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Enhanced Marginal Sensitivity Model and Bounds

Yi Zhang, Wenfu Xu, Zhiqiang Tan

TL;DR

The paper develops the Enhanced Marginal Sensitivity Model (eMSM), augmenting the traditional Marginal Sensitivity Model with an outcome-based sensitivity constraint to tighten bounds under unmeasured confounding. It derives sharp population bounds for the mean potential outcome $\mu^1$ under eMSM, provides conditions and a practical specification for parameter choices, and establishes doubly robust, calibratable estimators and confidence intervals for these bounds. By connecting eMSM to DV methods, the authors show how DV bounds can be viewed as unions of eMSMs with compatible parameters, and they derive sharp DV bounds via this framework. The approach is demonstrated through two real-data applications (RHC and NHANES), where eMSM yields notably narrower sensitivity intervals than MSM and yields robust conclusions under plausible levels of hidden confounding, illustrating its practical value for causal inference in observational studies.

Abstract

Sensitivity analysis is important to assess the impact of unmeasured confounding in causal inference from observational studies. The marginal sensitivity model (MSM) provides a useful approach in quantifying the influence of unmeasured confounders on treatment assignment and leading to tractable sharp bounds of common causal parameters. In this paper, to tighten MSM sharp bounds, we propose the enhanced MSM (eMSM) by incorporating another sensitivity constraint that quantifies the influence of unmeasured confounders on outcomes. We derive sharp population bounds of expected potential outcomes under eMSM, which are always narrower than the MSM sharp bounds in a simple and interpretable way. We further discuss desirable specifications of sensitivity parameters related to the outcome sensitivity constraint, and obtain both doubly robust point estimation and confidence intervals for the eMSM population bounds. The effectiveness of eMSM is also demonstrated numerically through two real-data applications. Our development represents, for the first time, a satisfactory extension of MSM to exploit both treatment and outcome sensitivity constraints on unmeasured confounding.

Enhanced Marginal Sensitivity Model and Bounds

TL;DR

The paper develops the Enhanced Marginal Sensitivity Model (eMSM), augmenting the traditional Marginal Sensitivity Model with an outcome-based sensitivity constraint to tighten bounds under unmeasured confounding. It derives sharp population bounds for the mean potential outcome under eMSM, provides conditions and a practical specification for parameter choices, and establishes doubly robust, calibratable estimators and confidence intervals for these bounds. By connecting eMSM to DV methods, the authors show how DV bounds can be viewed as unions of eMSMs with compatible parameters, and they derive sharp DV bounds via this framework. The approach is demonstrated through two real-data applications (RHC and NHANES), where eMSM yields notably narrower sensitivity intervals than MSM and yields robust conclusions under plausible levels of hidden confounding, illustrating its practical value for causal inference in observational studies.

Abstract

Sensitivity analysis is important to assess the impact of unmeasured confounding in causal inference from observational studies. The marginal sensitivity model (MSM) provides a useful approach in quantifying the influence of unmeasured confounders on treatment assignment and leading to tractable sharp bounds of common causal parameters. In this paper, to tighten MSM sharp bounds, we propose the enhanced MSM (eMSM) by incorporating another sensitivity constraint that quantifies the influence of unmeasured confounders on outcomes. We derive sharp population bounds of expected potential outcomes under eMSM, which are always narrower than the MSM sharp bounds in a simple and interpretable way. We further discuss desirable specifications of sensitivity parameters related to the outcome sensitivity constraint, and obtain both doubly robust point estimation and confidence intervals for the eMSM population bounds. The effectiveness of eMSM is also demonstrated numerically through two real-data applications. Our development represents, for the first time, a satisfactory extension of MSM to exploit both treatment and outcome sensitivity constraints on unmeasured confounding.

Paper Structure

This paper contains 35 sections, 10 theorems, 194 equations, 34 figures, 23 tables.

Table of Contents

  1. Abstract.
  2. Key words and phrases.
  3. Introduction
  4. Setup and marginal sensitivity model
  5. Proposed sensitivity model and population bounds
  6. Enhanced MSM
  7. Population sharp bounds
  8. Recommended specification of sensitivity parameters
  9. Sample estimation
  10. Comparison of bounds for binary outcomes
  11. DV and related results
  12. Comparison and improvement
  13. Empirical applications
  14. Application to RHC
  15. Application to NHANES
  16. ...and 20 more sections

Key Result

Proposition 1

The sharp upper bound $\nu^{1+}_{\text{\scriptsize{eMSM}}} (X)$ in eq:condtightup for $\nu^1(X) = E(Y^1 | T=0,X)$ can be determined as where $\tau(X)=\{\Lambda_2(X)-1\}/\{\Lambda_2(X)-\Lambda_1(X)\}$, $\rho_{\tau} (y,q)$ is the check function, and $q^*_{1,\tau}=q^*_{1,\tau}(X)$ is the $\tau(X)$-quantile of $P_Y(\cdot| T=1,X)$ as in eq:MSM-bound-mu--eq:MSM-bound-nu. The sharp upper bound $\mu^{1+}

Figures (34)

  • Figure 1: Graphical representation of eMSM constraints conditional on $X$.
  • Figure 2: Effect of worst-case confounder $U$ on $Y^1$, conditioned on $X$, according to Corollary \ref{['cor:eMSM-Q']}. Density functions of $Y|T=1$ (solid black), $Y^1|U=1$ (dashed blue), and $Y^1|U=0$ (dotdash red) are plotted. $\Lambda_2=\Lambda_1^{-1}=2$, and vertical dotted line marks the $2/3$-quantile of $Y|T=1$. The parameters $(\Delta_1,\Delta_2)$ are set such that $\psi_{1+}=1$ (left) or $\psi_{1+}=0.7$ (right).
  • Figure 3: Bounds of $\mu^1=E(Y^1)$ under DV model, conditioned on $X$. The original DV bounds \ref{['eq:dv-ab']} (solid black), the improved bounds \ref{['eq:Sjosharp-ab']} due to Sjölander (dotted red), and the sharp bounds from Proposition \ref{['prop:DVcompare']} (dashed blue) are compared. By row, $p_1\in\{0.5,0.7,0.9\}$ increases from top to bottom. By column, $\Lambda_1^{-1}=\Lambda_2\in\{1.2,1.5,2\}$ increases from left to right. The treatment assignment is balanced with $P(T=1)=0.5$ in all plots.
  • Figure 4: $\Lambda = 1$
  • Figure 5: $\Lambda = 1.2$
  • ...and 29 more figures

Theorems & Definitions (10)

  • Proposition 1: Sharp upper bounds
  • Corollary 1: Worst-case unmeasured confounding
  • Proposition 2: Sharp lower bounds
  • Proposition 3: Sharp bounds under recommended specification
  • Proposition 4: Doubly robust, relaxed upper bound
  • Proposition 5: Sharp bounds for DV model
  • Lemma S1
  • Lemma S2
  • Lemma S3
  • Proposition S1: Sharp bounds for DV model