Why Treatment-Confounder Feedback Is a Special Case
In longitudinal studies, a treatment can change a patient’s later health status, and that status can then influence the next treatment as well as the outcome. For example, an initial therapy may change a symptom score; the symptom score may influence whether therapy continues and may also predict the final outcome. The symptom score is then both a consequence of earlier treatment and a confounder of the later treatment-outcome relationship.
This pattern is called treatment-confounder feedback. It complicates ordinary adjustment: omitting the later symptom score can leave confounding of the later treatment, while adjusting for it in a conventional outcome regression can remove part of the effect of the earlier treatment that operates through the symptom score. G-estimation addresses the sequence by using each treatment’s assignment history in its own moment equation and blipping down treatment effects in reverse time order.
The essential distinction is the timing of the histories. \(L_1\) belongs in the history used to model or condition on \(A_2\), because it is measured before \(A_2\). It does not belong in the pre-treatment history \(H_1\) for \(A_1\). Thus the same covariate can be a necessary confounder for one treatment decision and a post-treatment variable for an earlier decision.
Assumptions and the History-Conditioned Moment
Suppose treatment is binary at each time. Write \(e_t(H_t)=P(A_t=1\mid H_t)\) for the conditional probability of treatment at time \(t\), and define the treatment residual \(R_t=A_t-e_t(H_t)\). The residual is centered within the observed treatment history: \(E(R_t\mid H_t)=0\). Its value records whether treatment was received more or less often than expected for people with that history.
A structural nested mean model specifies treatment blips: the causal contrasts associated with changing treatment at a particular time, given the history before that treatment. In a simple additive model, write the blip at time \(t\) as \(\gamma_t(A_t,H_t;\psi)=\psi_t(H_t)A_t\). Here \(\psi_t(H_t)\) is the modeled effect, on the outcome scale, of setting \(A_t\) from 0 to 1 for that history. It may be constant or depend on pre-treatment history.
For a candidate parameter \(\psi\), define a blipped-down outcome recursively, starting at the final treatment time:
In a correctly specified model, the appropriate blipped-down outcome at time \(t\) removes the modeled causal blip for treatment at that time (and, in the backward sequence, blips at later times). Under consistency and sequential exchangeability, it is mean-independent of \(A_t\) given \(H_t\) when \(\psi\) is the true blip parameter. Sequential exchangeability here means that, conditional on each \(H_t\), treatment assignment at time \(t\) is independent of the relevant counterfactual outcomes under the treatment strategies being compared. Positivity requires that each treatment option being compared have positive conditional probability for histories in the target population. A correctly specified blip model and these identification assumptions support the moment equation
This is a conditional equation: it applies within histories, not merely after averaging over a population with different treatment probabilities. In practice, one can use treatment-model probabilities to form \(R_t\) and solve sample analogues of the moment equations. The treatment model must represent \(P(A_t=1\mid H_t)\) adequately for this residual construction.
Why Feedback Does Not Break the Moment Logic
The key is that sequential exchangeability is assessed at each treatment decision using the history available immediately before that decision. At time 2, that history includes \(A_1\) and \(L_1\). Thus treatment at time 2 is compared among people with the same relevant observed history, even though \(L_1\) may itself have been changed by \(A_1\). At time 1, exchangeability instead uses \(H_1=L_0\).
The moment follows directly from conditional mean independence. At the true blip parameter, suppose \(E\{U_t(\psi)\mid A_t,H_t\}=E\{U_t(\psi)\mid H_t\}\). Then:
The first equality uses \(E(A_t-e_t(H_t)\mid H_t)=0\); the second uses the conditional mean independence of the blipped-down outcome from current treatment. This argument does not require \(L_1\) to be unaffected by \(A_1\). Rather, \(L_1\) is included when forming the time-2 residual, while the time-1 moment uses the time-1 history.
Worked Example: Identifying the Relevant History at Each Visit
Consider a hypothetical two-visit study. Patients have baseline severity \(L_0\), receive an initial treatment \(A_1\), and then have a symptom score \(L_1\) measured before the second treatment \(A_2\). Suppose the initial treatment can improve the symptom score, and clinicians use that score to decide whether to continue treatment. The score also predicts the final outcome \(Y\).
Time 1. The treatment history is \(H_1=L_0\). If the conditional probability of initial treatment is \(e_1(L_0)\), the residual is \(A_1-e_1(L_0)\). The later symptom score is not part of this history because it was measured after \(A_1\).
Time 2. The treatment history is \(H_2=(L_0,A_1,L_1)\). If the conditional probability of second treatment is \(e_2(L_0,A_1,L_1)\), the residual is \(A_2-e_2(L_0,A_1,L_1)\). Including \(A_1\) and \(L_1\) allows the comparison at the second treatment decision to reflect the way treatment assignment depends on the evolving history.
These equations do not say that the symptom score is irrelevant to the earlier treatment’s effect. It may lie on a pathway from \(A_1\) to \(Y\). Instead, the time-2 equation uses it to address confounding of \(A_2\), and the time-1 equation avoids treating it as a baseline adjustment variable for \(A_1\).
Backward Blipping in a Two-Time Model
For an additive two-time model with constant blips \(\psi_1\) and \(\psi_2\), the recursive transformation is
First, the time-2 moment uses \(U_2\) and \(H_2\). After estimating the later blip, the time-1 moment uses the outcome with both treatment blips removed and \(H_1\). This order matters: leaving the later treatment effect in the outcome used for the earlier equation can make the earlier treatment residual associate with outcome variation attributable to later treatment.
Worked Example: Solving the Later-Time Moment Within History Strata
Suppose a hypothetical dataset has four patients in two \(H_2\) strata, defined here by a low or high pre-\(A_2\) symptom score. Within each stratum, two patients have treatment probability \(e_2(H_2)=0.5\), one received \(A_2=1\) and one received \(A_2=0\). Their final outcomes are:
| Symptom-score stratum | \(A_2\) | \(Y\) |
|---|---|---|
| Low | 1 | 8 |
| Low | 0 | 5 |
| High | 1 | 12 |
| High | 0 | 9 |
Assume a common additive time-2 blip \(\psi_2\). In each stratum the residuals \(A_2-e_2(H_2)\) are \(0.5\) for the treated patient and \(-0.5\) for the untreated patient. The pooled sample moment is
Setting this sample moment to zero gives \(0.5(6-2\psi_2)=0\), so \(\hat{\psi}_2=3\). The model’s estimated additive contrast is therefore 3 outcome units for the second treatment, conditional on the modeled history. The same result holds within each stratum: the treated-minus-untreated outcome difference is 3 in the low stratum and 3 in the high stratum. The illustration assumes the treatment probabilities and common-blip model are appropriate; the calculation alone cannot establish those assumptions.
Worked Example: Using the Blipped-Down Outcome for the Earlier Treatment
Now consider a separate hypothetical four-patient dataset with a common baseline history, \(e_1(H_1)=0.5\), and a later blip already estimated as \(\hat{\psi}_2=3\). The values below include a post-\(A_1\) symptom score that differs by initial treatment. For the time-1 equation, compute \(U_2=Y-3A_2\); do not include the symptom score in the time-1 treatment residual.
| Patient | \(A_1\) | Post-\(A_1\) symptom | \(A_2\) | \(Y\) | \(U_2=Y-3A_2\) |
|---|---|---|---|---|---|
| 1 | 1 | Elevated | 1 | 11 | 8 |
| 2 | 0 | Low | 0 | 5 | 5 |
| 3 | 1 | Elevated | 0 | 10 | 10 |
| 4 | 0 | Low | 1 | 7 | 4 |
The time-1 moment uses \(U_1=U_2-\psi_1A_1\) and residuals \(A_1-0.5\). Its sample equation is
The root is \(\hat{\psi}_1=4.5\) outcome units. The later treatment blip was removed before estimating the earlier one. In this simple example, the time-1 comparison is centered on the baseline history, not the post-treatment symptom score. That choice allows the earlier treatment’s effect through the symptom score to remain part of its blip contrast, under the stated structural model and identification assumptions.
What Conditioning Does—and Does Not—Mean
Conditioning on \(H_t\) in a treatment residual is not the same as fitting one conventional regression of \(Y\) on all treatments and all measured covariates. At each time, the history defines the treatment-assignment comparison for that decision. Later histories can include variables affected by earlier treatment, because those variables precede later treatment. The earlier-treatment equation still uses its own pre-treatment history after later treatment effects have been blipped down.
The method does not make feedback harmless under any circumstances. If an important common cause of \(A_t\) and the relevant counterfactual outcome is unmeasured within \(H_t\), sequential exchangeability can fail. If a treatment option is absent for some histories, the corresponding contrast is not supported there by the observed data. A poorly specified blip model or treatment probability model can also make the sample moment misleading. G-estimation addresses the particular difficulty of time-varying confounding affected by past treatment; it does not eliminate the need to justify its assumptions.
Common Mistake
- Wrong reasoning: “Because \(L_1\) is affected by \(A_1\), it must be excluded from every treatment model.” Correction: \(L_1\) is included in \(H_2\), since it precedes \(A_2\) and can confound the \(A_2\)-outcome relationship. It is excluded from \(H_1\), where it is post-treatment.
- Wrong reasoning: “Once the moment conditions on \(L_1\), the earlier treatment effect through \(L_1\) has been removed.” Correction: \(L_1\) enters the later-treatment residual, not the earlier-treatment history. The time-1 moment uses the blipped-down outcome and pre-\(A_1\) history, so the earlier blip can include effects operating through later covariates, as defined by the model.
- Wrong reasoning: “A zero pooled residual-outcome product proves the model is correct.” Correction: A sample moment can be zero at an estimate even when assumptions or model forms are wrong. Identification requires the stated causal assumptions and a suitable blip specification.
Check Your Understanding
Use the treatment timing and moment equations to answer each question.
- In a two-time study, which variables belong in \(H_2\) if \(L_1\) is measured after \(A_1\) but before \(A_2\)?
- Why can \(L_1\) be included in the time-2 treatment residual while excluded from the time-1 treatment history?
- With \(e_2(H_2)=0.3\), what is the residual \(A_2-e_2(H_2)\) for a patient with \(A_2=1\), and for one with \(A_2=0\)?
- For an additive two-time model, write the blipped-down outcome used in the time-1 moment.
- Name two assumptions needed for the history-conditioned moment equations to identify the causal blips.