Sequential Exchangeability for Longitudinal Treatments

Why Exchangeability Must Be Sequential

In a longitudinal study, treatment decisions are made repeatedly. A patient's current treatment may depend on current disease severity, which may itself have been affected by earlier treatment. The question is therefore not simply whether treatment groups were comparable at baseline. At each decision time, we ask whether the people receiving different treatment options are comparable with respect to their future outcomes under specified treatment regimes, after accounting for what has happened and been measured so far.

Sequential exchangeability gives this question a precise form. It is an assumption about treatment assignment conditional on the observed history before each decision. It does not say that treatment is independent of all patient characteristics, nor that every treatment wave is randomized without qualification. It says that, within a sufficiently detailed history, the current treatment decision carries no further information about the counterfactual outcomes of interest.

Histories and the Assumption

Index treatment decisions by \(t=0,\ldots,K\). Let \(A_t\) denote treatment assigned at time \(t\), and let \(L_t\) denote covariates measured before \(A_t\). Thus \(L_0\) is baseline information, while \(L_t\) for \(t>0\) is measured after earlier treatment decisions and before the current one. Write \(\bar A_{t-1}=(A_0,\ldots,A_{t-1})\) for prior treatments and \(\bar L_t=(L_0,\ldots,L_t)\) for covariates observed up to the current decision. The pre-treatment history is

H_t=(\bar L_t,\bar A_{t-1}).

At \(t=0\), there are no prior treatments, so \(H_0=L_0\). Let \(d\) denote a treatment regime, such as a fixed sequence or a rule that chooses treatment using observed history, and let \(Y^d\) denote the outcome that would be observed if that regime were followed.

Definition: Sequential exchangeability holds for a set of regimes if, at each treatment time \(t\), and for histories consistent with the regime through time \(t-1\), the current treatment \(A_t\) is conditionally independent of the regime-specific future outcome \(Y^d\) given the observed pre-treatment history \(H_t\). In notation,

A_t \perp Y^d \mid H_t,

for each relevant \(t\) and \(d\), interpreted within histories in which the regime's earlier treatment decisions agree with the observed treatment history.

The conditional independence concerns potential outcomes, not merely outcomes measured after treatment. In words, among individuals with the same measured covariate and treatment history, the current treatment assignment does not distinguish those with better or worse outcomes under regime \(d\). In a fuller longitudinal formulation, the same idea can be required for the collection of potential outcomes under the regimes of interest, and for future covariate outcomes as well as the final outcome.

The condition is local in time: it is assessed before each treatment decision using information available by that time. A baseline-only statement can miss confounding that arises later. Conversely, conditioning on a covariate measured after the current treatment would not describe the pre-decision history and can change the causal question.

What Conditional Independence Implies

For a binary outcome, conditional independence means that within a fixed history, the probability of the regime-specific outcome does not depend on the treatment actually assigned at that decision. The same implication holds for conditional means when those means exist. The following derivation makes the meaning explicit.

A_t \perp Y^d \mid H_t \quad\Longrightarrow\quad P(Y^d=1\mid A_t=a,H_t=h) = P(Y^d=1\mid H_t=h).

By the definition of conditional independence, the conditional joint distribution factors into the product of its conditional marginal distributions:

P(Y^d=1,A_t=a\mid H_t=h) = P(Y^d=1\mid H_t=h)P(A_t=a\mid H_t=h).

When \(P(A_t=a\mid H_t=h)>0\), divide both sides by that probability. The result is the displayed equality. Thus, within history \(h\), the regime-specific outcome distribution is the same across current treatment assignments. The probability condition in this derivation only allows the conditional probability to be written; it is not a substitute for the substantive exchangeability assumption.

This is a statement about the counterfactual outcome \(Y^d\). In an observational data set, each person follows only one treatment path, so the regime-specific outcomes for other paths are not all observed. Consequently, the conditional independence assumption cannot generally be checked just by comparing observed outcomes in treatment groups.

Worked Example: Checking the Meaning Within a History

Consider a hypothetical population at one decision time, all with the same pre-treatment history \(H_t=h\). Suppose the regime-specific binary outcome \(Y^d\) is known in this constructed example. Among 100 people assigned \(A_t=1\), 30 would have \(Y^d=1\); among 100 assigned \(A_t=0\), 30 would have \(Y^d=1\). Does this comparison agree with conditional exchangeability for this history and outcome?

The conditional risks are

P(Y^d=1\mid A_t=1,H_t=h)=\frac{30}{100}=0.30, \qquad P(Y^d=1\mid A_t=0,H_t=h)=\frac{30}{100}=0.30.

Within this history, the risk is equal across the two treatment assignments. The difference is \(0.30-0.30=0\), which is consistent with the implication of conditional exchangeability for this binary outcome. If the entire conditional distributions—not just these binary risks—were equal, that would provide the corresponding distributional equality in this constructed population.

This example is not a practical test using ordinary observed data: it assumes \(Y^d\) is known for everyone, including people who did not follow \(d\). In real studies, those counterfactual outcomes are unavailable. The example illustrates what the assumption means, not a method for verifying it.

Why the History Must Include Evolving Confounders

A covariate can be both affected by earlier treatment and predictive of later treatment and outcome. Such a variable may be essential to the history for a later treatment decision, even though adjusting for it in a conventional outcome regression can raise separate complications. Sequential exchangeability focuses on the comparison at each decision: the history must contain the measured information that makes the current treatment assignment conditionally independent of future regime-specific outcomes.

A treatment history alone is usually not enough if clinicians also use current disease severity, symptoms, or biomarkers to choose treatment. Likewise, current covariates alone may be insufficient if earlier treatment affects later decisions or predicts the future outcome. The relevant history is the combination of observed covariate history and prior treatment history.

Worked Example: Treatment Selection by a Time-Varying Confounder

Suppose 200 hypothetical patients are equally divided between low and high disease severity before a treatment decision. Within each severity group, treatment is assigned at random. The probability of treatment is 0.20 for low-severity patients and 0.80 for high-severity patients. Under a fixed regime, the hypothetical risks of the outcome are 0.10 and 0.20 for low-severity patients under the untreated and treated regimes, respectively; for high-severity patients, they are 0.50 and 0.60.

Within each severity stratum, treatment assignment is random by construction, so it is independent of the regime-specific outcome in that stratum. The numbers assigned to treatment are 20 of the 100 low-severity patients and 80 of the 100 high-severity patients. The expected number of outcomes among treated patients is

$$ 20(0.20)+80(0.60)=4+48=52. $$

There are 100 treated patients, giving an expected observed risk of \(52/100=0.52\). Among untreated patients, 80 are low severity and 20 are high severity. Their expected number of outcomes is

80(0.10)+20(0.50)=8+10=18, \qquad \frac{18}{100}=0.18.

The observed treated-minus-untreated risk difference is therefore \(0.52-0.18=0.34\), even though within each severity stratum treatment assignment is exchangeable. The regime-specific population risks are \(0.5(0.10)+0.5(0.50)=0.30\) under no treatment and \(0.5(0.20)+0.5(0.60)=0.40\) under treatment, a difference of \(0.10\). The discrepancy arises because treatment is more common in the high-severity group, which has worse outcomes under either regime.

Thus, exchangeability need not hold marginally. In this example, conditioning on the severity history is essential; omitting it leaves treatment associated with prognosis. In an actual longitudinal study, the relevant severity measure might itself have been affected by earlier treatment, which is why the history must be updated at each decision.

Dynamic Regimes and Decision-Time Comparisons

A dynamic regime specifies treatment as a function of the history available at a decision. For example, a rule might assign treatment when a measured biomarker exceeds a threshold and withhold it otherwise. Sequential exchangeability is assessed at each time using the history before that time. For a regime \(d\), the relevant comparison at time \(t\) is among individuals whose observed histories have followed the regime up to \(t-1\); at their current history, compare the current assignment with the regime-specific future outcome.

This interpretation prevents a common misreading. The assumption does not say that treatment is independent of the covariates used by a dynamic rule. Treatment may strongly depend on those covariates. It says that, after conditioning on the measured history, there is no remaining dependence between the current assignment and the future outcome that would occur under the regime being considered.

Worked Example: Applying the Assumption to a Biomarker Rule

Suppose a two-decision regime assigns treatment at the second decision if the current biomarker is at least 50 units, and does not assign it if the biomarker is below 50 units. Consider 80 hypothetical patients with the same measured history before that decision, including the same prior treatment and a biomarker of 55 units. The rule therefore calls for treatment at this history. In a constructed population where regime-specific outcomes are known, 10 of the 40 patients actually treated and 10 of the 40 not treated have the outcome under this regime.

The two conditional risks are

\frac{10}{40}=0.25 \quad\text{and}\quad \frac{10}{40}=0.25.

The equality is consistent with sequential exchangeability for this history and the specified regime: knowing whether a patient actually received treatment at this decision does not distinguish their outcome under the regime. Since the biomarker is 55 units, following the regime would assign treatment at this decision regardless of the observed assignment. The example does not imply that the rule itself is randomized, or that the equality could be established from ordinary data where each patient's counterfactual outcome under the rule is not observed.

If, instead, the treated and untreated patients at the same recorded history differed in unmeasured clinical deterioration that predicted the regime-specific outcome, the equality could fail. A biomarker-based rule does not ensure exchangeability merely because its recorded threshold is explicit; the full measured history must account for the factors that guide actual treatment and predict future outcomes.

Interpreting and Assessing the Assumption

Sequential exchangeability is often described informally as “no unmeasured confounding at each treatment time.” That phrase is useful only when its timing and conditioning set are made explicit. It concerns the treatment decision at time \(t\), conditional on \(H_t\), and potential outcomes under the regimes of interest. It is not a claim that there are no unmeasured predictors of the outcome at all; rather, any unmeasured factor that jointly informs current treatment and the regime-specific outcome can threaten the condition.

The assumption is substantive. Researchers use knowledge of the clinical decision process, treatment timing, data quality, and measured prognostic factors to judge whether the histories are adequate. Observed balance in measured covariates can be informative about the recorded history, but it cannot establish independence from unmeasured variables or from counterfactual outcomes. Nor does exchangeability alone guarantee that a causal effect can be recovered: additional assumptions are needed to connect observed data to the counterfactual quantities.

State the decision time. Specify which treatment \(A_t\) is being considered and what was measured before it.
Write the history. Include the relevant covariates through \(L_t\) and prior treatments through \(A_{t-1}\).
Name the counterfactual target. Identify the regime-specific outcome \(Y^d\) to which the assumption applies.
Consider the assignment process. Ask what clinicians or participants knew and used when choosing treatment, including evolving severity and prior response.

Common Mistakes

Using baseline exchangeability as if it covered every treatment wave. Baseline comparability does not address a later treatment decision guided by updated severity. State the condition at each \(t\), conditional on \(H_t\).
Conditioning only on prior treatment. If current covariates influence both treatment and outcome, a treatment-only history can leave confounding. The history includes both \(\bar A_{t-1}\) and \(\bar L_t\).
Claiming the assumption is verified because measured groups look balanced. Balance in recorded variables cannot establish independence from unmeasured factors or unobserved regime-specific outcomes. It can support an assessment, not prove the assumption.
Confusing exchangeability with random treatment assignment marginally. Treatment can depend strongly on observed history and still be conditionally exchangeable after that history is accounted for.
Calling a future observed outcome the counterfactual outcome. \(Y^d\) is the outcome under a specified regime, not necessarily the outcome actually observed for a person who followed a different treatment path.

Key takeaway: Sequential exchangeability is a time-specific conditional independence assumption: given the measured covariate and treatment history before each decision, current treatment is independent of the relevant future regime-specific outcome. Its credibility depends on whether that history adequately captures the information guiding treatment and predicting those outcomes.

Check Your Understanding

Answer each question by referring to the treatment decision, history, and counterfactual outcome involved.

With \(L_0,L_1\) measured before their corresponding treatments \(A_0,A_1\), write the history \(H_1\) used for the second treatment decision.
In words, what does \(A_t \perp Y^d\mid H_t\) assert about people with the same history?
Why can treatment groups differ in their marginal regime-specific outcome risks even when they are exchangeable within levels of a time-varying covariate?
Why does balance on all recorded covariates not prove sequential exchangeability?
A dynamic rule assigns treatment based on a biomarker measured before \(A_t\). Should that biomarker be part of \(H_t\), and what does exchangeability say after conditioning on it?