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PUATE: Efficient ATE Estimation from Treated (Positive) and Unlabeled Units
abstractThe estimation of average treatment effects (ATEs), defined as the difference in expected outcomes between treatment and control groups, is a central topic in causal inference. This study develops semiparametric efficient estimators for ATE in a setting where only a treatment group and an unlabeled group—consisting of units whose treatment status is unknown—are observed. This scenario constitutes a variant of learning from positive and unlabeled data (PU learning) and can be viewed as a special case of ATE estimation with missing data. For this setting, we derive the semiparametric efficiency bounds, which characterize the lowest achievable asymptotic variance for regular estimators. We then construct semiparametric efficient ATE estimators that attain these bounds. Our results contribute to the literature on causal inference with missing data and weakly supervised learning.
Introduction
The estimation of average treatment effects (ATEs), defined as the difference in expected outcomes between treatment and control groups, is a fundamental problem in causal inference Imbens2015causalinference. Estimating ATEs enables researchers to quantify the causal impact of a treatment, intervention, or policy on an outcome of interest. This problem is of paramount importance across various fields, including economics, epidemiology, and the computer sciences.
Standard ATE estimation typically assumes access to both treatment and control groups, along with complete information on treatment assignment. However, in many practical situations, this assumption does not hold. In some cases, only a treatment group and an unknown group—comprising units for which treatment assignment is unobserved—are available. Such scenarios arise in various applications, including recommendation systems with implicit feedback, electronic health records, and marketing campaigns, where the absence of explicit treatment information poses significant challenges for causal inference.
For example, in a recommendation system, if a user buys a product from a website, we can infer that the user visited the site. However, if the purchase occurs in a physical store, we cannot determine whether the user visited the website. If building an online website is regarded as the treatment, this situation implies that users who bought the product in-store belong to an unknown group.
\paragraph{Content of this study.}
We address the problem of ATE estimation using only a treatment group and an unknown group. This setting is closely related to learning from positive and unlabeled data Sugiyama2022machinelearning, where the goal is to train a classifier using only positive and unlabeled instances. In our context, the challenge lies in efficiently estimating ATEs using the treatment (positive) and unknown groups. We refer to our setting and methodology as PUATE.
For this problem, we first derive semiparametric efficiency bounds, which are theoretical lower bounds on the asymptotic variance of regular estimators under the given data-generating processes (DGPs).\footnote{For regular estimators, see p.366 in Vaart1998asymptoticstatistics.} These bounds serve as benchmarks for evaluating estimator performance. As part of this derivation, we compute the efficient influence function, which provides insight into the construction of efficient estimators.
Using the efficient influence function, we develop semiparametric efficient ATE estimators that are $\sqrt{n}$-consistent and whose asymptotic variance achieves the efficiency bounds. These estimators are thus optimal under the semiparametric framework.
In this study, we consider two DGPs relevant to the PU setup: the censoring setting and the case-control setting Elkan2008learningclassifiers,duPlessis2015convexformulation. In the censoring setting, we are given a single dataset in which some units have missing treatment information while others are confirmed to have received treatment. In the case-control setting, we are provided with two datasets: one containing treated units and another comprising units with unknown treatment status.
Specifically, our contributions are as follows:
itemize[noitemsep, topsep=0pt, leftmargin=0.40cm]
• We formulate the ATE estimation problem with missing data using the PU learning framework.
• We derive efficiency bounds under both the censoring and case-control settings.
• We propose novel efficient estimators.
• We establish connections between ATE estimation with missing data and PU learning.
• We also propose alternative candidate estimators.
This study is organized as follows. Section (ref) formulates our problem. Section (ref) introduces simple ATE estimators, which are later shown to be inefficient. In Section (ref), we derive efficiency bounds, propose an efficient estimator, and establish its asymptotic properties under the censoring setting. Due to the space limitation, we show the ATE estimation in the case-control setting briefly in Section (ref) and mainly in Appendix (ref). Section (ref) presents simulation studies. We introduce related work in Appendix (ref). The details of PU learning methods in Appendix (ref)
Problem setting
Potential outcomes and parameter of interest
We consider binary treatments, $1$ and $0$. For each treatment $d \in \{1, 0\}$, there exists a potential outcome $Y(d) \in {\mathbb{R}}$. The outcome is observed only when the corresponding treatment is assigned to a unit. Each unit has $p$-dimensional covariates $X \in {\mathcal{X}} \subset {\mathbb{R}}^p$. This setting is called the Neyman-Rubin causal model Neyman1923surapplications,Rubin1974estimatingcausal.
We denote by $P_0 \in {\mathcal{P}}$ a distribution of $Y(d)$, $X$, and other random variables introduced below, which we call the true distribution, where ${\mathcal{P}}$ is the set of distributions. For simplicity, we assume that $P_0$ has a density. We denote the conditional density of $Y(d)$ given $X$ by $p_{Y(d), 0}(y(d) \mid X)$, and the marginal density of $X$ by $\zeta_0(x)$.
We consider $n$ units, each of which receives either treatment or control. Let $Y_i(d)$ and $X_i$ be i.i.d. copies of $Y(d)$ and $X$. Throughout this study, for a random variable $R$, let $R_i$ be its i.i.d. copy under $P_0$. If unit $i$ receives treatment $d$, we observe $Y_i(d)$ but not the counterfactual outcome.
We refer to the group of units who receive treatment $1$ as the treatment group and those who receive treatment $0$ as the control group. As formulated in the next subsection, we consider a scenario where part of the treatment group and a mixture of the treatment and control groups are observable, where the treatment indicator is unobservable. We refer to this mixed group as the unknown group.
Our objective is to estimate the ATE under $P_0$ using observed data, defined as $\tau_0 \coloneqq {\mathbb{E}}\left[Y(1) - Y(0)\right]$,
where ${\mathbb{E}}$ denotes the expectation under $P_0$.
Observations with two DGPs
In our setting, the observations are non-standard. We can only observe part of the treatment group and the unknown group, a mixture of the treatment and control groups. This setting is a variant of PU learning. PU learning encompasses two settings: the censoring setting and the case-control setting. In the censoring setting, we consider a single dataset with i.i.d. observations, where treatment labels contain missing values. Specifically, while part of the treatment group is observed, the mixture of the treated and control groups is also present. In the case-control setting, we observe two independent datasets: one consisting solely of the treatment group and the other comprising the unknown group. The censoring and case-control settings are also referred to as one-sample and two-sample settings, respectively Niu2016theoreticalcomparisons. The case-control setting can also be regarded as a form of stratified sampling, studied in Imbens1996efficientestimation and Wooldridge2001asymptoticproperties.
Censoring setting
In the censoring setting, we observe a single dataset ${\mathcal{D}}$, defined as follows:
align*[align* omitted — 149 chars of source]
where $O_i \in \{1, 0\}$ is an observation indicator with the observation probability $\pi_0(O \mid X)$, $Y_i$ is defined as
\[Y_i \coloneqq \mathbbm{1}[O_i = 1] Y_i(1) + \mathbbm{1}[O_i = 0] \widetilde{Y},\]
$\widetilde{Y}_i$ is the observation of the unknown groups and defined as
\[\widetilde{Y}_i \coloneqq \mathbbm{1}[D_i = 1] Y_i(1) + \mathbbm{1}[D_i = 0] Y_i(0),\]
$D_i \in \{1, 0\}$ is a (unobserved) treatment indicator, and we denote the conditional probability of $D_i$ given $X$ and $O_i = 0$ as $g_0(D \mid X) = {\mathbb{P}}(D \mid X, O = 0)$. We refer to $g_0(d \mid X)$ as the censoring propensity score.
Here, the density $p_0(x, o, y)$ is given as $p_0(x, o, y) = \zeta_0(x)\pi_0(o \mid x)p_{Y,0}(y \mid x)$,
where $p_{Y, 0}$ is the density of $Y$.
Case-control setting
In the case-control setting, we observe two stratified datasets, ${\mathcal{D}}_{{\mathrm{T}}}$ and ${\mathcal{D}}_{{\mathrm{U}}}$:
align*[align* omitted — 428 chars of source]
where $m$ and $l$ are fixed sample sizes of each dataset such that $m + l = n$, $X_{{\mathrm{T}}, j}$ represents the covariates of the treatment group, $Y_{{\mathrm{U}}, k}$ is the observed outcome defined as
\[Y_{{\mathrm{U}}, k} = \mathbbm{1}[D_k = 1]Y_k(1) + \mathbbm{1}[D_k = 0]Y_k(0),\]
and $D_k \in \{1, 0\}$ is a treatment indicator with probability $e_0(D \mid X)$. We refer to $e_0(d \mid X = x)$ as the case-control propensity score.
The densities $p_{{\mathrm{T}}, 0}(x, y(1))$ and $p_{{\mathrm{U}}, 0}(x, y_{{\mathrm{U}}})$ satisfy $p_{{\mathrm{T}}, 0}(x, y(1))= \zeta_{{\mathrm{T}}, 0}(x)p_{Y(1), 0}(y(1)\mid X = x)$ and $p_{{\mathrm{U}}, 0}(x, y_{{\mathrm{U}}}) = \zeta_0(x)p_{Y_{\mathrm{U}}, 0}(y_{{\mathrm{U}}}\mid X = x)$ respectively,
where $p_{Y_{\mathrm{U}}, 0}(y_{{\mathrm{U}}} \mid X = x)$ denotes the density of $Y_{{\mathrm{U}}, k}$ given $X = x$, and
$\zeta_{{\mathrm{T}}, 0}(x)$ represents the density of the covariates $X$ in the treatment group.
Although the ATE estimation in the case-control setting is also crucially important, due to the space limitation, we show the main results almost in Appendix (ref).
Difference between the two settings
figure[figure omitted — 267 chars of source]
We illustrate the concept of the censoring and case-control settings in Figure (ref).
A summary of the differences is provided below:
description[noitemsep, topsep=0pt, leftmargin=0.40cm]
• A single dataset is observed, containing partial treatment information and a mixture of treated and control groups.
• Two stratified datasets are observed—one consisting of the treatment group and the other comprising the unknown group.
The key distinction lies in the randomness of treatment group observations. In the censoring setting, the observation of the treatment group is a random event, where the observation indicator $O_i$ follows a probability $P(O\mid X)$. In contrast, in the case-control setting, the label observation is deterministic, and the treatment and unknown groups are drawn independently. This difference impacts the estimator design and efficiency bounds.
Note that the definitions and meanings of the propensity scores in the censoring and case-control settings are also different, and its difference stems from the definition of the observations indicators.
\paragraph{Notation.}
We summarize the notations above and introduce new notations. Let ${\mathbb{E}}$, ${\mathbb{P}}$, and $\mathrm{Var}$ be an expectation operator, a probability law, and a variance operator. For both settings, let us define $\mu_{{\mathrm{T}}, 0}(X) \coloneqq {\mathbb{E}}[Y(1)\mid X]$, $\mu_{{\mathrm{C}}, 0}(X) \coloneqq {\mathbb{E}}[Y(0)\mid X]$, and let $\tau_0(X) \coloneqq \mu_{{\mathrm{T}}, 0}(X) - \mu_{{\mathrm{C}}, 0}(X)$ be the conditional ATE. If a function $f$ depends on the true distribution $P_0$, we denote it by $f_0$
In the censoring setting, we use $\pi_0(o \mid X) \coloneqq {\mathbb{P}}(O = o\mid X)$, $g_0(d\mid X) \coloneqq {\mathbb{P}}(D = d\mid X, O = 0)$, and $\nu_0 \coloneqq {\mathbb{E}}[\widetilde{Y}\mid X, O = 0]$. Here, ${\mathbb{E}}[\mathbbm{1}[O = 1]Y\mid X] = \pi_0(O = 1 \mid X)\mu_{{\mathrm{T}}, 0}(X)$ and ${\mathbb{E}}[\mathbbm{1}[O = 0]Y\mid X] = \pi_0(0 \mid X)\nu_0(X) = \pi_0(0 \mid X)g_0(1\mid X)\mu_{{\mathrm{T}}, 0}(X) + \pi_0(0 \mid X)g_0(0\mid X)\mu_{{\mathrm{C}}, 0}(X)$ hold under Assumption (ref), defined later.
In the case-control setting, we use $e_0(d\mid X) \coloneqq {\mathbb{P}}(D = d \mid X)$ and $\mu_{{\mathrm{U}}, 0}(X) \coloneqq {\mathbb{E}}[Y_{{\mathrm{U}}}\mid X]$. Here, $\mu_{{\mathrm{U}}, 0}(X) = e_0(1\mid X)\mu_{{\mathrm{T}}, 0}(X) + e_0(0\mid X)\mu_{{\mathrm{C}}, 0}(X)$ holds under Assumption (ref). Let $r_0(X) \coloneqq \frac{\zeta_0(X)}{\zeta_{{\mathrm{T}}, 0}(X)}$ be the density ratio between the covariate densities.
Example of ATE estimators in the censoring setting
In this section, as a preliminary, we suggest a simple ATE estimator in the censoring. In Appendix (ref), we show the detailed result of the estimator and also propose a simple ATE estimator in the case-control setting. Note that as discussed in the subsequent subsections, the estimators are not efficient, i.e., there exist ATE estimators whose asymptotic variance is smaller.
The Inverse probability weighting estimator
We first consider the censoring setting. We begin by the arguments from the estimation of the propensity score. To estimate the propensity score, we employ the result in PU learning. Elkan2008learningclassifiers addresses PU learning in the censoring setting. In that work, they show that under the Selected Completely At Random (SCAR) assumption defined below Elkan2008learningclassifiers. This assumption is analogy of the Missing Completely A Random assumption (MCAR), which is common with the
missing data literature Little2002statisticalanalysis,Rubin1974estimatingcausal,Bekker2020learningfrom. Several works such as Bekker2018learningfrom attempt to relax the SCAR assumption, but to relax the assumption, we usually require additional assumptions.
assumption[SCAR]
It holds that ${\mathbb{P}}(D = 1, O = d \mid X) = {\mathbb{P}}(D = 1 \mid X) {\mathbb{P}}(O = d \mid D = 1)$.
From the assumption, we have $\pi_0(1\mid x) = {\mathbb{P}}(D = 1 \mid X) {\mathbb{P}}(O = d \mid D = 1)$ since ${\mathbb{P}}(D = 0, O = d\mid X) = 0$ holds by definition of the DGP.
Thus, under this assumption, the propensity scores $g_0(d \mid x)$ and $e_0(d \mid x)$ can be estimated using PU learning methods duPlessis2015convexformulation,Elkan2008learningclassifiers. If we know their true values, such assumptions are unnecessary. Note that the censoring PU learning studies aim to estimate ${\mathbb{P}}(D = d\mid X)$ not $g_0(d \mid X) = {\mathbb{P}}(D = d\mid X, O = 0)$. However, once we obtain an estimate of ${\mathbb{P}}(D = 1\mid X)$ and $\pi_0(0\mid X)$, we can obtain $g_0(1\mid X)$ from $g_0(1\mid X) = {\mathbb{P}}(O = 0 \mid D = 1){\mathbb{P}}(D = 1\mid X) / \pi_0(0 \mid X)$.
We further make the following unconfoundedness and common support assumptions, which are common in ATE estimation.
assumption[Unconfoundedness in the censoring setting]
The potential outcomes $(Y(1), Y(0))$ are independent of treatment assignment given covariates: $(Y(1), Y(0)) \rotatebox[origin=c]{90}{$\models$} (O, D) \mid X$.
assumption[Common support in the censoring setting]
There exists a constant $c$ independent of $n$ such that for all $x \in {\mathcal{X}}$, $\pi_0(d\mid x), g_0(d\mid x), \zeta_{{\mathrm{T}}, 0}(x), \zeta_0(x) > c$ hold.
Under these assumptions, the ATE $\tau_0$ is estimable by replacing the following two quantities with sample approximation: ${\mathbb{E}}[Y(1)] = {\mathbb{E}}\left[\frac{\mathbbm{1}[O = 1]Y}{\pi_0(1 \mid X)}\right]$ and ${\mathbb{E}}[Y(0)] = {\mathbb{E}}\left[\frac{\mathbbm{1}[O = 0]Y}{g_0(0\mid X)\pi_0(0 \mid X)}\right] - {\mathbb{E}}\left[\frac{g_0(1\mid X)\mathbbm{1}[O = 1]Y}{g_0(0\mid X)\pi_0(1 \mid X)}\right]$,
where $g_0$ and $\pi_0$ can be estimated using observations, and expectations can be approximated by sample averages. Such an estimator is a variant of the inverse probability weighting (IPW) estimator and shown in Remark (ref).
Toward efficient estimators
Thus, we can estimate the ATE in the censoring setting (and in the case-control setting, as shown in Appendix (ref)). However, it is unclear whether it is efficient; that is, the (asymptotic) variance is sufficiently small. In the subsequent subsections, we investigate efficient estimators and develop efficiency bounds, which work as a lower bound for the regular estimators. The efficiency bound also suggest the construction of efficient estimators, and we actually propose an estimator whose asymptotic variance aligns with the efficiency bound.
Semiparametric efficient ATE estimation under the censoring setting
This section presents a method for ATE estimation under the censoring setting. First, we derive the efficiency bound in Section (ref). Then, we propose our estimator in Section (ref) and show the asymptotic normality in Section (ref). Finally, in Section (ref), we discuss issues related to the estimation of the propensity score.
Efficient influence function and efficiency
First, we derive the efficiency bound for regular estimators, which provides a lower bound on asymptotic variances. The efficiency bound is characterized via the efficient influence function Vaart1998asymptoticstatistics, derived as follows (Proof is provided in Appendix (ref)):
lemmaIf Assumptions (ref)--(ref), then the efficient influence function is given as $\Psi^{\mathrm{cens}}(X, O, Y; \mu_{{\mathrm{T}}, 0}, \nu_0, \pi_0, g_0, \tau_0)$, where
\begin{align*}
&\Psi^{\mathrm{cens}}(X, O, Y; \mu_{{\mathrm{T}}}, \nu_0, \pi_0, g_0, \tau_0) \coloneqq S^{\mathrm{cens}}(X, O, Y; \mu_{{\mathrm{T}}, 0}, \nu_0, \pi_0, g_0) - \tau_0,\\
&S^{\mathrm{cens}}(X, O, Y; \mu_{{\mathrm{T}}, 0}, \nu_0, \pi_0, g_0) \coloneqq \frac{\mathbbm{1}[O = 1]\big(Y - \mu_{{\mathrm{T}}, 0}(X)\big)}{\pi_0(1\mid X)} - \frac{\mathbbm{1}[O = 0]\big(Y - \nu_0(X)\big)}{g_0(0\mid X)\pi_0(0\mid X)}\\
&\ \ \ +\frac{g_0(1\mid X)\mathbbm{1}[O = 1]\big(Y - \mu_{{\mathrm{T}}, 0}(X)\big)}{g_0(0\mid X)\pi_0(1\mid X)} + \mu_{{\mathrm{T}}, 0}(X) - \frac{1}{g_0(0\mid X)}\nu_0(X) + \frac{g_0(1\mid X)}{g_0(0\mid X)}\mu_{{\mathrm{T}}, 0}(X).
\end{align*}
Here, note that the efficient influence function depends on unknown $\mu_{{\mathrm{T}}, 0}, \nu_0, \pi_0, g_0$, which are referred to as nuisance parameters. Since the efficient influence function satisfies the equation ${\mathbb{E}}\left[\Psi^{\mathrm{cens}}(X, O, Y; \mu_{{\mathrm{T}}, 0}, \nu_0, \pi_0, g_0, \tau_0)\right] = 0$, if the nuisance parameters are known and the exact expectation is computed, we can obtain $\tau_0$ by solving for $\tau_0$ that satisfies ${\mathbb{E}}\left[\Psi^{\mathrm{cens}}(X, O, Y; \mu_{{\mathrm{T}}, 0}, \nu_0, \pi_0, g_0, \tau_0)\right] = 0$. Thus, the efficient influence function provides significant insights for constructing an efficient estimator. Furthermore, the accuracy of the estimation of the nuisance parameters affects the estimation of $\tau_0$, the parameter of interest.
From Theorem 25.20 in Vaart1998asymptoticstatistics, Lemma (ref) yields the following result about the efficiency bound.
theorem[Efficiency bound in the censoring setting]
If Assumptions (ref)--(ref), then the asymptotic variance of any regular estimator is lower bounded by
\begin{align*}
V^{\mathrm{cens}} &\coloneqq {\mathbb{E}}\left[\Psi^{\mathrm{cens}}(X, O, Y; \mu_0, \nu_0, \pi_0, g_0)^2\right]\\
&= {\mathbb{E}}\left[\left(1 - \frac{g_0(1\mid X)}{g_0(0\mid X)}\right)^2\frac{\mathrm{Var}(Y(1)\mid X)}{\pi_0(1\mid X)} + \frac{\mathrm{Var}(\widetilde{Y}\mid X)}{g_0(0\mid X)^2\pi_0(1\mid X)} + \Big(\tau_0(X) - \tau_0\Big)^2\right].
\end{align*}
We say that an estimator is efficient if its asymptotic variance aligns with $V^{\mathrm{cens}}$.
Semiparametric efficient estimator
Based on the efficient influence function, we propose an ATE estimator defined as $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n \coloneqq \frac{1}{n}\sum^n_{i=1}S^{\mathrm{cens}}(X_i, O_i, Y_i; \widehat{\mu}_{{\mathrm{T}}, n, i}, \widehat{\nu}_{n, i}, \widehat{\pi}_{n, i}, \widehat{g}_{n, i})$,
where $\widehat{\mu}_{{\mathrm{T}}, n, i}$, $\widehat{\nu}_{n, i}$, $\widehat{\pi}_{n, i}$ and $\widehat{g}_{n, i}$ are estimators of $\mu_{{\mathrm{T}}, 0}$, $\nu_0$, $\pi_0$, and $g_0$. Note that the estimators can depend on $i$.
This estimator is an extension of the augmented inverse probability weighting estimator, also called a doubly robust estimator Bang2005doublyrobust.
remark[Estimation equation]
There exist several intuitive explanations for $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$. One of the typical explanations is the one from the viewpoint of the estimation equation.
Given $\widehat{\mu}_{{\mathrm{T}}, n, i}$, $\widehat{\nu}_{n, i}$, $\widehat{\pi}_{n, i}$, and $\widehat{g}_{n, i}$, the estimator $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$ is obtained by solving the following equation: $\frac{1}{n}\sum^n_{i=1}\Psi^{\mathrm{cens}}(X_i, O_i, Y_i; \widehat{\mu}_{{\mathrm{T}}, n, i}, \widehat{\nu}_{n, i}, \widehat{\pi}_{n, i}, \widehat{g}_{n, i}, \widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n) = 0$.
Such a derivation of the efficient estimator as the estimation equation approach Schuler2024introductionmodern.
Consistency and double robustness
First, we prove the consistency result; that is, $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n \xrightarrow{{\mathrm{p}}} \tau_0$ holds as $n\to \infty$. We can obtain this result relatively easily compared to the asymptotic normality. We make the following assumption that holds for most estimators of the nuisance parameters.
assumptionAs $n \to \infty$, $\big\|\widehat{g}_{n, i} - g_0\big\|_2 = o_p(1)$ holds. Additionally, either of the followings holds for all $i\in \{1,2,\dots,n\}$:
\begin{itemize}[noitemsep, topsep=0pt, leftmargin=0.40cm]
• $\big\|\widehat{\pi}_{n, i} - \pi_0\big\|_2 = o_p(1)$.$\ \ \ \ \ \ \ $ {\labelitemi}$\ $ $\big\|\widehat{\nu}_{n, i} - \nu_0\big\|_2 = o_p(1)$ and $\big\|\widehat{\mu}_{{\mathrm{T}}, n, i} - \mu_{{\mathrm{T}}, 0}\big\|_2 = o_p(1)$.
\end{itemize}
For the estimation of the censoring propensity score, we can employ the existing PU learning methods in the censoring setting, such as Elkan2008learningclassifiers. Note that we can also apply methods for the case-control PU learning, such as duPlessis2015convexformulation, since, as the classification problem, the case-control setting is more general than the censoring setting Niu2016theoreticalcomparisons. Note that the case-control PU learning methods typically require the class prior ${\mathbb{P}}(D = 1)$, which can be estimated under several additional assumptions, even if we do not know it duPlessis2014classprior,Ramaswamy2016mixtureproportion,Kato2018alternateestimation.
Then, the following consistency result holds.
theorem[Consistency in the censoring setting]
If Assumptions (ref)--(ref), and (ref) hold, then $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n \xrightarrow{{\mathrm{p}}} \tau_0$ holds as $n\to \infty$.
Double robustness. There exists double-robustness structure such that given $\big\|\widehat{g}_{n, i} - g_0\big\|_2 = o_p(1)$, if either $\big\|\widehat{\pi}_{n, i} - \pi_0\big\|_2 = o_p(1)$ or $\big\|\widehat{\nu}_{n, i} - \nu_0\big\|_2 = o_p(1)$ and $\big\|\widehat{\mu}_{{\mathrm{T}}, n, i} - \mu_{{\mathrm{T}}, 0}\big\|_2 = o_p(1)$ holds, then $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n \xrightarrow{{\mathrm{p}}} \tau_0$ holds. Here, note that we need to estimate the propensity score consistently to estimate the ATE and the double robustness holds between the estimators of the observation probability $\pi_0$ and the expected outcomes $\mu_{{\mathrm{T}}, 0}$ and $\nu_0$.\footnote{In the standard setting, the double robustness holds between the estimators of the propensity score and the expected outcome.} This is because, in our setting, the treatment indicator is unobservable. Under this setting, to identify the ATE, we need to use the propensity score and cannot avoid its estimation.
Asymptotic normality
Next, we prove the asymptotic normality. Unlike consistency, we need to make a stronger assumption on the nuisance estimators, especially for the propensity score.
To establish the asymptotic normality or $\sqrt{n}$-consistency of the estimator, it is necessary to control the complexity of the estimators of the nuisance parameter. One of the simplest approaches is to assume the Donsker condition, but it is well known that the Donsker condition does not hold in several cases, such as high-dimensional regression settings. In such cases, asymptotic normality can still be established through sample splitting, a technique in this field Klaassen1987consistentestimation, which has been recently refined by Chernozhukov2018doubledebiased as cross-fitting.
Cross-fitting.
Cross-fitting is a variant of sample splitting Chernozhukov2018doubledebiased. We randomly partition ${\mathcal{D}}$ into $L > 0$ folds (subsamples), and for each fold $\ell \in {\mathcal{L}} \coloneqq \{1,2,\dots, L\}$, the nuisance parameters are estimated using all other folds. We estimate $\mu_{{\mathrm{T}}, 0}, \nu_0, \pi_0$, assuming that the propensity score $g_0$ is known. Let us denote the estimators in fold $\ell \in {\mathcal{L}}$ as $\widehat{\mu}^{(\ell)}_{{\mathrm{T}}, n}, \widehat{\nu}^{(\ell)}_n, \widehat{\pi}^{(\ell)}_n$. Let ${\mathcal{I}}^{(\ell)}$ be the set of the sample index belonging to fold $\ell$.
Various estimation methods and models can be employed, including neural networks and Lasso, provided they satisfy the convergence rate conditions specified in Assumption (ref). We later relax the assumption of a known propensity score. It is important to note that issues related to the propensity score estimation cannot be fully addressed even with cross-fitting. The pseudocode is in Algorithm (ref).
algorithm[algorithm omitted — 1,154 chars of source]
Asymptotic normality.
We describe the results only for the case with cross-fitting, but similar results hold for the case when we assume the Donsker condition.
We make the following assumptions.
assumptionThe propensity score $g_0$ is known ($\widehat{g}_{n, i} = g_0$).
assumptionFor each $\ell \in {\mathcal{L}}$, as $n\to\infty$, the followings hold:
\begin{itemize}[noitemsep, topsep=0pt, leftmargin=0.40cm]
• $\big\|\pi_0(d\mid X) - \widehat{\pi}^{(\ell)}_n(d\mid X)\big\|_2 = o_p(1)$ for $d\in\{1, 0\}$, $\big\| \mu_{{\mathrm{T}}, 0}(X) - \widehat{\mu}^{(\ell)}_{{\mathrm{T}}, n}(X) \big\|_2 = o_p(1)$, and $\big\| \nu_{0}(X) - \widehat{\nu}^{(\ell)}_{n}(X) \big\|_2 = o_p(1)$.
• $\big\|\pi_0(d\mid X) - \widehat{\pi}^{(\ell)}_n(d\mid X)\big\|_2\big\| \mu_{{\mathrm{T}}, 0}(X) - \widehat{\mu}^{(\ell)}_{{\mathrm{T}}, n}(X) \big\|_2 = o_p(n^{-1/2})$ for $d \in \{1, 0\}$.
• $\big\|\pi_0(0\mid X) - \widehat{\pi}^{(\ell)}_n(0\mid X)\big\|_2\big\| \nu_{0}(X) - \widehat{\nu}^{(\ell)}_{n}(X) \big\|_2 = o_p(n^{-1/2})$.
\end{itemize}
Then, we construct the estimator as
$\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n \coloneqq \frac{1}{n}\sum_{\ell \in {\mathcal{L}}}\sum_{i \in {\mathcal{I}}^{(\ell)}}S^{\mathrm{cens}}(X_i, O_i, Y_i; \widehat{\mu}^{(\ell)}_{{\mathrm{T}}, n}, \widehat{\nu}^{(\ell)}_n, \widehat{\pi}^{(\ell)}_n, g_0)$ and show the asymptotic normality holds as follows:
theorem[Asymptotic normality in the censoring setting]
Consider the censoring setting. Suppose that Assumptions (ref)--(ref), (ref), and (ref) hold; that is, $\widehat{g}_{n, i} = g_0$, and $\widehat{\mu}_{{\mathrm{T}}, n, i} = \widehat{\mu}^{(\ell)}_{{\mathrm{T}}, n}$, $\widehat{\nu}_{n, i} = \widehat{\nu}^{(\ell)}_n$, and $\widehat{\pi}_{n, i} = \widehat{\pi}^{(\ell)}_n$ are constructed via cross-fitting with certain convergence rates. Then, we have
\begin{align*}
\sqrt{n}\left(\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n - \tau_0\right) \xrightarrow{{\mathrm{d}}} \mathcal{N}(0, V^{\mathrm{cens}})\ \ as\ \ n\to \infty.
\end{align*}
The proof is provided in Appendix (ref).
The asymptotic variance of $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$ matches the efficiency bound. Therefore, Theorem (ref) also implies that the estimator $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$ is asymptotically efficient.
We discuss the other candidates of ATE estimators below.
remark[Inefficiency of the Inverse Probability Weighting (IPW) estimator]
We can define the IPW estimator as
$\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{IPW}}_n \coloneqq \frac{1}{n}\sum^n_{i=1}\left(\frac{\mathbbm{1}[O_i = 1]Y_i}{\widehat{\pi}_{n, i}(1\mid X_i)} - \frac{\mathbbm{1}[O_i = 0]Y_i}{\widehat{g}_{n, i}(0\mid X_i)\widehat{\pi}_{n, i}(0\mid X_i)} +\frac{\widehat{g}_{n, i}(1\mid X_i)\mathbbm{1}[O_i = 1]Y_i}{\widehat{g}_{n, i}(0\mid X_i)\widehat{\pi}_{n, i}(1\mid X_i)}\right)$.
Compared to our proposed efficient estimator, this estimator does not use the conditional outcome estimators Horvitz1952generalization. If $g_0$ and $\pi_0$ are known, this estimator is unbiased. However, it incurs a large asymptotic variance, given as $V^{\mathrm{IPW}} \coloneqq {\mathbb{E}}\Big[\left(1 - \frac{g_0(1\mid X)}{g_0(0\mid X)}\right)^2\frac{{\mathbb{E}}[Y(1)^2\mid X]}{\pi_0(1\mid X)}+ \frac{{\mathbb{E}}[\widetilde{Y}^2\mid X]}{g_0(0\mid X)^2\pi_0(1\mid X)}\Big]$.
Here, it holds that $V^{\mathrm{IPW}} \geq V^{\mathrm{cens}}$, where the equality holds when $\mu_{{\mathrm{T}}, 0}(x) = 0$ and $\nu_0 = 0$ hold for all $x$. Thus, the IPW estimator is inefficient compared to $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$. Additionally, if $\pi_0$ is unknown, the IPW estimator requires more restrictive conditions for the asymptotic normality than $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$.
remark[Direct Method (DM) estimator]
Another candidate is a DM estimator, defined as
$
\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{IPW}}_n \coloneqq \widehat{\mu}_{{\mathrm{T}}, n, i}(X) - \frac{1}{\widehat{g}_{n, i}(0\mid X)}\nu(X) + \frac{\widehat{g}_{n, i}(1\mid X)}{\widehat{g}_{n, i}(0\mid X)}\widehat{\mu}_{{\mathrm{T}}, n, i}(X)$,
which is also referred to as a naive plug-in estimator. The asymptotic normality significantly depends on the properties of the estimators $\widehat{\mu}_{{\mathrm{T}}, n, i}$ and $\widehat{g}_{n, i}$. Additionally, the DM method is known to be sensitive to model misspecification.
Unknown propensity score
We have assumed that the propensity score $g_0$ is known. This is because we cannot establish $\sqrt{n}$-consistency even if we assume the Donsker condition or employ cross-fitting if $g_0$ is estimated. However, this assumption can be relaxed by utilizing an additional dataset to estimate $g_0$.
Several practical scenarios exist. For instance, consider that the following additional dataset is available:
${\mathcal{D}}^{\mathrm{aux}} \coloneqq \left\{\big(X_{i'}, O_{i'}\big)\right\}^{n^{\mathrm{aux}}}_{i'=1},\ \big(X_{i'}, O_{i'}\big) \sim \zeta_0(x)\pi_0(o\mid x)$.
Such a dataset can be less costly since it does not have the outcome data.
Let $\widehat{g}_{n^{\mathrm{aux}}}$ be an estimator obtained from ${\mathcal{D}}^{\mathrm{aux}}$ and consider the following assumption:
assumptionIt holds that $\|\widehat{g}_{n^{\mathrm{aux}}} - g_0\|_2 = o_p(1)$ as $n^{\mathrm{aux}} \to \infty$.
If $n^{\mathrm{aux}}$ approaches infinity independently of $n$, under Assumption (ref), we can establish the asymptotic normality without assuming the propensity score is known.
corollary[Asymptotic normality in the censoring setting]
Consider the censoring setting. Suppose that Assumptions (ref)--(ref), (ref), and (ref) hold. Then, it holds that $\sqrt{n}\left(\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n - \tau_0\right) \xrightarrow{{\mathrm{d}}} \mathcal{N}(0, V^{\mathrm{cens}})$ as $n\to \infty$.
We can also use $\left\{\left(X_i, O_i\right)\right\}^n_{i=1}$ from ${\mathcal{D}}$ to estimate $g_0$ with ${\mathcal{D}}^{\mathrm{aux}}$. The inclusion of $\left\{\left(X_i, O_i\right)\right\}^n_{i=1}$ can improve empirical performance.
Another practical scenario involves an auxiliary dataset with treatment indicators and missing outcomes, given as ${\mathcal{D}}^{\mathrm{aux}'} \coloneqq \big\{\big(X_{i'}, D_{i'}\big)\big\}^{n^{\mathrm{aux}}}_{i'=1},\ \big(X_{i'}, D_{i'}\big) \sim \zeta_0(x)g_0(o\mid x)$.
table[table omitted — 1,277 chars of source]
figure[figure omitted — 238 chars of source]
Semiparametric efficient ATE estimation under the case-control setting
Here, we briefly introduce the ATE estimator in the case-control setting. More detailed results are shown in Appendix (ref).
We define
align*[align* omitted — 644 chars of source]
as an ATE estimator in the case-control setting.
Here, $\widehat{\mu}_{{\mathrm{T}}, n, i}$, $\widehat{\mu}_{{\mathrm{U}}, n, i}$, $\widehat{e}_{n, i}$, and $\widehat{r}_{n, i}$ are estimators of $\mu_{{\mathrm{T}}, 0}$, $\mu_{{\mathrm{U}}, 0}$, $e_0$, and $r_0$, where $m$ and $l$ denote the dependence on each dataset.
For the estimator, we show the following theorem, which is an informal version of Theorem (ref)
theorem[Asymptotic normality in the case-control setting (Informal)]
Fix $\alpha \in (0, 1)$. For $n > 0$,
consider the case-control setting with sample sizes $m, l$ such that $m = \alpha n$ and $l = (1-\alpha)n$. If the case-control propensity score $e_0$ and the density ratio are known ($r_0$ $\widehat{e}_{n, i} = e_0$ and $\widehat{r}_{n, i} = r_0$), and $\widehat{\mu}_{{\mathrm{T}}, n, i} = \widehat{\mu}^{(\ell)}_{{\mathrm{T}}, m}$, $\widehat{\mu}_{{\mathrm{U}}, n, i} = \widehat{\mu}^{(\ell)}_{{\mathrm{U}}, l}$ are consistent estimators constructed via cross-fitting. Then, under regularity conditions (see Theorem (ref)), we have $\sqrt{n}\left(\widehat{\tau}^{\mathrm{cc}\mathchar`-\mathrm{eff}}_{n} - \tau_0\right) \xrightarrow{{\mathrm{d}}} \mathcal{N}(0, V^{\mathrm{cc}})$ as $n\to \infty$, where $V^{\mathrm{cc}} > 0$ is the efficiency bound defined in Theorem (ref).
Simulation studies
This section investigates the empirical performance of the proposed estimators. We also show the experimental results using semi-synthetic data in Appendix (ref).
Censoring setting
We generate synthetic data under the censoring setting, where the covariates $X$ are drawn from a multivariate normal distribution as $X \sim \zeta_0(x)$,
where $\zeta_0(x)$ is the density of $\mathcal{N}(0, I_p)$, and $I_p$ denotes the $(p\times p)$ identity matrix. We set $p = 3$. Set ${\mathbb{P}}(D\mid X) = \mathrm{trunc}(\mathrm{sigmoid}(X^\top \beta), 0.1, 0.9)$,
where $\beta$ is a coefficient sampled from $\mathcal{N}(0, 0.5I_{p})$, and $\mathrm{trunc}(t, a, b)$ truncates $t$ by $a$ and $b$ ($a < b$). Treatment $D$ is sampled from the probability. The observation indicator $O$ is generated from a Bernoulli distribution with probability $c$ if $D_i = 1$ and $O_i = 0$ if $D_i = 0$. Here, $c$ is generated from a uniform distribution with support $[0, 1]$.
The outcome is generated as $Y = X^\top \beta + 1.1 + \tau_0 \cdot D + \varepsilon$,
where $\varepsilon \sim \mathcal{N}(0,1)$, where we set $\tau_0 = 3$.
The nuisance parameters are estimated using linear regression and (linear) logistic regression. We compared our proposed estimator, $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$, with the other candidates, the IPW estimator $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{IPW}}_n$ and the DM estimator $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{DM}}_n$, defined in Remarks (ref) and (ref), respectively. Note that all of these estimators are proposed by us, and our goal is not to confirm $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$ outperforms the others, while our recommendation is $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$. We consider both cases where the propensity score is either estimated using the method proposed by Elkan2008learningclassifiers or assumed to be known.
We set $n = 3000$.
We conduct $5000$ trials and report the empirical mean squared errors (MSEs) and biases for the true ATE and the coverage ratio (Cov. ratio) computed from the confidence intervals in Table (ref). We also present the empirical distributions of the ATE estimates in Figure (ref).
As the theory suggests, $\widehat{\tau}^{\mathrm{cens}\mathchar`-\mathrm{eff}}_n$ exhibits smaller MSEs compared to other methods. Interestingly, when the propensity score is estimated, the MSEs decrease, a phenomenon reported in existing studies. The coverage ratio is also accurate. The empirical distribution of the ATE estimates demonstrates the asymptotic normality.
Case-control setting
In the case-control setting, covariates for the treatment and unknown groups are generated from different $p$-dimensional normal distributions: $X_{{\mathrm{T}}} \sim \zeta_{{\mathrm{T}}, 0}(x)$ and $X \sim \zeta_0(x) = e_0(1)\zeta_{{\mathrm{T}}, 0}(x) + e_0(0)\zeta_{{\mathrm{C}}}(x)$, where we set $p=3$, $\zeta_{{\mathrm{T}}, 0}(x)$ and $\zeta_{{\mathrm{C}}}(x)$ are the densities of normal distributions $\mathcal{N}(\mu_p\bm{1}_p, I_p)$ and $\mathcal{N}(\mu_n\bm{1}_p, I_p)$, $\mu_p = 0.5$ and $\mu_n = 0$, $\bm{1}_p = (1\ 1\ \cdots\ 1)^\top$, and $e_0(1)$ is the class prior set as $e_0(1) = 0.3$. By definition, the propensity score $e_0(d\mid x)$ is given as $e_0(1\mid x) = e_0(1) \zeta_{{\mathrm{T}}, 0}(x) / \zeta_0(x)$. The outcome is generated similarly to the censoring setting $Y = X^\top \beta + 1.1 + \tau_0 D + \varepsilon$, where $\tau_0 = 3$.
We set $m = 1000$ and $l = 2000$ and compute the same evaluation metrics as in the censoring setting. Although logistic regression is used, the propensity score model is misspecified, while the expected conditional outcome follows a linear model.
Overall, $\widehat{\tau}^{\mathrm{cc}\mathchar`-\mathrm{eff}}_n$ demonstrates robust performance in terms of MSE, bias, and coverage ratio. The poor performance of the IPW estimator is attributed to model misspecification.
We investigate non-linear settings in Appendix (ref).
Conclusion
In this study, we investigated PUATE, the problem of ATE estimation in the presence of missing treatment indicators. We formulated the problem using the censoring and case-control settings, inspired by PU learning. For each setting, we derived the efficiency bound and developed an efficient estimator. Our analysis revealed that achieving asymptotic normality and efficiency. Future research directions include extending our approach to the semi-supervised setting, handling additional missing values, and the relaxation of assumptions regarding the missingness mechanism.
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\onecolumn