Marginal Structural Models

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What is a Marginal Structural Model?

Marginal Structural Models (MSMs) represent a fundamental conceptual and methodological advance in causal inference from observational data. Introduced by Robins and colleagues in the late 1990s, MSMs provide a rigorous framework for estimating causal effects while avoiding the limitations of traditional regression approaches.

Marginal Structural Model (MSM): A model for the marginal mean of the counterfactual (potential) outcome as a function of treatment: $$E[Y^a] = \beta_0 + \beta_1 a$$

For binary treatment, this yields:

$E[Y^0] = \beta_0$ (mean counterfactual outcome if no one treated)
$E[Y^1] = \beta_0 + \beta_1$ (mean counterfactual outcome if everyone treated)
$\beta_1 = E[Y^1] - E[Y^0]$ (Average Treatment Effect)

Why the name "Marginal Structural Model"?

The terminology reflects three important features:

Marginal: The model describes the marginal (population-average) mean of the counterfactual outcome, averaged over the distribution of confounders - NOT conditioned on them.
Structural: The model describes a causal relationship (the effect of treatment on outcome), not merely a statistical association. The parameters have causal interpretations.
Model: It is indeed a model - we specify the functional form relating treatment to outcome (though for binary treatment, the model is typically saturated and makes no functional form assumptions).

Example: Smoking Cessation and Weight Gain
In the NHEFS study, the MSM for weight change is: $$E[Y^a] = \beta_0 + \beta_1 a$$ where $a$ indicates quitting smoking (1) vs. not quitting (0).

Interpretation:

$\beta_0 = 1.78$ kg: Average weight change if the entire population continued smoking
$\beta_1 = 3.44$ kg: Causal effect of quitting - the additional weight gain caused by quitting
$E[Y^1] = 5.22$ kg: Average weight change if everyone quit

Key contrasts with standard regression:

Feature	Standard Regression	MSM
Target	$E[Y	A,L]$
Estimation	OLS/ML	IP-weighted regression
Interpretation	Conditional on confounders	Marginal (population-average)
Causal	Only under strong assumptions	Yes, under exchangeability
Confounders in model	Yes (adjust by conditioning)	No (adjust by weighting)

The Key Insight: MSMs allow us to model causal effects directly - we write down a model for counterfactual outcomes rather than observed outcomes conditional on confounders. This conceptual shift is profound: we're no longer asking "what is Y when A=1 and L=l?" but rather "what would Y be if we set A=1 for everyone?"

Reference: Hernan MA, Robins JM. What If (2020), Chapter 12 introduces MSMs. The foundational paper is Robins JM, Hernan MA, Brumback B. "Marginal structural models and causal inference in epidemiology." Epidemiology. 2000;11(5):550-560.

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