Structural Nested Models Versus Marginal Structural Models

Two Models, Two Causal Targets

Structural nested models and marginal structural models can both address time-varying treatment and confounding affected by earlier treatment. Their central difference is not simply the estimation algorithm: they parameterize different causal quantities. A structural nested model describes the effect of changing treatment at a particular decision time, conditional on the history at that time. A marginal structural model describes the mean outcome under a treatment regime in a specified population.

That distinction matters when treatment effects vary across histories. A conditional blip can describe that variation directly, while a marginal regime mean averages over a population distribution. Even when both models are correctly specified, their coefficients need not be numerically equal or answer the same question.

The counterfactual quantities require a well-defined treatment regime and outcome. For identification from observed data, standard conditions include consistency, no interference between individuals, sequential exchangeability given measured history, and positivity: each treatment level under consideration must have positive probability at histories to which the target applies. These conditions do not make the two model targets identical; they support identification of the causal quantities each model parameterizes.

What a Blip Parameter Represents

A blip is a contrast attached to a treatment decision. In a simple one-time setting with binary treatment \(A\), history \(H\), and outcome \(Y\), a conditional mean blip can be written as \(E[Y^{1}-Y^{0}\mid H]\). An SNM might model this as \(\gamma(H,A;\psi)=A\{\psi_0+\psi_1 H\}\), where \(\psi\) indexes the conditional effect. In a longitudinal setting, blips are defined sequentially: later treatment effects are removed from the outcome before isolating the contrast at an earlier decision. The treatment history in a blip model is therefore part of its interpretation, not just a convenient set of predictors.

G-estimation uses a model for treatment assignment given history and searches for blip parameters that make a suitable blipped-down outcome satisfy conditional moment restrictions. In a one-time binary-treatment illustration, let \(\pi(H)=P(A=1\mid H)\), and let \(U(\psi)=Y-\gamma(H,A;\psi)\) be the outcome after removing the modeled treatment blip. A moment condition has the form \(E[(A-\pi(H))U(\psi)q(H)]=0\), where \(q(H)\) is a chosen function of history. The treatment model supplies \(\pi(H)\); the blip model supplies \(U(\psi)\). This does not require fitting a conventional regression of the observed outcome on treatment and history as the primary estimator of the blip.

In longitudinal applications, the treatment model is sequential: it describes \(P(A_t\mid H_t)\) at each decision time. Under a correctly specified blip model and the identification conditions, g-estimation uses treatment residuals such as \(A_t-E[A_t\mid H_t]\) in conditional moment equations. The sequential construction removes later treatment effects when estimating earlier blips. The exact moment equations depend on the treatment type and chosen blip functions; the key requirement is that the modeled blipped-down outcome obey the stipulated conditional restrictions.

What a Marginal Structural Model Represents

An MSM instead places a model directly on the mean outcome under regimes. For example, a two-time additive MSM might specify \(E[Y^{a_1,a_2}]=\beta_0+\beta_1a_1+\beta_2a_2\). Here \(\beta_1\) and \(\beta_2\) describe changes in the population marginal mean when a regime component changes, under the model’s assumptions. They are not automatically history-specific effects. An MSM can include regime interactions or baseline effect modifiers, but its defining target remains marginal over the covariate distribution unless a different target population is explicitly specified.

A common estimation strategy is inverse-probability weighting. The treatment model estimates the probability of each observed treatment given history; the resulting weights create a pseudo-population in which measured history is balanced across treatment choices, under correct specification and positivity. One then fits the marginal outcome model in that weighted population. The MSM itself is an outcome model for counterfactual regime means; the weighted estimator need not also fit a conditional outcome regression. Other strategies, such as g-computation, do require an outcome regression, and augmented estimators may use both treatment and outcome models.

How the Targets Relate—and Why They Differ

Under a compatible SNM, regime contrasts can be assembled from sequential blips. The basic justification is telescoping: compare two regimes by changing one treatment decision at a time, and sum the resulting sequential contrasts. For a two-time regime, for example, the contrast between \((1,1)\) and \((0,0)\) can be decomposed as the change from \((0,0)\) to \((1,0)\), followed by the change from \((1,0)\) to \((1,1)\). Taking expectations gives the marginal regime contrast, with each conditional blip averaged over the appropriate history distribution.

The averaging distribution is important. A blip conditional on \(H_t\) is not generally equal to a marginal MSM coefficient. The history distribution at a later time may itself depend on earlier treatment, so a later blip averaged under different earlier regimes can produce different marginal contrasts. Equality of a blip parameter and an MSM coefficient requires additional structure, such as a simple additive model with no relevant effect modification and a matching scale and target population.

Modeling Requirements in Practice

Both approaches need a credible account of treatment assignment when treatment is not randomized. Their requirements differ in what that model is used to do and what outcome model is specified.

In either approach, misspecification can undermine the result. A poor treatment model can invalidate g-estimation moments or produce inappropriate weights; extreme weights also signal limited information and can make weighted estimates unstable. A poorly chosen blip function may fail to represent conditional treatment effects, while a misspecified MSM may fail to represent the marginal regime means of interest. Diagnostics and sensitivity analyses should therefore be tied to the particular model and estimator rather than treated as interchangeable checks.

Common Mistake