ρ-GNF: A Copula-based Sensitivity Analysis to Unobserved Confounding Using Normalizing Flows

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 is an element of[-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.