Marginal Sensitivity Model (MSM) in Causal Inference

• Marginal Sensitivity Model (MSM) is a framework for evaluating causal effects when unmeasured confounding challenges the unconfoundedness assumption.
• It replaces point identification with partial identification, deriving sharp bounds on causal parameters through controlled departures from observed-data treatment mechanisms.
• MSM and its extensions address binary, continuous, and time-varying treatments, providing robust tools for policy learning and sensitivity analysis in observational studies.

Marginal Sensitivity Model (MSM) denotes a class of sensitivity models for causal inference with unmeasured confounding. In the standard potential-outcomes framework, MSM replaces point identification under unconfoundedness with partial identification over a class of full-data distributions that are consistent with the observed data and satisfy a user-specified restriction on the shift induced by latent confounders in treatment assignment. The resulting causal parameters, such as mean potential outcomes and the average treatment effect, are therefore characterized by sharp bounds rather than a single identified value. In the recent literature, MSM appears in its classical binary-treatment form and in extensions to continuous, multivalued, time-varying, and distributional settings (Zhang et al., 20 May 2025, Zhang et al., 2022, Jesson et al., 2022, Frauen et al., 2023).

1. Classical formulation

The standard setup uses observed covariates $X$, binary treatment $T\in\{0,1\}$, potential outcomes $Y^0,Y^1$, and observed outcome $Y=TY^1+(1-T)Y^0$. Under unconfoundedness, $(Y^0,Y^1)\perp T\mid X$, quantities such as $\mu_0=E(Y^0)$, $\mu_1=E(Y^1)$, and $\mathrm{ATE}=\mu_1-\mu_0$ are point identified; MSM is designed for the case in which this assumption may fail because of unmeasured confounding (Zhang et al., 20 May 2025).

The original marginal sensitivity model of Tan (2006) is written as a bound on the density ratio

$$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$

where $\Lambda_2(X)\ge 1>\Lambda_1(X)>0$. A latent-confounder formulation introduces an unmeasured covariate $T\in\{0,1\}$0 such that

$T\in\{0,1\}$1

and constrains

$T\in\{0,1\}$2

This is equivalently an odds-ratio shift constraint on treatment assignment induced by $T\in\{0,1\}$3; the papers note that MSM and this latent-confounder version induce the same sharp bounds for $T\in\{0,1\}$4 (Zhang et al., 20 May 2025).

An alternative but closely related parameterization uses the observed propensity score $T\in\{0,1\}$5 and the full-data propensity score $T\in\{0,1\}$6. In that form, the MSM imposes

$T\in\{0,1\}$7

equivalently

$T\in\{0,1\}$8

This makes explicit that $T\in\{0,1\}$9 bounds the worst-case odds-ratio distortion between the observed and full-data treatment mechanisms, with $Y^0,Y^1$0 corresponding to exact unconfoundedness (Zhang et al., 2022).

In adjacent literatures the acronym “MSM” often denotes marginal structural model rather than marginal sensitivity model. Several papers explicitly warn against this terminological ambiguity; in the present sense, MSM refers to the Rosenbaum/Tan-style or Robins-style sensitivity-analysis framework for departures from ignorability or selection bias, not to longitudinal marginal structural models (Nab et al., 2019).

2. Partial identification and sharp bounds

Sensitivity analysis under MSM treats causal inference as an optimization over full-data distributions $Y^0,Y^1$1 that are compatible with the observed-data law and satisfy the sensitivity restriction. The target quantities are then only partially identified, and one derives lower and upper sharp bounds. For $Y^0,Y^1$2, if

$Y^0,Y^1$3

then the sharp upper bound has the form

$Y^0,Y^1$4

with an analogous lower bound (Zhang et al., 20 May 2025).

For the standard binary-treatment MSM reviewed in the deMSM paper, the upper conditional bound is

$Y^0,Y^1$5

where

$Y^0,Y^1$6

and $Y^0,Y^1$7 is the quantile-loss term built from the $Y^0,Y^1$8-quantile of $Y^0,Y^1$9. This exhibits a characteristic feature of MSM sharp bounds: deviations from the unconfoundedness reference value $Y=TY^1+(1-T)Y^0$0 are governed by extremal quantile-loss expressions (Zhang et al., 20 May 2025).

The $Y=TY^1+(1-T)Y^0$1-based formulation makes the same structure explicit in an optimization language. With

$Y=TY^1+(1-T)Y^0$2

the mean-one constraint

$Y=TY^1+(1-T)Y^0$3

is combined with interval restrictions induced by $Y=TY^1+(1-T)Y^0$4, and sharp bounds on $Y=TY^1+(1-T)Y^0$5 are obtained by optimizing

$Y=TY^1+(1-T)Y^0$6

The resulting optimizer is an extreme two-point weighting rule: $Y=TY^1+(1-T)Y^0$7 where $Y=TY^1+(1-T)Y^0$8 is a conditional quantile threshold. This quantile-threshold form explains the connection to a Neyman–Pearson argument and is one of the main reasons the literature treats MSM bounds as sharp and tractable (Zhang et al., 2022).

A bound is sharp if every value within the bound is attainable under some full-data distribution satisfying observed-data compatibility, consistency, latent ignorability, and the sensitivity constraints. The deMSM paper states this directly and constructs an explicit full-data distribution $Y=TY^1+(1-T)Y^0$9 that attains the bounds, showing that the bound is the best possible under the model, not merely conservative (Zhang et al., 20 May 2025).

3. Refinements and alternative sensitivity metrics

A major line of development starts from the observation that standard MSM restricts only the treatment side of confounding. The enhanced MSM (eMSM) adds an outcome sensitivity constraint

$(Y^0,Y^1)\perp T\mid X$0

where $(Y^0,Y^1)\perp T\mid X$1. The paper shows that eMSM sharp bounds for expected potential outcomes are always narrower than the MSM sharp bounds, and under a recommended specification the additional outcome sensitivity is summarized by a scalar shrinkage parameter $(Y^0,Y^1)\perp T\mid X$2, with $(Y^0,Y^1)\perp T\mid X$3 recovering MSM and $(Y^0,Y^1)\perp T\mid X$4 recovering the no-confounding-in-mean benchmark (Zhang et al., 11 Apr 2025).

The distributionally enhanced marginal sensitivity model (deMSM) replaces eMSM’s mean-outcome constraint with a distributional constraint on potential outcomes: $(Y^0,Y^1)\perp T\mid X$5 Together with the treatment-probability-shift constraint, this implies an MSM with tighter implied sensitivity parameters. The paper emphasizes four related facts: deMSM is a submodel of an implied MSM; it reduces to the usual MSM when the outcome constraint is vacuous; its sharp bounds coincide with MSM bounds under the implied parameters; and the bounds are symmetric in the treatment and outcome sensitivity parameter pairs because both are expressed on the same density-ratio scale (Zhang et al., 20 May 2025).

A different critique targets MSM’s use of an $(Y^0,Y^1)\perp T\mid X$6-type restriction. The $(Y^0,Y^1)\perp T\mid X$7- and $(Y^0,Y^1)\perp T\mid X$8-sensitivity paper argues that the MSM bound on the logit difference may render the analysis conservative because it is determined by the worst case that may not occur very often. It therefore proposes an alternative model based on the second moment of the propensity score ratio: $(Y^0,Y^1)\perp T\mid X$9 together with $\mu_0=E(Y^0)$0 and the same mean-one constraint. In this formulation, MSM quantifies worst-case conditional strength of unmeasured confounding, whereas the $\mu_0=E(Y^0)$1 model quantifies average confounding strength (Zhang et al., 2022).

The literature also contains a risk-ratio-based modification. For binary or multivalued treatment, the modified model imposes

$\mu_0=E(Y^0)$2

with $\mu_0=E(Y^0)$3. This changes the sensitivity scale from odds ratios to risk ratios while retaining computational tractability through linear fractional programming or linear programming (Basit et al., 2023).

4. Generalizations beyond a single binary treatment

For continuous-valued interventions, the continuous treatment-effect marginal sensitivity model (CMSM) replaces the propensity score by the conditional treatment density. With latent confounder $\mu_0=E(Y^0)$4,

$\mu_0=E(Y^0)$5

This is presented as a direct continuous-treatment generalization of MSM ideas: propensity score becomes treatment density, and odds-ratio or inverse-propensity restrictions become density-ratio or likelihood-ratio restrictions. The resulting estimands are the conditional average potential outcome $\mu_0=E(Y^0)$6 and the average potential outcome $\mu_0=E(Y^0)$7, rather than binary-treatment ATEs (Jesson et al., 2022).

The generalized marginal sensitivity model (GMSM) pushes this much further. For each treatment-to-variable relation $\mu_0=E(Y^0)$8, it bounds the distribution shift in the latent confounder: $\mu_0=E(Y^0)$9 This formulation yields a unified sensitivity framework for discrete, continuous, and time-varying treatments, and for mediation, path analysis, and distributional effects. In the weighted GMSM, the choice $\mu_1=E(Y^1)$0 recovers the standard binary-treatment MSM, while $\mu_1=E(Y^1)$1 yields the continuous-treatment CMSM and the longitudinal LMSM (Frauen et al., 2023).

The same paper interprets causal sensitivity analysis as a distribution shift in the latent confounders while evaluating the causal effect of interest. In the special case of a single binary treatment, its sharp bounds for (conditional) average treatment effects coincide with recent optimality results for causal sensitivity analysis, which positions the classical MSM as one special case within a broader distribution-shift framework (Frauen et al., 2023).

MSM has also been embedded in downstream decision problems. In policy learning under unobserved confounding, the model defines an uncertainty set $\mu_1=E(Y^1)$2 of counterfactual distributions satisfying the odds-ratio restriction

$\mu_1=E(Y^1)$3

and robust policy criteria are defined as worst-case welfare or worst-case policy improvement over that set. In the binary-treatment case, the resulting robust criteria admit sharp closed-form expressions built from the same quantile-threshold structure that underlies sharp MSM bounds (Jin et al., 28 Jul 2025).

5. Estimation, computation, and inference

At the population level, MSM bounds are derived through extremal quantile-loss expressions; estimation therefore turns on nuisance quantities such as treatment mechanisms, conditional quantiles, and conditional outcome laws. In the $\mu_1=E(Y^1)$4 and $\mu_1=E(Y^1)$5 frameworks, the literature develops one-step semiparametric estimators based on efficient influence functions, together with $\mu_1=E(Y^1)$6-fold cross-fitting and multiplier bootstrap simultaneous confidence bands for sensitivity curves indexed by $\mu_1=E(Y^1)$7, $\mu_1=E(Y^1)$8, or related parameters (Zhang et al., 2022).

For continuous treatments, the CMSM paper emphasizes scalability to high-dimensional, large-sample observational data through uncertainty-aware deep models. Its estimation strategy combines neural conditional density or outcome modeling with a constrained optimization induced by the bounded-weight sensitivity model, and uses percentile bootstrap for finite-sample uncertainty. The method is designed to estimate pointwise lower and upper bounds on dose-response functions rather than only a single scalar estimand (Jesson et al., 2022).

The recent GMSM work proposes a scalable algorithm that first estimates observational conditional laws for mediators and outcomes, then computes sharp bounds by applying the extremal shift formulas. For continuous outcomes, the implementation uses conditional normalizing flows and an importance-sampling estimator; the paper highlights that once nuisance models are fit, the importance-sampling bound computation for expectation is $\mu_1=E(Y^1)$9, versus $\mathrm{ATE}=\mu_1-\mu_0$0 for earlier grid-search-based approximations (Frauen et al., 2023).

Refinements of the classical model can often reuse MSM machinery. The deMSM paper states that deMSM population sharp bounds coincide with MSM sharp bounds under implied parameters $\mathrm{ATE}=\mu_1-\mu_0$1, so estimation can use machinery already developed for MSM, including doubly robust point estimation of population bounds and confidence intervals accounting for sampling uncertainty. The eMSM paper similarly derives doubly robust point estimation and asymptotically valid Wald confidence intervals for its population bounds under recommended parameterizations (Zhang et al., 20 May 2025, Zhang et al., 11 Apr 2025).

6. Calibration, applications, and limitations

Calibration of sensitivity parameters remains central because $\mathrm{ATE}=\mu_1-\mu_0$2, $\mathrm{ATE}=\mu_1-\mu_0$3, $\mathrm{ATE}=\mu_1-\mu_0$4, $\mathrm{ATE}=\mu_1-\mu_0$5, and related quantities are not identifiable from the observed data alone. One recurring strategy is calibration by observed covariates: for the classical MSM, the $\mathrm{ATE}=\mu_1-\mu_0$6 and $\mathrm{ATE}=\mu_1-\mu_0$7 paper calibrates $\mathrm{ATE}=\mu_1-\mu_0$8 by leaving out an observed covariate $\mathrm{ATE}=\mu_1-\mu_0$9, refitting the propensity model, and comparing the most extreme odds ratio between the propensity score with and without $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$0. The same paper argues that this inherits MSM’s pessimism because it focuses on the maximum odds-ratio distortion rather than an average-case measure of confounding (Zhang et al., 2022).

Recent refinements try to make parameter choice more interpretable. The deMSM paper recommends constant sensitivity parameters

$$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$1

and especially suggests the simple symmetric calibration $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$2. Under this choice, the paper interprets deMSM through a base quantile level and a shrinkage level, and notes that if $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$3, then the quantile level is $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$4 and the deMSM deviation is $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$5 of the corresponding MSM deviation (Zhang et al., 20 May 2025).

Applications span a wide range of domains. CMSM has been applied to climatological impacts of human emissions on cloud properties using satellite observations from the past 15 years, where continuous interventions and hidden confounding are both central (Jesson et al., 2022). The $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$6 and $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$7 paper studies the ATE of high fish consumption on blood mercury and uses that example to compare worst-case and average-case sensitivity curves (Zhang et al., 2022). The eMSM paper reports two real-data applications, including the RHC study and an NHANES fish consumption study, and emphasizes that eMSM intervals are always narrower than MSM intervals for the same treatment sensitivity parameters (Zhang et al., 11 Apr 2025). Policy-learning work has used MSM-based uncertainty sets in the JTPA study and Head Start program to show how policy recommendations evolve with $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$8 (Jin et al., 28 Jul 2025).

Several limitations recur across the literature. First, identification remains partial and depends on external judgment about sensitivity parameters rather than data-driven identification. Second, overlap and positivity remain critical; the CMSM paper stresses that positivity is particularly fragile in the continuous-treatment regime, and the risk-ratio-based MSM paper reports that performance deteriorates under lack of overlap (Jesson et al., 2022, Basit et al., 2023). Third, the classical $$\lambda(Y^1,X)=\frac{dP(Y^1\mid T=0,X)}{dP(Y^1\mid T=1,X)} \in [\Lambda_1(X),\Lambda_2(X)],$$9-type MSM may be conservative because it controls the maximum pointwise discrepancy over all covariate-outcome configurations rather than an average strength of hidden confounding (Zhang et al., 2022). Finally, some refinements are narrower in scope than the classical model: the deMSM paper centers its main text on $\Lambda_2(X)\ge 1>\Lambda_1(X)>0$0, with extensions to $\Lambda_2(X)\ge 1>\Lambda_1(X)>0$1 and ATE in the supplement (Zhang et al., 20 May 2025).

Taken together, these developments place MSM at the center of modern sensitivity analysis for unmeasured confounding. Its defining feature is not a single formula but a modeling principle: encode controlled violations of ignorability by bounding how far the latent-adjusted treatment mechanism, or a related distributional object, can depart from its observed-data counterpart, and then derive sharp or near-sharp ignorance regions for the causal target. Subsequent work has preserved that principle while changing the sensitivity scale, adding outcome-side constraints, extending the treatment space, and integrating estimation with modern machine learning and robust decision-making (Zhang et al., 20 May 2025, Zhang et al., 2022, Jesson et al., 2022, Frauen et al., 2023)