Sensitivity Analysis to Unobserved Confounding with Copula-based Normalizing Flows

TL;DR

This paper introduces a copula-based normalizing flow method, $ ho$-GNF, for sensitivity analysis in causal inference, allowing estimation of causal effects under unobserved confounding with quantifiable bounds.

Contribution

The paper presents a novel $ ho$-GNF model that captures unobserved confounding effects and provides a Bayesian framework for sensitivity analysis in causal inference.

Findings

The $ ho$-curve bounds the average causal effect under different confounding strengths.

The Bayesian extension yields credible intervals for causal effects.

Experiments demonstrate the method's effectiveness on simulated and real data.

Abstract

We propose a novel method for sensitivity analysis to unobserved confounding in causal inference. The method builds on a copula-based causal graphical normalizing flow that we term $\rho$-GNF, where $\rho \in [-1,+1]$ is the sensitivity parameter. The parameter represents the non-causal association between exposure and outcome due to unobserved confounding, which is modeled as a Gaussian copula. In other words, the $\rho$-GNF enables scholars to estimate the average causal effect (ACE) as a function of $\rho$, accounting for various confounding strengths. The output of the $\rho$-GNF is what we term the $\rho_{curve}$, which provides the bounds for the ACE given an interval of assumed $\rho$ values. The $\rho_{curve}$ also enables scholars to identify the confounding strength required to nullify the ACE. We also propose a Bayesian version of our sensitivity analysis method. Assuming a prior over the sensitivity parameter $\rho$ enables us to derive the posterior distribution over the ACE, which enables us to derive credible intervals. Finally, leveraging on experiments from simulated and real-world data, we show the benefits of our sensitivity analysis method.