Treatment Effect Boundaries

• Treatment effect boundaries are the structural limits that define the range of potential causal effects consistent with observed data and underlying assumptions.
• They serve as essential tools for sensitivity analysis, enabling researchers to gauge the impact of unobserved confounding and partial identification in causal inference.
• These boundaries guide methodological improvements and policy targeting by interpreting heterogeneity and addressing sample selection and model misspecification.

Treatment effect boundaries are the fundamental limits or structural thresholds that characterize the range, location, or support of causal effects in a population. They delineate the set of treatment effects consistent with the observed data, model assumptions, and possible unobserved confounding. The treatment effect boundary perspective is prevalent in modern causal inference and partial identification, encompassing sharp nonparametric bounds, boundaries induced by sample selection, regions where parametric identification fails, or explicit regime transitions in spatio-temporal models. These boundaries serve as key objects for sensitivity analysis, robust inference, policy targeting, and interpretation of heterogeneity, especially when the observed data or research design falls short of full point identification.

1. Foundations of Partial Identification and Boundaries

Treatment effect boundaries arise fundamentally from the fact that, except in special scenarios (e.g., randomized trials with perfect compliance), portions of the potential outcome space are unobserved. Under these circumstances, the joint distribution of potential outcomes is not fully identified by the observable data.

• Nonparametric Bounds: Without strong assumptions beyond SUTVA and bounded outcomes, the average treatment effect (ATE) β = E[Y(1) – Y(0)] is partially identified. For instance, for Y ∈ [0,1], the observed data only guarantee that β lies in the interval

$\left[ a - b - p_1,\; a - b + p_0 \right],$

where $a = \mathbb{E}[Y(1)|Z=1]\,Pr(Z=1)$, $b = \mathbb{E}[Y(0)|Z=0]\,Pr(Z=0)$, and $p_1 = Pr(Z=1)$, $p_0 = Pr(Z=0)$ (Richardson et al., 2015).

• Ignorance Region: The identified set or “ignorance region” encompasses all treatment effect values consistent with the observed data and maintained assumptions. Every element in this region is supported by a full data distribution compatible with the marginal constraints but varying on the unobserved (joint) structure.
• Role of Assumptions: Imposing additional assumptions—monotonicity, exclusion, mean independence—can tighten or collapse these boundaries, but always at the risk of violating model validity (Richardson et al., 2015, Vikström et al., 2017, Possebom, 2019). The trade-off is between informativeness and credibility.

2. Sensitivity Analysis and Boundary Specification

Sensitivity analysis systematically explores how treatment effect boundaries change as additional, often untestable, assumptions are varied.

• Modeling Unmeasured Confounding: In observational studies, sensitivity parameters explicitly bound the possible bias from unobserved U. For example, the $\Gamma$–Cornfield condition

$$\frac{1}{\Gamma} \leq \frac{P(Z=1|x,u)}{P(Z=0|x,u)} \frac{P(Z=0|x,\tilde u)}{P(Z=1|x,\tilde u)} \leq \Gamma$$

defines an extrinsic boundary on the impact of hidden bias. This is formalized through loss minimization or dual representation strategies, yielding tight, minimax-optimal bounds for the CATE and ATE (Yadlowsky et al., 2018).

• Longitudinal and Mediated Settings: Sensitivity boundaries are constructed either by varying bias functions (as in marginal structural models) or by treating direct and indirect effects as only set-identified, typically via linear programming under bounds or explicit perturbation of inverse probability weights (Richardson et al., 2015, Huber et al., 2020).
• Sample Selection and Principal Stratification: In principal strata or “doomed” group analyses, sharp boundaries are derived by algebraic decomposition of observed and missing potential outcomes, often parameterized by sensitivity parameters quantifying selection or mediation departure (Richardson et al., 2015, Possebom, 2019, Heiler, 2022, Lee et al., 6 Nov 2024).

3. Heterogeneity, Selection, and Distributional Boundaries

Boundaries also characterize heterogeneity in treatment effects (HTE), especially under sample selection, endogeneity, or monotonicity.

• Sample Selection with Monotonicity/Mean Dominance: When the outcome is observed only for select subpopulations (the “always-observed”), the marginal treatment effect (MTE) is not point-identified. Bounds are constructed by “sandwiching” the counterfactual using known support assumptions and possibly, mean dominance relations to rule out extreme adverse cases (Possebom, 2019).
• Heterogeneous Effect Regions: When the observation or selection mechanism is itself affected by treatment (as in non-ignorable attrition), conditional bounds on HTE can be estimated as functions of policy-relevant covariates using debiased/double machine learning estimators. These methods construct valid misspecification-robust confidence bands over the bounding functions (Heiler, 2022).
• Distributional Boundaries and Risk: Distributional summaries of treatment effects—such as conditional value at risk (CVaR)—are bounded by the corresponding CATE distribution. These functionals provide tight boundaries for the risk to subpopulations, operationalized as

$$\mathrm{CVaR}_{\alpha}(ITE) \leq \mathrm{CVaR}_{\alpha}(CATE)$$

(Kallus, 2022).

• Sharp Probabilistic Bounds for Individual Effects: In the binary outcome/treatment case, the marginal pmf of the ITE is bounded sharply by Fréchet–Hoeffding inequalities, with the interval width reflecting the intrinsic indeterminacy of joint potential outcomes (Zhang et al., 9 Jun 2025).

4. Structure, Smoothing, and Boundary Estimation

Several approaches explicitly define, estimate, and visualize treatment effect boundaries in structured, high-dimensional, or dynamic settings.

• Spatial and Temporal Treatment Effect Boundaries: Using reaction–diffusion models, spatial and temporal boundaries are formalized as critical distances $d^*$ and times $\tau^*$ at which local treatment effects decay below specified thresholds. These are explicit structural parameters governed by diffusion ($\lambda$) and decay ($\delta$) rates, identified via direct regression and residual decay estimation (Kikuchi, 1 Oct 2025).
• Boundary Discontinuity Designs (BDD): In multi-score or geographic RD designs, the treatment effect function along a continuous boundary $B$ is estimated using local polynomials. Here, naive distance-based estimators suffer from order-h bias near kinks in the boundary, whereas bivariate estimators in the original score space circumvent this bias and deliver optimal convergence (Cattaneo et al., 8 May 2025).
• Smooth Approximations and Efficient Inference: When non-overlap or non-smoothness precludes standard semiparametric estimation, smoothing techniques (e.g., logistic or kernel approximations) convert non-smooth bounds into smooth, pathwise differentiable functionals. Targeted Minimum Loss-Based Estimation (TMLE) leverages the efficient influence function for valid, robust inference; multiplier bootstrap yields uniformly valid confidence sets (Susmann et al., 24 Sep 2025).
• Plausible Bounds in Dynamic Settings: For temporal treatment paths, cumulative or restricted plausible bounds identify intervals over functionals (e.g., average effect or regularized smooth path) rather than uniform pointwise intervals, using post-selection inference to control for data-driven model selection (Freyaldenhoven et al., 17 May 2025).

5. Inference and Limitations Arising from Boundaries

Inference in the presence of treatment effect boundaries demands explicit accounting for two sources of uncertainty: sampling variability and partial identification.

• Ignorance and Uncertainty Regions: Confidence intervals are constructed as uncertainty regions over the identified set, often by forming intervals

$$UR_p(\beta) = \left[\hat{\beta}_l - c_\alpha\frac{\widehat{\sigma}_l}{\sqrt{n}},\, \hat{\beta}_u + c_\alpha\frac{\widehat{\sigma}_u}{\sqrt{n}}\right]$$

where $\hat{\beta}_l, \hat{\beta}_u$ are the estimated lower and upper boundaries and $c_\alpha$ is the appropriate critical value (Richardson et al., 2015, Heiler, 2022).

• Non-Overlap and Smooth Bounds: When the overlap assumption fails, ATE is only partially identifiable. Non-overlap bounds combine trimmed effects in the overlap subpopulation with sharp worst-case extrapolation in the non-overlap region; these can be efficiently estimated and deliver higher power than doubly robust estimators, provided the non-overlap subpopulation is not too large (Susmann et al., 24 Sep 2025).
• Partial Identification Persists in Large Samples: For the ITE, prediction intervals do not collapse with increasing sample size—valid prediction intervals must hold for all joint distributions consistent with the marginals, which typically remain wide even in randomized trials, unless the marginals are extremely unbalanced or additional structure is imposed (Zhang et al., 9 Jun 2025).
• Trade-off Between Assumptions and Informativeness: Broad, uninformative boundaries are the price of minimal assumptions. Introducing credible, often context-specific restrictions (e.g., monotonicity, exclusion, or sufficient treatment values) sharpens the boundaries but can lead to model misspecification if not adequately justified (Richardson et al., 2015, Lee et al., 6 Nov 2024).

6. Applications and Broader Implications

Treatment effect boundaries play a central role across diverse empirical settings:

• Observational Studies: Permit robust statements about treatment effect direction or nullity without strict unconfoundedness, via nonparametric or sensitivity bounds (Richardson et al., 2015, Yadlowsky et al., 2018, Kallus, 2022).
• Randomized Experiments with Attrition/Noncompliance: Deliver principal strata and complier average causal effect bounds, with the width reflecting information loss due to missingness or noncompliance (Richardson et al., 2015, Possebom, 2019).
• Heterogeneity and Policy Targeting: Guide which subpopulations can be reliably targeted given partial identification, support risk-averse policies by quantifying effect uncertainty for specific quantiles or regions (Kallus, 2022, Heiler, 2022).
• Dynamic and Spatial Policies: Identify the effective reach or persistence of interventions, link spatial/temporal scaling laws, and inform design of geographically or temporally delimited policies (Kikuchi, 1 Oct 2025, Cattaneo et al., 8 May 2025).
• Contrast Between Fisher and Neyman Nulls: Boundaries clarify when evidence suffices to reject an average effect null (Neyman) but not an individual-level effect null (Fisher), with implications for individualized decision-making and the interpretation of “no sharp null” results (Zhang et al., 9 Jun 2025).

7. Limitations and Directions for Further Research

Several challenges and limitations persist:

• Reliance on Unverifiable Assumptions: Gratifyingly sharp boundaries often require assumptions (monotonicity, mean dominance, selection stability) whose validity may be difficult to assess, making sensitivity analysis indispensable (Richardson et al., 2015, Possebom, 2019).
• Computational Complexity: Large-scale partial identification problems (e.g., high-dimensional sample selection or mediation models using linear programming or stochastic causal programming) can be computationally demanding, though algorithmic advances (e.g., TMLE, debiased/double machine learning, efficient linear programming) have improved scalability (Yadlowsky et al., 2018, Padh et al., 2022, Susmann et al., 24 Sep 2025).
• Interpretability of Bounds: When bounds remain wide or uninformative even after accounting for all credible restrictions, this points to irreducible ignorance in the research design—highlighting the value of experimental over observational designs, or the need for additional data collection to close boundaries (Zhang et al., 9 Jun 2025).
• Extension to Multivalued and Continuous Treatments: Treatment effect boundaries for continuous or multivalued exposures require generalizations of binary-focused techniques, often relying on quantile-based or functional constraints and efficient estimation strategies to ensure computational tractability and sharpness (Baitairian et al., 4 Nov 2024, Lee et al., 6 Nov 2024).

Treatment effect boundaries thus offer a rigorous, technically precise framework for quantifying what can be learned—and what cannot—from incomplete or imperfect causal research designs, providing both caution and actionable sensitivity diagnostics for empirical and policy analysis.