Why Audit the Whole Timeline?
In a longitudinal study, each treatment decision may depend on covariates measured after earlier treatment. A credible assumption audit therefore has to follow the timeline: identify what is known at each decision, which counterfactual quantities are relevant, and whether the observed data support the treatment choices being compared. A treatment that is well represented at baseline can still be unsupported in a later history.
Consider three treatment decisions, followed by an outcome. Let \(L_0\) be baseline covariates, \(A_0\) the first treatment, \(L_1\) covariates measured next, \(A_1\) the second treatment, \(L_2\) the next covariates, and \(A_2\) the third treatment. The final outcome is \(Y\). Treatment and covariate histories are ordered as observed; \(L_t\) is measured before \(A_t\).
The superscripts specify the treatment history that precedes a potential quantity. For example, \(L_2^{1,0}\) is the covariate value that would be measured after \(A_0=1,A_1=0\); \(Y^{1,0,1}\) is the outcome under the full sequence \((1,0,1)\). These are individual-level counterfactual quantities, not additional observed columns.
Three Assumptions, Matched to the Timeline
Consistency does not say that a person’s observed outcome equals every potential outcome. It links the observed outcome only to the counterfactual under that person’s realized sequence. The same treatment label must also represent a sufficiently well-defined intervention: if materially different versions are mixed under one label, the potential outcome indexed by that label may not be well-defined.
More explicitly, the future vector at decision \(t\) includes the potential covariates not yet measured and the potential outcomes under the treatment sequences under consideration. For example, at \(t=1\), it includes \(L_2^{a_0,a_1}\) and \(Y^{a_0,a_1,a_2}\) for relevant continuations. This is a substantive no-unmeasured-confounding assumption conditional on the recorded history; it is not established merely by observing many covariates.
Positivity is history-specific. A marginal treatment proportion between zero and one does not guarantee that both treatments occur within every relevant history. For a target sequence, the condition concerns its successive choices on histories generated under its earlier choices, not every logically imaginable history.
Worked Example: Map Observations to Counterfactuals
Worked Example: Map Observations to Counterfactuals
Suppose two people have the following records. The column \(Y^{000}\) is shown to distinguish a counterfactual value from the observed outcome; it is not generally observed for people who did not receive \(000\).
| Person | \(L_0\) | \((A_0,A_1,A_2)\) | Observed \(L_1,L_2,Y\) | \(Y^{000}\) |
|---|---|---|---|---|
| P | 0 | (1,0,1) | 1, 0, 8 | Unobserved |
| Q | 1 | (0,0,0) | 0, 1, 4 | 4 |
For P, consistency gives \(L_1=L_1^1=1\), \(L_2=L_2^{1,0}=0\), and \(Y=Y^{1,0,1}=8\). It does not give \(Y^{000}=8\); that value is unobserved. For Q, consistency gives \(L_1=L_1^0=0\), \(L_2=L_2^{0,0}=1\), and \(Y=Y^{000}=4\). These equalities illustrate the exact links consistency supplies, not a way to recover unobserved counterfactuals.
The histories are also distinct: Q’s record supports the observed history \((L_0=1,A_0=0,L_1=0,A_1=0,L_2=1)\) before \(A_2\). P’s record supports \((L_0=0,A_0=1,L_1=1,A_1=0,L_2=0)\). Treatment support at one of these histories says nothing by itself about support at the other.
A Worked Identification Result
The assumptions can be audited against a specific target, such as the mean outcome under a fixed sequence \(\bar a\). The result below makes clear which covariate distributions are needed. In particular, the exchangeability condition includes future potential covariates as well as outcomes; outcome exchangeability alone would not justify replacing their distributions with observed conditional distributions.
where \(\bar l_t=(l_0,\ldots,l_t)\) and \(\bar a_t=(a_0,\ldots,a_t)\). For discrete covariates, the integral is replaced by a sum. The factors are observed-data distributions evaluated along the specified treatment sequence.
Proof. Begin with the baseline distribution: \(L_0\) is measured before treatment, so its distribution under the fixed sequence is \(f(l_0)\). At decision 0, sequential exchangeability makes \(A_0\) independent, conditional on \(L_0\), of the future potential covariates and outcomes under the relevant continuations. Thus the distribution of \(L_1^{a_0}\) conditional on \(L_0=l_0\) is the same as its distribution among those with \(L_0=l_0,A_0=a_0\). Consistency identifies the latter distribution with the observed conditional distribution \(f(l_1\mid l_0,a_0)\). Positivity ensures this conditional distribution is defined on the histories used.
Now fix \(l_0,l_1\) along the sequence. At decision 1, sequential exchangeability conditional on the observed history makes the distribution of \(L_2^{a_0,a_1}\) equal to its distribution among people with that history and \(A_1=a_1\). By consistency, this is \(f(l_2\mid l_0,l_1,a_0,a_1)\). Positivity again supplies support. At decision 2, exchangeability for \(Y^{\bar a}\), conditional on the history and past treatments, gives its conditional mean as the mean among those with \(A_2=a_2\); consistency identifies that mean as the observed conditional mean of \(Y\). Multiplying the baseline and covariate transition distributions and integrating this terminal conditional mean yields the displayed formula. \(\square\)
The derivation follows the observed histories decision by decision. It does not infer the distribution of a future counterfactual covariate from exchangeability for the outcome alone. That distinction matters whenever a treatment affects later covariates.
Worked Example: Check Sequential Support Numerically
Worked Example: Check Sequential Support Numerically
Suppose \(X=L_0\) is binary, with \(P(X=0)=0.5\). In the history \(X=0\), let \(P(A_0=1\mid X=0)=0.4\). Among that history with \(A_0=1\), suppose \(P(L_1=1\mid X=0,A_0=1)=0.6\). At the resulting history, let \(P(A_1=1\mid X=0,A_0=1,L_1=1)=0.25\). Further suppose \(P(L_2=0\mid X=0,A_0=1,L_1=1,A_1=1)=0.5\), and \(P(A_2=1\mid X=0,A_0=1,L_1=1,A_1=1,L_2=0)=0.2\). Calculate the probability of this observed path.
By the chain rule, its probability is the product of the listed conditional probabilities:
Each treatment choice on this particular path has positive probability, so the path is supported. This calculation does not establish positivity everywhere: one must check the relevant treatment probabilities at every history that can arise under the target sequence. Nor does positive support establish exchangeability; that is a separate causal assumption.
Worked Example: Find a Hidden Support Failure
Worked Example: Find a Hidden Support Failure
A study reports that 60% of participants received treatment \(A_1=1\), so both values appear frequently overall. But among participants with \(L_0=0,A_0=1,L_1=1\), every participant received \(A_1=1\): the estimated conditional probability is \(P(A_1=1\mid L_0=0,A_0=1,L_1=1)=1\).
For any target sequence requiring \(A_1=0\) at this history, the conditional probability of the required choice is \(P(A_1=0\mid L_0=0,A_0=1,L_1=1)=0\). Sequential positivity therefore fails on this history if it occurs with positive probability under the target’s preceding treatments. The overall 60% treatment rate cannot repair the missing conditional support. An audit should report the history and decision where the failure occurs rather than only a marginal treatment count.
How to Audit and Interpret the Findings
List each covariate measurement before the treatment decision it informs, then place the outcome after the final decision.
For each time-varying covariate and the outcome, record which preceding treatment history defines its potential value.
Check consistency, conditional exchangeability for the future quantities needed, and positive probability of each required treatment on relevant histories.
Observed histories can reveal empty treatment cells or inconsistent records; they cannot prove that all confounders were measured or that exchangeability holds.
Useful diagnostics include tabulating treatment counts within clinically meaningful histories, examining whether some histories have only one treatment, and checking whether treatment labels conceal distinct versions. Sparse cells and estimated probabilities near zero warn of unstable comparisons even when exact positivity failure is not evident. These checks diagnose observed-data problems; they do not certify the causal assumptions.
The identifying formula can be implemented through standardization (the g-formula) or related weighting approaches. Standardization models the covariate transitions and outcome mean; weighting instead reweights observed histories according to treatment probabilities. A sequentially randomized study is a useful reference because random assignment can support exchangeability by design, but it still requires appropriate treatment support and consistent treatment definitions. None of these approaches compensates for a structural zero or unmeasured confounding merely by changing the calculation.
A common mistake is to treat a positive marginal treatment rate as proof of positivity, or to treat consistency as if it revealed every counterfactual outcome. Both errors ignore the history-specific, sequence-specific nature of the assumptions. A second mistake is to use exchangeability for \(Y^{\bar a}\) alone to substitute observed distributions for future covariates in a g-formula derivation. The correction is to state exchangeability for those future covariates too, or to use a derivation whose assumptions directly justify each conditional step.
The assumptions have different empirical status. Consistency can be threatened by ambiguous treatment versions and can sometimes be investigated by protocol review. Positivity can be assessed from treatment support, though finite samples make near-violations difficult to distinguish from exact zeros. Exchangeability is fundamentally untestable from the observed data alone: measured balance is not proof that no unmeasured common causes remain. The audit makes the assumptions explicit; it cannot turn them all into observable facts.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For the sequence \((0,1,0)\), what does consistency say about the observed outcome?
- Why is an overall treatment proportion strictly between zero and one insufficient to establish sequential positivity?
- At decision \(A_1\), which kind of counterfactual quantities must be covered by the exchangeability condition used in the g-formula derivation?
- In the support example, calculate \(P(A_1=0\mid L_0=0,A_0=1,L_1=1)\). What does its value imply for a target requiring \(A_1=0\) there?
Solutions. (1) It says \(Y=Y^{0,1,0}\), and likewise \(L_1=L_1^0\) and \(L_2=L_2^{0,1}\); it says nothing about outcomes under other sequences. (2) Positivity is conditional on histories, so an action can have positive marginal frequency while being absent in a relevant stratum. (3) It must cover the future covariates and outcomes needed for the target continuation, including \(L_2^{a_0,a_1}\) and \(Y^{a_0,a_1,a_2}\). (4) The probability is \(1-1=0\); the required treatment is unsupported at that history, so positivity fails there when the history is relevant to the target.