Why Exchangeability Matters for Cumulative Incidence
A treatment can affect whether a particular event occurs by changing the rate of that event, the rate of a competing event, or both. For example, a therapy might reduce disease recurrence but increase death before recurrence. A causal contrast in recurrence cumulative incidence therefore concerns the probability of recurrence by a specified time under each treatment strategy—not simply a comparison of recurrence among people who remain alive or event-free.
The causal question is whether people receiving different treatments would, apart from treatment, have had comparable potential competing-risk outcomes after accounting for measured baseline characteristics. Conditional exchangeability states this comparability as an assumption. It does not follow merely from fitting a regression model or observing similar measured characteristics.
The joint formulation says that, within levels of \(L\), treatment assignment carries no further information about the potential first-event time and type. It is stronger than an assumption about one single cause-and-time indicator. The horizon-specific version can be sufficient when the estimand concerns only that indicator; it should not automatically be treated as an assumption about the entire event process.
This assumption concerns assignment to the treatment strategies being compared. Other requirements are distinct. For instance, interpreting observed outcomes as outcomes under the treatment actually received requires consistency, and causal identification also requires that relevant treatment options occur across the covariate strata being studied. Those conditions do not replace exchangeability.
Which Baseline Variables Belong in \(L\)?
A baseline confounder is a pre-treatment characteristic that helps explain both treatment assignment and the potential event outcomes. The required conditioning set is not a universal checklist: it depends on how treatment is chosen and on the event processes in the particular setting. The substantive goal is to make the potential outcome independent of treatment within levels of the measured baseline covariates.
For a cumulative incidence of cause \(k\), it is not enough to consider only variables that predict cause \(k\) directly. A baseline factor that affects a competing event may alter whether cause \(k\) can occur first, and may also influence treatment assignment. Such a factor can therefore confound the cause-\(k\) cumulative-incidence contrast even if it has little direct effect on cause \(k\)'s event process.
- Potential confounders: baseline disease severity, age, prior health, or clinician and site characteristics when they influence both treatment choice and subsequent event processes.
- Prognostic variables alone: a variable that predicts outcomes but does not relate to treatment assignment need not be included to make exchangeability true, although it may be useful for precision or modeling.
- Post-treatment variables: do not add a variable measured after treatment merely because it predicts the outcome. It may be affected by treatment, so conditioning on it can change the causal question or introduce bias.
- Selection variables: conditioning on event-free survival after treatment is not a substitute for baseline adjustment. Survival can be affected by treatment and by prognostic factors, so selecting survivors can create noncomparability.
The relevant baseline set must be justified using subject-matter knowledge about treatment selection and all event types that contribute to the outcome. A variable need not be a cause of the event of interest alone to matter. Conversely, including every available variable is not a sound rule: adjustment for a collider or a post-treatment variable can make an analysis less credible rather than more credible.
Worked Example: Choosing Baseline Confounders in a Treatment Study
Hypothetical setting. A clinic compares two treatments for a chronic condition. The outcome is recurrence by 2 years, with death before recurrence treated as a competing event. Before treatment, clinicians record disease severity, kidney function, age, and hospital. Clinicians tend to choose treatment 1 for severe disease and impaired kidney function; these features also predict recurrence and death. Hospitals differ in treatment preference and in follow-up care that may affect both event types.
Reasoning. Severity and kidney function are plausible confounders because each influences treatment choice and potential event outcomes. Hospital may also be needed: it predicts treatment assignment and may predict the event process. Age belongs in \(L\) if it contributes to treatment selection and potential outcomes; if it predicts outcomes but does not affect treatment choice, it is prognostic but not required for exchangeability on that account alone.
Death must be considered in this reasoning. A factor associated with death can change the chance of recurrence before death. Thus, for a recurrence cumulative-incidence contrast, the question is not only whether the recorded factors predict recurrence, but whether they jointly suffice to remove treatment-outcome dependence for the competing-risk outcome.
The proposed set \(L=(\text{severity},\text{kidney function},\text{hospital},\text{age})\) is defensible only if those variables, as measured, are sufficient to make treatment assignment conditionally independent of the relevant potential outcomes. The study description makes this plausible for some variables; it cannot establish that no important confounder was omitted.
What the Assumption Implies
Conditional exchangeability connects a counterfactual risk to outcomes observed among people with the corresponding treatment, within baseline covariate strata. The justification is a direct use of conditional independence followed by averaging over the baseline covariate distribution of the target population.
The third line follows from conditional exchangeability: conditioning additionally on \(A=a\) does not change the conditional distribution of the potential outcome. The last line uses consistency, so that the observed event time and type among those who received \(a\) agree with their potential event time and type under \(a\). For continuous or high-dimensional \(L\), the sum is replaced by an average or integral over its distribution.
This identity is a justification, not a complete analysis recipe. It presumes that the relevant covariate strata and observed treatment groups are defined, and it does not resolve inadequate treatment overlap, censoring, measurement error, or model misspecification. In particular, the equality does not make an unmeasured confounder disappear.
Worked Example: A Conditional Risk Contrast and Its Marginal Meaning
Hypothetical data-generating illustration. Suppose \(L\) has two levels, low and high baseline risk, with 60% of the target population at low risk and 40% at high risk. Assume conditional exchangeability, consistency, and adequate treatment representation within both levels. The observed cause-1 cumulative incidences by 1 year are:
| Baseline risk level | Population proportion | Risk under treatment 0 | Risk under treatment 1 |
|---|---|---|---|
| Low | 0.60 | 0.10 | 0.08 |
| High | 0.40 | 0.30 | 0.24 |
By the conditional-exchangeability identity, treatment-specific risks in each stratum represent the corresponding potential risks. For treatment 0, the target-population cumulative incidence is $$ F_1^0(1)=0.60(0.10)+0.40(0.30)=0.06+0.12=0.18. $$ For treatment 1, it is $$ F_1^1(1)=0.60(0.08)+0.40(0.24)=0.048+0.096=0.144. $$ Thus the standardized risk difference, treatment 1 minus treatment 0, is \(0.144-0.18=-0.036\), or a 3.6 percentage-point lower 1-year cause-1 cumulative incidence under treatment 1 in this hypothetical target population. The exchangeability assumption is what permits the observed within-stratum risks to stand in for potential risks; arithmetic alone does not justify that causal interpretation.
Competing Events and the Scope of Exchangeability
A common error is to define exchangeability only for the event of interest while ignoring treatment assignment’s relationship with the competing event. The cumulative incidence of cause \(k\) is the probability that cause \(k\) occurs first by time \(t\). A competing event can remove someone from the possibility of subsequently experiencing cause \(k\) as the first event. Consequently, a baseline factor that confounds treatment assignment and the competing-event process can compromise a cause-\(k\) contrast.
Worked Example: A Competing-Event Predictor Can Be a Confounder
Hypothetical setting. A treatment is prescribed more often to patients with a baseline condition that also increases death before recurrence. The target is recurrence cumulative incidence by 3 years, and death is the competing event. Suppose the baseline condition does not materially increase the recurrence rate among people still alive, but it does increase mortality and affects treatment selection.
The condition remains a potential confounder for recurrence cumulative incidence. The causal outcome is not merely a recurrence rate among survivors; it is the probability of recurrence occurring first by 3 years. If the condition changes the chance of dying first, it can change that probability. Since it also predicts treatment, treatment groups can differ in their potential recurrence cumulative incidence even after restricting attention to recurrence-related biology.
Therefore, the condition should be considered in the baseline exchangeability set unless there is a defensible reason it is not associated with the relevant potential outcomes or treatment assignment. Restricting analysis to people who survive would not solve the problem: survival occurs after treatment and is part of the competing-risk process.
How to Assess the Assumption—and What It Cannot Do
Exchangeability is not testable from the observed data alone. A dataset can show that treatment groups are balanced on recorded covariates, but this cannot demonstrate balance on unmeasured potential-outcome predictors. Nor does a nonsignificant association between a covariate and treatment prove that the covariate is irrelevant. The assessment is substantive: specify how treatment was chosen, map plausible baseline causes of treatment and each relevant event process, and explain why the measured set is adequate.
Several warning signs call for caution:
- Important treatment-selection factors were not measured, or were measured only after treatment began.
- Clinician preference, access to care, or disease severity plausibly affects both treatment and competing events but is absent from \(L\).
- The analysis conditions on post-treatment survival or another variable affected by treatment.
- Covariates are selected solely by automated significance testing, rather than by their causal role and timing.
- The conclusion is described as causal even though the exchangeability argument is only “no measured differences” between treatment groups.
Exchangeability also does not guarantee that a chosen model correctly estimates conditional risks, that censoring is handled appropriately, or that treatment strategies are sufficiently represented in every baseline stratum. It is one causal assumption among several. The later methods for standardization and weighting use it as a foundation; neither method can repair a substantively false exchangeability assumption merely by changing the estimator.
Check Your Understanding
Answer each question using the definitions and reasoning above. Brief solutions follow.
- A baseline factor predicts death before the event of interest and also influences treatment choice. Why can it matter for exchangeability for the event-of-interest cumulative incidence?
- A biomarker measured one month after treatment starts predicts recurrence. Should it automatically be included in the baseline set \(L\)? Explain.
- In the two-stratum example, what is the treatment-1 minus treatment-0 risk difference by 1 year?
- Does showing balance on every recorded baseline covariate establish conditional exchangeability? Why or why not?
Solutions. (1) Death can prevent the event of interest from occurring first, so a factor that affects death can affect that cause’s cumulative incidence; because it also predicts treatment, it can induce confounding. (2) No. It is measured after treatment begins and may be affected by treatment, so conditioning on it can alter the causal question or introduce bias. (3) The risks are \(0.144\) and \(0.18\), so \(0.144-0.18=-0.036\): 3.6 percentage points lower under treatment 1 in the hypothetical target population. (4) No. Recorded balance gives no guarantee about unmeasured common causes, and exchangeability is a substantive assumption rather than a property established by observed balance alone.