$\rho$-GNF: A Copula-based Sensitivity Analysis to Unobserved Confounding Using Normalizing Flows

TL;DR

This paper introduces $ ho$-GNF, a novel sensitivity analysis method using copulas and normalizing flows to quantify unobserved confounding effects on causal estimates, providing tighter bounds than existing methods.

Contribution

The paper develops $ ho$-GNF, a copula-based normalizing flow approach that models unobserved confounding and estimates causal effect bounds as a function of confounding strength.

Findings

$ ho$-GNF produces narrower ACE bounds than traditional methods.

It effectively estimates confounding strength needed to nullify causal effects.

The method is validated on simulated and real-world datasets.

Abstract

We propose a novel sensitivity analysis to unobserved confounding in observational studies using copulas and normalizing flows. Using the idea of interventional equivalence of structural causal models, we develop $\rho$-GNF ($\rho$-graphical normalizing flow), where $\rho{\in}[-1,+1]$ is a bounded sensitivity parameter. This parameter represents the back-door non-causal association due to unobserved confounding, and which is encoded with a Gaussian copula. In other words, the $\rho$-GNF enables scholars to estimate the average causal effect (ACE) as a function of $\rho$, while accounting for various assumed strengths of the unobserved confounding. The output of the $\rho$-GNF is what we denote as the $\rho_{curve}$ that provides the bounds for the ACE given an interval of assumed $\rho$ values. In particular, the $\rho_{curve}$ enables scholars to identify the confounding strength required to nullify the ACE, similar to other sensitivity analysis methods (e.g., the E-value). Leveraging on experiments from simulated and real-world data, we show the benefits of $\rho$-GNF. One benefit is that the $\rho$-GNF uses a Gaussian copula to encode the distribution of the unobserved causes, which is commonly used in many applied settings. This distributional assumption produces narrower ACE bounds compared to other popular sensitivity analysis methods.