G-Estimation at a Single Treatment Decision

From a Proposed Blip to an Estimating Equation

With one treatment decision, the central computational task in g-estimation is to find the blip parameter that removes the treatment effect from the observed outcome. If the parameter is correct, the resulting blipped-down outcome has the same conditional mean in the treated and untreated groups at each covariate history. This turns a counterfactual effect model into an equation that can be evaluated using observed data.

Let \(L\) denote covariates measured before treatment, \(A\in\{0,1\}\) the treatment, and \(Y\) the observed outcome. Write \(Y^a\) for the outcome that would be observed under treatment level \(a\). Consider the additive structural nested mean model with a single scalar parameter \(\psi\):

The blip function is \(\gamma(a,l;\psi)=\psi a\). It is additive because the model specifies a mean difference on the outcome scale, and it has no effect modification by \(L\). G-estimation at this decision searches for the value of \(\psi\) that makes the observed outcome minus its proposed treatment blip conditionally unrelated, in mean, to treatment.

The identifying assumptions used here are consistency (\(Y=Y^A\)), conditional exchangeability (\(Y^a\perp A\mid L\) for \(a=0,1\)), and positivity (\(0<P(A=1\mid L=l)<1\) for covariate values in the target population). Assume the relevant conditional means exist and the additive blip model is correct. Consistency and exchangeability identify the observed conditional mean in each treatment group with the corresponding potential-outcome mean. Positivity ensures that both treatment levels are represented at the histories where the contrast is to be identified.

The Conditional Moment and Its Derivation

Set \(e(L)=P(A=1\mid L)\). Under the assumptions above, define \(m_a(l)=E[Y\mid A=a,L=l]\). The model and exchangeability give

Consequently, at the true parameter \(\psi_0\), the conditional means of the blipped-down outcome agree:

This equality is the treatment-history conditional moment condition for a single decision. An equivalent form uses the propensity residual \(A-e(L)\). Since \(E[A-e(L)\mid L]=0\), equal conditional means of \(U(\psi_0)\) across the two treatment groups imply \(E[(A-e(L))U(\psi_0)\mid L]=0\). Conversely, because \(A\) is binary and positivity holds, that conditional residual equation implies the two conditional means are equal.

To verify the conditional equation directly, note that

For uniqueness and the estimator formula, first calculate the two conditional expectations involving treatment:

The first identity follows by splitting over \(A=1\) and \(A=0\): its left side is \(e(L)\{1-e(L)\}m_1(L)-\{1-e(L)\}e(L)m_0(L)\). The second follows because \(A^2=A\). Since the model gives \(m_1(L)-m_0(L)=\psi_0\), taking expectations yields

Thus the scalar moment equation is linear in \(\psi\), and its coefficient is strictly positive when \(E[e(L)\{1-e(L)\}]>0\). Solving it gives the population identification formula

This derivation also clarifies the scope of the result: the conditional equation is stronger than one averaged scalar equation. A single scalar moment identifies the common \(\psi\) under the stated homogeneous blip model; if the conditional treatment effect varies with \(L\), positive and negative discrepancies can cancel in the averaged equation.

Worked Calculations

Implementation, Diagnostics, and Alternatives

In a sample, if \(e(L_i)\) is available, the empirical moment equation is

provided the denominator is nonzero. The population justification requires the propensity \(e(L)\) to represent the conditional treatment probability. Estimating that probability in practice introduces additional conditions; the formula alone does not guarantee validity if the treatment model is inappropriate.

Several checks are useful before interpreting the result. Examine treatment probabilities across covariate histories: probabilities near zero or one make \(e(L)\{1-e(L)\}\) small, so those histories contribute little information about the blip. Check whether there are treated and untreated observations across relevant ranges of \(L\), and whether the sample denominator is small. Also inspect whether the conditional contrasts appear compatible with a common additive effect. A single pooled equation may equal zero even when stratum-specific discrepancies have opposite signs.

A common mistake is to treat a zero pooled moment as proof that every conditional moment is correct. The equation \(E[(A-e(L))U(\psi)]=0\) averages over \(L\); it can be zero through cancellation. The correction is to distinguish the model-based scalar estimating equation from the stronger conditional claim \(E[U(\psi)\mid A,L]=E[U(\psi)\mid L]\). The latter is the model implication at the true parameter, while the scalar equation is a way to estimate a common parameter under the assumed blip model.

G-estimation is not the only way to estimate a marginal treatment contrast. The parametric g-formula models outcome means and standardizes them over covariates; inverse-probability weighting reweights treatment groups using treatment probabilities. In contrast, this one-decision g-estimation equation explicitly blips down the outcome and uses the treatment-assignment residual. These approaches rely on different modeling steps, but none repairs a lack of positivity, unmeasured confounding, or a substantively incorrect target model automatically.

The method here is deliberately limited to a binary treatment at one decision and a constant additive mean blip. It does not establish that individual effects are constant, and it does not specify a model for effect modification. Its principal value is conceptual and practical: it makes the structural nested mean model estimable through a transparent conditional moment, while exposing how overlap and model specification govern the information in that moment.