Identification Under Sequential Exchangeability

What Identification Requires

A regime-specific counterfactual mean describes the average outcome that would occur if everyone followed a specified treatment regime. The longitudinal g-formula expresses that mean using observed conditional distributions. But the formula is not justified by algebra alone: its interpretation as a counterfactual mean depends on assumptions linking observed treatment and outcomes to the counterfactual quantities.

Consider treatments \(A_0,\ldots,A_K\), baseline covariates \(L_0\), and time-varying covariates \(L_1,\ldots,L_K\), where \(L_t\) is measured before treatment \(A_t\). The final outcome \(Y\) is measured after \(A_K\). A fixed regime is \(\bar a=(a_0,\ldots,a_K)\). Write \(Y^{\bar a}\) for the outcome under that regime and \(L_t^{\bar a}\) for the covariate at time \(t\) under the regime. The observed history just before \(A_t\) is \(H_t=(L_0,A_0,L_1,\ldots,A_{t-1},L_t)\).

Definition: The regime-specific counterfactual mean is \(\psi(\bar a)=E[Y^{\bar a}]\). It is identified from the observed data if its value is uniquely determined by the observed-data distribution under stated assumptions. Identification is a population-level property; it does not by itself specify an estimator or guarantee that a fitted statistical model is adequate.

The three assumptions have different jobs. Consistency links a person's observed outcome to the counterfactual outcome for the treatment actually received. Sequential exchangeability rules out treatment selection based on relevant future counterfactuals after conditioning on observed history. Positivity ensures the observed data contain treatment assignments needed to learn about the regime at histories the regime can produce.

Assumptions for a Fixed Regime

Consistency: If a person follows \(\bar a\), then \(Y=Y^{\bar a}\), and the observed covariate history equals the corresponding counterfactual history. More generally, if the observed treatment history through time \(t\) equals the regime prefix \((a_0,\ldots,a_t)\), observed covariates and outcomes up to that point agree with their counterfactual values under that prefix. This condition presumes a sufficiently well-defined treatment: different versions represented by the same treatment value must not have distinct relevant effects.

Sequential exchangeability: At each time \(t\), for every regime prefix compatible with the observed history, \(A_t\) is conditionally independent of the future counterfactual covariates and outcome under that regime, given \(H_t\). In particular, for a fixed \(\bar a\), this requires \(A_t \perp (L_{t+1}^{\bar a},\ldots,L_K^{\bar a},Y^{\bar a})\mid H_t\), with the covariate terms omitted when \(t=K\). The condition is required at each treatment decision, not merely at baseline.

Positivity for \(\bar a\): At each time \(t\), the conditional probability \(P(A_t=a_t\mid H_t=h)\) is positive for almost every history \(h\) that can occur under the regime prefix through time \(t-1\). Equivalently, the regime must not require a treatment at a history that has zero probability of receiving it in the observed data. The condition concerns histories relevant to this regime, not necessarily every conceivable history.

These are assumptions about the data-generating process, not conclusions established by a statistical test. Exchangeability is a substantive no-unmeasured-confounding condition. Positivity is a support condition. A sample can show sparse treatment observations in a history stratum, warning of practical instability, but finite data cannot prove that a population probability is strictly positive.

The Observed-Data G-Formula

Let \(p(l_0)\) denote the baseline covariate distribution, and let \(p(l_t\mid \bar l_{t-1},\bar a_{t-1})\) denote the observed conditional distribution of \(L_t\) among people whose treatment history is the specified regime prefix. Define \(m(\bar l_K,\bar a)=E[Y\mid \bar L_K=\bar l_K,\bar A_K=\bar a]\). For discrete covariates, the g-formula is a sum; for continuous covariates, the corresponding terms are integrated with respect to their conditional distributions.

\psi(\bar a) = \int m(\bar l_K,\bar a)\, p(l_0) \prod_{t=1}^{K}p(l_t\mid \bar l_{t-1},\bar a_{t-1}) \,d\bar l_K .

Here \(\bar A_K=(A_0,\ldots,A_K)\) and \(\bar a\) specifies all treatment values, including \(a_K\) in the outcome regression. The covariate-transition factors stop at \(L_K\): there is no \(L_{K+1}\) in this setup. The formula averages the outcome regression over covariate histories generated by the regime, rather than over the covariate distribution among everyone observed under their naturally occurring treatments.

Theorem: Suppose consistency and sequential exchangeability hold for \(\bar a\), and positivity holds at every treatment decision on histories reached under the regime. Suppose the conditional distributions and conditional outcome means in the formula are well-defined (using regular conditional distributions where needed), and \(E[|Y^{\bar a}|]<\infty\). Then the observed-data g-formula equals \(E[Y^{\bar a}]\).

Proof. Begin with the distribution of \(L_0^{\bar a}\), which is the baseline distribution \(p(l_0)\), because the regime does not alter baseline covariates. Suppose a regime-prefix history has been generated through \(L_{t-1}^{\bar a}\). Sequential exchangeability says that, conditional on the corresponding observed history, receiving \(A_{t-1}=a_{t-1}\) does not select people according to their future counterfactual covariate. Consistency identifies the observed \(L_t\) among those with that history and treatment with \(L_t^{\bar a}\). Positivity ensures this conditional observed distribution is available. Thus the next covariate has conditional distribution \(p(l_t\mid \bar l_{t-1},\bar a_{t-1})\). Applying this argument successively for \(t=1,\ldots,K\) gives the regime distribution of the covariate path as \(p(l_0)\prod_{t=1}^K p(l_t\mid\bar l_{t-1},\bar a_{t-1})\).

At the final decision, sequential exchangeability and positivity allow the conditional mean of \(Y^{\bar a}\) at each such history to be represented by \(E[Y\mid\bar L_K=\bar l_K,\bar A_K=\bar a]=m(\bar l_K,\bar a)\), with consistency identifying the observed outcome for those who followed the full regime. Taking the expectation of this conditional mean over the regime covariate-path distribution gives the displayed integral. Integrability makes this expectation well-defined. Hence the formula equals \(E[Y^{\bar a}]\). \(\square\)

Worked Applications

Worked Example: A Two-Treatment Regime Mean

Suppose there are two binary treatments, \(A_0,A_1\), and one binary intermediate covariate \(L_1\). Consider regime \((1,1)\). Assume the observed conditional probabilities among the regime-matching treatment histories are \(P(L_1=1\mid A_0=1)=0.6\) and \(P(L_1=0\mid A_0=1)=0.4\). The conditional outcome means under the full treatment history are \(E[Y\mid A_0=1,L_1=1,A_1=1]=0.8\) and \(E[Y\mid A_0=1,L_1=0,A_1=1]=0.3\). There are no baseline covariates.

The g-formula averages the two outcome means using the covariate distribution under \(A_0=1\):

E[Y^{(1,1)}] =0.6(0.8)+0.4(0.3) =0.48+0.12 =0.60.

The probabilities sum to one, \(0.6+0.4=1\), and the resulting mean lies between the two conditional means, \(0.3\) and \(0.8\). The calculation identifies the regime mean only if the assumptions hold, including positive probability of both treatments at the histories required by this regime.

Worked Example: Averaging Over Baseline Covariates

Let baseline \(L_0\) be binary, with \(P(L_0=1)=0.25\) and \(P(L_0=0)=0.75\). Suppose the g-formula, including any intermediate covariate averaging, gives a regime-specific conditional mean of \(0.40\) for \(L_0=1\) and \(0.20\) for \(L_0=0\).

The target is the mean for the baseline population as a whole, so the two conditional means receive baseline-population weights:

E[Y^{\bar a}] =0.25(0.40)+0.75(0.20) =0.10+0.15 =0.25.

The weights are the baseline proportions, not the proportions of these strata among people who happened to follow \(\bar a\). Using the latter would generally change the target population and would not perform the standardization in the g-formula.

Worked Example: A Positivity Failure Can Destroy Identification

Let \(H\) be a binary baseline covariate with \(P(H=1)=P(H=0)=0.5\). In the observed data, treatment \(A\) is always 0 when \(H=1\), while among \(H=0\) treatment is assigned with positive probability. Consider the mean outcome under the intervention \(A=1\). The observed data reveal \(Y^0=0\) when \(H=1\), but reveal nothing about \(Y^1\) in that stratum.

Construct two possible full-data distributions. In both, retain exactly the same observed treatment and outcome distributions. In the first, set \(Y^1=0\) for everyone with \(H=1\); in the second, set \(Y^1=1\) for everyone with \(H=1\). These choices do not change any observed record because nobody with \(H=1\) receives \(A=1\). Leave all other counterfactual and observed quantities identical between the two distributions.

The two regime means differ by the contribution from \(H=1\). If the observed, identified conditional mean among \(H=0\) under treatment is \(0.2\), the first distribution gives \(E[Y^1]=0.5(0)+0.5(0.2)=0.1\), while the second gives \(E[Y^1]=0.5(1)+0.5(0.2)=0.6\). Both distributions produce the same observed data and can satisfy consistency and conditional exchangeability; they differ only in an unobserved counterfactual quantity. Since \(P(A=1\mid H=1)=0\), positivity fails, and the observed data do not uniquely determine \(E[Y^1]\).

Why the Conditions Matter in Practice

The theorem is an identification result, not a claim that a fitted g-formula will be accurate in a particular sample. The observed conditional distributions and outcome regression must still be estimated. If a parametric model imposes incorrect functional forms, the fitted formula may not approximate the identified functional. If a required history-treatment combination is very rare, estimation can be unstable even when its population probability is technically positive.

Useful warning signs include empty or nearly empty treatment-by-history cells, fitted treatment probabilities near zero for the regime action, and large changes in estimates when a small number of observations or modeling choices are changed. These signs do not diagnose unmeasured confounding. Conversely, good balance or reasonable prediction diagnostics do not establish sequential exchangeability. The earlier material on positivity diagnostics and sensitivity to model misspecification addresses related empirical checks, but neither replaces the causal assumptions.

The g-formula and inverse-probability weighting are alternative ways to use observed data under causal identification assumptions. The g-formula represents the regime mean through covariate-transition distributions and an outcome regression; inverse-probability weighting reweights observed individuals according to treatment probabilities. Their estimation behavior can differ, but neither method repairs failure of exchangeability or lack of support for the target regime. In particular, no choice of outcome model can recover a treatment-specific outcome in a history stratum that has no observed recipients without adding extrapolation assumptions.

Common pitfall: “The treatment model has no zero probabilities, so the regime effect is identified.” This is incomplete. Positivity must hold at the relevant histories, and consistency and sequential exchangeability must also hold. Another error is to regard a good-fitting outcome regression as proof of exchangeability. Model fit concerns observed-data patterns; exchangeability concerns the relationship between treatment and counterfactual outcomes conditional on history.

Identification is also regime-specific. A history may have adequate treatment support for one regime and inadequate support for another. The regime-specific positivity condition therefore cannot be replaced by a single statement that treatment occurred in both arms overall. The relevant question is whether each specified treatment action is observed with positive probability in the histories the regime itself can generate.

Check Your Understanding

Answer each question by applying the identification argument, not by assuming that a formula is valid merely because it can be calculated.

Which assumption links an observed outcome to a counterfactual outcome when the observed treatment history matches a regime?
In the two-treatment example, what would the regime mean be if the conditional means were \(0.7\) for \(L_1=1\) and \(0.2\) for \(L_1=0\), with the same covariate probabilities?
Why is positivity needed for the observed conditional outcome mean in a history stratum?
If treatment is randomized at baseline but later treatment depends on a time-varying covariate, does baseline randomization alone establish sequential exchangeability at the later decision?

Solutions. (1) Consistency supplies the link: for people following the regime, their observed outcome equals their regime counterfactual outcome. (2) The mean is \(0.6(0.7)+0.4(0.2)=0.42+0.08=0.50\). (3) If the regime treatment never occurs in that stratum, the observed data contain no outcomes under that treatment there; distinct counterfactual outcome distributions can then agree on all observed data. (4) No. Later treatment assignment must also be conditionally independent of the relevant future counterfactuals given the observed history at that later decision; baseline randomization does not guarantee this.