Why Define a Group by Both Potential Event Outcomes?
After randomization, an intercurrent event such as treatment discontinuation or rescue medication may occur in one arm more often than the other. A tempting question is whether treatment works among patients who do not experience that event. But “did not experience the event” is observed after assignment: it can select different kinds of patients in the two arms. A causal question about a common group needs a group definition that does not depend on which assignment a patient happened to receive.
Principal stratification defines such groups using each patient’s potential event status under both treatment assignments. Membership is conceptually fixed before assignment, even though each patient’s membership is not fully observed. The focus here is the effect among patients who would avoid the event under either assignment.
Definition and Target Effect
Let \(A\in\{0,1\}\) denote randomized assignment, with \(0\) the control and \(1\) the treatment. Let \(E(a)\) be the potential indicator of the specified intercurrent event under assignment \(a\), coded \(1\) if the event occurs and \(0\) if it does not. Let \(Y(a)\) be the potential endpoint under assignment \(a\), measured on the prespecified endpoint scale. The four possible pairs \((E(0),E(1))\) define the principal strata.
This is a population-level causal contrast within a counterfactually defined subgroup, not a comparison of the patients observed to be event-free in each randomized arm. As with other causal contrasts, its interpretation depends on defining the endpoint and event precisely. In particular, \(Y(a)\) must be well-defined for the target stratum under assignment \(a\), and the meaning of a positive difference depends on the endpoint’s direction.
The usual randomized-trial assumptions still matter. Random assignment makes \(A\) independent of the potential outcomes and event statuses in expectation; consistency links observed outcomes and event statuses to the potential values under the assigned treatment; and no interference means one patient’s assignment does not change another patient’s potential outcomes. These assumptions support causal interpretation, but they do not reveal both potential event statuses for any one patient.
Worked Example: Defining the Effect in a Hypothetical Trial
Suppose a hypothetical trial concerns a symptom score in which higher values are better. Among 200 patients, imagine the potential event-status strata and their potential-outcome means were known as follows. This is a conceptual illustration: an actual parallel-group trial cannot observe both event statuses and both outcomes for each patient.
| Stratum \((E(0),E(1))\) | Number of patients | Mean \(Y(0)\) | Mean \(Y(1)\) |
|---|---|---|---|
| Always event-free \((0,0)\) | 100 | 6 | 8 |
| Event only under control \((1,0)\) | 30 | 5 | 9 |
| Event only under treatment \((0,1)\) | 10 | 7 | 3 |
| Event under both assignments \((1,1)\) | 60 | 4 | 4 |
The target stratum is the first row. Its mean difference is
Thus, in this hypothetical population, the mean score under treatment would be 2 points higher than under control among patients who would avoid the event in either arm. The contrast is not the difference between the mean outcomes of all patients who avoid the event in their assigned arm. For example, event-free patients under treatment include the 100 always-event-free patients and the 30 patients in the \((1,0)\) stratum. Event-free patients under control include the 100 always-event-free patients and the 10 patients in the \((0,1)\) stratum. Those are different mixtures.
What Randomization Identifies—and What It Does Not
Randomization allows estimation of each arm’s marginal event risk, \(P(E(0)=1)\) and \(P(E(1)=1)\). It does not identify the joint distribution of \(E(0)\) and \(E(1)\), because only one of these two potential event outcomes is observed for each patient. In particular, it does not generally identify how many patients belong to the always-event-free stratum.
Write \(q=P(E(0)=0,E(1)=0)\), the prevalence of that stratum. The event-free margins are \(P(E(0)=0)\) and \(P(E(1)=0)\). The intersection cannot exceed either margin, and its probability cannot be less than the sum of the margins minus one (with zero as a lower limit). Equivalently:
These are bounds on stratum prevalence, not bounds on the treatment effect within the stratum. Outcome information and additional assumptions are needed to identify or constrain that effect.
Worked Example: Bounding the Always-Event-Free Proportion
In a hypothetical randomized trial, 30% of control patients and 20% of treatment patients experience the event. The corresponding event-free proportions are 0.70 and 0.80. With no further assumptions, the always-event-free proportion satisfies
so \(0.50\le q\le0.70\). The margins therefore allow anything from 50% to 70% of patients to be event-free under both assignments.
Now add the assumption of monotonicity \(E(1)\le E(0)\) for every patient: treatment cannot cause the event in someone who would avoid it under control. Under this assumption, anyone with \(E(0)=0\) must also have \(E(1)=0\). Thus the always-event-free stratum has prevalence \(P(E(0)=0)=0.70\). The 0.10 difference between the event-free margins is the proportion in the \((1,0)\) stratum, and 0.20 are in the \((1,1)\) stratum. This identifies stratum prevalence under monotonicity, but it still does not by itself identify the outcome effect within the stratum.
Identification Requires More Than a Stratum Label
An observed event-free group is not generally the target principal stratum. In the treatment arm, observed event-free patients have \(E(1)=0\), which can include both \((0,0)\) and \((1,0)\) patients. In the control arm, observed event-free patients have \(E(0)=0\), which can include both \((0,0)\) and \((0,1)\) patients. Comparing these observed subgroups can therefore mix different principal strata.
Additional assumptions can sometimes link observed data to the target effect. One example is monotonicity, which rules out one of the two discordant strata. Another is principal ignorability: conditional on specified baseline covariates, potential outcomes in one treatment arm do not differ in mean across certain principal strata. This is a substantive assumption about latent group membership and outcomes, not a consequence of randomization. Its plausibility must be considered in the clinical setting.
Worked Example: A Point Estimate Under Explicit Assumptions
Consider hypothetical data from a trial with 100 patients per arm. There are 80 event-free patients under treatment, whose mean symptom score is 8.0 points, and 60 event-free patients under control, whose mean is 6.0 points. Assume monotonicity \(E(1)\le E(0)\). Then all 60 control patients who avoid the event belong to the always-event-free stratum, and the remaining 20 treatment patients who avoid the event belong to the \((1,0)\) stratum.
The control-arm mean of 6.0 points estimates the mean \(Y(0)\) in the target stratum, subject to the trial’s sampling uncertainty. But the treatment-arm mean of 8.0 points is a mixture of the target stratum’s mean and the \((1,0)\) stratum’s mean. To use it as the target stratum’s treatment mean, impose the additional assumption that, among patients with \(E(1)=0\), mean \(Y(1)\) is the same for those with \(E(0)=0\) and \(E(0)=1\). Under that assumption, the treated event-free mean represents the always-event-free mean under treatment. The resulting contrast is
This is an assumption-based estimate, not a result identified by randomization alone. If the 20 patients in the \((1,0)\) stratum would have different treatment outcomes from the always-event-free patients, the treatment-arm event-free mean would not equal the target stratum’s mean.
Bayesian Modeling, Diagnostics, and Limitations
A Bayesian analysis can represent the unknown stratum proportions and stratum-specific outcome distributions in a joint model. Because stratum membership is latent, the observed-data likelihood generally combines contributions from multiple possible strata. Priors and modeling restrictions can express assumptions about the unobserved dependence between \(E(0)\) and \(E(1)\), or about outcomes across strata. The resulting uncertainty is conditional on those assumptions; a prior does not make the missing cross-world information appear in the data.
Useful diagnostics include checking whether the observed event rates and outcome distributions are reproduced by the fitted model, examining whether estimates change substantially under plausible alternative assumptions about stratum membership, and assessing sensitivity to prior choices for weakly identified quantities. A narrow posterior interval does not guarantee strong empirical identification: it may reflect restrictive assumptions or informative priors. Report the assumptions and sensitivity results alongside the estimate.
The principal-stratum approach differs from the treatment-policy strategy, which targets effects according to randomized assignment regardless of the event, and from a hypothetical strategy, which asks about outcomes under a specified scenario in which the event would not occur. It also differs from a composite strategy, which incorporates the event into the endpoint definition. Each answers a different clinical question. Principal stratification is especially demanding because the target group is defined by joint potential outcomes that are never jointly observed.
Common Mistake and Practical Interpretation
The estimand matters when the clinical question specifically concerns patients who would avoid an event under either treatment. Its limitation is equally important: membership cannot be verified patient by patient, and identification depends on assumptions beyond the randomized design. If those assumptions are not credible or results are highly sensitive to them, a different estimand may provide a more defensible answer to the clinical question.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- What potential event-status pair defines the always-event-free principal stratum?
- Event risks are 0.25 under control and 0.15 under treatment. Give the no-assumption bounds for the always-event-free proportion.
- Why does randomization not identify the joint distribution of \(E(0)\) and \(E(1)\)?
- Under monotonicity \(E(1)\le E(0)\), what is the always-event-free prevalence if the control event-free proportion is 0.75?
Solutions. (1) The defining pair is \((E(0),E(1))=(0,0)\). (2) The bounds are \(\max(0,1-0.25-0.15)=0.60\) and \(\min(0.75,0.85)=0.75\), so the prevalence is between 0.60 and 0.75. (3) Each patient reveals the event status under the assigned treatment only; the other potential status is unobserved, so randomization identifies the margins but not their within-patient association. (4) Under monotonicity, \(E(0)=0\) implies \(E(1)=0\), so the always-event-free prevalence equals the control event-free proportion, 0.75.