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

Yao Zhang, Qingyuan Zhao

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.

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

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 -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 -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 -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.
Paper Structure (40 sections, 20 theorems, 186 equations, 3 figures, 2 tables)

This paper contains 40 sections, 20 theorems, 186 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Marginal sensitivity models
  3. Example: model comparison
  4. Our contributions
  5. Preliminary: assumption & notation
  6. Related works
  7. -analysis
  8. Lagrangian formulation
  9. Sensitivity value formulation
  10. Comparison: -analysis vs. -analysis
  11. Semiparametric inference
  12. One-step estimation: -model
  13. One-step estimation: -model
  14. Simultaneous confidence bands
  15. Simulation study
  16. ...and 25 more sections

Key Result

Proposition 1

Under assumption:first, the program equ:double_ipw with $\lambda>0$ is solved by where $g(X,Y) = (\xi_X - Y ) \mathbbold{1}_{ \{Y\leq \xi_X \} }$ and $\xi_X$ is the unique root of the following function that is strictly increasing:

Figures (3)

  • Figure 1: Measure of confounding strength in the $L^{\infty}$- and $L^{2}$-models. As the unmeasured confounding becomes effective with a large probability ($\mathbb P\{X=0\}$ in our simulation), the measure of confounding strength in the $L^2$-model increases, while the measure in the $L^\infty$-model shows only small variation.
  • Figure 2: Sensitivity curves of the ATE of fish consumption on blood mercury in the real-data study in \ref{['sect:real_data']}. The shaded regions are 90% lower confidence bands from multiplier bootstrap. Our $L^2$-analysis can verify if the ATE is robust against unmeasured confounding that is independent of observed confounders $X$ and has an additive effect on the potential outcomes $Y(0)$ and $Y(1)$.
  • Figure 3: Illustration of program solutions.

Theorems & Definitions (40)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 1
  • Proposition 7
  • Theorem 2
  • Theorem 3
  • ...and 30 more