Sensitivity Analysis via the Proportion of Unmeasured Confounding

Abstract

In observational studies, identification of ATEs is generally achieved by assuming that the correct set of confounders has been measured and properly included in the relevant models. Because this assumption is both strong and untestable, a sensitivity analysis should be performed. Common approaches include modeling the bias directly or varying the propensity scores to probe the effects of a potential unmeasured confounder. In this article, we take a novel approach whereby the sensitivity parameter is the “proportion of unmeasured confounding”: the proportion of units for whom the treatment is not as good as randomized even after conditioning on the observed covariates. We consider different assumptions on the probability of a unit being unconfounded. In each case, we derive sharp bounds on the average treatment effect as a function of the sensitivity parameter and propose nonparametric estimators that allow flexible covariate adjustment. We also introduce a one-number summary of a study’s robustness to the number of confounded units. Finally, we explore finite-sample properties via simulation, and apply the methods to an observational database used to assess the effects of right heart catheterization. Supplementary materials for this article are available online.

KEYWORDS:

• Observational study
• Optimization
• Partial identification
• Semiparametric theory

Supplementary Materials

The supplementary materials contain proofs of all results along with discussions of possible extensions to the sensitivity analysis model proposed and some additional data analysis.

Acknowledgments

The authors thank Sivaraman Balakrishnan, Colin Fogarty, Marshall Joffe, Alan Mishler, Pratik Patil, and members of the Causal Group at Carnegie Mellon University for helpful discussions.

Notes

1 As pointed out by an anonymous reviewer, the mixture model (1) could be generalized to $\mathbb{Q}={\sum }_{j=1}^J{ϵ}_j{\mathbb{Q}}_j$, where each Q_j is a distribution on the counterfactuals capturing different degrees of the confounding. While richer sensitivity analyses can yield more nuanced conclusions, the large number of parameters whose plausibility range would need to be assessed ($J-1$ in this case) may hinder their applications in many settings. For instance, consider the toy example above, with $X=\varnothing$ and $Y,A,U\in \{0,1\}$ for simplicity. Suppose that $\mathbb{P}(U=1|A)={\gamma }_0+{\gamma }_{1}A$ and ${\mathbb{Q}}_s(Y^a=1|A,U)={\alpha }_{1}s+(1-s)({\alpha }_2+{\alpha }_{3}U)$, for some constants γ and α. Then, ${\mathbb{E}}_{{\mathbb{Q}}_0}(Y^1-Y^0)={\mathbb{E}}_{{\mathbb{Q}}_1}(Y^1-Y^0)=0$ and ${\mathbb{E}}_{{\mathbb{Q}}_1}(Y^1|A=1)-{\mathbb{E}}_{{\mathbb{Q}}_1}(Y^0|A=0)=0$, but ${\mathbb{E}}_{{\mathbb{Q}}_0}(Y^1|A=1)-{\mathbb{E}}_{{\mathbb{Q}}_0}(Y^0|A=0)={\alpha }_3{\gamma }_1$, which is generally nonzero.

2 In principle, one could construct the empirical moment condition after performing the rearrangement procedure of Chernozhukov, Fernandez-Val, and Galichon (2009). Whether or not the rearrangement is done, we expect the inference about ϵ₀ to be equivalent asymptotically and vary minimally in finite samples.

3 To incorporate finite sampling uncertainty in sensitivity analyses, one-number summaries of a study’s robustness are generally computed as the values of the sensitivity parameter(s) such that a α-level confidence interval for the ATE under no unmeasured confounding includes the null value. Choosing different αs to estimate the ATE with no residual confounding may then yield different conclusions regarding the study’s robustness to unmeasured confounding, despite the latter being a separate inferential task. Constructing a confidence interval for ϵ₀ directly bypasses this issue.

4 In the context of the toy example of Section 1.1, U and S indicate whether the parents are smokers and whether they would smoke at home, respectively, X₁ and X₂ may be measures of the parents’ education level and income, respectively, A indicates adolescent alcohol consumption and Y indicates the occurrence of liver disease.

5 Available at http://biostat.mc.vanderbilt.edu/wiki/Main/DataSets.

Additional information

Funding

Edward Kennedy gratefully acknowledges financial support from NSF grant DMS1810979.