Meta-Learners for Partially-Identified Treatment Effects Across Multiple Environments

TL;DR

The paper tackles partial identification of conditional average treatment effects (CATE) when data come from multiple environments and standard assumptions (overlap, unconfoundedness) may fail. By treating the environment as an instrumental variable, it derives environment-aware bounds for the CATE and proposes model-agnostic meta-learners to estimate these bounds, including naïve plug-in and two-stage WB/CB learners with theoretical guarantees such as consistency and double robustness. Empirical results on synthetic and real-world data show that the proposed bounds are valid and can be tightened by cross-environment information, with cross-environment, doubly robust learners often performing best in complex settings. The methods generalize to IV settings like randomized trials with non-compliance and offer a practical pathway for robust causal inference in heterogeneous environments, while highlighting careful interpretation under IV assumptions.

Abstract

Estimating the conditional average treatment effect (CATE) from observational data is relevant for many applications such as personalized medicine. Here, we focus on the widespread setting where the observational data come from multiple environments, such as different hospitals, physicians, or countries. Furthermore, we allow for violations of standard causal assumptions, namely, overlap within the environments and unconfoundedness. To this end, we move away from point identification and focus on partial identification. Specifically, we show that current assumptions from the literature on multiple environments allow us to interpret the environment as an instrumental variable (IV). This allows us to adapt bounds from the IV literature for partial identification of CATE by leveraging treatment assignment mechanisms across environments. Then, we propose different model-agnostic learners (so-called meta-learners) to estimate the bounds that can be used in combination with arbitrary machine learning models. We further demonstrate the effectiveness of our meta-learners across various experiments using both simulated and real-world data. Finally, we discuss the applicability of our meta-learners to partial identification in instrumental variable settings, such as randomized controlled trials with non-compliance.

Figures (6)

• Figure 1: Intuition for our bounds. Left: Two propensity scores $\pi^e_1(x) = \mathbb{P}(A^e = 1 \mid X^e = x)$ and $\pi^j_1(x) = \mathbb{P}(A^j = 1 \mid X^j = x)$ corresponding to different environments $e$ and $j$ are plotted over observed confounders $X$. Large values of $\pi^e_1(x)$ and $\pi^j_1(x)$ correspond to a high probability of receiving treatment and vice versa. Right: CATE $\tau(x)$ together with bounds depending on violations of overlap and unconfoundedness. In region $\mathcal{A}_1$, no overlap violations occur, leading to wide bounds for the CATE $\tau$ due to potential unobserved confounding. In region $\mathcal{A}_2$, overlap violations occur at opposite ends across environments, leading to tight bounds for $\tau$. In region $\mathcal{A}_3$, overlap violations occur on the same end across environments, leading to wide bounds for $\tau$ due to a lack of data for treated individuals.
• Figure 2: Causal graphs for different environments $e$ and $j$. We assume that the causal structure between $e$ and $j$ remains unchanged but we allow for different treatment assignment mechanisms (propensity scores) $\pi^e_a(x)$ and $\pi^j_a(x)$. The dotted arrows indicate potential unobserved confounding.
• Figure 3: Comparison of estimation methods for bounds based on synthetic dataset 2 (predicted bounds $mean \pm 3std$ over 5 runs). Left: Oracle bounds for within- (WB) and cross- (CB) environments. Center: Estimated bounds by the naïve plug-in learner. Right: Estimated bounds by our two-stage meta-learners (here: WB-learner with CB-DR-learner).
• Figure 4: Insights from real-world data: Effect of comorbidity on mortality in COVID-19 patients in Brazil. Left: Estimated bounds by naïve plug-in learner. Right: Estimated bounds by our two-stage meta-learners (here: WB-learner and CB-PI-learner).
• Figure 5: Comparison of estimation methods for tightest predicted bounds based on synthetic dataset 2 (predicted bounds $mean \pm 3std$ over 5 runs). Left: Oracle bounds for within- (WB) and cross- (CB) environments. Center: Estimated bounds by the naïve plug-in learner. Right: Estimated bounds by our two-stage meta-learners (here: WB-learner with CB-DR-learner).

Theorems & Definitions (9)

• Lemma 4.1
• proof
• Theorem 4.2
• proof
• Theorem 5.1: Consistency and double robustness
• proof
• proof
• proof
• proof