Why Post-Randomization Selection Changes the Question
Suppose a trial compares an experimental treatment with control, and some participants experience a specified event after randomization, such as discontinuing assigned treatment or starting rescue therapy. An analysis restricted to participants who did not experience the event may appear to compare like with like. It does not necessarily do so: treatment assignment can affect who remains in that observed group. The event-free participants in one arm may therefore differ systematically from those in the other arm, even when the original assignment was randomized.
A principal-stratum estimand makes a different, precisely defined comparison. It targets participants who would avoid the event under both treatment assignments. This is a causal subgroup defined by counterfactual event statuses, not simply the participants observed to avoid the event in their assigned arm.
The event, time horizon, treatment conditions, outcome, and summary measure must be specified. For example, \(S(a)=1\) might mean no treatment discontinuation before week 12, while \(Y(a)\) might be a week-12 symptom score. The principal stratum is defined relative to those particular treatment conditions and that event definition. If the outcome is not meaningful or defined after the event, restricting the target to participants who would avoid it under either assignment can ensure that both potential outcomes in the contrast are defined. It does not, by itself, make the stratum observable.
This estimand differs from comparing participants with \(S=1\) in each observed arm. A randomized trial reveals \(S(A)\), but for each participant it reveals only the event status under the assigned treatment. It never reveals both \(S(0)\) and \(S(1)\). Randomization makes treatment assignment independent of the potential outcomes and event statuses in expectation; it does not reveal which event-free participants belong to the always-event-free stratum.
Worked Example: Defining and Calculating the Target Effect
Consider a hypothetical trial in which the outcome is a symptom score from 0 to 100, with lower scores better. Suppose the target stratum makes up 50% of the randomized population. In this stratum, the mean potential score is 50 under control and 42 under experimental treatment.
The event-free status under both assignments is \(S(0)=S(1)=1\). The effect is therefore the difference in mean potential scores within this same latent subgroup:
The interpretation is that, among participants who would avoid the specified event under either assignment, the experimental treatment lowers the mean symptom score by 8 points relative to control. This is a subgroup-specific causal contrast; it is not the effect in all randomized participants and not the difference between the observed event-free arm means.
Why Observed Event-Free Groups Do Not Identify the Stratum
The four possible principal strata are always-event-free \((1,1)\), event-free only under control \((1,0)\), event-free only under treatment \((0,1)\), and event-free under neither \((0,0)\). In the control arm, observed event-free participants come from the first and second strata. In the treatment arm, they come from the first and third. Thus, the observed event-free groups generally have different mixtures.
This can be expressed as a mixture problem. Assuming randomized assignment and consistency, the mean among observed control participants who avoid the event is the mean of \(Y(0)\) over those with \(S(0)=1\). That group includes both \((1,1)\) and \((1,0)\). The corresponding treatment mean includes \((1,1)\) and \((0,1)\). Neither observed mean isolates the target stratum unless additional assumptions or information separate the mixtures.
These are not, in general, \(E[Y(0)\mid S(0)=S(1)=1]\) and \(E[Y(1)\mid S(0)=S(1)=1]\). Conditioning on an observed post-randomization event status can select different potential-outcome subgroups across arms. This is why randomization alone does not turn an event-free comparison into the always-event-free causal effect.
Worked Example: The Selected-Group Contrast Can Differ
In a hypothetical randomized population, suppose the principal-stratum proportions and relevant potential-outcome means are as follows. The score is measured only when a participant avoids the event under that assignment, so entries for assignments under which a stratum would experience the event are not needed.
| Principal stratum \((S(0),S(1))\) | Population proportion | Mean \(Y(0)\), if event-free | Mean \(Y(1)\), if event-free |
|---|---|---|---|
| Always-event-free \((1,1)\) | 0.50 | 50 | 58 |
| Event-free only under control \((1,0)\) | 0.10 | 80 | Not defined |
| Event-free only under treatment \((0,1)\) | 0.30 | Not defined | 90 |
| Event-free under neither \((0,0)\) | 0.10 | Not defined | Not defined |
Among control-assigned participants, 0.60 of the population would be event-free: 0.50 always-event-free and 0.10 event-free only under control. Their mean score is
Among treatment-assigned participants, 0.80 would be event-free: 0.50 always-event-free and 0.30 event-free only under treatment. Their mean score is
The observed event-free contrast is \(70-55=15\) points, favoring control because higher scores are worse. But the target effect among always-event-free participants is \(58-50=8\) points, also favoring control. The numerical agreement in direction here is incidental: the two contrasts compare different groups and need not agree in size or direction.
Identification Requires More Than Randomization
The proportion event-free in each arm identifies the marginal probabilities \(P(S(0)=1)\) and \(P(S(1)=1)\), in a large randomized trial. It does not identify the joint probability \(P(S(0)=1,S(1)=1)\), nor which observed participants belong to the joint stratum. Knowing two marginal probabilities does not determine how the two potential event statuses are paired within individuals.
For example, if \(P(S(0)=1)=0.70\) and \(P(S(1)=1)=0.80\), the always-event-free proportion could range from 0.50 to 0.70. The lower endpoint follows because at least \(0.70+0.80-1=0.50\) must be event-free under both statuses; the upper endpoint cannot exceed the smaller marginal, 0.70. These bounds concern stratum size, not the stratum-specific outcome effect, which remains unidentified from the two observed event-free means alone.
One possible route uses covariates measured before randomization. A particularly strong assumption, often called principal ignorability, says that within levels of those covariates, the relevant potential outcome does not differ according to the participant's unobserved status under the other assignment. In general, a further assumption or model is still needed to identify principal-stratum membership probabilities from the observed data.
A useful special case combines principal ignorability with monotonicity. For an event-avoidance indicator, \(S(1)\geq S(0)\) asserts that treatment cannot cause a participant to experience the event if that participant would avoid it under control. This rules out the \((1,0)\) stratum. Under monotonicity, every control-assigned participant who avoids the event is always-event-free. However, treated participants who avoid the event can still be either always-event-free or event-free only under treatment. Principal ignorability is then needed to relate their observed outcomes to those of the always-event-free members.
Worked Example: Identification With Covariates Under Explicit Assumptions
Consider a hypothetical trial with a baseline covariate \(X\) having two equally common groups, low and high risk. The event-free probabilities and observed event-free outcome means are:
| Baseline group | \(P(S(0)=1\mid X)\) | \(P(S(1)=1\mid X)\) | Control mean among event-free | Treatment mean among event-free |
|---|---|---|---|---|
| Low risk | 0.40 | 0.70 | 40 | 50 |
| High risk | 0.80 | 0.90 | 60 | 80 |
Assume randomization and consistency; monotonicity \(S(1)\geq S(0)\); and principal ignorability for the treatment outcome, so that within each baseline group the mean treatment outcome among those with \(S(1)=1\) equals the mean among those who are always-event-free. Under monotonicity, control event-free participants are all always-event-free, so the control means in the table give the stratum-specific control outcome means directly.
The overall always-event-free proportions contributed by the two groups are \(0.50(0.40)=0.20\) for low risk and \(0.50(0.80)=0.40\) for high risk. Thus, the stratum proportion is \(0.20+0.40=0.60\), and its covariate weights are \(0.20/0.60=1/3\) and \(0.40/0.60=2/3\). Standardize each arm's outcome means to those weights:
Under the stated assumptions, the estimated effect is a 16.67-point higher mean score under treatment in the always-event-free stratum. Since higher scores are worse in this example, that contrast favors control. The calculation is not justified by the observed data alone: monotonicity and principal ignorability are substantive, untestable assumptions, and different assumptions could give a different answer.
Assumptions, Diagnostics, and Alternatives
A principal-stratum interpretation requires care about both the causal setup and the identification strategy. At minimum, specify the randomized treatment conditions and the event definition consistently for every participant; assume consistency between assigned treatment and the corresponding potential outcome; and assume no interference between participants when interpreting individual potential outcomes. Randomization supports comparisons by assignment, but does not establish monotonicity, principal ignorability, or the joint distribution of potential event statuses.
Some checks can reveal problems in an analysis, even though they cannot verify its cross-world assumptions. Examine event rates by arm and across baseline covariates. If a monotonicity assumption says treatment cannot reduce event avoidance, an observed pattern strongly contrary to that direction is a warning that the assumption or event coding may be untenable. Assess covariate overlap and whether estimated stratum weights are driven by very few observations. Report the sensitivity of conclusions to plausible departures from the untestable assumptions. No diagnostic using observed trial data can directly confirm that a participant belongs to the always-event-free stratum.
The estimand should also be compared with alternatives according to the clinical question. A treatment-policy estimand includes outcomes regardless of the event after randomization. A composite estimand incorporates the event into the outcome definition. A while-on-treatment estimand focuses on outcomes while participants remain on treatment. The principal-stratum estimand instead asks about treatment effects in a latent subgroup defined by how participants would experience the event under both assignments. These targets answer different questions; no one is automatically the correct substitute for another.
The main limitation is interpretability and identification: “people who would avoid the event either way” is a coherent causal target, but membership cannot be determined for an individual from a parallel-group trial. The resulting effect may also apply to a subgroup that is difficult to describe clinically. If the purpose is to inform a decision for all eligible patients, the principal-stratum effect alone may not be sufficient. State the target population and assumptions explicitly, and avoid presenting a model-dependent estimate as if it were a directly observed randomized contrast.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Define the always-event-free stratum using potential event statuses. Why is it not the same as the event-free participants observed in either one arm?
- Suppose \(P(S(0)=1)=0.60\) and \(P(S(1)=1)=0.75\). Give the possible range for \(P(S(0)=1,S(1)=1)\).
- Under monotonicity \(S(1)\geq S(0)\), which principal stratum is ruled out? Does monotonicity alone identify the always-event-free treatment effect?
- In a hypothetical analysis, the observed event-free mean is 45 under control and 52 under treatment. Is the always-event-free effect necessarily 7 outcome units? Explain.
Solutions. (1) The stratum is defined by \(S(0)=1\) and \(S(1)=1\). An observed event-free group reveals only \(S(A)=1\), so it can include participants who would experience the event under the other assignment. (2) The joint probability is at least \(0.60+0.75-1=0.35\) and at most \(\min(0.60,0.75)=0.60\), so its range is 0.35 to 0.60. (3) Monotonicity rules out \((1,0)\), event-free only under control. It does not alone identify the treated outcome mean within the always-event-free stratum, because treated event-free participants can include \((0,1)\) participants. (4) No. The difference \(52-45=7\) compares two observed groups that may contain different mixtures of principal strata; further assumptions are required to interpret it as the always-event-free effect.