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Data Fusion for Partial Identification of Causal Effects
abstractData fusion techniques integrate information from heterogeneous data sources to improve learning, generalization, and decision-making across data sciences. In causal inference, these methods leverage rich observational data to improve causal effect estimation, while maintaining the trustworthiness of randomized controlled trials. Existing approaches often relax the strong "no unobserved confounding" assumption by instead assuming exchangeability of counterfactual outcomes across data sources. However, when both assumptions simultaneously fail---a common scenario in practice---current methods cannot identify or estimate causal effects. We address this limitation by proposing a novel partial identification framework that enables researchers to answer key questions such as: Is the causal effect positive/negative? and How severe must assumption violations be to overturn this conclusion? Our approach introduces interpretable sensitivity parameters that quantify assumption violations and derives corresponding causal effect bounds. We develop doubly robust estimators for these bounds and operationalize breakdown frontier analysis to understand how causal conclusions change as assumption violations increase. We apply our framework to the Project STAR study, which investigates the effect of classroom size on students’ third-grade standardized test performance. Our analysis reveals that the Project STAR results are robust to simultaneous violations of key assumptions, both on average and across various subgroups of interest. This strengthens confidence in the study's conclusions despite potential unmeasured biases in the data.
Introduction
Modern evidence-based decision-making increasingly relies on combining information from various sources -- a practice known as data fusion. From integrating satellite imagery across multiple spatial resolutions tang2016combined,yan2021large to merging genetic markers with electronic health records hall2016merging,conroy2023uk,zawistowski2023michigan, data fusion enables more robust, generalizable, and efficient analysis meng2020survey. In causal inference, data fusion has emerged as a popular paradigm, recently recognized as one of the top ten research directions for advancing the field mitra2022future,bareinboim2016causal. Data fusion approaches in causal inference have focused on generalizing or transporting evidence from experimental studies degtiar2023review,pearl2015generalizing,dahabreh2019generalizing,lu2019causal, precisely estimating heterogeneous causal effects brantner2023methods,yang2023elastic, improving efficiency rosenman2023combining,lin2025combining, and mitigating estimation bias kallus2018removing,colnet2024causal, among other things.
Consider the example of the Project STAR study, which investigated the impact of class size on students’ academic performance. The dataset comprises both a randomized controlled trial (RCT) -- where students were randomly assigned to different classroom sizes -- and an observational cohort where students self-selected into classrooms mosteller1995tennessee, project_star_data. While the experimental data likely ensures internal validity, it may suffer from limited external validity or generalizability von2018does,justman2018randomized. Conversely, the observational data better reflects real-world settings but may suffer from unobserved confounding athey2020combining,parikh2023double. Merging the two sources can yield more precise and externally valid treatment effect estimates under milder, partially testable assumptions parikh2023double,wu2022integrative. Specifically, the average treatment effect (ATE) becomes identifiable if either the RCT generalizes well to the target population or the observational data satisfies no unmeasured confounding (NUC) lin2025combining,yang2023elastic.
However, if both assumptions fail -- i.e., the RCT lacks external validity and the observational data is confounded -- then the treatment effect is no longer point-identifiable, even in the limit of infinite data. In such settings, classical estimators break down. Nevertheless, researchers may still answer important questions like: Is the treatment effect positive? or How severe must assumption violations be to overturn this conclusion? These are questions of partial identification, where the goal is to estimate a plausible range or bounds on the treatment effect rather than a single point estimate cornfield1959smoking,manski2003partial.
While a rich literature on partial identification exists, many approaches rely on strong distributional or parametric assumptions and often ignore opportunities to tighten bounds by leveraging multiple datasets rosenbaum1983assessing,blackwell2014selection,ding2016sensitivity,bonvini2022sensitivity,nguyen2017sensitivity, nguyen2018sensitivity, nie2021covariate, colnet2022causal, dahabreh2023sensitivity,huang2022sensitivity. This creates a critical gap in sensitivity analysis frameworks that are both flexible and informative when combining experimental and observational data.
Contributions. We propose a general framework for partially identifying treatment effects by integrating complementary strengths of experimental and observational studies. Our key contributions are:
enumerate[leftmargin=*]
• We introduce interpretable sensitivity parameters, $\gamma$ and $\rho$, that quantify the extent of external validity violations and unmeasured confounding, respectively.
• We develop a double machine learning estimator based on the efficient influence function (EIF) for estimating treatment effect bounds as a function of $(\gamma, \rho)$, without relying on strong distributional or parametric assumptions.
• We operationalize an efficient breakdown frontier analysis, which characterizes regions in the $(\gamma, \rho)$ space where the treatment effect remains conclusively positive (or negative) -- allowing for assessment of the robustness of causal conclusions under simultaneous assumption violations.
Our framework enables comprehensive sensitivity analyses in data fusion settings, helping researchers transparently explore the consequences of assumption violations. By leveraging the internal validity of RCTs and the representativeness of observational studies, our approach yields tighter, more interpretable bounds on treatment effects and offers a principled way to assess robustness.
Preliminaries
We consider the setting where we have a sample $\mathcal{D}_n = \{1, \dots, n\}$ of $n$ units across an experimental cohort ($\mathcal{D}_e$) and an observational study ($\mathcal{D}_o$) drawn identically and independently from $\mathcal{P}$. For each unit $i \in \mathcal{D}_n = \mathcal{D}_e \cup \mathcal{D}_o$, $S_i = \mathbbm{1}[i \in \mathcal{D}_e]$ is a binary experimental cohort indicator, $T_i \in \{0,1\}$ is the binary treatment indicator, $Y_i$ is the observed outcome, and $\mathbf{X}_i$ is the vector of pretreatment covariates. We assume the outcome space is bounded and positive, but note that, without loss of generality, our approach can be applied to all bounded outcome scenarios by simply shifting the outcome domain. $Y_i(0)$ and $Y_i(1)$ denote the two potential outcomes for unit $i$. We assume the stable unit treatment value assumption (SUTVA), which ensures no interference between units and a single version of each treatment, as well as the consistency assumption, so that $Y_i = T_i Y_i(1) + (1 - T_i) Y_i(0)$.
Typically, one is interested in using these datasets to estimate the following two standard estimands, namely the average treatment effect (ATE): $\tau = \mathbb{E}_{\mathcal{P}}[Y(1) - Y(0)],$ and the conditional average treatment effect (CATE): $\tau(\mathbf{x}) = \mathbb{E}_{\mathcal{P}}[Y(1) - Y(0) \mid \mathbf{X}=\mathbf{x}].$
We assume the following standard conditions hold:
assumption(Treatment Positivity). For $s\in\{0,1\}$ and all $\mathbf{x}$, $\exists c > 0$ such that\newline
$c < P(T = 1 \mid \mathbf{X} = \mathbf{x}, S=s) < 1-c$.
assumption(Study Positivity). For all $\mathbf{x}$, $\exists c > 0$ such that $c < P(S = 1 \mid \mathbf{X} = \mathbf{x}) \leq 1$
assumption(Internal Validity of the Experiment). $(Y(0), Y(1)) \perp\!\!\!\!\perp T \mid \mathbf{X} = \mathbf{x}, S = 1$
However, we acknowledge the possibility of unobserved confounders that concurrently influence $S$, $T$, and $Y$. Due to such unobserved confounding, the following exchangeability assumptions, which are standard in the literature, may fail to hold:
assumption(No Unobserved Confounding (NUC) in the Observational Data).\newline
$(Y(0), Y(1)) \perp\!\!\!\!\perp T \mid \mathbf{X} = \mathbf{x}, S = 0$
assumption(Study Exchangeability). $(Y(0), Y(1)) \perp\!\!\!\!\perp S \mid \mathbf{X} = \mathbf{x}$
In this paper, we explicitly consider scenarios in which (ref) and (ref) assumptions are simultaneously violated, thereby challenging the point identifiability of $\tau$ and $\tau(\mathbf{x})$.
Discussion of Assumptions. (ref) is the standard treatment positivity assumption, ensuring overlap between treated and control groups. (ref) and (ref) are structurally equivalent, differing only in the sample subset (experimental vs. observational units). Internal validity in RCTs is generally accepted due to randomization, whereas NUC is stronger, as treatment may depend on unobserved confounders. Combine experimental and observational samples requires the additional (ref) and (ref) assumptions. (ref) states that each unit must have a nonzero probability of being an experimental unit and is necessary to ensure overlap between the two study cohorts. (ref) is the study exchangeability assumption, which states that, conditional on covariates, potential outcomes are exchangeable across studies. Like NUC, it can be a strong assumption, as study participation may depend on unobservables.
Quantifying Assumption Violations
We introduce two additional terms, $\rho$ and $\gamma$, that separately quantify violations of (ref) and (ref), respectively. The value of $\rho \geq 0$ quantifies the level of unobserved confounding in the observational data, corresponding to a violation of (ref) when $\rho > 0$. The value of $\gamma \geq 0$ quantifies the difference in potential outcomes between the RCT data and the observational data, corresponding to a violation of (ref) when $\gamma > 0$. Both terms report the level of violation as relative measures of the observed outcomes. For example, $\rho=0.2$ corresponds to the setting that unobserved confounding in the observational dataset affects outcomes by 20%. In Section (ref), we formally define $\rho$ and $\gamma$, expand on their interpretation, and employ them as sensitivity parameters for partial identification of $\tau$ and $\tau(\mathbf{x})$.
Breakdown Frontiers
masten2020inference introduce an approach for visualizing how conclusions about a parameter of interest vary as a set of assumptions are relaxed. We leverage this framework to plot how treatment effect conclusions change as we relax (ref) and (ref) via our sensitivity parameters $\rho$ and $\gamma$.
\Needspace{16\baselineskip}
wrapfigure[wrapfigure omitted — 243 chars of source]
Figure (ref) illustrates a breakdown frontier plot constructed by bounding the treatment effect for various $(\rho, \gamma)$ pairs. The x-axis represents violations of (ref), expressed as percentages corresponding to $(100\times\rho)$%. Likewise, the y-axis represents violations of (ref), expressed as $(100\times\gamma)$%. As a result, the bottom-left of the plot corresponds to stronger assumptions (small $\rho, \gamma$), and the top-right to weaker assumptions (large $\rho, \gamma$). For each ($\rho, \gamma$) pair, we estimate upper and lower bounds on the treatment effect. These estimates divide the plot into four regions: (i) Conclusive: The point estimates of the upper and lower bounds are both positive (or both negative), and both confidence intervals exclude zero at the chosen confidence level masten2020inference. (ii) Tentative: The point estimates of the upper and lower bounds are both positive (or both negative), but at least one of the corresponding confidence intervals includes zero. (iii) Inconclusive: The point estimates of the upper and lower bounds are not the same sign. (iv) \textbf{\textit{Incompatible}}: The sensitivity parameter values lie outside the admissible range for this dataset; they imply assumptions that contradict observed discrepancies between the study groups (see Section (ref)). We note that Figure (ref) is an example for illustration, and that the size and shape of the regions vary by dataset.
Relevant Literature
Data Fusion for Causal Inference. Data fusion methods leverage randomized trials to mitigate unmeasured confounding in observational data and have become central to causal inference. Broadly, approaches differ based on whether treatments and outcomes are observed only in the trial degtiar2023review or in both datasets brantner2023methods, lin2024data. Our setting aligns with the latter. A more detailed review is provided in Appendix \ref*{appdx:ext-lit-review}.
Partial Identification. Partial identification (ID) and sensitivity analysis frameworks are widely used to assess robustness to assumption violations in observational studies cornfield1959smoking, rosenbaum1983assessing, liu2013introduction, ding2016sensitivity, bonvini2022sensitivity. In data fusion, most sensitivity approaches focus on violations of study exchangeability when observational treatments or outcomes are missing nguyen2017sensitivity, nguyen2018sensitivity, nie2021covariate, colnet2022causal, dahabreh2023sensitivity, huang2022sensitivity.
In settings where treatments and outcomes are available from the observational cohort (like ours), both non-unmeasured confounding (NUC) and study exchangeability must be addressed. Most existing work assumes exchangeability and focuses on NUC violations lin2024data, lin2025combining, triantafillou2023learning, chen2021minimax, oberst2022understanding, kallus2018removing, yang2020improved, rosenman2023combining, yang2023elastic. yang2023elastic and parikh2023double propose tests for assumption violations: the former attributing test failures to NUC, the latter recognizing that failures may stem from either assumption, though requiring knowledge of which one fails—a difficult task in practice.
Partial Id w/ Data Fusion. While partial ID and sensitivity methods are well-developed for fusion without observational treatments/outcomes, they remain sparse when these are available. Related work includes partial ID approaches in contextual bandits joshi2023towards and structural causal modeling with qualitative knowledge zhang2022partial. Most closely related, yu2024using develop a two-parameter sensitivity analysis for fusion settings, addressing NUC and study positivity, but assuming study exchangeability.
Partial Identification
In this section, we present a general framework for partial identification of treatment effects under simultaneous violations of the no unmeasured confounding assumption ((ref)) and study exchangeability ((ref)). When either assumption fails, point identification of the average treatment effect (ATE) and conditional average treatment effect (CATE) becomes impossible.
To quantify the degree of these violations, we introduce two interpretable parameters: $\rho$, capturing the extent of unmeasured confounding in the observational cohort, and $\gamma$, capturing the extent of study exchangeability violation between the experimental and observational populations.
We define:
\[\small
\rho := \sup_{\mathbf{x}, t}
\left|1 -
\frac{
\colorbox{boxorange}{\boxed{\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=0, T=1-t]}}
}{
\colorbox{boxteal}{\boxed{\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=0, T=t]}}
}
\right| \text{, }
\gamma := \sup_{\mathbf{x}, t}
\left|1 -
\frac{
\colorbox{boxorange}{\boxed{\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=0]}}
}{
\colorbox{boxteal}{\boxed{\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=1]}}
}
\right|.
\]
The blue denominator terms represent quantities point-identifiable without assumptions (ref) or (ref), whereas the orange numerator terms involve counterfactual quantities that are not directly observed.
If (ref) or (ref) holds, then $\rho=0$ or $\gamma=0$, respectively, and treatment effects are point-identifiable. Conversely, dissimilarity between $\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=0, T=1-t]$ and $\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=0, T=t]$ reflects unobserved confounding (i.e., a violation of (ref)) and leads to nonzero values of $\rho$. Similarly, dissimilarity between $\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=0]$ and $\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=1]$ reflects study selection bias (i.e. a violation of (ref)) and corresponds to nonzero values of $\gamma$. Both $\rho$ and $\gamma$ are relative measures of dissimilarity; for example, $\rho=1$ implies a 100% difference between the observed outcome expectation and the counterfactual counterpart. Note that while $\rho$ and $\gamma$ could be extended to functions of $\mathbf{x}$ and $t$, we conservatively treat $\rho$ and $\gamma$ as scalars, taking the supremum over all covariate-treatment profiles $(\mathbf{x}, t)$. This enables tractable, worst-case sensitivity analyses.
We now focus on using $\rho$ and $\gamma$ to bound $\tau$ and $\tau(\mathbf{x})$ when (ref) and (ref) are simultaneously violated. For parsimony, we define the following estimable quantities: (i) study selection score: $g_s(\mathbf{x}) = \mathbb{P}_{\mathcal{P}}(S = s \mid \mathbf{X} = \mathbf{x})$, (ii) treatment propensity score: $e_t(\mathbf{x}, s) = \mathbb{P}_{\mathcal{P}}(T = t \mid \mathbf{X} = \mathbf{x}, S = s)$, and (iii) expected outcome: $\mu(\mathbf{x}, s, t) = \mathbb{E}_{\mathcal{P}}[Y \mid \mathbf{X} = \mathbf{x}, S = s, T = t].$ Using the law of iterated expectation over study selection, we express:
equation[equation omitted — 427 chars of source]
where we invoke (ref) to replace the potential outcome expectation $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}, S=1]$ with the observed outcome expectation $\mu(\mathbf{x}, 1, t)$, since treatment is randomized in the experimental study ($S=1$). In contrast, without (ref) or (ref), the term $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}, S=0]$ remains unidentifiable. However, we can leverage $\rho$ and $\gamma$ to construct sharp, identifiable bounds on $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}, S=0]$, and thereby obtain bounds on the overall potential outcome $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}]$.
Towards this, we define two estimable functions that upper bound $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}, S=0]$:
\[
v(\mathbf{x},t,\gamma) := (1+\gamma)\mu(\mathbf{x},1,t), \quad
w(\mathbf{x},t,\rho) := e_t(\mathbf{x},0)\mu(\mathbf{x},0,t) + e_{1-t}(\mathbf{x},0)(1+\rho)\mu(\mathbf{x},0,t).
\]
The function $v(\mathbf{x}, t, \gamma)$ is derived from experimental study data. It is based on the relative deviation of $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}, S=0]$ from the identifiable quantity $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}, S=1] = \mu(\mathbf{x},1,t)$, as governed by the parameter $\gamma$. Conversely, the function $w(\mathbf{x}, t, \rho)$ is derived from observational study data. It combines the identifiable component $\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=0, T=t] = \mu(\mathbf{x},0,t)$, weighted by the treatment propensity, $e_t(\mathbf{x},0)$, with a term that inflates the counterfactual component $\mathbb{E}_{\mathcal{P}}[Y(t)\mid \mathbf{X}=\mathbf{x}, S=0, T=1-t]$ according to the parameter $\gamma$.
In Lemma (ref), we combine the upper bounds $v(\mathbf{x}, t, \gamma)$ and $w(\mathbf{x}, t, \rho)$, along with their lower bound counterparts $v(\mathbf{x}, t, -\gamma)$ and $w(\mathbf{x}, t, -\rho)$, to construct tight, identifiable bounds on $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}]$. Specifically, we replace the unidentifiable term $\mathbb{E}_{\mathcal{P}}[Y(t) \mid \mathbf{X}=\mathbf{x}, S=0]$ in Equation (ref) with the $\min$ of the two upper bounds and the $\max$ of the two lower bounds.
restatable[Conditional Potential Outcome Bounds]{lemma}{potoutlemma}
Suppose (ref)-(ref) hold. Then for any $t \in \{0,1\}$ and given $\mathbf{x}$, if $v(\mathbf{x},t,-\gamma) \leq w(\mathbf{x},t,\rho)$ and $w(\mathbf{x},t,-\rho) \leq v(\mathbf{x},t,\gamma)$,
the conditional potential outcome satisfies
\begin{gather*}
\mathbb{E}_{\mathcal{P}}[Y(t)\mid\mathbf{X}=\mathbf{x}] \in [l(\mathbf{x},t,\rho,\gamma), u(\mathbf{x},t,\rho,\gamma)], \;\; where \\
l(\mathbf{x},t,\rho,\gamma) = g_1(\mathbf{x})\mu(\mathbf{x},1,t) + g_0(\mathbf{x})\max\left\{w(\mathbf{x},t,-\rho), v(\mathbf{x},t,-\gamma)\right\}, \\
u(\mathbf{x},t,\rho,\gamma) = g_1(\mathbf{x})\mu(\mathbf{x},1,t) + g_0(\mathbf{x})\min\left\{w(\mathbf{x},t,\rho), v(\mathbf{x},t,\gamma)\right\}.
\end{gather*}
A full derivation of the results leading to Lemma (ref) is provided in Appendix \ref*{appdx:pid-proofs}. Building on this results, we next derive bounds on the conditional and average treatment effects.
restatable[Treatment Effect Bounds]{theorem}{tebounds}
Suppose (ref)-(ref) hold and that for each $\mathbf{x}$ and $t \in \{0,1\}$, $v(\mathbf{x},t,-\gamma) \leq w(\mathbf{x},t,\rho)$ and $w(\mathbf{x},t,-\rho) \leq v(\mathbf{x},t,\gamma)$.
Then, the conditional average treatment effect satisfies:
$l(\mathbf{x},1,\rho,\gamma) - u(\mathbf{x},0,\rho,\gamma) \leq \tau(\mathbf{x}) \leq u(\mathbf{x},1,\rho,\gamma) - l(\mathbf{x},0,\rho,\gamma)$.
Further, if this holds for all $\mathbf{x}$ such that $\mathbb{P}_{\mathcal{P}}(\mathbf{X}=\mathbf{x}) > 0$, then the average treatment effect satisfies:
$\mathbb{E}_{\mathcal{P}}[l(\mathbf{X},1,\rho,\gamma) - u(\mathbf{X},0,\rho,\gamma)] \leq \tau \leq \mathbb{E}_{\mathcal{P}}[u(\mathbf{X},1,\rho,\gamma) - l(\mathbf{X},0,\rho,\gamma)]$.
We derive doubly robust estimators for the treatment effect bounds established in Theorem (ref) in the next section. Before proceeding, we discuss infeasibility conditions in Remark (ref).
remark[(In)compatible $\rho$ and $\gamma$.]
Every $(\rho, \gamma)$ pair corresponds to a data-generating process that could, in principle, have produced the observed data. However, some values of $(\rho, \gamma)$ imply assumptions that conflict with what we observe. To illustrate this, consider a scenario where $\exists (\mathbf{x}, t)$ such that the conditional expectations of the outcome differ across study groups, i.e. $\left| \mathbb{E}_{\mathcal{P}}[Y \mid \mathbf{X} = \mathbf{x}, S=1, T=t] - \mathbb{E}_{\mathcal{P}}[Y \mid \mathbf{X} = \mathbf{x}, S=0, T=t] \right| = \Delta(t) > 0$. parikh2023double shows that $\Delta(t) > 0$ implies that (ref) and/or (ref) is violated. Therefore, setting $(\rho, \gamma) = (0,0)$--- which implies both assumptions hold---contradicts the observed difference $\Delta(t) > 0$.
More broadly, the bounds in Lemma (ref) and Theorem (ref) are valid only when both $v(\mathbf{x},t,-\gamma) \leq w(\mathbf{x},t,\rho)$ and $w(\mathbf{x},t,-\rho) \leq v(\mathbf{x},t,\gamma)$; a violation makes the parameters incompatible. Because checking the inequalities at every $(\mathbf{x}, t)$ is infeasible in most settings, we test them in expectation over $\mathbf{X}$ for each treatment arm. The null distribution is estimated with a resampling test that keeps the fitted propensity and outcome models fixed, as generating resamples that satisfy the null and re-estimate these models is non-trivial. This may label some pairs incompatible that a full bootstrap would not. Intuition behind and estimation of (in)compatibility are discussed further in Appendix \ref*{appdx:incompatible}.
Semiparametrically Efficient Estimation
We now turn to the problem of estimating the bounds identified in Section (ref). Our goal is to construct doubly robust estimators that offer both statistical efficiency and robustness to model misspecification chernozhukov2018double. However, a key challenge arises: the presence of non-differentiable $\max$ and $\min$ operators in our estimands makes it intractable to directly derive the efficient influence functions (EIFs) needed for such estimators. To address this issue, in Section (ref) we introduce smooth approximations to the bounds using the Boltzmann operator in place of the $\max$ and $\min$ functions. These approximations enable the derivation of EIFs and, in turn, the construction of bias-corrected estimators in Section (ref).
In this section, we focus on bounds for $\tau$, but note that similar steps apply to deriving efficient estimators for bounds on $\tau(\mathbf{x})$.
Smooth Bounds
For any $x_1,x_2\in\mathbbm{R}$, the Boltzmann operator is of the form $\lambda_1 x_1 + \lambda_2 x_2$ where
equation*[equation* omitted — 174 chars of source]
This operator is similar to the popular softmax function in that $\lambda_1 + \lambda_2 = 1$ and their relative magnitudes are linked to $x_1$ and $x_2$. However, it differs in its incorporation of the hyperparameter $\alpha$ which causes $\lambda_1 x_1 + \lambda_2 x_2$ to approach $\max(x_1, x_2)$ as $\alpha\rightarrow \infty$ and $\min(x_1, x_2)$ as $\alpha\rightarrow -\infty$.
In Lemma (ref), we show that the Boltzmann operator can be used to construct smooth approximations of our partial identification bounds. Specifically, we replace the functions $l(\mathbf{X}, t, \rho, \gamma)$ and $u(\mathbf{X}, t, \rho, \gamma)$ from Lemma (ref) with $b(\mathbf{X}, t, \rho, \gamma, \alpha)$, which uses the Boltzmann operator in place of the $\max$ and $\min$ functions. We then establish treatment effect bounds using these smooth approximations---analogous to Theorem (ref)---and show that, as $\alpha$ increases, these estimates converge to the $\max$ and $\min$ bounds.
restatable[name=Smooth Bounds]{lemma}{smoothboundslemma}
Consider a setting where (ref)-(ref) hold but (ref) and (ref) may not. Define
\begin{equation*}
b(\mathbf{X}, t, \rho, \gamma, \alpha) :=
g_1(\mathbf{X})\mu(\mathbf{X},1,t) +
g_0(\mathbf{X})\left\{
\lambda_1(\mathbf{X}, t, \rho, \gamma, \alpha) v + \lambda_2(\mathbf{X}, t, \rho, \gamma, \alpha) w
\right\}, where
\end{equation*}
\[
\lambda_1(\mathbf{X}, t, \rho, \gamma, \alpha) = \frac{\exp(\alpha v)}{\exp(\alpha v) + \exp(\alpha w)}, \quad
\lambda_2(\mathbf{X}, t, \rho, \gamma, \alpha) = \frac{\exp(\alpha w)}{\exp(\alpha v) + \exp(\alpha w)},
\]
and \( v = v(\mathbf{X}, t, \gamma) \), \( w = w(\mathbf{X}, t, \rho) \).
Then for any $\alpha > 0$, $\rho$, and $\gamma$ such that $\forall t\in\{0,1\}$ and $\forall \mathbf{x}$ for which $\mathbb{P}_{\mathcal{P}}(\mathbf{X} = \mathbf{x}) >0$, $b(\mathbf{x}, t, -\rho, -\gamma, \alpha) \leq b(\mathbf{x}, t, \rho, \gamma, -\alpha)$, it follows that
\begin{equation*}
\mathbb{E}_{\mathcal{P}}[b(\mathbf{X}, 1, -\rho,-\gamma, \alpha) - b(\mathbf{X}, 0, \rho,\gamma, -\alpha)]
\leq \tau \leq
\mathbb{E}_{\mathcal{P}}[b(\mathbf{X},1,\rho,\gamma, -\alpha) - b(\mathbf{X},0,-\rho,-\gamma, \alpha)],
\end{equation*}
and
\begin{align*}
\lim_{\alpha\rightarrow\infty}\mathbb{E}_{\mathcal{P}}[b(\mathbf{X}, 1, -\rho,-\gamma, \alpha) - b(\mathbf{X}, 0, \rho,\gamma, -\alpha)] &= \mathbb{E}_{\mathcal{P}}[l(\mathbf{X}, 1, \rho,\gamma) - u(\mathbf{X}, 0, \rho,\gamma)], \\
\lim_{\alpha\rightarrow\infty}\mathbb{E}_{\mathcal{P}}[b(\mathbf{X},1,\rho,\gamma, -\alpha) - b(\mathbf{X},0,-\rho,-\gamma, \alpha)] &= \mathbb{E}_{\mathcal{P}}[u(\mathbf{X},1,\rho,\gamma) - l(\mathbf{X},0,\rho,\gamma)].
\end{align*}
Efficient Estimators
Having established smooth, differentiable approximations of our bounds, we now derive their corresponding EIFs, which form the basis for the bias-corrected estimators. We begin by defining $\theta(t,\rho,\gamma,\alpha) := \mathbb{E}_{\mathcal{P}}[b(\mathbf{X}, t, \rho, \gamma, \alpha)]$. Using this, we can express the bounds on $\tau$ as
equation*[equation* omitted — 194 chars of source]
where each bound is written as a difference between two instances of $\theta$ with different parameter settings. Therefore, once we establish an EIF for the general form $\theta(t,\rho,\gamma,\alpha)$, we can obtain EIFs for both the upper and lower bounds by leveraging the linearity of EIFs.
Following the approach of schuler2024moderncausalinference, we derive the EIF for $\theta(t,\rho,\gamma,\alpha)$, denoted by $\phi(Z; t,\rho,\gamma,\alpha)$ with $Z = (\mathbf{X}, S, T, Y)$. The full, centered EIF is given below with each term tagged by a superscript \((\cdot)\) for reference. For brevity, we omit the explicit arguments of $v$, $w$, $\lambda_1$, and $\lambda_2$, which match those of $\phi$.
\[
aligned\phi(Z; t,\rho,\gamma,\alpha)
&= \Bigg[S\mu(\mathbf{X},1,t) + (1 - S)\{\lambda_1 v + \lambda_2 w\}\Bigg]^{(i)}
+ \Bigg[\frac{S\mathbb{I}(T=t)}{e_t(\mathbf{X},1)}\{Y - \mu(\mathbf{X},1,t)\}\Bigg]^{(ii)}\\
&\quad+ \Bigg[S(1+\gamma)\{\lambda_1 + \alpha\lambda_1\lambda_2(v - w)\}
\frac{\mathbb{I}_t(T)g_0}{e_t(\mathbf{X},1)g_1}\{Y - \mu(\mathbf{X},1,t)\}\Bigg]^{(iii)}\\
&\quad+ \Bigg[(1 - S)\{\lambda_2 + \alpha\lambda_1\lambda_2(w - v)\}
\biggl\{\frac{\mathbb{I}_t(T)}{e_t(\mathbf{X},0)}\{Y - \mu(\mathbf{X},0,t)\}(1 + \rho e_{1-t})\\
& + \rho\,\mu(\mathbf{X},0,t)(\mathbb{I}_{1-t}(T) - e_{1-t})\biggr\}\Bigg]^{(iv)}
- \theta(t,\rho,\gamma,\alpha)^{(v)}.
\]
The five superscripted terms correspond to: (i) A plug-in term from the experimental and observational samples. (ii) A correction term for $g_1(\mathbf{X})\mu(\mathbf{X},1,t)$, using experimental samples. (iii) A correction term for $g_0(\mathbf{X})\lambda_1 v$, using experimental samples. (iv) A correction term for $g_0(\mathbf{X})\lambda_2 w$, using observational samples. (v) A centering term, $-\theta(t,\rho,\gamma,\alpha)$, to ensure $\mathbb{E}_{\mathcal{P}}[\phi(Z; t,\rho,\gamma,\alpha)] = 0$.
We will use the EIF for the generic $\theta(t,\rho,\gamma,\alpha)$ to obtain EIFs for the lower and upper bounds. Denoting the lower bound estimand as $\theta_{LB}(\rho,\gamma,\alpha) = \theta(1,-\rho,-\gamma, \alpha) - \theta(0, \rho, \gamma, -\alpha)$, and the upper bound as $\theta_{UB}(\rho,\gamma,\alpha) = \theta(1,\rho,\gamma,-\alpha) - \theta(0, -\rho, -\gamma, \alpha)$, their EIFs are given by
align*[align* omitted — 230 chars of source]
We use these EIFs to construct bias-corrected estimators for the lower and upper bounds. Let $\hat{g}_s$, $\hat{e}_t$, and $\hat{\mu}$ denote the estimated study selection, treatment propensity, and outcome regression functions used to compute $\theta_{LB}$ and $\theta_{UB}$. These are commonly referred to as nuisance functions, as they are not themselves of interest but are necessary for estimation. We collectively denote them by $\hat{\eta} = (\hat{g}_s, \hat{e}_t, \hat{\mu})$, where the hat symbol $\hat{\cdot}$ indicates an estimated quantity. The bias-corrected estimator allows these components to be estimated with flexible machine learning models, which helps protect against model misspecification, while still enabling valid inference chernozhukov2018double. The form of the lower bound estimator is
align*[align* omitted — 217 chars of source]
where $\hat{\theta}_{LB}^{plugin}(\rho,\gamma,\alpha; \hat{\eta})$ is the plug-in estimate and $\hat{\phi}_{LB}(Z_i; \rho,\gamma,\alpha; \hat{\eta})$ is the corresponding centered EIF evaluated at each sample $Z_i = (\mathbf{X}_i, S_i, T_i, Y_i)$. The estimator for the upper bound is defined analogously using the corresponding plug-in and EIF components.
To ensure valid inference, we employ cross-fitting and assume standard convergence conditions on nuisance functions, leading to asymptotic normality kennedy2024semiparametric,chernozhukov2018double,rudolph2024improvingefficiencytransportingaverage,schuler2024moderncausalinference. These properties enable our estimator to adapt to complex data-generating processes while maintaining statistical validity. We present full implementation details in Algorithm (ref) (Appendix \ref*{appdx:algorithms}) and formally state the asymptotic properties of our estimators in Theorem (ref).
I
restatable[name=Asymptotic Properties]{theorem}{asymptoticsthm} If (ref)-(ref) are satisfied, then for any $\alpha > 0$, $\rho$, and $\gamma$ such that $\forall t\in\{0,1\}$ and $\forall \mathbf{x}$ for which $\mathbb{P}_{\mathcal{P}}(\mathbf{X} = \mathbf{x}) >0$, $b(\mathbf{x}, t, -\rho, -\gamma, \alpha) \leq b(\mathbf{x}, t, \rho, \gamma, -\alpha)$, we have that $\sqrt{n}\left(\hat{\theta}_{LB}^{bc}(\rho,\gamma,\alpha; \hat{\eta}) - \theta_{LB}(\rho,\gamma,\alpha) \right) \xrightarrow{d} \mathcal{N}(0, \sigma_{LB}^2)$ and $\sqrt{n}\left(\hat{\theta}_{UB}^{bc}(\rho,\gamma,\alpha; \hat{\eta}) - \theta_{UB}(\rho,\gamma,\alpha) \right) \xrightarrow{d} \mathcal{N}(0, \sigma_{UB}^2)$,
where $\sigma_{LB}^2 = Var[\phi_{LB}(Z; \rho,\gamma,\alpha)]$ and $\sigma_{UB}^2 = Var[\phi_{UB}(Z; \rho,\gamma,\alpha)]$.
Theorem (ref) leverages results on estimators derived from EIFs kennedy2024semiparametric,chernozhukov2018double and establishes that our estimators for the partial identification bounds are asymptotically unbiased under standard regularity conditions. Variance can be estimated using the sample variance of the estimated influence functions or via resampling methods such as the bootstrap, enabling valid confidence interval construction. We note that while larger values of $\alpha$ yield closer approximations to the non-smooth bounds, they may also lead to estimator instability in small samples due to the increasingly steep gradients of the smoothed function near the $\max$/$\min$ crossover point.
Experimental Results
In this section, we bring together the partial identification bounds developed in Section (ref), the estimators derived in Section (ref), and the breakdown frontier plots from masten2020inference introduced in Section (ref) to demonstrate how our sensitivity parameters, $\rho$ and $\gamma$, enable comprehensive sensitivity analysis. We begin with synthetic data to illustrate key properties of our framework under varying data generating processes. We then return to the Project STAR study from the \hyperref[sec:intro]{Introduction}, examining the robustness of treatment effect estimates in the presence of unobserved confounding.
Simulation Study
We consider a data generating process with an unobserved confounder, $U$, which simultaneously affects study selection $(S)$, treatment assignment among observational units $(T | S=0)$, and outcomes $(Y)$. We generate a baseline dataset with a positive treatment effect, as well as four variants where we (a) increase the treatment effect, (b) decrease the treatment effect, (c) increase the amount of unobserved confounding, and (d) decrease the amount of unobserved confounding. We plot the breakdown frontiers for each of these datasets in Figure (ref).
figure[figure omitted — 521 chars of source]
We observe that the conclusive region (dark green) expands as the magnitude of the treatment effect increases. Conversely, weaker effects lead to a smaller conclusive region. We also observe that as the amount of unobserved confounding shrinks, so does the incompatible region (black), while greater confounding enlarges it. This behavior demonstrates how the breakdown frontier plot effectively summarizes the strength of evidence for a conclusive treatment effect by incorporating both the effect size and the observed discrepancies between observational and experimental data.
Specific details of the data generating process are provided in Appendix \ref*{appdx:sim-setup}, and an algorithm for constructing the breakdown frontier plot is included in Appendix \ref*{appdx:bf-algo}. The procedure involves a handful of hyperparameters, including the minimum and maximum values of $\rho$ and $\gamma$, the confidence level, and the $\alpha$ scale used in the Boltzmann operator.
Project STAR
Project STAR was a large-scale study conducted in Tennessee to investigate the effect of class size on student learning outcomes mosteller1995tennessee, project_star_data. The experimental cohort included 11,601 students randomly assigned to one of two groups: small classes (13–17 students) ($T=0$) and regular classes (22–25 students) ($T=1$). An observational cohort of 1,780 students---assigned to the same class size types but without randomization---was also available. Demographic data, including gender, race, birth year, birth month, and free lunch eligibility, were collected for both groups. Learning outcomes were measured using standardized test administered from kindergarten through third grade. Our analysis focuses on test scores from third grade.
Figure (ref)(a) presents a breakdown frontier analysis of the Project STAR ATE, varying the sensitivity parameters $\rho$ and $\gamma$. The incompatible region at small values of both parameters aligns with prior findings on unmeasured confounding in the dataset von2018does, justman2018randomized, athey2020combining, parikh2023double. In contrast to existing estimation approaches, which require assuming either (ref) or (ref) holds, our framework enables investigation of causal effects under simultaneous violations of both. The analysis show that as long as study exchangeability violations remain below 5%, there is conclusive evidence of a positive ATE---even under substantial NUC violations. Given that scores range from 486–745 (mean 618), this suggests that study selection bias would need to shift outcomes by over 30 points on average to render the results inconclusive.
Beyond the population ATE, our framework supports subgroup comparisons. Figure (ref)(b) shows breakdown frontier plots for students who enrolled in kindergarten before age six (left) and at six or older (right). Consistent with simulation insights, the positive treatment effect is more robust to assumption violations for the older subgroup, suggesting a larger benefit for older entrants. Developmental Psychology describes significant changes in cognitive development around the typical kindergarten entry age piaget1964cognitive, and education research has shown that students who begin kindergarten at an older age tend to experience early learning advantages datar2006does. While neither directly addresses class size, these findings provide context for why older students may be better positioned to benefit from the learning environment of smaller classes---a hypothesis further supported by our analysis. The older subgroup also exhibits a larger incompatible region, potentially reflecting additional unmeasured confounding related to delayed school entry (ages six to eight) and its influence on study participation, class assignment, and outcomes.
figure[figure omitted — 392 chars of source]
Conclusion
Causal inference methods for data fusion typically assume either study exchangeability or NUC. Our work addresses settings where both assumptions may be violated, filling a gap in partial identification and sensitivity analysis. We introduce interpretable sensitivity parameters--- $\gamma$ for external validity violations and $\rho$ for unmeasured confounding---that enable transparent robustness assessments. We derive treatment effect bounds under these parameters and develop double machine learning estimators.
We use breakdown frontier plots to visualize regions where treatment effects remain conclusively positive or negative. Applications to synthetic data and Project STAR highlight our method's utility. In the Project STAR analysis, we find that the positive effect of small class sizes is robust under substantial violation of both (ref) and (ref). Subgroup analysis further reveals heterogeneity in this robustness, with stronger conclusions for students who enrolled at older ages.
Limitations & Future Work. Our framework supports a single binary treatment and does not handle multiple experimental or observational datasets. Extending it to continuous treatments or dynamic regimes is a promising direction. Large values of $\alpha$ can cause instability, particularly in small samples, due to steep gradients in the smoothed approximations. Our approach to identifying incompatible $(\rho, \gamma)$ values uses a simplified resampling test that does not account for uncertainty from estimating nuisance functions (Appendix \ref*{appdx:incompatible}). While sufficient for visual diagnostics, improving this test is a direction for future work. Finally, our framework supports sensitivity analysis across $(\rho, \gamma)$ values but does not prescribe how to select them. Although our use of relative measures helps, domain expertise is needed to interpret plausible violation levels. Appendix \ref*{appdx:bf-algo} discusses guidance and computational considerations for selecting breakdown frontier plot parameters.
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