A Two-Period Exchangeability Assessment

Why Audit Each Decision Separately?

In a two-period observational study, treatment is assigned at two different times. Between those decisions, a covariate may be measured that is affected by the first treatment and may influence the second. An exchangeability assessment therefore has to answer two questions, not one: what information was available at each assignment, and which potential outcomes must be independent of that assignment given that information?

This tutorial makes that assessment explicit. It is not enough to say that treatment is “adjusted for covariates” or that the study is “exchangeable.” The first assignment and the second assignment have different observed histories, and the relevant potential outcomes must be stated for each.

The Two Decisions and Their Histories

Use the following notation. Let \(L_0\) denote measured baseline covariates, \(A_0\) the first treatment, \(L_1\) covariates measured after \(A_0\) but before the second treatment, \(A_1\) the second treatment, and \(Y\) the final outcome. For simplicity, treatment values in the examples will be 0 or 1.

Definition: The observed history immediately before the first treatment is \(H_0=L_0\). The observed history immediately before the second treatment is \(H_1=(L_0,A_0,L_1)\). The first history contains baseline information; the second also contains the earlier treatment and the intermediate covariates measured before the second assignment.

For a treatment history \((a_0,a_1)\), write \(L_1^{a_0}\) for the intermediate covariate that would be observed under first treatment \(a_0\), and \(Y^{a_0,a_1}\) for the final outcome under treatment history \((a_0,a_1)\). Under consistency, the observed \(L_1\) equals \(L_1^{A_0}\), and the observed \(Y\) equals \(Y^{A_0,A_1}\). The potential outcomes for the two possible values of \(A_1\) are the relevant counterfactual outcomes at the second decision, holding the prior treatment history fixed.

Definition: In a two-period study, sequential exchangeability requires both of the following conditions, for each treatment history and among histories where the conditioning events have positive probability:

At the first decision, \(A_0\) is conditionally independent of the potential intermediate covariates and final outcomes given \(L_0\):

A_0 \perp\!\!\!\perp \left\{L_1^{a_0},Y^{a_0,a_1}:a_0,a_1\in\{0,1\}\right\} \mid L_0.

At the second decision, among individuals with \(A_0=a_0\), \(A_1\) is conditionally independent of the potential final outcomes given the observed history:

A_1 \perp\!\!\!\perp \left\{Y^{a_0,a_1}:a_1\in\{0,1\}\right\} \mid L_0,A_0=a_0,L_1,\qquad a_0\in\{0,1\}.

The first condition includes potential \(L_1\) as well as potential final outcomes: the first assignment precedes both. The second condition concerns potential final outcomes under the remaining treatment choices, conditional on the history actually observed before \(A_1\). These are statements about conditional distributions, not merely about whether treatment and outcome have a marginal association.

A Practical Assessment Ledger

For a real study, write down the decision-specific information before judging whether the assumptions are plausible. The following ledger separates the two questions.

Decision	Observed history before assignment	Potential quantities relevant to exchangeability	Condition to assess
First treatment \(A_0\)	\(L_0\)	\(L_1^{a_0}\) and \(Y^{a_0,a_1}\) across treatment histories	No residual association between \(A_0\) and these potential quantities within levels of \(L_0\)
Second treatment \(A_1\)	\((L_0,A_0,L_1)\)	\(Y^{a_0,a_1}\) across possible \(a_1\), for the observed prior \(a_0\)	No residual association between \(A_1\) and these potential outcomes within the second-decision history

This ledger helps prevent a common mismatch: checking independence from final outcomes at both times but forgetting that the first assignment can also affect the intermediate covariate. It also prevents using baseline covariates alone as the second-decision history when \(L_1\) was measured before \(A_1\).

Worked Example: Listing the Requirements in a Study

Suppose a study records baseline disease severity \(L_0\), assigns an initial treatment \(A_0\), measures a biomarker \(L_1\) six weeks later, assigns a second treatment \(A_1\), and records a clinical outcome \(Y\). What must be considered at each treatment decision?

First decision. The observed history is severity \(L_0\). The relevant potential quantities include the biomarker under each initial treatment, \(L_1^0\) and \(L_1^1\), and the outcome under each two-treatment history, such as \(Y^{0,0}\), \(Y^{0,1}\), \(Y^{1,0}\), and \(Y^{1,1}\). The assessment asks whether, within levels of \(L_0\), the choice of \(A_0\) is independent of those potential quantities.

Second decision. For someone with \(A_0=a_0\), the observed history is \((L_0,a_0,L_1)\). The relevant potential outcomes are \(Y^{a_0,0}\) and \(Y^{a_0,1}\). The assessment asks whether, among people with that observed history, the choice of \(A_1\) is independent of these outcomes.

The biomarker belongs in the second history because it is measured before the second assignment. The actual value of \(A_1\) does not belong in that history: it is the assignment whose exchangeability is being assessed.

What Conditional Independence Implies

A useful way to interpret each condition is through conditional distributions. If treatment is conditionally independent of the relevant potential outcomes, then the distribution of those potential outcomes is the same across treatment choices within the specified history. For integrable outcomes, the same is true of their conditional means.

Result: Under first-decision exchangeability, for any \(a_0,a_1\) and any \(l_0\) with positive probability,

E\!\left[Y^{a_0,a_1}\mid A_0=a,L_0=l_0\right] = E\!\left[Y^{a_0,a_1}\mid L_0=l_0\right]

for each treatment value \(a\) for which the conditional expectation is defined. Under second-decision exchangeability, for each \(a_0,a_1,l_0,l_1\) with positive probability,

E\!\left[Y^{a_0,a_1}\mid A_1=a,L_0=l_0,A_0=a_0,L_1=l_1\right] = E\!\left[Y^{a_0,a_1}\mid L_0=l_0,A_0=a_0,L_1=l_1\right].

Proof. Conditional independence means that, given the stated history, the conditional distribution of the potential outcome does not depend on the treatment assignment. At the first decision, for any measurable set \(B\),

P\!\left(Y^{a_0,a_1}\in B\mid A_0=a,L_0=l_0\right) = P\!\left(Y^{a_0,a_1}\in B\mid L_0=l_0\right).

Thus the two conditional distributions agree. If \(Y^{a_0,a_1}\) is integrable, integrating the identity over its outcome values gives equality of conditional means. The same argument, conditioning instead on \((L_0,A_0=a_0,L_1)\) and comparing values of \(A_1\), proves the second equality. \(\square\)

This result is an interpretation of the assumption, not a way to verify it from observed outcomes: for each person, only the outcome under the treatment history actually received is observed. A model can fit observed data well without establishing equality of counterfactual distributions.

Worked Example: Checking Exchangeability in a Specified Model

Let \(L_0\) be binary. Suppose \(A_0\), conditional on \(L_0\), is a fair coin independent of all potential quantities. Define \(L_1^{a_0}=a_0\) and \(Y^{a_0,a_1}=L_0+2a_0+a_1\). At the second decision, suppose \(A_1\), given the observed history, is generated by a fresh coin with probability

P(A_1=1\mid L_0,A_0,L_1)=0.2+0.2L_0+0.2A_0.

First decision. Within either value of \(L_0\), \(A_0\) has probability \(1/2\) of being 1 and is independent of the potential quantities. For example, at \(L_0=1\), \(Y^{0,1}=2\) and \(Y^{1,1}=4\) are fixed potential values, while \(P(A_0=1\mid L_0=1)=1/2\). The same assignment probability applies regardless of those values. The first exchangeability condition holds.

Second decision. Consider \(L_0=1,A_0=0,L_1=0\). The assignment probability is \(0.2+0.2(1)+0.2(0)=0.4\). Here \(Y^{0,0}=1\) and \(Y^{0,1}=2\) are fixed, and the fresh coin does not depend on them. In particular, \(P(A_1=1\mid L_0=1,A_0=0,L_1=0,Y^{0,1}=2)=0.4\), equal to the assignment probability given the history alone. At \(L_0=1,A_0=1,L_1=1\), the probability is \(0.2+0.2+0.2=0.6\); it again does not depend on the potential outcomes. The second condition holds as well.

This example verifies the conditions from a specified data-generating model. In an observational study, the counterfactual values in these calculations are not all observed, so the same direct check is not available.

A Failure at One Decision Can Be Hidden by the Other

Worked Example: The First Decision Fails, the Second Holds

Let \(L_0\) and \(L_1\) be constant, and let an unrecorded binary variable \(U\) satisfy \(P(U=1)=P(U=0)=1/2\). Set \(A_0=U\), define \(Y^{a_0,a_1}=U+2a_0+a_1\), and assign \(A_1\) by an independent fair coin.

First decision. With the constant \(L_0\), the relevant potential outcome \(Y^{0,0}=U\) equals 1 when \(A_0=1\) and 0 when \(A_0=0\). Consequently,

P(Y^{0,0}=1\mid A_0=1,L_0)=1,\qquad P(Y^{0,0}=1\mid A_0=0,L_0)=0.

These probabilities differ, so \(A_0\) is not independent of \(Y^{0,0}\) given \(L_0\). First-decision exchangeability fails.

Second decision. Given the history \((L_0,A_0,L_1)\), the potential outcomes are fixed functions of \(U\), and \(U=A_0\). The independent fair coin used for \(A_1\) has probability \(1/2\) of assigning treatment regardless of those potential outcomes. Thus, within each such history, \(A_1\) is independent of the relevant potential outcomes. Second-decision exchangeability holds.

The example demonstrates why an overall statement that “the study is exchangeable” is too imprecise. The conditions must be assessed separately. A condition holding at one decision does not repair a failure at the other.

Why Observed Data Cannot Prove the Assumption

The conditions concern potential outcomes, most of which are counterfactual for each individual. The following construction shows that even identical observed distributions can correspond to different exchangeability conclusions.

Result: The observed distribution alone need not determine whether first-decision exchangeability holds.

Proof by construction. Let \(L_0\) and \(L_1\) be constant, and let \(A_1\) be an independent fair coin in both models. In Model 1, set \(A_0\) to a fair coin and define \(Y^{a_0,a_1}=a_0\). Then the potential outcomes are constants, so they are independent of \(A_0\); first-decision exchangeability holds. The observed outcome is \(Y=A_0\), by consistency.

In Model 2, let \(U\) be a fair binary variable, set \(A_0=U\), and define \(Y^{a_0,a_1}=U\) for every treatment history. Now \(Y=U=A_0\) is also true for every observed individual, but \(A_0\) is associated with the potential outcome \(Y^{0,0}=U\). First-decision exchangeability fails.

In both models, \(A_0\) is fair, \(A_1\) is an independent fair coin, and \(Y=A_0\). Therefore their observed joint distributions of \((A_0,A_1,Y)\) agree, even though the first exchangeability conclusion differs. No diagnostic computed only from those observed variables can distinguish the two models. \(\square\)

Diagnostics, Alternatives, and Limits

An assessment should combine a careful record of timing with substantive knowledge about treatment assignment. Useful questions include:

Were all covariates in the stated history measured before that treatment decision?
Could a factor affecting both treatment choice and a relevant potential outcome have been omitted from the history?
Was an intermediate covariate incorrectly treated as though it were measured before the first treatment?
Does the proposed explanation for treatment assignment change between the first and second decisions?

Observed comparisons of treatment groups within measured histories can reveal discrepancies and suggest model problems, but a lack of observed imbalance does not prove conditional independence from unobserved potential outcomes. Also, exchangeability is distinct from positivity: even if the independence assumptions are plausible, the study must separately have treatment support for the histories of interest. That support question is not settled by this exchangeability audit.

The main alternative is to assert a single joint independence condition for the entire treatment sequence. Such a statement can be stronger than needed and can obscure which decision creates concern. Sequential assessment instead matches each treatment to its own history and its own relevant potential outcomes. Conversely, assessing only baseline exchangeability is incomplete when a second treatment is assigned after \(L_1\) has been measured.

The central limitation is unavoidable: counterfactual independence is not directly observable. A defensible assessment therefore states the assumptions precisely, explains why the measured history is adequate at each decision, and reports uncertainty about unmeasured factors rather than treating a fitted model as proof.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

In the notation used here, what is the observed history before \(A_1\), and why is \(A_1\) excluded from it?
Which potential quantities are relevant to exchangeability at the first decision?
If \(A_0\) is exchangeable given \(L_0\), does that establish exchangeability of \(A_1\) given \((L_0,A_0,L_1)\)? Explain.
In the failure example, calculate \(P(Y^{0,0}=1\mid A_0=1,L_0)\) and \(P(Y^{0,0}=1\mid A_0=0,L_0)\). What do they show?
Can a balance check using observed data prove sequential exchangeability? Give the reason.

Solutions. (1) The history is \((L_0,A_0,L_1)\); \(A_1\) is excluded because it is the treatment assignment being assessed, not information available before that assignment. (2) The first condition concerns \(L_1^{a_0}\) and \(Y^{a_0,a_1}\) across treatment histories. (3) No. The second condition is a separate conditional independence statement about \(A_1\) and potential outcomes within the second-decision history. (4) The probabilities are 1 and 0, respectively, so the conditional distributions differ and first-decision exchangeability fails. (5) No. Balance checks use observed variables, whereas exchangeability concerns potential outcomes that are not all observed; different causal models can produce the same observed distribution.