of the joint density $p(t,\bm{s})$.
Under Assumptions (ref) and (ref), the dose-response curve $m(t)$ and its derivative $\theta(t)$ are identifiable as: $$ m(t)=\mbox{$\mathbb{E}$}\left[\mu(t,\bm{S})\right] \quad \text{ and } \quad \theta(t) = \mbox{$\mathbb{E}$}\left[\frac{\partial}{\partial t} \mu(t,\bm{S})\right], $$ respectively. In (ref) and (ref), we first study nonparametric inference on $m(t)$ and $\theta(t)$ this positivity condition. Later, in (ref) and (ref), we examine the identification and estimation issues as well as our proposed inference method without positivity.
Before discussing our estimation strategy on the derivative effect curve $\theta(t)$, we first review the existing approaches for estimating the dose-response curve $m(t)$ via kernel smoothing when the positivity condition is valid. Specifically, under Assumptions (ref) and (ref), there are three major estimation strategies for $t\mapsto m(t)=\mbox{$\mathbb{E}$}\left[Y(t)\right]$ with observed data $\left\{(Y_i,T_i,\bm{S}_i)\right\}_{i=1}^n$ listed as follows.
$\bullet$ {\bf Regression Adjustment (RA) Estimator:} Since $m(t)$ coincides with the form $\mbox{$\mathbb{E}$}\left[\mu(t,\bm{S})\right]$ under Assumptions (ref) and (ref), it leads to a plug-in estimator as:
where $\widehat{\mu}(t,\bm{s})$ is a (consistent) estimator of the conditional mean outcome function $\mu(t,\bm{s})$.
$\bullet$ {\bf Inverse Probability Weighting (IPW) Estimator:} The IPW estimator follows from the rationale that $m(t) = \mbox{$\mathbb{E}$}\left[\frac{Y \cdot \mathbbm{1}_{\{T=t\}}}{p_{T|\bm{S}}(t|\bm{S})} \right]$ under Assumptions (ref) and (ref) in the discrete treatment setting. When the treatment variable $T \in \mathcal{T}$ is continuous, one common approach is to smooth the indicator function $\mathbbm{1}_{\{T=t\}}$ by a kernel function $K:\mathbb{R}\to [0,\infty)$, yielding the following IPW estimator as:
where $h>0$ is a smoothing bandwidth and $\widehat{p}_{T|\bm{S}}(t|\bm{s})$ is a (consistent) estimator of the conditional density $p_{T|\bm{S}}(t|\bm{s})$. In practice, without loss of its consistency, one can implement a self-normalized IPW estimator (ref) of $m(t)$ as shown in (ref) to reduce the variance of (ref). Note that there are other approaches than kernel smoothing to smoothly approximate the non-regular target parameter $m(t)$, such as the series method outlined in Section 2.4 of luedtke2024one.
$\bullet$ {\bf Doubly Robust (DR) Estimator:} The above RA estimator (ref) can be combined with the IPW estimator (ref) to obtain the following DR estimator as:
where $\widehat{\mu}(t,\bm{s})$ and $\widehat{p}_{T|\bm{S}}(t,\bm{s})$ are (consistent) estimators of $\mu(t,\bm{s})$ and $p_{T|\bm{S}}(t,\bm{s})$ respectively. For completeness, we state and prove the asymptotic properties of the above estimators in (ref).
In this section, analogous to the estimation of $m(t)$ in (ref), we study three different methods for estimating the derivative effect curve $t\mapsto \theta(t)=\frac{d}{dt}\mbox{$\mathbb{E}$}\left[Y(t)\right]$ with kernel smoothing under Assumptions (ref) and (ref). Notably, both the IPW and DR estimators of $\theta(t)$ are novel contribution to the existing literature and exhibit distinct insights.
$\bullet$ {\bf Regression Adjustment (RA) Estimator:} Assumption (ref)(d), together with other conditions in (ref) and (ref), guarantees the identification of $\theta(t)$ as $\mbox{$\mathbb{E}$}\left[\frac{\partial}{\partial t} \mu(t,\bm{S})\right]$ and provides a natural RA estimator as:
where $\widehat{\beta}(t,\bm{s})$ is a (consistent) estimator of $\beta(t,\bm{s})=\frac{\partial}{\partial t} \mu(t,\bm{s})$.
$\bullet$ {\bf Inverse Probability Weighting (IPW) Estimator:} Inspired by the nonparametric derivative estimator in mack1989derivative, we propose the following IPW estimator of $\theta(t)$ as:
where $K:\mathbb{R}\to [0,\infty)$ is a kernel function with $\kappa_2=\int u^2 K(u)\,du$, $h>0$ is a smoothing bandwidth, and $\widehat{p}_{T|\bm{S}}(t|\bm{s})$ is a (consistent) estimator of the conditional density $p_{T|\bm{S}}(t|\bm{s})$. One can implement the self-normalized IPW estimator (ref) of $\theta(t)$ in (ref) to reduce the variance of (ref).
$\bullet$ {\bf Doubly Robust (DR) Estimator:} To achieve the doubly robust property like $\widehat{m}_{\mathrm{DR}}(t)$ in (ref) (see also (ref)), we propose the following DR estimator of $\theta(t)$ as:
where $\widehat{\mu}(t,\bm{s}),\widehat{\beta}(t,\bm{s}), \widehat{p}_{T|\bm{S}}(t,\bm{s})$ are (consistent) estimators of $\mu(t,\bm{s}),\beta(t,\bm{s}), p_{T|\bm{S}}(t,\bm{s})$, respectively. We discuss how these nuisance functions can be estimated in (ref). The key insight of why $\widehat{\theta}_{\mathrm{DR}}(t)$ in (ref) embraces the doubly robust property is that we leverage a local polynomial approximation fan1996local to push the residual of the IPW component in (ref) to at least second order before combining with the RA component. In other words, it can be shown that the Neyman orthogonality holds as $h\to 0$ neyman1959optimal,neyman1979c,chernozhukov2018double. As pointed out in Remark (ref), we need to compute the inverse probability weights at the sample points as $\frac{1}{\widehat{p}_{T|\bm{S}}(T_i|\bm{S}_i)}, i=1,...,n$ for the above DR estimator (ref). If we otherwise compute the inverse probability weights at the (query) points as $\frac{1}{\widehat{p}_{T|\bm{S}}(t|\bm{S}_i)}$ for $i=1,...,n$, then the resulting $\widehat{\theta}_{\mathrm{DR}}(t)$ will be asymptotically biased even when both of the conditional density model $p_{T|\bm{S}}$ and the outcome model $\mu,\beta$ are correctly specified. Finally, we also outline a self-normalized version of (ref) in (ref) for stabilizing its variance.
We introduce some regularity conditions for our subsequent theoretical analysis. Let $\mathcal{J} \subset \mathcal{T}\times \mathcal{S}$ be the support of the joint density $p(t,\bm{s})$, $\mathcal{J}^{\circ}$ be the interior of $\mathcal{J}$, and $\partial\mathcal{J}$ be the boundary of $\mathcal{J}$.
Assumptions (ref) and (ref) are common smoothness conditions for derivative estimation with kernel smoothing methods gasser1984estimating,mack1989derivative,wand1994kernel,wasserman2006all. These assumptions can be relaxed by the H\"older continuity condition. The uniform lower bound on $p_T(t)$ within its support $\mathcal{T}$ in Assumption (ref)(c) is only needed when we establish the uniform consistency of our proposed estimators and identify the derivative effect curve $\theta(t)$ when the positivity condition is violated. Assumption (ref)(a,b) are more like properties than regularity conditions on those commonly used kernel functions, such as the triangular kernel $K(u)=(1-|u|)\,\mathbbm{1}_{\{|u|\leq 1\}}$ and Epanechnikov kernel $K(u)=\frac{3}{4}(1-|u|)\, \mathbbm{1}_{\{|u|\leq 1\}}$. Finally, the VC-type condition in Assumption (ref)(c) is only required when we are interested in the uniform consistency of our proposed estimators over $\mathcal{T}$.
The following theorem summarizes the asymptotic properties of our proposed DR estimator $\widehat{\theta}_{\mathrm{DR}}(t)$, and we defer its proof and other consistency results for $\widehat{\theta}_{\mathrm{RA}}(t), \widehat{\theta}_{\mathrm{IPW}}(t)$ in (ref).
As established by (ref), the proposed estimator $\widehat{\theta}_{\mathrm{DR}}(t)$ achieves doubly robust consistency for $\theta(t)$, provided that either the conditional density model $\bar{p}_{T|\bm{S}}$ or the outcome model $\bar{\mu},\bar{\beta}$ is correctly specified. Unlike the DR estimator $\widehat{m}_{\mathrm{DR}}(t)$ of the dose-response curve $m(t)$, which only requires the specification of $\mu(t,\bm{s})$ in the outcome model, the DR estimator $\widehat{\theta}_{\mathrm{DR}}(t)$ of the derivative effect $\theta(t)$ necessitates specifying both $\mu(t,\bm{s})$ and its partial derivative $\beta(t,\bm{s})=\frac{\partial}{\partial t}\mu(t,\bm{s})$ in the outcome model. This added complexity is essential for accurately estimating derivatives.
We require in (ref) and other subsequent results that $\widehat{\mu},\widehat{\beta},\widehat{p}_{T|\bm{S}}$ are obtained from a data sample independent of $\left\{(Y_i,T_i,\bm{S}_i)\right\}_{i=1}^n$. This requirement avoids the need for uniform entropy conditions on $\bar{\mu}, \bar{p}_{T|\bm{S}}$ imposed by kennedy2017non. When no additional data sample is available, these nuisance function estimators $\widehat{\mu},\widehat{\beta},\widehat{p}_{T|\bm{S}}$ can still be estimated using cross-fitting techniques, allowing for valid construction of the associated estimators of $\theta(t)$; see (ref) for the detailed procedures. Importantly, the established rates of convergence in (ref) remain unchanged for the cross-fitted estimators.
Finally, the estimation bias of $\widehat{\theta}_{\mathrm{DR}}(t)$ is of order $O(h^2)$ due to the use of a standard second-order kernel. While a lower-bias estimator could be derived under Assumptions (ref) and (ref) by employing a higher-order kernel, we still recommend our proposed estimator $\widehat{\theta}_{\mathrm{DR}}(t)$ for practical use, as higher-order kernels introduce greater complexity and are more challenging to implement.
To leverage the asymptotic normality of $\widehat{\theta}_{\mathrm{DR}}(t)$ for pointwise inference on $\theta(t)$ in practice, we need to address two additional challenges: (i) estimate the asymptotic variance $V_{\theta}(t)$; and (ii) select a proper bandwidth parameter $h>0$.
For challenge (i), we estimate $V_{\theta}(t)$ in (ref) by the sample variance of the influence function $\phi_{h,t}$ or the asymptotic linear form as:
The cross-fitted version of $\widehat{V}_{\theta}(t)$ can be found in (ref) of (ref). Notice that the second part $\sqrt{h^3}\left[\widehat{\beta}(t,\bm{S}_i) - \widehat{\theta}_{\mathrm{DR}}(t) \right]$ in (ref) is asymptotically negligible. We keep this part mainly for a more conservative estimate of the asymptotic variance $V_{\theta}(t)$ to guarantee a better empirical coverage of the resulting pointwise confidence interval.
For challenge (ii), the optimal bandwidth that minimizes the asymptotic mean squared error of $\widehat{\theta}_{\mathrm{DR}}(t)$ is of order $O\left(n^{-\frac{1}{7}}\right)$. However, to construct a valid Wald-type confidence interval, an undersmoothing bandwidth $h$ is typically required for the first-order bias of $\widehat{\theta}_{\mathrm{DR}}(t)$ to be asymptotically negligible, i.e., $h^2\sqrt{nh^3}=o(1)$ wasserman2006all. Therefore, we recommend choosing the bandwidth $h$ to be of order $O\left(n^{-\frac{1}{5}}\right)$, aligning with the outputs of standard bandwidth selection methods for nonparametric regression wand1994kernel,li2004cross.
Finally, the $(1-\tau)$-level confidence interval for $\theta(t)$ is thus given by $\left[\widehat{\theta}_{\mathrm{DR}}(t) \pm q_{1-\frac{\tau}{2}} \sqrt{\frac{\widehat{V}_{\theta}(t)}{nh^3}} \right]$, where $q_{1-\frac{\tau}{2}}$ is the $\left(1-\frac{\tau}{2}\right)$ quantile of the standard normal distribution $\mathcal{N}(0,1)$.
We now study the nonparametric efficiency of our DR estimator $\widehat{\theta}_{\mathrm{DR}}(t)$, providing extra insights into its formulation. As discussed in (ref), a key challenge in deriving a nonparametric efficiency bound for $\theta(t)$ is its lack of pathwise differentiability bickel1998efficient,diaz2013targeted, which implies that the efficient influence function relative to a nonparametric model does not always exist. Our solution is to derive the efficient influence function for a smooth functional $\mbox{$\mathrm{P}$}\mapsto \varpi_{h,t}(\mbox{$\mathrm{P}$}):= \mathbb{E}\left[\frac{Y \cdot \left(\frac{T-t}{h}\right) K\left(\frac{T-t}{h}\right)}{h^2\cdot \kappa_2 \cdot p_{T|\bm{S}}(T|\bm{S})}\right]$ for a fixed bandwidth $h>0$ van2018cv,takatsu2022debiased. This functional, defined by the IPW form of $\theta(t)$, smoothly approximates $\theta(t)$ with $\varpi_{h,t}(\mbox{$\mathrm{P}$}) - \theta(t) = O(h^2)$ as shown in (ref) and remains pathwise differentiable for any fixed $h>0$. We establish in the following theorem that our DR estimator $\widehat{\theta}_{\mathrm{DR}}(t)$ attains the same asymptotic variance $V_{\theta}(t)$ as the (approximated) one obtained from the efficient influence function of $\varpi_{h,t}(\mbox{$\mathrm{P}$})$, up to a finite-sample bias of order $O(h^2)$.
The proof of (ref) is in (ref). We recommend defining the DR estimator of $\theta(t)$ as (ref) or (ref) rather than the one-step estimator based on the efficient influence function (ref) because of two reasons. First, evaluating the integral in (ref) is computationally challenging in practice. Second, the approximation error of (ref) or (ref) relative to (ref) is of the same order $O(h^2)$ as the approximation error of (ref) to the target parameter $\theta(t)$.
This section discusses the general identification issue on the dose-response curve $t\mapsto m(t)=\mbox{$\mathbb{E}$}\left[Y(t)\right]$ and its derivative effect curve $t\mapsto \theta(t)=\frac{d}{dt}\mbox{$\mathbb{E}$}\left[Y(t)\right]$ when the positivity condition (Assumption (ref)) is violated. We propose an additive structural assumption on the outcome model in (ref) to address the identification issue. However, even under this additive confounding model (ref), the IPW and DR estimators of $m(t)$ and $\theta(t)$ remain inconsistent without the positivity condition due to the support discrepancy. To resolve this inconsistency, we leverage techniques from nonparametric set estimation to propose our bias-corrected IPW and DR estimators.
When the positivity condition (Assumption (ref)) fails to hold, the conditional mean outcome (or regression) function $\mu(t,\bm{s}) = \mathbb{E}(Y|T=t,\bm{S}=\bm{s})$ is not well-defined in those regions of $\mathcal{T}\times\mathcal{S}$ that lie outside the support $\mathcal{J}$ of the joint density $p(t,\bm{s})$. Hence, the G-computation formulae $\mbox{$\mathbb{E}$}\left[\mu(t,\bm{S})\right]$ and $\mbox{$\mathbb{E}$}\left[\frac{\partial}{\partial t} \mu(t,\bm{S})\right]$ are ill-defined and cannot be used to identify $m(t)$ and $\theta(t)$, respectively.
Similarly, identifying $m(t)$ and $\theta(t)$ through the IPW formulae requires the positivity condition as well, because we demonstrate in the proofs of (ref) and Proposition (ref) that
Therefore, it is impossible in general to identify the causal dose-response curve $t\mapsto m(t)=\mbox{$\mathbb{E}$}\left[Y(t)\right]$ and its derivative effect curve $t\mapsto \theta(t) = \frac{d}{dt}\mbox{$\mathbb{E}$}\left[Y(t)\right]$ without further identification or structural assumptions when the positivity condition is violated.
While the identifications of $m(t)$ and $\theta(t)$ are infeasible without positivity in general, they are indeed identifiable under an additive structural assumption on the potential outcome model as $Y(t) = \bar{m}(t) + \eta(\bm{S}) +\epsilon$ for any $t\in \mathcal{T}$ zhang2024nonparametric, which, under the consistency condition (Assumption (ref)(a)), is equivalent to the following additive confounding model
where $\bar{m}:\mathcal{T}\to \mathbb{R}$ and $\eta:\mathcal{S}\to \mathbb{R}$ are deterministic functions, $\mbox{$\mathbb{E}$}(\epsilon|T,\bm{S})=0$, $\mathrm{Var}(\epsilon|T,\bm{S}) > \sigma^2 >0$, and $\mbox{$\mathbb{E}$}|\epsilon|^{2+c_1} <\infty$ as in Assumption (ref)(c). Such an additive model is a common working model in the context of spatial statistics paciorek2010importance,schnell2020, where the covariates $\bm{S}\in \mathcal{S}\subset \mathbb{R}^d$ consist of spatial locations or other spatially correlated confounding variables. More broadly, it also appears in the literature of nonparametric stone1985additive and high-dimensional statistics meier2009high,guo2019decorrelated.
Under model (ref), the dose-response curve $m(t)$ and its derivative $\theta(t)$ become
They are identifiable from the observable data through the formulas
For completeness, we also summarize this identification theory as Proposition (ref) in (ref). As a result, the RA estimator of $\theta(t)$ under model (ref) without assuming the positivity condition is given by
where $\widehat{\beta}(t,\bm{s})$ and $\widehat{F}_{\bm{S}|T}(\bm{s}|t)$ are (consistent) estimators of of $\beta(t,\bm{s}) = \frac{\partial}{\partial t} \mu(t,\bm{S})$ and the conditional cumulative distribution function (CDF) $\mbox{$\mathrm{P}$}_{\bm{S}|T}(\bm{s}|t):= F_{\bm{S}|T}(\bm{s}|t)$, respectively. By (ref), the integral RA estimator of $m(t)$ under model (ref) can be written as:
Both estimators (ref) and (ref) are consistent even when the positivity condition is violated zhang2024nonparametric; see also (ref) and (ref). In the sequel, we will discuss both the challenges and solutions for extending these RA estimators to IPW and DR estimators of $\theta(t)$ and $m(t)$ under model (ref).
Although the causal quantities $m(t)$ and $\theta(t)$ are identifiable under the additive confounding model (ref), the IPW formulae (ref) are indeed biased without positivity due to the support discrepancy between the conditional density $p_{\bm{S}|T}(\bm{s}|t)$ for $t\in \mathcal{T}$ and the marginal density $p_{\bm{S}}(\bm{s})$. To examine these biases, we can equivalently analyze the following oracle IPW estimators of $m(t)$ and $\theta(t)$ defined as:
where the estimated conditional density $\widehat{p}_{T|\bm{S}}(t|\bm{s})$ is replaced by the true one $p_{T|\bm{S}}(t|\bm{s})$.
The proof of Proposition (ref) is in (ref). We emphasize that the IPW estimators in (ref) have two layers of bias. First, if $\frac{A_h}{h}\to 0$ as $h\to 0$ (see also Remark (ref) below), then the results in Proposition (ref) will imply that
where we recall that $m(t)=\bar{m}(t) + \mbox{$\mathbb{E}$}\left[\eta(\bm{S})\right]$ and $\theta(t)=\bar{m}'(t)$ from (ref). Second, if $\frac{A_h}{h}$ does not converge to 0, then the bias of $\widetilde{\theta}_{\mathrm{IPW}}(t)$ will be larger or even diverging to infinity as $h\to 0$. In reality, the estimation biases or inconsistencies of IPW estimators in (ref) are due to the discrepancy between the conditional support $\mathcal{S}(t)$ of $p_{\bm{S}|T}(\bm{s}|t)$ and the marginal support $\mathcal{S}$ of $p_{\bm{S}}(\bm{s})$. To correct for the bias of IPW estimators, it is necessary to address the geometric discrepancy, a solution to which will be elaborated upon in (ref).
Finally, since both RA and IPW estimators cannot be used to identify and estimate $m(t)$ and $\theta(t)$ due to identification and inconsistency issues, the previously studied DR estimators (ref) and (ref) will be pointless without the positivity condition.
In this section, we present our solution for addressing the estimation biases of IPW estimators for the dose-response curve $m(t)$ and its derivative $\theta(t)$, as described in (ref), when the positivity condition (Assumption (ref)) is violated. Specifically, our proposed IPW and DR estimators for $\theta(t)$ under the additive confounding model (ref) rely on a consistent estimation of the interior region of the support of the conditional density $p_{\bm{S}|T}(\bm{s}|t)$. Our approach establishes a connection between the classical support estimation problem and a contemporary causal inference challenge, namely the dose-response curve estimation problem.
Recall from (ref) and Proposition (ref) that the oracle IPW estimator of $\theta(t)$ is the sample average of the IPW quantity $\Xi_t(Y,T,\bm{S}) = \frac{Y\left(\frac{T-t}{h}\right) K\left(\frac{T-t}{h}\right)}{h^2 \cdot \kappa_2 \cdot p_{T|\bm{S}}(T|\bm{S})}$, and it is biased for estimating the quantity of interest $\theta(t)=\bar{m}'(t)$ even under model (ref). In particular, $\mbox{$\mathbb{E}$}\left[\Xi_t(Y,T,\bm{S}) \right]$ converges to $\bar{m}'(t)\cdot \rho(t)$ as $h\to 0$ under some mild regularity conditions, where $\rho(t)=\mbox{$\mathrm{P}$}\left(\bm{S}\in \mathcal{S}(t)\right)$ for any $t\in \mathcal{T}$. The first step toward removing the bias of $\mathbb{E}\left[\Xi_t(Y,T,\bm{S})\right]$ is to decouple the quantity of interest $\theta(t)=\bar{m}'(t)$ from the nuisance function $\rho(t)$. To this end, we consider a modified IPW quantity defined as:
in which we multiply the original IPW quantity $\Xi_t(Y,T,\bm{S})$ by a density ratio $\frac{p_{\bm{S}|T}(\bm{S}|t)}{p_S(\bm{S})}$. The following proposition demonstrates that the remaining bias in $\mathbb{E}\left[\widetilde{\Xi}_t(Y,T,\bm{S})\right]$ can be disentangled from the quantity of interest $\theta(t)=\bar{m}'(t)$ in an additive form.
Proposition (ref) reveals that the estimation bias of the modified IPW quantity (ref) results from the support discrepancy between $\mathcal{S}(t)$ and the integration range $\mathcal{S}(t+uh)$ for a given integration variable $u\in \mathbb{R}$; see (ref) for an illustration. As shown in Proposition (ref), this additive bias may not always shrink at the rate $O(h^2)$ as $h\to 0$. To further reduce the bias of the modified IPW quantity (ref) to $O(h^2)$ without assuming positivity, we address the support discrepancy of (ref) by restricting the conditional density $p_{\bm{S}|T}(\bm{s}|t)$ to its interior region, defining it as $p_{\zeta}(\bm{s}|t)$, and refining (ref) as:
Essentially, the only requirement for defining the $\zeta$-interior conditional density $p_{\zeta}(\bm{s}|t)$ is that its support satisfies the following condition:
Here, we propose two approaches for defining $p_{\zeta}(\bm{s}|t)$ and leave other options to interested readers.
{\bf 1. Support Shrinking Approach:} Let $\mathcal{S}(t) \ominus \zeta = \left\{\bm{s}\in \mathcal{S}(t): \inf_{\bm{x}\in \partial \mathcal{S}(t)}\left|\left| \bm{s}-\bm{x} \right|\right|_2 \geq \zeta\right\}$ denote the set of interior points of $\mathcal{S}(t)$ that are at least a distance $\zeta$ away from the boundary $\mathcal{S}(t)$. Then, we define the $\zeta$-interior conditional density with $\zeta>0$ being a tuning parameter as:
This interior density is indeed the conditional density $p_{\bm{S}|T}(\bm{s}|t)$ restricted to the interior of its support $\mathcal{S}(t)$. Its estimator $\widehat{p}_{\zeta}(\bm{s}|t)$ can be constructed using a support estimator $\widehat{\mathcal{S}}(t)$ and constraining the conditional density estimator $\widehat{p}_{\bm{S}|T}(\bm{s}|t)$ within the region $\widehat{\mathcal{S}}(t)\ominus \zeta$.
{\bf 2. Level Set Approach:} Let $\mathcal{L}_{\zeta}(t) = \left\{\bm{s}\in \mathcal{S}(t): p_{\bm{S}|T}(\bm{s}|t) \geq \zeta\right\}$ be the $\zeta$-upper level set of the conditional density $p_{\bm{S}|T}(\bm{s}|t)$. Then, we define the $\zeta$-interior conditional density as:
The level set approach restricts the conditional density $p_{\bm{S}|T}(\bm{s}|t)$ to the high-density region, which is generally located away from the support boundary. We may construct the estimator $\widehat p_{\zeta}(\bm{s}|t)$ using a level set estimator $\widehat{\mathcal{L}}_{\zeta}(t) = \left\{\bm{s}\in \mathcal{S}(t): \widehat{p}_{\bm{S}|T}(\bm{s}|t) \geq \zeta\right\}$ and constraining $\widehat{p}_{\bm{S}|T}(\bm{s}|t)$ to $\widehat{\mathcal{L}}_{\zeta}(t)$.
We further specialize condition (ref) for the above two approaches by introducing the following smoothness condition on the conditional support $\mathcal{S}(t)$.
To some extent, Assumption (ref) can be viewed as a Lipschitz condition of the conditional support $\mathcal{S}(t)$. It can be satisfied when the Euclidean norm of the gradient $\left|\left| \nabla_{\bm{s}} p_{\bm{S}|T}(\bm{s}|t) \right|\right|_2$ is bounded away from 0 at the boundary of $\mathcal{S}(t)$ cadre2006kernel. This assumption allows us to ignore the boundary discrepancy as long as we do not evaluate our IPW quantity (ref) near the boundary; see (ref) for a graphical illustration.
The proof of Proposition (ref) is in (ref). This result demonstrates that the expectation of our newly modified IPW quantity $\widetilde{\Xi}_{t,\zeta}(Y,T,\bm{S})$ in (ref) converges to the quantity of interest $\theta(t)=\bar{m}'(t)$ in the standard order $O(h^2)$ as $h\to 0$ under the additive confounding model (ref). Notice that the tuning parameter $\zeta=\zeta_n >0$ in (ref) is allowed to converge to 0 as $n\to \infty$, as long as the condition $h=h_n < \frac{\zeta_n}{A_0}$ holds under Assumption (ref).
Given this newly modified IPW quantity (ref), we propose the bias-corrected IPW estimator of $\theta(t)$ without the positivity condition as:
where $\widehat{p}(t,\bm{s})$ is a consistent estimator of the joint density $p(t,\bm{s})$ and $\widehat{p}_{\zeta}(\bm{s}|t)$ is an estimated $\zeta$-interior conditional density.
Finally, we combine the modified RA estimator (ref) with our bias-corrected IPW estimator (ref) to propose our bias-corrected DR estimator of $\theta(t)$ as:
Notice that for the RA component of $\widehat{\theta}_{\mathrm{C,DR}}(t)$, we replace the original conditional CDF estimator $\widehat{F}_{\bm{S}|T}$ in (ref) with the estimated $\zeta$-interior conditional density $\widehat{p}_{\zeta}$. This modification is necessary because the IPW component of $\widehat{\theta}_{\mathrm{C,DR}}(t)$ is defined through $\widehat{p}_{\zeta}$. Both the RA and IPW components need to match up with each other in the definition of $\widehat{\theta}_{\mathrm{C,DR}}(t)$ for its consistency.
The following theorem summarizes the asymptotic properties of our DR estimator (ref) of $\theta(t)$ under the additive confounding model (ref) without assuming the positivity condition. We again defer its proof and other consistency results for our RA (ref) and IPW (ref) estimators of $\theta(t)$ to (ref).
Similar to our discussions in (ref) and (ref), $V_{C,\theta}(t)$ in (ref) attains the nonparametric efficiency bound derived from a smooth functional $\mbox{$\mathrm{P}$}\mapsto \varpi_{C,h,t}(\mbox{$\mathrm{P}$}):= \mathbb{E}\left[\frac{Y \cdot \left(\frac{T-t}{h}\right) K\left(\frac{T-t}{h}\right) \cdot \bar{p}_{\zeta}(S|t)}{h^2\cdot \kappa_2 \cdot p(T,\bm{S})}\right]$, and we can estimate it by $$\widehat{V}_{C,\theta}(t) = \frac{1}{n} \sum_{i=1}^n \left\{\phi_{C,h,t}\left(Y_i,T_i,\bm{S}_i;\widehat{\mu}, \widehat{\beta}, \widehat{p}, \widehat{p}_{\zeta}\right) + \sqrt{h^3}\left[\int \widehat{\beta}(t,\bm{s}) \cdot \widehat{p}_{\zeta}(\bm{s}|t)\, d\bm{s} - \widehat{\theta}_{\mathrm{C,DR}}(t) \right]\right\}^2$$ and choose the bandwidth $h$ to be of order $O\left(n^{-\frac{1}{5}}\right)$ to ensure valid inference. As a corollary, we can plug either IPW (ref) or DR (ref) estimators into our integral formula (ref) to obtain the integral IPW or DR estimators of the dose-response curve $m(t)$ under model (ref). We establish the asymptotic theory for these integral estimators in Corollary (ref) of (ref).
In this section, we evaluate the finite-sample performances of our proposed estimators of $\theta(t)=\frac{d}{dt}\mbox{$\mathbb{E}$}\left[Y(t)\right]$ in (ref) and compare them with the finite-difference approach in colangelo2020double under the positivity condition through simulation studies and an analysis of the Job Corps program in the United States. Furthermore, we compare the bias-corrected estimators of $\theta(t)$ in (ref) with their counterparts via simulation studies when the positivity condition is violated.
We generate i.i.d. observations $\{(Y_i,T_i,\bm{S}_i)\}_{i=1}^n$ from the following data-generating model as in colangelo2020double,klosin2021automatic:
where $F_{\mathcal{N}(0,1)}$ is the CDF of $\mathcal{N}\left(0, 1\right)$, $\bm{\xi}=(\xi_1,...,\xi_d)^T \in \mathbb{R}^d$ has its entry $\xi_j=\frac{1}{j^2}$ for $j=1,...,d$ as well as $\Sigma_{ii}=1$, $\Sigma_{ij}=0.5$ when $|i-j|=1$, and $\Sigma_{ij}=0$ when $|i-j|>1$ for $i,j=1,...,d$. Here, $d=20$ unless stated otherwise. The dose-response curve is thus given by $m(t)= 1.2t+t^2$, and our parameter of interest is the derivative effect curve $\theta(t)=1.2+2t$.
We evaluate our proposed estimators of $\theta(t)$ in (ref) alongside the finite-difference estimator by colangelo2020double with 5-fold cross-fitting. In particular, we replicate their finite-difference estimators using their neural network (NN) and kernel neural network (KNN) models for estimating the nuisance functions $\mu(t,\bm{s})$ and $p_{T|\bm{S}}(t|\bm{s})$, which yield their best performances. Additionally, similar to the setups in colangelo2020double,klosin2021automatic, we use the Epanechnikov kernel $K(u)=\frac{3}{4}(1-|u|)\, \mathbbm{1}_{\{|u|\leq 1\}}$ under a bandwidth choice $h=1.25\, \widehat{\sigma}_T\cdot n^{-\frac{1}{5}}$, where $\widehat{\sigma}_T$ is the sample standard deviation of $\{T_1,...,T_n\}$. Furthermore, for our proposed estimators, the nuisance functions $\mu(t,\bm{s})$ and $\beta(t,\bm{s})$ are estimated by neural network models as well, while $p_{T|\bm{S}}(t|\bm{s})$ is estimated by either the method of kernel density estimation (KDE) on residuals or the approach of regressing kernel-smoothed outcomes (RKS); see (ref) for details. To prevent division by zero, all estimated conditional density values $\widehat{p}_{T|\bm{S}}(T_i|\bm{S}_i), i=1,...,n$ smaller than 0.001 are set to this value. For comparison, we also implement our proposed DR estimator of $\theta(t)$ under the true conditional density (“True”). All our DR estimators are self-normalized as described in (ref) to reduce their variances. The nominal levels of all the yielded pointwise confidence intervals are set to 95%.
The simulation results are shown in (ref) for various sample sizes, where the estimation biases, root mean square errors (RMSEs), and coverage rates of confidence intervals are calculated by averaging over 1000 Monte Carlo replications. Additional results when the bandwidth parameter varies or cross-fitting is not employed are in (ref) and (ref). Unlike prior studies in colangelo2020double,klosin2021automatic, which focus solely on $t=0$, our comparative simulations evaluate 81 treatment values across $t\in [-2,2]$. Overall, our proposed DR estimators, using either true or KDE-estimated conditional density values, outperform the finite-difference methods of colangelo2020double in terms of estimation bias while maintaining comparable RMSE. When it comes to statistical inference, the confidence intervals from our DR estimators consistently show better empirical coverages than those from colangelo2020double. These performance advantages of our DR estimators arise from directly estimating and inferring $\theta(t)$ without requiring a step-size parameter for finite-difference approximations.
We now assess the finite-sample performances of our bias-corrected IPW and DR estimators of $\theta(t)$ in (ref) and compare them with those counterparts in (ref) when the positivity condition is violated. To this end, we generate i.i.d. data $\{(Y_i, T_i,S_i)\}_{i=1}^n$ from the following data-generating model
where $E\sim \mathrm{Uniform}[-0.3,0.3]$ is an independent treatment variation and $\epsilon\sim \mathcal{N}(0,1)$ is an independent noise variable. The marginal supports of $T$ and $S$ are $\mathcal{T}=[-1.3, 1.3]$ and $\mathcal{S}=[-1,1]$ respectively, while the joint support of $(T,S)$ only covers a thin band region of the product space $\mathcal{T}\times \mathcal{S}$; see Figure 1 in zhang2024nonparametric for illustration. The true derivative effect curve is thus given by $\theta(t)=3t^2 + 2t$.
We evaluate our bias-corrected estimators of $\theta(t)$ in (ref) on the simulated dataset, alongside those estimators from (ref) that assumes the positivity condition. All these estimators are assessed with 5-fold cross-fitting. Again, we use the Epanechnikov kernel $K(u)=\frac{3}{4}(1-|u|)\, \mathbbm{1}_{\{|u|\leq 1\}}$ under a bandwidth choice $h=2\,\widehat{\sigma}_T\cdot n^{-\frac{1}{5}}$. For those estimators assuming positivity, we estimate the nuisance functions $\mu(t,s)$ and $\beta(t,s)$ by neural network models in (ref) and utilize the true conditional density function $p_{T|S}$ evaluated at the observations $\{(T_i,S_i)\}_{i=1}^n$. For the bias-corrected estimators, we estimate the joint density $p(t,s)$ and conditional density $p_{S|T}(s|t)$ using kernel density estimation with a Gaussian kernel $K(u)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{u^2}{2}\right)$. The estimated interior densities $\widehat{p}_{\zeta}(S_i|t), i=1,...,n$ are computed via the trimming method outlined in Remark (ref). All the estimators are self-normalized as described in (ref) to reduce their variances, and the nominal levels of all the yielded pointwise confidence intervals are set to 95%.
The simulation results for different sample sizes are presented in (ref), where the estimation biases, root mean square errors (RMSEs), and coverage rates of confidence intervals are calculated by averaging over 1000 Monte Carlo replications. Additional results when the bandwidth parameter varies or cross-fitting is not employed are in (ref) and (ref). The bias-corrected IPW estimator (ref) effectively reduces the estimation biases of the standard IPW estimator (ref) of $\theta(t)$ across when the positivity condition is violated. Furthermore, the bias-corrected DR estimator (ref) achieves comparable biases and RMSEs to its standard counterpart (ref), even when (ref) uses the oracle conditional density $p_{T|S}$. Notably, the confidence intervals yielded by the bias-corrected DR estimator (ref) exhibit better coverage probabilities compared to its counterpart (ref). These findings support the theoretical properties of our proposed bias-corrected IPW and DR estimators in (ref). Nonetheless, the bias-corrected RA estimator (ref) remains the preferred choice when it comes to estimation accuracy due to its simplicity under violations of the positivity condition.
We demonstrate the applicability of our proposed DR estimators for $\theta(t)$ by extending the analysis of colangelo2020double on the Job Corps program in the United States (U.S.). This program aims at providing academic and vocational training to U.S. legal residents aged 16--24 who come from low-income households schochet2001national. The data used in our analysis originated from the National Job Corps Study, which conducted some randomized experiments on first-time applicants in the 48 contiguous states and the District of Columbia between November 1994 and February 1996 schochet2008does.
Numerous studies have examined the causal effects of the Job Corps program from various angles flores2009identification,flores2012estimating,huber2014identifying,lee2018partial,huber2020direct,lee2024lee. Following colangelo2020double, we analyze the relationship between employment outcomes and the duration of academic and vocational training, focusing on the derivative effect curve $\theta(t)=\frac{d}{dt}\mbox{$\mathbb{E}$}\left[Y(t)\right]$. The data sample includes 4,024 individuals who received at least 40 hours of training. The outcome variable $Y$ represents the proportions of weeks employed in the second year following the program assignment, and the treatment variable $T$ is the total hours of academic and vocational training received. The covariate vector $\bm{S}$, comprising 49 socioeconomic characteristics, ensures the validity of the ignorability assumption flores2012estimating; see Table 4 in huber2020direct for detailed descriptions of the covariates. Before applying derivative effect estimation methods, categorical covariates were converted to dummy variables, and all variables were standardized to have mean 0 and variance 1.
We apply our proposed DR estimator (ref) with the same setup as in (ref) to the standardized data, extending the range of queried treatment values from $[320,1840]$ to $[40, 4000]$. For consistency, we use the same bandwidth parameter $h=223$ and apply the neural network model for conditional density estimation as in colangelo2020double. The estimated derivative effect curves with 95% confidence intervals under 5-fold cross-fitting are shown in (ref). Overall, our DR estimator produces similar patterns to the finite-difference estimates from colangelo2020double. However, our confidence intervals are more conservative and include 0 for nearly all treatment values, suggesting insufficient evidence to confirm the program’s effectiveness. Additional results when cross-fitting is not employed are shown in (ref).
In summary, this paper studies nonparametric DR inference methods for the derivative function of the dose-response curve with and without the positivity condition. We establish the asymptotic properties of our proposed estimators under mild conditions, permitting the use of machine learning methods for nuisance function estimation with cross-fitting. Furthermore, our identification theory and refinements of IPW and DR estimators without positivity open up a novel link between the dose-response curve inference challenge and the nonparametric set estimation problem. Simulation studies and empirical applications demonstrate the advantages of our DR estimator over the existing finite-difference method for derivative effect inference. This work also highlights several avenues for future research.
{\bf 1. Bias correction for DR estimators:} As shown in (ref) and (ref), our DR estimators of $\theta(t)$ contain bias terms of order $O(h^2)$. These biases become asymptotically negligible when the bandwidth is chosen as $h\asymp n^{-\frac{1}{5}}$ that matches up the standard rate of convergence for nonparametric regression. To guarantee valid inference, an alternative approach is to explicitly estimate and correct these bias terms, as demonstrated by calonico2018effect,cheng2019nonparametric,takatsu2022debiased. A rigorous investigation of this bias-corrected approach for our DR estimators would be a valuable direction for future research.
{\bf 2. Derivative estimation in other causal contexts:} Our proposed DR inference methods for $\theta(t)$ can be naturally extended to conduct inference on other causal estimands of interest, such as the instantaneous causal effect $\frac{d}{d t} \mbox{$\mathbb{E}$}\left[Y(t)|\bm{S}=\bm{s}\right]$ stolzenberg1980measurement,ratkovic2023estimation or the marginal direct and indirect effects in causal mediation analysis huber2020direct,xu2021multiply.
We thank Alex Luedtke and Jon A. Wellner for their helpful comments. YZ is supported in part by YC's NSF grant DMS-2141808. YC is supported by NSF grants DMS-1952781, 2112907, 2141808, and NIH U24-AG07212.
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