Additive Structural Mean Models On The Risk-Difference Scale

Why Use an Additive Structural Mean Model?

For a binary outcome, a difference in risks is often directly interpretable: a treatment that changes risk from 20% to 35% raises it by 15 percentage points. An additive structural mean model (SMM) expresses a treatment effect on this risk-difference scale while using an instrument to address treatment endogeneity. The model does not make treatment assignment random. Instead, it combines a model for the treatment effect with assumptions that make the instrument informative about that effect.

The central parameter is measured in probability units. Its interpretation depends on the model and the IV assumptions: a fitted additive effect is not automatically the average treatment effect for everyone in a population. This tutorial defines the model, derives its instrument-based moment condition, and makes clear what extra assumptions support a particular interpretation.

This is a structural restriction on counterfactual outcomes, not simply a regression of observed \(Y\) on \(A\). It says that the conditional mean treatment effect is constant across the conditioning groups in the model. It does not require every individual to have exactly the same treatment effect; it restricts a conditional mean. Consistency, \(Y=Y^A\), connects the observed outcome to the potential outcome under the treatment received.

From the Model to an Instrument Moment

To obtain an instrument-based estimate, suppose the following conditions hold. First, consistency holds and the instrument satisfies exclusion: changing \(Z\) does not directly change the outcome, apart from any change it causes in treatment. Second, conditional on \(X\), the instrument is independent of the potential outcomes and potential treatment assignments. Third, the instrument is relevant: treatment probability varies with \(Z\) for at least some values of \(X\). Finally, the additive SMM is correctly specified with a common \(\psi\). These conditions are substantive, not consequences of observing an association between \(Z\), \(A\), and \(Y\).

Define the outcome after subtracting the modeled treatment effect as \(Y-\psi A\). By consistency and the SMM,

Averaging over \(A\) conditional on \(Z,X\) gives \(E(Y-\psi A\mid Z,X)=E(Y^0\mid Z,X)\). Under instrument independence and exclusion, the mean of \(Y^0\) does not vary with \(Z\) conditional on \(X\), so

Thus, the instrument should not predict the mean of the adjusted outcome \(Y-\psi A\), conditional on \(X\). One corresponding population moment is

With no covariates and a binary instrument, this moment is equivalent to equating the instrument-group differences in the outcome and treatment, after scaling the latter by \(\psi\). If the first stage is nonzero, solving the equation yields the ratio below. With covariates, estimation uses conditional or appropriately weighted moments; the simple unadjusted ratio need not be suitable when the instrument groups differ in \(X\).

The derivation is direct: the moment condition implies that the mean of \(Y-\psi A\) is the same in the two instrument groups. Subtracting the \(Z=0\) equation from the \(Z=1\) equation gives

Rearranging gives the stated ratio. This equation establishes an estimand only when the assumptions and model hold. If treatment effects differ across people, or the instrument is invalid, the ratio may still be calculable but need not equal the constant \(\psi\) in the model.

Interpreting the Estimate Carefully

Under a correctly specified constant-effect SMM, \(\psi\) is the model’s common conditional mean risk difference. This is stronger than saying that an instrument is valid. In settings where treatment effects vary, a binary-instrument Wald ratio can instead have a complier-average interpretation under additional assumptions such as monotonicity (no individual whose treatment moves in the opposite direction from the instrument). That is a different interpretation and does not make the constant-effect model true. Without the constant-effect restriction or assumptions that support another specific estimand, the ratio should not be labelled the population average treatment effect.

The model is additive on the risk scale, not on the odds or log-odds scale. It also has a feasibility implication: predicted risks must remain between 0 and 1 in the covariate and treatment groups to which the model is applied. A constant risk difference can be implausible across groups with very different baseline risks, even when it summarizes an effect adequately over a narrower range.

Diagnostics, Alternatives, and Limitations

Diagnostics cannot prove the IV assumptions, particularly exclusion. They can reveal warning signs. Report the first-stage difference and its uncertainty; a small denominator makes the ratio unstable, so minor sampling variation in either contrast can produce a large change in \(\hat{\psi}_{IV}\). Inspect instrument and treatment distributions across covariate groups, and assess whether the proposed adjustment accounts for important differences in instrument assignment. Consider whether a direct pathway from the instrument to the outcome is plausible. Finally, check whether fitted risks remain in the unit interval and whether one constant risk difference is credible across the intended population.

A common mistake is to interpret a large Wald ratio as evidence of a large treatment effect. The ratio can be large because the first-stage denominator is close to zero, not because the outcome contrast is large. Another mistake is to treat a statistically strong association between \(Z\) and \(A\) as proof that \(Z\) is independent of potential outcomes or satisfies exclusion. Relevance is necessary, but it does not establish instrument validity.

The additive SMM is useful when risk differences are the intended scale and a constant conditional mean effect is plausible. It is not the only IV approach. A linear IV analysis can produce a similar ratio in the binary-instrument setting, but its causal interpretation still depends on the relevant assumptions and estimand. A control-function approach specifies a model for outcome dependence on unobserved treatment selection and requires its own structural assumptions. A conventional regression of \(Y\) on \(A\) estimates an association unless treatment assignment is adequately addressed; it does not become an IV analysis by including \(Z\) as an ordinary predictor.

This model’s main limitations are substantive and statistical. Exclusion and instrument independence cannot usually be fully tested from observed data. Weak relevance undermines precision and can make ordinary large-sample approximations unreliable. A constant additive effect can be too restrictive when effects vary, and risk differences can behave poorly when baseline risks approach 0 or 1. When those concerns are serious, the analyst should state which alternative estimand and assumptions are being used rather than silently changing the interpretation of the additive SMM estimate.