Demystifying Proximal Causal Inference

Abstract

Proximal causal inference (PCI) has emerged as a promising framework for identifying and estimating causal effects in the presence of unobserved confounders. While many traditional causal inference methods rely on the assumption of no unobserved confounding, this assumption is likely often violated. PCI addresses this challenge by relying on an alternative set of assumptions regarding the relationships between treatment, outcome, and auxiliary variables that serve as proxies for unmeasured confounders. We review existing identification results, discuss the assumptions necessary for valid causal effect estimation via PCI, and compare different PCI estimation methods. We offer practical guidance on operationalizing PCI, with a focus on selecting and evaluating proxy variables using domain knowledge, measurement error perspectives, and negative control analogies. Through conceptual examples, we demonstrate tensions in proxy selection and discuss the importance of clearly defining the unobserved confounding mechanism. By bridging formal results with applied considerations, this work aims to demystify PCI, encourage thoughtful use in practice, and identify open directions for methodological development and empirical research.

Figures (5)

• Figure 1: An example of a directed acyclic graph (DAG) illustrating the relationships between the two types of proxy variables and the unobserved confounders, treatment, and outcome. Arrows represent allowed dependence, while a lack of arrow represents an independence assumption. For example, the lack of arrow between $Z$ and $Y$ represents the assumption that $Z\perp\!\!\!\perp Y\mid U,A,X$ (A.\ref{['condZ']}). However, the arrow from $Z$ to $A$ reflects that $Z$ may or may not be a cause of $A$. Note that this is not the only DAG that can also satisfy PCI assumptions. See Supplemental Table A.1 in tchetgen_tchetgen_introduction_2024 for examples of other DAGs that do (and do not) satisfy PCI assumptions.
• Figure 2: A example causal diagram where there are multiple ways to define a set of unobserved confounders $U$ that satisfy $Y(a) \perp\!\!\!\perp A \mid U,X$ (latent unconfoundedness, A.\ref{['latent']}): as any of the three unobserved variables individually ($U=U_1,U=U_2,$ or $U=U_3$), any pair of them (e.g. $U=\{U_1,U_2\}$), or all three ($U=\{U_1,U_2,U_3\}$).
• Figure 3: This diagram lists the core assumptions of PCI and summarizes the assumptions of two complementary identification approaches: (i) via the Outcome Bridge Function miao_identifying_2018tchetgen_tchetgen_introduction_2024, and (ii) via the Treatment Bridge Function cui_semiparametric_2024. For each approach, the required bridge function assumption, completeness conditions, and formulas for average treatment effect (ATE) identification are displayed. Note that in both approaches, one completeness condition is required for identification, while another completeness condition can be used with regularity conditions miao_identifying_2018cui_semiparametric_2024 to justify that a bridge function exists. Variations on these paths are presented in miao_confounding_2024 (see Remark \ref{['remark:miao2024']}) and cui_semiparametric_2024.
• Figure 4: Illustrative example showing how alternative representations of the unobserved confounder $U$ affect the plausibility of proximal causal inference (PCI) assumptions. (A) Initial directed acyclic graph (DAG) proposed by researchers. (B) Corrected DAG including an additional unmeasured variable $U_2$. (C) Researcher 1’s reformulation, treating $U_1$ as the sole unobserved confounder; replacing $U_2$ with bidirectional arrows to indicate the additional unobserved source of dependence between variables. (D) Researcher 2’s reformulation, defining $U_{1,2} = (U_1, U_2)$ as a joint construct.
• Figure 5: A DAG that depicts an example of a traditional latent variable model, with four measurements. These can also be viewed as disconnected proxies.

Theorems & Definitions (17)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Remark 9
• Remark 10