Identification of Blip Parameters Under Sequential Exchangeability

What It Means to Identify a Blip Parameter

A structural nested mean model (SNMM) describes how the mean outcome changes when treatment at one decision time is changed, while the rest of the treatment regime is specified. Identification asks whether the distribution of observed treatment, covariate, and outcome data determines that change. It is a population-level question, distinct from whether a particular estimator has small bias or variance.

Consider decision times \(t=1,\ldots,K\). Let \(L_t\) be the covariate history measured before treatment \(A_t\), and write \(H_t=(\bar L_t,\bar A_{t-1})\) for the observed history available at that decision. Let \(Y^{\bar a}\) denote the outcome that would be observed under treatment regime \(\bar a=(a_1,\ldots,a_K)\). Similarly, \(L_t^{\bar a_{t-1}}\) is the covariate value at time \(t\) under the preceding treatment history. The reference treatment is denoted \(0\), and \(\bar 0\) is the regime that assigns the reference treatment at every time.

This definition makes the target precise: it is a conditional mean contrast for specified counterfactual regimes and a specified history. In a recursive SNMM, later blips are used to remove treatment effects from the outcome before an earlier blip is assessed. The earlier material on sequential blip-down establishes that recursive construction; here the question is whether the counterfactual means, and therefore the blips, are determined by observed data.

Assumptions and the Identification Formula

The assumptions concern the connection between observed and counterfactual data. For clarity, suppose there is no loss to follow-up; censoring would require additional assumptions and methods.

• Consistency. If a person follows regime \(\bar a\), then the observed outcome equals \(Y^{\bar a}\), and covariates observed up to time \(t\) equal the corresponding counterfactual covariates under the treatment history actually received. The treatment versions represented by each \(a_t\) must be well defined.
• Sequential exchangeability. At each \(t\), conditional on \(H_t\), treatment assignment is independent of the counterfactual covariates and outcomes under every regime of interest. In notation, \(A_t\) is conditionally independent of \(\{L_{s}^{\bar a_{s-1}}:s>t,\;Y^{\bar a}: \bar a\}\) given \(H_t\).
• Positivity. For each regime and history used in the target contrast, every treatment value required by that regime has positive conditional probability given the history. For discrete treatments, this means \(P(A_t=a_t\mid H_t=h_t)>0\) on the relevant support.

For a fixed regime \(\bar a\), the longitudinal g-formula identifies its outcome mean by combining the observed covariate transition distributions under that regime with the observed outcome distribution. To make the factorization explicit, suppose the covariates and treatment are discrete. Then

For continuous covariates, replace the sums by integrals. Conditional regime means are identified by the corresponding conditional version of this same factorization, restricted to histories with positive probability under the specified past regime. These identified means give the blip contrast: calculate the mean under \((\bar a_{t-1},a_t,\bar 0)\), calculate it under \((\bar a_{t-1},0,\bar 0)\), and subtract.

Proof. Fix a regime \(\bar a\). By consistency, among individuals whose observed treatment history is \(\bar a\), the observed outcome and covariates equal the counterfactual outcome and covariates under that regime. Sequential exchangeability permits the conditional distribution of each next covariate under the intervention to be replaced by its observed conditional distribution given the preceding history and treatment. Positivity ensures that each such observed conditional distribution is defined for the treatment values required by \(\bar a\). Applying the law of total expectation successively over \(L_1,\ldots,L_K\) gives the displayed product formula for \(E[Y^{\bar a}]\). The same argument conditional on the past-regime history identifies the conditional means used in the blip definition, on their supported histories. Their difference is therefore identified. Finally, if two parameter values \(\psi_1,\psi_2\) yield the same blip function on that support, injectivity implies \(\psi_1=\psi_2\); thus the parameter itself is identified. \(\square\)

Worked Identification Calculations

What the Assumptions Do—and Do Not—Guarantee

The theorem separates three distinct obstacles. Consistency connects the observed outcome to the counterfactual outcome for the treatment actually received. Sequential exchangeability rules out unmeasured confounding conditional on the recorded history. Positivity ensures the data include the treatment choices needed to learn each regime-specific quantity. None of these assumptions alone is enough: exchangeability does not supply outcomes for a treatment level that never occurs, and positivity does not eliminate confounding.

There are two further qualifications. First, identification is only on the relevant history support. If a history can occur under one past regime but not another, the conditional contrast at that history may not be identified for both regimes. Second, identification of the entire blip function is not automatically identification of a finite-dimensional parameter. The map from \(\psi\) to \(\gamma_t(\cdot;\psi)\) must distinguish parameter values on the observed support. A model whose covariate modifier varies only outside that support cannot identify its coefficient.

A useful alternative route is the inverse-probability-weighted marginal structural model (MSM), which reweights observations to represent treatment regimes. The g-formula directly models covariate transitions and outcomes; weighting instead models treatment probabilities. Both rely on appropriate exchangeability and positivity conditions, and both can be unstable when treatment probabilities are near zero. The present argument identifies the regime means and blip contrasts; it does not choose an estimator or derive the g-estimation moment conditions.

In practice, inspect estimated treatment probabilities and regime-specific history counts. Empty or sparse treatment-history combinations are warnings about positivity, not merely inconveniences for computation. Large standard errors or extreme weights can indicate weak practical support. Diagnostics cannot prove exchangeability, which is an assumption about unmeasured causes, and a well-fitting model cannot repair a structural lack of support. Parametric extrapolation can produce a number where positivity fails, but that number is supplied by the extrapolation assumptions rather than identified nonparametrically from the observed data.

Common Pitfall

Wrong reasoning: “The observed treatment groups have different mean outcomes, so the difference identifies the blip.” This conclusion generally ignores both time-varying covariates and treatment assignment. A raw group difference need not compare the same history distribution, and treatment groups may differ because of confounding.

Correction: First specify the two regimes in the blip contrast. Then identify each regime-specific conditional mean using consistency, sequential exchangeability, and positivity—for example, through the longitudinal g-formula—and take their difference. Finally, check that the proposed SNMM maps this function to one unique parameter value on the supported histories.