G-Estimation For Binary Structural Mean Models

Why G-Estimation Blips Down the Outcome

With a binary outcome, a treatment effect on the log-odds scale cannot generally be estimated by treating the instrument as if it were an ordinary covariate in a logistic regression. G-estimation instead asks which proposed treatment effect, when removed from the observed outcome risk, leaves a risk that is compatible with the instrument being independent of the outcome under no treatment. The crucial operation is called blipping down.

Let $Y$ be a binary outcome, $A$ a binary treatment, $Z$ an instrument, and $L$ measured pre-treatment covariates. Write $Y^a$ for the potential outcome under treatment level $a$. The parameter $\psi$ describes the treatment effect on the log-odds scale. This is a structural parameter: it is not automatically the coefficient from an ordinary logistic regression of $Y$ on $A$.

Definition: A logistic structural mean model specifies the conditional mean contrast $$ \operatorname{logit} E(Y^a\mid A,Z,L)-\operatorname{logit} E(Y^0\mid A,Z,L)=\psi a,\qquad a\in\{0,1\}. $$ Here $\operatorname{logit}(p)=\log\{p/(1-p)\}$. Under consistency, the observed conditional risk $p(A,Z,L)=E(Y\mid A,Z,L)$ obeys $$ \operatorname{logit} p(A,Z,L)=\operatorname{logit} E(Y^0\mid A,Z,L)+\psi A. $$ For a common-effect model, the same $\psi$ applies across the covariate and instrument strata represented in the model.

For a proposed value $\psi$, the model implies a no-treatment, or blipped-down, risk:

g_\psi(A,Z,L)=\operatorname{expit}\{\operatorname{logit}p(A,Z,L)-\psi A\}, \qquad \operatorname{expit}(x)=\frac{1}{1+\exp(-x)}.

The quantity $g_\psi$ is a conditional mean implied by the structural model. It is not an observed binary outcome and does not reveal any one person's $Y^0$. Its role is to produce, under the correct model and assumptions, the conditional mean of the untreated potential outcome given the observed variables.

Assumptions and the G-Estimating Equation

The instrument must satisfy the usual instrumental-variable requirements, conditional on $L$. In particular, it must be independent of the relevant potential outcomes and potential treatment choices given $L$, must affect the outcome only through treatment (the exclusion restriction), and must predict treatment in the population or strata used for estimation. Consistency links the observed outcome to the potential outcome under received treatment. Positivity requires meaningful support for instrument and treatment values within covariate strata being analyzed. The structural mean model must also adequately represent the effect on the chosen log-odds scale.

Instrument independence implies that the mean untreated potential outcome does not vary with $Z$ conditional on $L$: $E(Y^0\mid Z,L)=E(Y^0\mid L)$. At the correct $\psi$, averaging the blipped-down conditional mean over the observed treatment distribution therefore gives no residual association between that risk and the instrument. A convenient population moment is:

E\left[\{Z-E(Z\mid L)\}g_\psi(A,Z,L)\right]=0.

With a binary instrument and no covariates, this is equivalent to equality of the average blipped-down risks across the two instrument groups. With covariates, residualizing $Z$ by its conditional mean makes the moment conditional on $L$, rather than relying on an unadjusted comparison that could reflect differences in covariate composition.

Key result: G-estimation chooses the value of $\psi$ that solves the instrument–blipped-risk moment condition. In a sample, estimate the observed risk model $p(A,Z,L)$, substitute its fitted risks into $g_\psi$, and find the root of the corresponding sample moment. Valid interpretation depends on the IV assumptions and the structural model, not on the equation alone.

Why the Moment Condition Works

The justification follows directly from the structural model and the law of total expectation. At the true parameter $\psi_0$, the model and consistency give:

g_{\psi_0}(A,Z,L) =\operatorname{expit}\{\operatorname{logit}E(Y\mid A,Z,L)-\psi_0 A\} =E(Y^0\mid A,Z,L).

Taking conditional expectations over the observed treatment within each $(Z,L)$ stratum gives:

E\{g_{\psi_0}(A,Z,L)\mid Z,L\} =E\{E(Y^0\mid A,Z,L)\mid Z,L\} =E(Y^0\mid Z,L).

Under instrument independence, the final expression equals $E(Y^0\mid L)$, which does not vary with $Z$. Therefore, conditional on $L$, the covariance between $Z$ and $g_{\psi_0}$ is zero. Multiplying the conditional covariance by the covariate distribution and averaging yields the stated population moment. This derivation also shows why the blipped risk must be averaged over treatment: the risk conditional on $A,Z,L$ need not itself be independent of $Z$.

Worked Example: Solving the G-Estimating Equation

Consider hypothetical data with a binary instrument $Z$, no covariates, and the following treatment and outcome counts. The sample sizes are large enough that the displayed cell proportions are treated as the fitted conditional risks for this calculation.

Instrument	Treatment status	Number	Number with $Y=1$	Observed risk
$Z=0$	$A=0$	825	165	0.2000
$Z=0$	$A=1$	275	75	0.2727
$Z=1$	$A=0$	440	88	0.2000
$Z=1$	$A=1$	660	180	0.2727

Treatment prevalence is $275/1100=0.25$ when $Z=0$, compared with $660/1100=0.60$ when $Z=1$, so the instrument is relevant in these data. Let $q(\psi)$ be the blipped-down risk among treated individuals. For the observed treated risk $75/275=180/660=3/11$, its odds are $(3/11)/(8/11)=3/8$. Thus:

q(\psi)=\operatorname{expit}\{\log(3/8)-\psi\}.

Among untreated individuals, the blipped-down risk remains $0.2000$, regardless of $\psi$, because $A=0$. The average blipped-down risk in each instrument group is therefore:

\begin{aligned} \bar g_0(\psi)&=0.75(0.2000)+0.25q(\psi),\\ \bar g_1(\psi)&=0.40(0.2000)+0.60q(\psi). \end{aligned}

Set the group averages equal. Subtracting the first from the second gives $0.35\{q(\psi)-0.2000\}=0$. Since $0.35\ne 0$, the root requires $q(\psi)=0.2000$. The logit of $0.2000$ is $\log(0.2/0.8)=\log(1/4)$. Solving for $\psi$:

\psi=\log(3/8)-\log(1/4)=\log(3/2)=0.4055.

The estimated structural odds ratio is $\exp(0.4055)=1.500$. Under the model and IV assumptions, this says that treatment multiplies the conditional odds of the outcome by 1.5 on the specified structural scale. It is not a claim that the outcome risk increases by 50%; an odds ratio and a risk ratio are different quantities.

Implementing G-Estimation in Practice

In an applied analysis, the conditional risk $p(A,Z,L)$ is usually unknown. Estimate it using an outcome-risk model that includes the observed treatment, instrument, and relevant covariates, with sufficient flexibility for their relationships when the sample allows. For each candidate $\psi$, calculate fitted blipped-down risks and evaluate the sample analogue of the moment. The estimate is a root of that equation. If several valid instruments are available, they can provide multiple moment conditions; estimation then requires a stated rule for combining them, and additional moments can support an overidentification check.

A practical sequence is:

Check the design.
Identify the instrument, treatment, outcome, covariates, and the substantive basis for relevance, independence, and exclusion.

Fit the observed risk.
Estimate $p(A,Z,L)$ without interpreting its treatment coefficient as the structural effect.

Blip down and solve.
For candidate values of $\psi$, compute $g_\psi$ and find where the instrument-based sample moment is zero.

Quantify uncertainty and inspect stability.
Use inference that accounts for estimation of the risk model, and examine the first stage, support, fitted risks, and behavior of the estimating equation.

Worked Example: Checking Instrument Relevance

In a separate hypothetical sample, 500 people have $Z=0$, of whom 245 receive treatment; another 500 have $Z=1$, of whom 255 receive treatment. The estimated treatment prevalences are $245/500=0.4900$ and $255/500=0.5100$. The first-stage difference is $0.5100-0.4900=0.0200$, only two percentage points.

For an approximate standard error of this difference, treating the two groups as independent:

SE=\sqrt{\frac{0.49(0.51)}{500}+\frac{0.51(0.49)}{500}} =\sqrt{0.0009996}=0.0316.

The difference is about $0.0200/0.0316=0.632$ standard errors from zero. This is weak evidence of a first-stage difference in this sample, and the small estimated difference warns that the G-estimate may be unstable or imprecise. It does not prove the instrument is irrelevant: the calculation is a sample diagnostic, while relevance is a population condition. Nor can a strong first stage establish instrument independence or exclusion.

Worked Example: Interpreting a Candidate Blip

Suppose a fitted conditional outcome risk for a treated person’s covariate and instrument stratum is $p=0.4000$, and consider the candidate structural effect $\psi=\log(2)=0.6931$. The observed-risk odds are $0.4/0.6=2/3$. Blipping down divides these odds by $\exp(\psi)=2$, giving $1/3$; the resulting risk is:

g_\psi=\frac{(2/3)/2}{1+(2/3)/2} =\frac{1/3}{4/3}=0.2500.

Thus the candidate model maps a treated conditional risk of $0.4000$ to an implied untreated conditional mean of $0.2500$. This calculation alone does not establish that $\psi$ is correct. The candidate is supported by G-estimation only if the resulting blipped-down risks satisfy the instrument moment across the relevant observations and covariate strata.

Diagnostics, Comparisons, and Limitations

A meaningful root is not guaranteed. Inspect the first-stage association within covariate strata, the overlap of treatment and instrument values, fitted risks near zero or one, and the estimating function over a plausible range of $\psi$. A flat function or multiple roots signals weak information or ambiguity; no root can signal incompatibility between the fitted risk model, structural model, and moment conditions. In finite samples, such behavior can also arise from estimation noise. Report it rather than selecting a convenient root without explanation.

G-estimation differs from ordinary logistic regression, which describes a conditional association in the observed data and does not by itself solve treatment confounding with an instrument. It also differs from residual-inclusion methods, which add a first-stage residual or generalized residual to an outcome model. These approaches can target different quantities and rely on different model specifications. G-estimation directly imposes the IV moment on a blipped-down risk, but it still depends on correct structural and IV assumptions and careful estimation of the observed risk.

The common mistake is to calculate a blipped-down risk for each person and call it that person's observed counterfactual $Y^0$. It is not: $g_\psi$ is a conditional mean, which may take any value between zero and one. A second mistake is to interpret $\psi$ as an ordinary logistic-regression coefficient or to assume that instrument balance after blipping proves the instrument valid. The moment is a consequence of the assumptions; observed balance cannot verify the untestable independence and exclusion restrictions.

Finally, the structural effect is tied to the specified scale and homogeneity assumption. If treatment effects differ across people or covariate strata, a single $\psi$ may be a poor model or may summarize an instrument- and population-dependent quantity. The fitted risk model, limited overlap, sparse outcome cells, and weak instruments can further affect the result. G-estimation provides a principled estimation strategy under its assumptions, not a remedy for a poorly supported instrument or a misspecified structural model.

Check Your Understanding

Answer each question, then compare with the solutions.

For $p=0.30$, $A=1$, and $\psi=\log(2)$, calculate the blipped-down risk.
Why does the G-estimating equation use an instrument residual such as $Z-E(Z\mid L)$?
Suppose $A=0$ and the fitted observed risk is $0.18$. What is the blipped-down risk for any value of $\psi$?
Does a strong association between $Z$ and $A$ establish the exclusion restriction? Explain.

Solutions. (1) The observed odds are $0.30/0.70=3/7$. Dividing by 2 gives odds $3/14$, so the risk is $(3/14)/(1+3/14)=3/17=0.1765$. (2) Conditional residualization removes differences in the instrument's distribution explained by $L$, so the moment tests the instrument association within covariate strata rather than conflating it with covariate composition. (3) When $A=0$, the term $\psi A$ is zero, so $g_\psi=0.18$. (4) No. It supports relevance, but exclusion concerns whether $Z$ affects $Y$ only through $A$, a separate substantive assumption.

Instrument	Treatment status	Number	Number with \(Y=1\)	Observed risk
\(Z=0\)	\(A=0\)	825	165	0.2000
\(Z=0\)	\(A=1\)	275	75	0.2727
\(Z=1\)	\(A=0\)	440	88	0.2000
\(Z=1\)	\(A=1\)	660	180	0.2727