Three assumptions, three different jobs
To identify the mean outcome under a treatment regime, we need more than a formula: we need assumptions that connect the regime’s counterfactual outcomes to observed data. In a longitudinal study, those assumptions have distinct roles. Consistency connects observed outcomes with potential outcomes under the treatment actually received. Sequential exchangeability makes treatment groups comparable after conditioning on measured history. Positivity ensures the treatment choices required by the regime occur with positive probability at the relevant histories.
Consider two treatment decisions. Let \(L_0\) denote baseline covariates, \(A_0\) the first treatment, \(L_1\) covariates measured after \(A_0\) and before the second treatment, \(A_1\) the second treatment, and \(Y\) the final outcome. For a fixed regime \((a_0,a_1)\), write \(Y^{a_0,a_1}\) for the outcome that would be observed under those treatments, and \(L_1^{a_0}\) for the intermediate covariates that would arise under the first treatment \(a_0\).
These statements presume the potential outcomes and treatment histories are well-defined, the relevant conditional distributions exist, and the variables are measurable. Positivity is stated only over histories relevant to the target regime; it does not require every treatment at every conceivable history. The identification formula below concerns a fixed treatment sequence. Dynamic regimes replace \(a_0,a_1\) with rules evaluated at the observed history, but the same three roles remain.
Where each assumption enters the identification formula
Under these assumptions, the regime-specific mean is identified from the observed-data distribution by the two-time g-formula. For continuous variables, the integrals below are with respect to the relevant conditional distributions; for discrete variables they are sums.
The formula averages the observed outcome mean among people who received the regime’s treatment at both decisions. The averaging distribution for \(L_1\) is the distribution that follows the first regime treatment, and the outer average uses the target population’s baseline distribution.
By iterated expectation, start with \(E[Y^{a_0,a_1}]=\int E[Y^{a_0,a_1}\mid L_0=l_0]f(l_0)\,dl_0\).
Exchangeability makes the counterfactual distribution under \(a_0\) match that among people with \(A_0=a_0\), conditional on \(L_0\). Positivity ensures this treatment group exists at the baseline histories being averaged. Consistency identifies its observed \(L_1\) distribution with that of \(L_1^{a_0}\).
Within each baseline stratum, integrate the conditional counterfactual outcome mean over \(L_1^{a_0}\). Its distribution is identified by \(f(l_1\mid l_0,A_0=a_0)\) using first-decision exchangeability and consistency.
Second-decision exchangeability replaces the counterfactual mean at each history with the mean among those with \(A_1=a_1\). Second-decision positivity ensures that this conditional mean is supported by observed data. Consistency then equates the observed outcome in that group to \(Y^{a_0,a_1}\). Substitution gives the stated formula.
This derivation is a bookkeeping device: each assumption licenses a particular replacement. In particular, positivity does not make treatment groups exchangeable, and exchangeability does not create observations at histories where the regime’s treatment never occurs.
Worked Example: Calculating a Two-Decision Regime Mean
Suppose \(L_0\) is constant and the target regime is \(A_0=1,A_1=1\). Among people with \(A_0=1\), \(P(L_1=1\mid A_0=1)=0.4\) and \(P(L_1=0\mid A_0=1)=0.6\). The observed conditional outcome means under \(A_1=1\) are 10 when \(L_1=1\) and 4 when \(L_1=0\). Assume the three identification conditions hold, with positive treatment probabilities at both histories.
The regime-specific mean is the weighted average over the intermediate history:
The weights sum to one, \(0.4+0.6=1\). Consistency gives the observed outcome means their counterfactual interpretation under the matching treatment history; exchangeability justifies using those observed treatment groups; positivity ensures both groups are available. The arithmetic alone cannot establish any of those conditions.
Why the assumptions cannot replace one another
The formula can be written down even when an assumption fails, but then at least one of its observed-data components may no longer represent the target counterfactual quantity. The following examples isolate two different failures. In each, the other conditions are not a substitute for the failed one.
Worked Example: Exchangeability Fails Despite Treatment Support
Let \(L_0\) and \(L_1\) be constant, and hold \(A_0\) fixed. An unmeasured binary variable \(U\) has probability \(0.5\) of being 1. At the second decision, \(P(A_1=1\mid U=1)=0.8\) and \(P(A_1=1\mid U=0)=0.2\). Define \(Y^{a_0,a_1}=U+2a_1\), and let observed outcomes follow consistency.
Treatment support is positive: the marginal probability of \(A_1=1\) is \(0.5(0.8)+0.5(0.2)=0.5\). But the regime-specific mean for \(A_1=1\) is
Among those observed with \(A_1=1\), the probability \(U=1\) is \(0.5(0.8)/0.5=0.8\), and the probability \(U=0\) is \(0.2\). Thus their observed mean is \(0.8(3)+0.2(2)=2.8\), not 2.5. Since \(U\) predicts both treatment and outcome, exchangeability given the recorded history fails. Consistency and positive treatment probability do not remove that confounding.
Worked Example: Positivity Fails at an Intermediate History
Suppose \(L_0\) is constant, \(A_0\) is randomized, and \(L_1\) is equally likely to be 0 or 1 after \(A_0=a_0\). At the second decision, treatment is deterministic: \(A_1=L_1\). The target regime is \((a_0,1)\). The observed outcome is 0 for both observed histories: \(Y=0\) when \((L_1,A_1)=(0,0)\) and when \((L_1,A_1)=(1,1)\).
At \(L_1=0\), the target treatment \(A_1=1\) has probability zero, so second-decision positivity fails. To see why the target mean is not determined by the observed data, consider two possible specifications of the unobserved outcome under \((L_1,A_1)=(0,1)\): one sets it to 0, and the other sets it to 10. Both agree with all observed outcomes and with consistency for observed treatment histories. Under regime \((a_0,1)\), the first specification gives mean \(0.5(0)+0.5(0)=0\); the second gives \(0.5(10)+0.5(0)=5\). Observed data cannot choose between them.
Even if exchangeability is maintained, it cannot supply outcomes for a treatment-history combination that never occurs. A positive probability of \(A_1=1\) overall would not be enough: support is needed specifically at the histories the regime reaches.
Diagnostics, alternatives, and limitations
The assumptions are not interchangeable, and they are not all testable from the observed data. Useful diagnostics can reveal problems, but cannot prove the assumptions. Check treatment counts and estimated treatment probabilities at the baseline and intermediate histories required by the regime. Empty or very small treatment groups flag unsupported or weakly supported histories. Review whether important predictors of both treatment and outcome were measured before each decision; an omitted time-varying confounder threatens exchangeability. For consistency, verify that the treatment definitions and timing used in the analysis correspond to the regime being interpreted.
The g-formula above is one identification representation. Inverse-probability weighting uses treatment probabilities to reweight observed participants, while outcome regression models the conditional means in the displayed formula. Under the identifying assumptions, these approaches target the same regime mean, though their estimation procedures differ. Neither alternative repairs a failure of exchangeability, consistency, or positivity. In particular, extreme estimated treatment probabilities can make weighting unstable and signal limited support; modeling may smooth estimates across sparse histories, but that extrapolation is not empirical identification.
The result is also limited to the specified population, treatment regime, covariate history, and outcome. It does not guarantee that a fitted statistical model is correct, that an estimate is precise, or that the treatment is sufficiently well-defined for consistency. These are separate concerns from the identification argument.
A common reasoning error
- Wrong reasoning: “We adjusted for all measured covariates and both treatments occurred in the sample, so the regime mean is identified.”
- Correction: Adjustment supports exchangeability only if the measured history suffices to control confounding at each decision. Treatment must have positive probability at the specific histories relevant to the regime, not merely somewhere in the sample. Consistency must also connect observed outcomes to the stated intervention. Check each assumption at the step where it is used.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- In the two-decision g-formula, which assumption licenses replacing a counterfactual outcome mean with an observed mean among people receiving \(A_1=a_1\)?
- Why is a positive marginal probability \(P(A_1=a_1)\) not sufficient for sequential positivity?
- In the first worked example, verify the regime mean and explain why it is not simply the outcome mean among all observed participants.
- Can consistency by itself identify a regime mean if some treatment history in that regime is never observed? Explain.
- Name one diagnostic for weak positivity and one limitation of that diagnostic.
Solutions. (1) Second-decision exchangeability, together with consistency to interpret the observed outcome. (2) The regime may require treatment at a history where the conditional treatment probability is zero, even if that treatment occurs elsewhere. (3) The calculation is \(10(0.4)+4(0.6)=6.4\); the population outcome mean mixes treatment histories and does not isolate the specified regime. (4) No. Consistency only links observed outcomes to the treatments actually received; it cannot reveal outcomes under an unobserved treatment-history combination. (5) Sparse or empty treatment groups at relevant histories can flag weak support. Such counts diagnose possible problems but cannot establish positivity for every target history or prove exchangeability.