Auditing Fairness under Unobserved Confounding

TL;DR

This work tackles the challenge of auditing fairness when unobserved confounding could influence both resource allocation and outcomes. It introduces a partial-identification framework that bounds the treatment rate among the needy, $P(T=1|Y(0)=1, D=1)$, without requiring complete identifiability, by leveraging pre-availability data to estimate baseline risk and covariate information to tighten bounds. The authors develop bias-corrected, cross-fitted estimators for the bounds, establish asymptotic normality under a margin condition, and extend the approach with a bounded confounding sensitivity model parameterized by $\gamma$. They validate the framework on Paxlovid allocation data and semi-synthetic/synthetic settings, showing meaningful inequities that persist under plausible unobserved confounding and demonstrating improved accuracy over plug-in methods. The work provides a principled tool for auditing fairness in real-world decision systems where unobserved risk factors are likely present, with direct implications for policy and resource distribution.

Abstract

Many definitions of fairness or inequity involve unobservable causal quantities that cannot be directly estimated without strong assumptions. For instance, it is particularly difficult to estimate notions of fairness that rely on hard-to-measure concepts such as risk (e.g., quantifying whether patients at the same risk level have equal probability of treatment, regardless of group membership). Such measurements of risk can be accurately obtained when no unobserved confounders have jointly influenced past decisions and outcomes. However, in the real world, this assumption rarely holds. In this paper, we show that, surprisingly, one can still compute meaningful bounds on treatment rates for high-risk individuals (i.e., conditional on their true, \textit{unobserved} negative outcome), even when entirely eliminating or relaxing the assumption that we observe all relevant risk factors used by decision makers. We use the fact that in many real-world settings (e.g., the release of a new treatment) we have data from prior to any allocation to derive unbiased estimates of risk. This result enables us to audit unfair outcomes of existing decision-making systems in a principled manner. We demonstrate the effectiveness of our framework with a real-world study of Paxlovid allocation, provably identifying that observed racial inequity cannot be explained by unobserved confounders of the same strength as important observed covariates.

Figures (6)

• Figure 1: A conceptual figure to build intuition for our main partial identification result (Theorem \ref{['theorem:no_assumption']}). For simplicity, let the treatment rate $P(T=0)=0.3$ and mortality rate $P(Y(0)=1)=0.3$. We denote upper (blue) / lower (red) bounds on treatment rate among the needy $P(T=1 | Y(0)=1, X)$. In the marginal case (left), we see that the bounds are vacuous. However, when we exploit covariate information (right) (e.g., selecting 2 samples from X=0 and 1 sample from X=1), we observe bounds that are much tighter.
• Figure 2: A causal graph consistent with Assumptions \ref{['asmp:covariate_stability']} and \ref{['asmp:stable_basline_risk']}, even given unobserved (light gray) confounders $C$. Dark gray variables are observed. This causal structure is sufficient, but not necessary, for our assumptions to hold: See \ref{['app:causal_swig']} for more details.
• Figure 3: Upper and lower bounds for treatment rate among the needy $P(T = 1 |Y(0) = 1, D=1, G=g)$ are computed for each racial group $g$, with varying values of $\gamma \in [1,1.5]$. The shaded area represents a 95% confidence interval.
• Figure 4: (Semi-Synthetic Data) Upper and lower bounds for treatment rate among the needy, with 95% confidence intervals, for each racial group $g$, with varying values of $\gamma \in [1,2]$ (true value of $\gamma = 1.5$).
• Figure 5: (Synthetic Data) Accuracy of our bias-corrected bounds compared to their plugin counterparts in capturing the true treatment rates. We use the derived 95% confidence interval from our bias-corrected estimates for both methods.

Theorems & Definitions (50)

• Definition 1: Treatment Rate Among the Needy
• Definition 2
• Proposition 1
• Theorem 4.1: Bounds under arbitrary unmeasured confounding
• Lemma 1
• Lemma 2
• Theorem 4.2: Asymptotic Normality of Estimators
• Definition 3
• Theorem 4.3: Bounds with $\gamma$
• Proposition 1