Marginal Structural Model Estimands for Treatment Histories

From Treatment Histories to Causal Targets

In longitudinal studies, treatment is given at multiple times, and treatment at one visit may affect both later treatment and the outcome. A causal question therefore needs to specify the whole treatment history being compared, not just whether a person was ever treated. It also needs to say whose outcomes are being averaged. A marginal structural model (MSM) is a way to represent causal outcome means across treatment histories; the causal targets themselves are defined before choosing that model.

Let \(t=0,\ldots,K\) index treatment times, let \(A_t\) denote treatment at time \(t\), and write \(\bar A_t=(A_0,\ldots,A_t)\) for the history through time \(t\). The complete history is \(\bar A_K\). For a specified history \(\bar a=(a_0,\ldots,a_K)\), let \(Y^{\bar a}\) denote the outcome that would be observed if treatment were set to that history. This is a potential outcome; for each individual, only the outcome under the treatment history actually followed is observed.

The subscript \(F^*\) matters. It identifies the population over which potential outcomes are averaged: for example, eligible patients in a defined health system during a specified period. The regime-specific mean is not generally a mean conditional on a particular covariate value. Rather, it averages across the target population’s covariate distribution. The same treatment histories can therefore have different marginal means, and different marginal contrasts, in different target populations.

A fixed history, such as “no treatment at either visit,” specifies each treatment value in advance. A dynamic regime instead specifies a rule that uses information available at each visit. Write \(g\) for such a rule and \(Y^g\) for the potential outcome under following it. Its regime-specific mean is \(\mu_{F^*}(g)=E_{F^*}[Y^g]\). A contrast between two dynamic regimes is defined in the same way as a contrast between fixed histories. Both are causal targets; the distinction is how the treatment sequence is specified.

Defining an Estimand Before a Model

An estimand is the precise quantity a study aims to learn. For longitudinal treatment questions, it should make clear the outcome, the treatment histories or regimes being compared, the target population, and the effect scale. For example, a mean difference in outcome units and a ratio of means are distinct estimands, even if they compare the same two regimes.

An MSM is a model for the marginal potential-outcome means as a function of treatment history. In general notation, it writes \(\mu_{F^*}(\bar a)=m(\bar a;\beta)\), where \(m\) is a chosen function and \(\beta\) is a parameter vector. If the model represents all the regime-specific means exactly, its parameters encode those means and contrasts. The model is a representation of the causal targets, not their definition. Merely fitting a model with treatment history as predictors does not make its coefficients causal.

A model may also be deliberately simpler than the full collection of regime-specific means. In that case, a parameter can summarize a pattern across histories rather than equal one particular contrast. If the model is not exactly correct, the precise meaning of its fitted parameters also depends on the fitting criterion and the distribution used to fit it. Thus a study should not silently equate a coefficient with a causal contrast: it must state the model and show how the target contrast is encoded by its parameters.

To interpret a regime-specific mean causally from observational data, additional conditions are needed. Consistency requires that, for an individual whose observed treatment history is \(\bar a\), the observed outcome equals \(Y^{\bar a}\). No interference requires that one individual’s treatment does not change another individual’s potential outcome, under the study’s unit definition. Sequential exchangeability requires that, at each treatment time, treatment assignment is conditionally independent of the relevant future potential outcomes given the measured history available before that treatment. Positivity requires that each treatment value specified by a regime has positive probability at the histories represented in the target population. These conditions concern identification from data; they do not alter the definition of the estimand.

How an MSM Represents the Means

A simple MSM can use a separate mean for each treatment history. For two visits with binary treatment, there are four histories: 00, 10, 01, and 11. An identity-link saturated model is \(\mu_{F^*}(a_0,a_1)=\beta_0+\beta_1a_0+\beta_2a_1+\beta_3a_0a_1\). “Saturated” here means that the model has enough parameters to represent any four cell means. The interaction term permits the effect of treatment at one visit to differ according to treatment at the other visit.

This example illustrates why an estimand and a model parameter are not interchangeable. The estimand “mean difference between 11 and 00” is defined by two potential-outcome means. The parameterization is one way of representing those means. A different model could use a different parameterization while representing the same contrast. In an unsaturated model, some means may be constrained by its form, so not every possible set of regime means can be represented.

The Target Population Is Part of the Estimand

Marginal means average over a population distribution. If that distribution changes, the average can change even when the stratum-specific potential-outcome means do not. This is not a defect in marginalization: it reflects that the causal question concerns a different population. The target population should therefore be specified before reporting a marginal contrast.

A Short Derivation: Model Contrasts as Parameter Contrasts

For a linear-in-parameters MSM, a contrast between two histories can be derived by subtracting their model predictions. Let \(\boldsymbol{x}(\bar a)\) be the model’s design vector for history \(\bar a\), so that \(\mu_{F^*}(\bar a)=\boldsymbol{x}(\bar a)^\mathsf{T}\beta\). For histories \(\bar a\) and \(\bar a'\):

Thus, once the target contrast and model are stated, its representation is determined by the difference between the two design vectors. In the saturated two-visit example, the design vectors for 11 and 00 differ in the columns for \(a_0\), \(a_1\), and \(a_0a_1\), so all three corresponding coefficients contribute. This algebra justifies reading a contrast from the model predictions rather than assuming it equals a single coefficient.

Common Mistake: Treating a Model Coefficient as the Target

A common error is to report a treatment-history coefficient as “the causal effect” without stating which histories it compares, the outcome scale, or the population being averaged over. That reasoning confuses a model’s chosen parameterization with the causal estimand. A coefficient may represent a contrast only for particular histories or under particular model coding; interactions can make the relevant contrast a combination of coefficients.

The correction is to define the desired contrast first, for example \(\mu_{F^*}(11)-\mu_{F^*}(00)\), then calculate it from the model predictions or the appropriate parameter contrast. Also state \(F^*\) and the units or scale. Finally, distinguish this target from its estimation: causal interpretation from observational data requires appropriate identification conditions, and a model’s presence alone does not establish them.