Why Identification Requires History-Specific Comparisons
In a longitudinal study, treatment decisions recur as a person’s health and treatment history evolve. A treatment at one visit may affect later covariates, and those covariates may affect both later treatment and the outcome. The causal question for a structural nested mean model is therefore not simply whether people who ever received treatment had different outcomes. It is whether, at a particular decision time, the treatment options can be compared among people with the same observed history, in a way that identifies the model’s nested treatment contrast.
The key assumptions are consistency, positivity, and sequential exchangeability. They do different jobs. Consistency connects observed data to potential outcomes under the treatment actually received. Positivity requires the treatment options being compared to occur for each relevant history. Sequential exchangeability rules out confounding of each treatment decision by unmeasured information, conditional on the observed past. Together, and with a correctly specified blip model when a parametric model is used, they identify the causal blip contrasts represented by the model. None of the assumptions can be replaced by merely adjusting for a long list of variables.
Histories, Potential Outcomes, and Blips
Suppose treatment is assigned at times \(t=1,\ldots,K\). Let \(A_t\) denote treatment at time \(t\), and let \(L_t\) denote covariates measured before that treatment decision. Write \(\bar A_t=(A_1,\ldots,A_t)\) and \(\bar L_t=(L_1,\ldots,L_t)\). The observed pre-treatment history at time \(t\) is \(H_t=(\bar L_t,\bar A_{t-1})\). A treatment history specifies the treatment received at every decision time. Let \(Y^{\bar a}\) be the outcome that would be observed under treatment history \(\bar a\).
A structural nested mean model describes treatment effects as a sequence of blips: contrasts between nested counterfactual outcomes whose treatment histories differ at a particular decision. For instance, a time-\(t\) blip compares treatment at \(t\) with no treatment at \(t\), given the same prior history; the precise nested contrast also follows the model’s convention for subsequent treatment. Earlier treatments and covariates are not interchangeable with the current treatment decision. In particular, a time-\(t\) blip is not automatically the raw difference in final outcomes between everyone treated and untreated at \(t\).
For the final treatment time, there are no later treatment decisions to account for. The history-specific blip can then be written as a conditional mean contrast between the two potential outcomes under the same past history and different values of \(A_K\). At an earlier time, later treatment effects must be handled according to the nested model; a raw outcome comparison can combine the time-\(t\) effect with effects of later treatment.
What the Assumptions Support
Each assumption licenses a particular link in the identification argument. The comparison is always tied to a treatment decision and its history, not to an unrestricted comparison of treatment groups.
| Assumption | History-specific comparison it supports | What it does not guarantee |
|---|---|---|
| Consistency | For people who actually followed a specified treatment history, their observed outcome represents the potential outcome under that history. | It does not make treatment groups comparable or supply outcomes for histories nobody followed. |
| Positivity | Within each relevant observed history, the data include a chance of receiving each treatment option being contrasted. | It does not ensure that treatment assignment is unconfounded or that the model is correct. |
| Sequential exchangeability | Among people with the same observed history, treatment groups can represent the corresponding potential-outcome means. | It does not follow from observing many covariates; it is an assumption about all common causes being adequately measured and conditioned on. |
Sequential exchangeability is imposed at each decision time, conditional on the history available before that decision. That history must include relevant past treatment and covariate information, including covariates affected by earlier treatment when they confound later treatment. The assumption does not say that later covariates are independent of earlier treatment. Nor does it say that treatment is unconfounded without conditioning on history.
Positivity is also local to the decision. A treatment option may be common overall yet absent for people with a particular history. A comparison for that history is then unsupported, even if the overall treatment groups are large. The relevant histories are those occurring in the target population and needed for the blip contrast.
Why These Assumptions Identify a Contrast
The basic argument can be seen at a final treatment decision, conditional on a history \(H_K=h\). Let \(Y^{\bar a_{K-1},a}\) denote the potential outcome under the same prior treatment history and current treatment \(a\). Sequential exchangeability says that, within this history, those assigned \(A_K=a\) represent the potential outcomes under that choice. Consistency then equates those potential outcomes with observed outcomes among people who actually received \(a\). Positivity ensures such people exist in the population for both choices.
The first equality is justified by exchangeability, and the second by consistency. Positivity is needed to define the observed conditional mean on the right. Applying the argument to \(a=1\) and \(a=0\) identifies the conditional blip:
For a full longitudinal regime, the same logic is applied successively at each treatment decision, while averaging over covariate histories under the regime being considered. This yields the longitudinal g-formula. In a discrete-covariate setting, one version is:
Here \(L_1\) is drawn from its baseline distribution, and the product represents the covariate-history distribution under the specified treatment regime. The formula is not a claim that observed covariates are unaffected by treatment. Instead, it uses their conditional distributions given the treatment history, then averages the outcome means over the covariate histories that would arise under the regime. Consistency, sequential exchangeability, and positivity are what allow these observed conditional distributions and means to represent the required counterfactual quantities.
For a structural nested model, the formula identifies the regime-specific means from which the model’s nested contrasts are defined. If the model represents those contrasts with parameters, the assumptions identify the causal quantities the parameters are intended to encode. They do not by themselves guarantee that a chosen functional form for the blip is correct or that its parameters can be uniquely recovered if distinct parameter values imply the same contrasts.
Worked Example: A Final-Time Blip Within Each History
Consider a hypothetical study with a binary final-visit treatment \(A_2\), a pre-treatment risk group \(L_2\), and a final outcome measured in symptom-score points. Suppose the observed conditional means are as follows.
| Risk group \(L_2\) | Mean outcome if \(A_2=1\) | Mean outcome if \(A_2=0\) |
|---|---|---|
| Lower risk | 8 points | 5 points |
| Higher risk | 14 points | 10 points |
Assume consistency, positivity for both treatments in each risk group, and sequential exchangeability conditional on \(H_2\), which includes the prior treatment and measured covariate history as well as \(L_2\). For illustration, suppose the displayed means are also conditional on the same prior treatment history.
In the lower-risk history, the identified final-time blip is \(8-5=3\) symptom-score points. In the higher-risk history, it is \(14-10=4\) points. For example, the first calculation is:
The interpretation is a higher mean symptom score under treatment by 3 points in the lower-risk history, under the stated assumptions and treatment comparison. The data do not show that treatment causes this difference merely because the two observed means differ. The interpretation depends on the assumptions, especially that the histories contain the confounders of the final treatment decision and outcome.
Worked Example: Standardizing Across Histories
Use the hypothetical history-specific means above, and suppose that in the target population half the people are in each risk group. The treatment is more common in the lower-risk group: \(P(A_2=1\mid L_2=\text{lower})=0.75\), while \(P(A_2=1\mid L_2=\text{higher})=0.25\). Assume the conditions for identifying both history-specific contrasts hold.
The mean outcome under treatment, standardized to the target population’s equal distribution of risk groups, is:
The corresponding mean under no treatment is:
Thus the standardized contrast is \(11-7.5=3.5\) points. By contrast, the crude observed treated mean weights the lower-risk group at 0.75 and the higher-risk group at 0.25:
Among untreated people, the lower-risk group’s share is 0.25 and the higher-risk group’s share is 0.75, so:
The crude difference is \(9.5-8.75=0.75\) points, not the standardized contrast of \(3.5\) points. The treatment groups have different histories, so the crude contrast mixes within-history treatment differences with differences in the distribution of risk. Standardizing compares the treatment options using the same target distribution of histories.
Worked Example: A Missing Treatment Option
Suppose a hypothetical study has a high-risk history \(H_2=h\) in which every person received treatment: \(P(A_2=1\mid H_2=h)=1\). Assume the outcome ranges from 0 to 20 points, and the observed mean among treated people with this history is 14 points.
Consistency identifies the mean under treatment for this history as 14 points. But no one with this history received no treatment, so the observed data provide no conditional mean \(E[Y\mid H_2=h,A_2=0]\). Sequential exchangeability cannot fill in a comparison group that does not exist. Positivity fails for the no-treatment option.
Even if the assumptions needed to transport observed outcomes were otherwise plausible, the no-treatment mean could be anywhere from 0 to 20 points based on the stated outcome range alone. The treatment-minus-no-treatment blip could therefore range from \(14-20=-6\) to \(14-0=14\) points. These are not a data-identified point estimate; they illustrate how little the observed treated outcomes alone determine. Extrapolating a model to supply the missing mean adds assumptions beyond positivity and exchangeability.
Common Mistake: Treating the Assumptions as Interchangeable
A frequent error is to say, “We adjusted for the covariates, so the treatment effect is identified.” Adjustment does not establish any of the three conditions automatically. If an important common cause of treatment and outcome is unmeasured, sequential exchangeability may fail. If some treatment option never occurs for a relevant history, positivity fails even when every measured confounder is included. If the treatment or outcome definition is ambiguous, consistency may fail because the observed treatment does not correspond to a single well-defined intervention.
A second error is to compare final outcomes by current treatment while ignoring treatment history. In longitudinal settings, people with the same current treatment can have different prior treatments and covariate histories. Moreover, earlier treatment may have changed covariates that affect current treatment and the outcome. A structural nested blip is a history-specific nested contrast; its identification requires the assumptions to hold at the decision time and for the particular treatment histories in that contrast.
Check Your Understanding
Use the definitions and comparisons in this tutorial to answer the following questions.
- At a particular treatment decision, what information belongs in the history on which sequential exchangeability is conditioned?
- Explain why consistency alone cannot identify a treatment contrast when nobody with a given history received one of the treatment options.
- In the standardization example, why did the crude outcome difference differ from the contrast standardized to equal risk-group proportions?
- For an earlier treatment time, why might the raw final-outcome difference fail to represent the structural nested blip at that time?
- If positivity fails for one treatment option within a history, what can the observed outcomes identify about the missing counterfactual mean without additional assumptions?