$L^{\infty}$- and $L^2$-sensitivity analysis for causal inference with unmeasured confounding

TL;DR

This paper introduces a new $L^2$-norm based sensitivity analysis method for causal inference that offers less conservative bounds and improved calibration over traditional $L^{ {infty}}$-based models, with practical estimators and confidence bands.

Contribution

It proposes an $L^2$-norm based sensitivity model, deriving sharp bounds, efficient estimators, and confidence bands, enhancing causal inference robustness against unmeasured confounding.

Findings

The $L^2$-based model yields tighter bounds than $L^{ {infty}}$-based models.

The proposed estimators are efficient and easy to compute.

Application to real data demonstrates improved calibration and tighter bounds.

Abstract

Sensitivity analysis for the unconfoundedness assumption is crucial in observational studies. For this purpose, the marginal sensitivity model (MSM) gained popularity recently due to its good interpretability and mathematical properties. However, as a quantification of confounding strength, the $L^{\infty}$-bound it puts on the logit difference between the observed and full data propensity scores may render the analysis conservative. In this article, we propose a new sensitivity model that restricts the $L^2$-norm of the propensity score ratio, requiring only the average strength of unmeasured confounding to be bounded. By characterizing sensitivity analysis as an optimization problem, we derive closed-form sharp bounds of the average potential outcomes under our model. We propose efficient one-step estimators for these bounds based on the corresponding efficient influence functions. Additionally, we apply multiplier bootstrap to construct simultaneous confidence bands to cover the sensitivity curve that consists of bounds at different sensitivity parameters. Through a real-data study, we illustrate how the new $L^2$-sensitivity analysis can improve calibration using observed confounders and provide tighter bounds when the unmeasured confounder is additionally assumed to be independent of the measured confounders and only have an additive effect on the potential outcomes.