Why Baseline Exchangeability Is Not Enough

Why the Baseline Condition Does Not Settle Later Decisions

A treatment may be exchangeable at baseline and still be followed by a later treatment decision that is related to prognosis. The reason is that patients’ histories can change after the first treatment: a variable measured before the second decision may both predict the outcome and influence the second treatment. If the first treatment affects that variable, it is not a baseline characteristic. Baseline exchangeability alone says nothing that guarantees how the later treatment was assigned.

Use two treatment times, \(A_0\) and \(A_1\), a post-baseline variable \(L_1\) measured before \(A_1\), and an outcome \(Y\). Write \(L_1^{a_0}\) for the value of \(L_1\) under baseline treatment \(a_0\), and \(Y^{a_0,a_1}\) for the outcome under the treatment history \((a_0,a_1)\). The observed history follows consistency: \(L_1=L_1^{A_0}\) and \(Y=Y^{A_0,A_1}\).

Definition: Baseline exchangeability, in a setting with baseline covariates \(X\), states that the initial treatment is independent of the potential outcomes under all treatment histories, conditional on \(X\): \(\{Y^{a_0,a_1}:a_0,a_1\}\mathrel{\perp\!\!\!\perp} A_0\mid X\). Later-decision exchangeability is a separate condition: at the second decision, \(A_1\) must be independent of the relevant potential outcomes conditional on the observed history available then, including \(X,A_0,L_1\). The baseline condition does not itself assert this later condition.

The distinction is about the treatment assignment mechanisms at two different times. Even if \(A_0\) is randomized, clinicians may choose \(A_1\) in response to \(L_1\). If \(L_1\) also predicts \(Y\), then a comparison of later-treated and untreated patients that ignores \(L_1\) can mix treatment differences with prognostic differences. That possibility is not ruled out by baseline exchangeability.

A Two-Period Example

The following example isolates the issue with binary variables and explicit probabilities. There is no informative baseline covariate \(X\). Let \(U\) be a binary baseline prognostic factor with \(P(U=1)=0.5\). Randomize \(A_0\), with \(P(A_0=1)=0.5\), independently of \(U\). Let the post-baseline variable and potential outcome be

L_1^{a_0}=\max(U,a_0), \qquad Y^{a_0,a_1}=\max(L_1^{a_0},a_1).

Thus \(L_1\) is affected by \(A_0\): when \(U=0\), it changes from 0 under \(a_0=0\) to 1 under \(a_0=1\). It also predicts the outcome. At the second decision, assign \(A_1\) with probability 0.8 when \(L_1=1\) and probability 0.2 when \(L_1=0\), using a random draw independent of \(U\) given \(L_1\). Both second-period treatment options therefore occur at either value of \(L_1\).

Worked Example: Baseline Randomization, Later Prognostic Imbalance

First check baseline exchangeability. The entire collection of potential outcomes is a function of \(U\), while \(A_0\) was randomized independently of \(U\). Consequently, for every measurable set \(B\) of potential-outcome vectors and \(a_0\in\{0,1\}\), \(P(\{Y^{a_0,a_1}\}\in B\mid A_0=a_0)=P(\{Y^{a_0,a_1}\}\in B)\). Baseline exchangeability holds.

Now restrict attention to people with \(A_0=0\). In this group, \(L_1=U\), and \(Y^{0,0}=U\). The probability of receiving \(A_1=1\) is 0.8 for \(U=1\) and 0.2 for \(U=0\). Since \(P(U=1\mid A_0=0)=0.5\),

P(A_1=1\mid A_0=0) =(0.8)(0.5)+(0.2)(0.5)=0.5.

Bayes’ rule gives

P(Y^{0,0}=1\mid A_1=1,A_0=0) =P(U=1\mid A_1=1,A_0=0) =\frac{(0.8)(0.5)}{0.5}=0.8,

whereas

P(Y^{0,0}=1\mid A_1=0,A_0=0) =P(U=1\mid A_1=0,A_0=0) =\frac{(0.2)(0.5)}{0.5}=0.2.

Thus, even within the baseline-treatment group, the later-treated and untreated groups have different distributions of a potential outcome. Baseline exchangeability did not make later treatment exchangeable conditional on \(A_0\) alone. The difference arises because \(A_1\) responds to \(L_1\), which carries prognostic information.

Deriving the Distortion in an Observed Comparison

The imbalance matters because observed outcomes in the two \(A_1\) groups are not, by themselves, the outcomes the same population would have had under each choice of \(A_1\). In the example, keep \(A_0=0\). Consistency gives \(Y=\max(L_1,A_1)\). When \(A_1=1\), the observed outcome is always 1. When \(A_1=0\), it equals \(L_1\).

Worked Example: The Observed Difference Is Not the Intervention Effect

The observed mean outcome among those receiving \(A_1=1\), with \(A_0=0\), is \(E[Y\mid A_1=1,A_0=0]=1\). Among those receiving \(A_1=0\), it is \(E[Y\mid A_1=0,A_0=0]=0.2\), because the calculation above shows that 20% of that group has \(L_1=1\). The observed difference is therefore \(1-0.2=0.8\).

Under an intervention setting \(A_1=1\) for everyone in the \(A_0=0\) group, \(Y^{0,1}=1\) for everyone, so \(E[Y^{0,1}]=1\). Under an intervention setting \(A_1=0\), \(Y^{0,0}=U\), so \(E[Y^{0,0}]=0.5\). The intervention contrast is \(1-0.5=0.5\), not the observed difference of 0.8. The excess 0.3 in the observed comparison comes from the greater prevalence of \(L_1=1\) among those selected for \(A_1=1\), not from the treatment effect.

This calculation does not claim that adjustment for \(L_1\) always suffices in real studies. In this deliberately simple example, the later-treatment random draw is independent of \(U\) conditional on \(L_1\), so conditioning on \(L_1\) removes this particular source of association. The point is that the baseline condition alone did not establish that fact.

What the Example Proves—and What It Does Not

Result: Baseline exchangeability does not logically imply that later treatment is exchangeable conditional only on baseline treatment history. In the example, baseline exchangeability holds, but the conditional distributions of \(Y^{0,0}\) differ between \(A_1=1\) and \(A_1=0\) within \(A_0=0\): the probabilities are 0.8 and 0.2.

The result follows directly from the verified calculations: if the later treatment were independent of \(Y^{0,0}\) given \(A_0\), then the two conditional probabilities in the result would be equal. Their unequal values disprove that independence. This is a counterexample to an implication, not a claim that later exchangeability must fail whenever baseline exchangeability holds.

Indeed, in this example the later assignment is randomized within levels of \(L_1\). Conditional on \(A_0,L_1\), the random draw used to assign \(A_1\) is independent of \(U\), and the potential outcomes are functions of \(A_0\) and \(U\). Since \(L_1\) determines \(U\) when \(A_0=0\), and \(L_1=1\) for all individuals when \(A_0=1\), the potential outcomes are fixed within each \((A_0,L_1)\) stratum. Later exchangeability conditional on that full history holds here. That is a feature of the example’s assignment mechanism, not a consequence of baseline exchangeability.

This distinction is useful when assessing a study. Baseline randomization supports a claim about the initial assignment; it does not certify the second assignment. For the later decision, one must assess whether treatment selection is adequately explained by the measured history available at that time. Earlier material on conditional independence gives the language for stating such conditions; the essential point here is to apply them separately at each decision.

Contrasts, Diagnostics, and Limits

Worked Example: If the Later Treatment Is Also Randomized

Keep the same \(U,L_1\), and potential outcomes, but assign \(A_1\) by a fair random draw independent of \(U\) and \(L_1\). Within \(A_0=0\), the observed mean for \(A_1=1\) is still 1, because \(Y=\max(L_1,1)=1\). For \(A_1=0\), randomization leaves \(P(L_1=1)=0.5\), so the observed mean is \(0.5\). The observed contrast is \(1-0.5=0.5\), equal to the intervention contrast \(E[Y^{0,1}]-E[Y^{0,0}]=1-0.5\). The treatment-affected prognostic variable remains present; what changed is the second treatment assignment mechanism.

In practice, compare the distributions of measured, pre-decision prognostic variables across later treatment groups, within relevant earlier treatment histories. Large differences in \(L_1\) are a warning that a baseline-only comparison may be misleading. A fitted treatment model can also reveal that later treatment probabilities vary sharply with measured history. These checks can identify measured imbalance or sparse support; they cannot verify independence from unmeasured causes of both treatment and outcome.

The main alternatives are to justify later randomization, or to make and defend an exchangeability assumption conditional on the observed history. Neither alternative follows just from baseline exchangeability. A model that adjusts only for baseline variables can miss a treatment-affected predictor of later treatment and outcome. Conversely, blindly conditioning on every measured variable is not a general remedy: the variable’s timing and causal role matter. The earlier material on time-varying confounding and regression bias motivates that caution; later methods develop how to use histories when the required assumptions are credible.

Finally, exchangeability and positivity are different requirements. In the numerical example, both values of \(A_1\) have positive probability at either value of \(L_1\), but that support does not remove the association between \(A_1\) and prognosis when \(L_1\) is ignored. Conversely, observing good balance on measured variables does not establish exchangeability if unmeasured prognostic factors still influence treatment.

Common Mistake

Wrong reasoning: “The first treatment was randomized, so the treatment groups are exchangeable; therefore the later-treated and untreated groups can also be compared as if randomized.” This carries a baseline property forward without examining the second assignment mechanism.
Correction: State the independence condition for the decision actually being studied and condition on the history available before that decision. A post-baseline variable such as \(L_1\) may be affected by \(A_0\), predict \(Y\), and influence \(A_1\). Baseline exchangeability alone neither removes that association nor guarantees that conditioning on measured history is sufficient.

Key takeaway: Exchangeability at baseline concerns the initial treatment assignment. It does not automatically extend to later treatment decisions. At each decision, ask what information shaped treatment choice and whether that information is related to the potential outcome.

Check Your Understanding

Answer each question, then check the solution given with it.

If \(A_0\) is independent of \(U\), and every \(Y^{a_0,a_1}\) is a function of \(U\), why does baseline exchangeability hold? Solution: The potential-outcome vector is a function of \(U\), so it is independent of \(A_0\) when \(U\) is independent of \(A_0\).
In the numerical example, what is \(P(Y^{0,0}=1\mid A_1=1,A_0=0)\)? Solution: It is \(0.8\), since the probability of \(U=1\) among those selected for \(A_1=1\) is \((0.8)(0.5)/0.5=0.8\).
What is the intervention contrast \(E[Y^{0,1}]-E[Y^{0,0}]\) in that example, and how does it compare with the observed contrast? Solution: The intervention contrast is \(1-0.5=0.5\); the observed contrast is \(1-0.2=0.8\), larger by 0.3.
Does the example show that exchangeability conditional on the full history \((A_0,L_1)\) must fail? Solution: No. Here the later treatment draw is independent within levels of \(L_1\), and the example’s potential outcomes are fixed within each \((A_0,L_1)\) stratum. The example shows that baseline exchangeability alone does not establish later exchangeability.
Why is positivity of both \(A_1\) values within each \(L_1\) value not enough to remove confounding in a baseline-only comparison? Solution: Positivity means both treatment options occur; it does not make their recipients prognostically comparable. In the example, treatment probabilities are positive, yet \(L_1\) and \(Y^{0,0}\) are imbalanced across later treatment groups.