Interpreting MSM Estimates Under Outcome-Model Misspecification

Why Outcome-Model Misspecification Changes the Target

Marginal structural models (MSMs) summarize outcomes under treatment histories. Weighting is intended to make comparisons across those histories less confounded, subject to the required causal assumptions. But weighting does not guarantee that a chosen outcome model can represent the resulting response pattern. If, for example, the true response depends on an interaction between treatments at two visits but the MSM omits that interaction, its fitted coefficients generally do not equal the parameters of a true model. They instead describe a weighted compromise among the histories included in the analysis.

That distinction matters for interpretation. A coefficient from a misspecified MSM may not represent the intended causal contrast, and a contrast calculated from several coefficients may be inaccurate even when the weights are appropriate. To interpret results, identify the weighted target, the restriction imposed by the outcome model, and the treatment-history contrast that the fitted model predicts.

In this tutorial, \(W\) denotes the product of the relevant treatment and, when needed, censoring weights. The weights define the target distribution implicit in the estimating equations. For causal interpretation, the weights and analysis must be appropriate for the chosen target population, and the usual identification conditions must hold: consistency, positivity, and sequential exchangeability (or the corresponding assumptions for the study design). Correct weight specification is also required for the intended weighted population to represent the causal target. These assumptions address confounding and selection; they do not make a restrictive MSM correct.

What a Weighted Estimating Equation Targets

For an identity-link MSM that is linear in its coefficients, write \(m(a;\beta)=X(a)^\mathsf{T}\beta\), where \(X(a)\) is the model’s design vector for history \(a\). Weighted least squares solves

At the population level, the corresponding parameter \(\beta^*\) solves \(E[W X(A)\{Y-X(A)^\mathsf{T}\beta^*\}]=0\). If \(E[W X(A)X(A)^\mathsf{T}]\) is nonsingular, this gives

This is a weighted projection: the model’s predictions are chosen to satisfy weighted residual equations. If the MSM is correct, the predictions can represent the target means at all histories in its scope. If it is not, \(\beta^*\) depends on both the true outcome pattern and the weighted distribution of treatment histories. Changing the weights can therefore change the projection even if the underlying response pattern stays the same.

Other links and estimating equations can have different algebraic forms, but the central idea persists: under misspecification, the population equation defines the target. It is not generally a coefficient in the true causal response function. The fitted model’s predicted contrast for histories \(a\) and \(a'\) is \(m(a;\beta^*)-m(a';\beta^*)\) on the model’s link scale when appropriate, or on the outcome scale for an identity-link model. Whether that contrast approximates \(\mu(a)-\mu(a')\) must be assessed, not assumed.

Deriving the Weighted Projection

For the identity-link case, the population weighted estimating equation also follows from minimizing weighted squared prediction error. Define \(L(\beta)=E[W\{Y-X(A)^\mathsf{T}\beta\}^2]\). Differentiating with respect to \(\beta\) gives

Thus a minimizer satisfies the weighted estimating equation. If the design has full rank, the expected weighted squared error is a convex quadratic with a unique minimizer. Under misspecification, this minimizer is the best weighted linear approximation in this squared-error sense; it need not equal the causal mean at any particular history. The weights determine which discrepancies contribute most to the approximation.

Worked Examples

How to Assess Consequential Functional-Form Errors

A model’s coefficients alone do not reveal whether its restriction is consequential. The same coefficient can contribute to several predicted history contrasts, and a contrast may be accurate by cancellation even when the component predictions are poor. A focused assessment compares model predictions with weighted outcome patterns at histories supported by the data, emphasizing the contrasts that matter substantively.

For example, with two binary treatments an additive identity-link MSM implies zero interaction contrast:

If the corresponding weighted outcome means show a substantial difference between these treatment effects, the additive restriction is questionable. More generally, for each important pair of histories, compare the model-predicted contrast with the estimated weighted contrast, and examine weighted residual patterns. Flexible alternatives can include interactions, nonlinear dose terms, or a more flexible representation of treatment history, provided the data support the added complexity. These checks do not establish causal identification; they evaluate whether the chosen model represents the weighted response pattern adequately.

A robust or clustered standard error addresses uncertainty in estimation under its stated sampling assumptions. It does not change the population estimating equation, make a misspecified model represent the true response, or remove the resulting discrepancy between a predicted and true causal contrast. A narrow interval around a pseudo-true parameter is still an interval about that parameter, not automatic evidence that the scientific contrast has been estimated accurately.

Common Mistake

• Wrong reasoning: “The treatment weights are correctly specified, so the weighted MSM coefficients must equal the causal effects.”
• Correction: Correct weights support the intended weighted causal comparison under the identification assumptions. The outcome-model form separately determines how that comparison is summarized. If the form is too restrictive, the weighted equations estimate a projection or other pseudo-true solution, and its predicted contrasts can differ from the causal contrasts.
• Another warning: Do not infer adequate fit from one contrast that happens to agree with its target. Inspect the predictions and relevant contrasts across histories, within the region of treatment-history support.