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Robust Weighted Triangulation of Causal Effects Under Model Uncertainty

Rohit Bhattacharya, Ina Ocelli, Ted Westling

Abstract

A fundamental challenge in causal inference with observational data is correct specification of a causal model. When there is model uncertainty, analysts may seek to use estimates from multiple candidate models that rely on distinct, and possibly partially overlapping, sets of identifying assumptions to infer the causal effect, a process known as triangulation. Principled methods for triangulation, however, remain underdeveloped. Here, we develop a framework for causal effect triangulation that combines model testability methods from causal discovery with statistical inference methods from semiparametric theory, while avoiding explicit model selection and post-selection inference problems. We propose a triangulation functional that combines identified functionals from each model with data-driven measures of model validity. We provide a bound on the distance of the functional from the true causal effect along with conditions under which this distance can be taken to zero. Finally, we derive valid statistical inference for this functional. Our framework formalizes robustness under causal pluralism without requiring agreement across models or commitment to a single specification. We demonstrate its performance through simulations and an empirical application.

Robust Weighted Triangulation of Causal Effects Under Model Uncertainty

Abstract

A fundamental challenge in causal inference with observational data is correct specification of a causal model. When there is model uncertainty, analysts may seek to use estimates from multiple candidate models that rely on distinct, and possibly partially overlapping, sets of identifying assumptions to infer the causal effect, a process known as triangulation. Principled methods for triangulation, however, remain underdeveloped. Here, we develop a framework for causal effect triangulation that combines model testability methods from causal discovery with statistical inference methods from semiparametric theory, while avoiding explicit model selection and post-selection inference problems. We propose a triangulation functional that combines identified functionals from each model with data-driven measures of model validity. We provide a bound on the distance of the functional from the true causal effect along with conditions under which this distance can be taken to zero. Finally, we derive valid statistical inference for this functional. Our framework formalizes robustness under causal pluralism without requiring agreement across models or commitment to a single specification. We demonstrate its performance through simulations and an empirical application.
Paper Structure (22 sections, 4 theorems, 44 equations, 5 figures, 1 table)

This paper contains 22 sections, 4 theorems, 44 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Causal Graph Preliminaries
  3. Motivating Scenarios
  4. New Triangulation Method
  5. Triangulation functional
  6. Inference Procedure
  7. Applying The Triangulation Procedure In Practice
  8. Multiple Candidate Adjustment Sets
  9. Backdoor, frontdoor, and IV
  10. Framingham Data Application
  11. Conclusion and Future Work
  12. Proof Of Theorem \ref{['thm:robustness']}
  13. Deriving The Partial Derivative Vector In Lemma \ref{['lem:partials']}
  14. Partial derivative of $\psi$ with respect to $\beta_k$
  15. Partial derivative of $\psi$ with respect to $\psi_k$
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose assumptions ${\cal A}$ used to justify the tests of model correctness hold, and let $\mathcal{C} \subseteq \{1, \dotsc, K\}$ and $\mathcal{I} = \{1, \dotsc, K\} \setminus \mathcal{C}$ be the subsets of indices $k$ such that model $\mathcal{M}_k$ is correct and incorrect, respectively. Then where $D_a = \left(\sum_{k \in \cal{C}} \delta_a(\beta_k)\right) / \left(\sum_{k \in \cal{I}} \delta

Figures (5)

  • Figure 1: (a) A hidden variable causal DAG. (b) Graph representing intervention on $M$ from which we can read the Verma constraint $Z \perp\!\!\!\perp Y \mid C$ in $P(V)/P(M\mid A, Z, C)$.
  • Figure 2: Causal DAGs used for motivating robust triangulation. (a) A causal DAG where uncertainty about what variables to adjust for may yield M-bias. (b, c, d) Under model uncertainty, analysts may wish to triangulate effects from each of these models that rely on qualitatively distinct assumptions---violations of these assumptions are shown via blue dashed edges.
  • Figure 3: Gaussian kernel approximation of $\mathbb{I}(\beta_k = 0)$.
  • Figure 4: Causal DAG used in our simulations to demonstrate robustness to (a) M-bias in some adjustment sets, and (b) misspecification of backdoor, frontdoor, or IV models.
  • Figure 5: Point estimates averaged over $200$ trials; shaded bands correspond to $2.5$ and $97.5$ percentiles of the estimates.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • proof
  • proof
  • proof