Principal-Stratum Interpretation and Identification

Why Define Groups by Two Potential Event Statuses?

Some treatment questions concern an outcome that is meaningful only for people with a particular event status. For example, a quality-of-life score at one year may be defined only for people alive at one year. Comparing the observed scores among survivors in the two randomized arms does not, by itself, compare the same kind of people: treatment may affect who survives. A principal-stratum estimand makes the target group explicit by defining membership through each person’s status under both treatment options.

This is a conceptual definition, not a claim that both statuses can be observed for one person. In a randomized trial, each participant receives one treatment, so only the event status under that assigned treatment is observed. The unobserved status under the other treatment is counterfactual. This distinction is central to interpreting—and assessing the identification of—a principal-stratum effect.

The always-status effect compares the two potential outcomes within the same conceptual group: people who would have the required status under either treatment. If \(Y\) is a score measured in points, the effect is expressed in points. Its causal interpretation relies on the potential-outcomes framework: treatment assignment is well defined, observed outcomes agree with potential outcomes under the treatment received (consistency), and one person’s assignment does not affect another person’s outcome (no interference). Randomization makes assignment independent of the potential statuses and outcomes in expectation; it does not reveal both potential statuses for any individual.

The definition must match the question. If \(S=1\) means alive at one year, the estimand is an effect among people who would survive to one year under either treatment. If \(S=1\) means no rescue medication before assessment, it is an effect among people who would avoid rescue under either treatment. These are different groups and different questions. The event status must be substantive and defined in advance; ordinary missing measurement is not automatically the same thing as the event defining a principal stratum.

What Randomization Identifies—and What It Does Not

Write \(S=S(A)\) for the observed status. Under randomization and consistency, the trial can estimate the marginal status probabilities \(P(S(0)=1)\) and \(P(S(1)=1)\), as well as the outcome distribution under each assigned treatment among participants with \(S=1\). But the joint distribution of \((S(0),S(1))\) is not observed: each participant reveals only one component of that pair. In particular, knowing the proportion with \(S(0)=1\) and the proportion with \(S(1)=1\) does not generally tell us which people belong to the always-status stratum.

This is a cross-world problem: stratum membership depends on potential statuses under mutually exclusive treatment assignments. Randomization supports comparisons of treatment groups, but it does not turn an unobserved potential status into observed data. Even if the stratum proportions were known, the treatment outcome among always-status participants would still need to be separated from outcomes among other people who share the observed status in one arm.

Why the Effect Usually Needs More Than Randomization

Under the monotonicity assumption in the example, control participants with \(S=1\) belong to the always-status stratum. Treatment participants with \(S=1\), however, include both always-status and treatment-protected participants. Thus the observed treatment-arm mean among those with \(S=1\) is a mixture. Without further information or assumptions, it does not reveal the mean for always-status participants alone.

The following calculation shows the problem. Suppose the observed mean outcome among control participants with \(S=1\) is 10 points, and the observed mean among treatment participants with \(S=1\) is 12 points. Keep the illustrative status probabilities \(0.70\) and \(0.80\), with monotonicity. The treatment-arm mean is a weighted average of the always-status and treatment-protected means. If the protected group’s treatment mean were 18 points, the implied always-status treatment mean would be:

The always-status effect would then be \(11.14-10=1.14\) points. But if the protected group’s treatment mean were 8 points instead, the same observed treatment-arm mean implies:

Both calculations produce the same observed arm-specific means and status probabilities, yet different always-status effects. The values are hypothetical and illustrate nonidentification: the observed summaries and monotonicity alone do not determine the stratum-specific outcome mean.

Additional Assumptions and Their Consequences

One possible additional assumption is principal ignorability, usually conditional on measured baseline covariates \(X\). In the always-status setting, a typical version states that, among people with \(S(a)=1\) and the same \(X\), the potential outcome \(Y(a)\) does not depend on the other potential status \(S(1-a)\). This makes the outcome mean among observed status-positive participants informative about the relevant principal stratum after accounting for \(X\). It is an untestable assumption about counterfactual groups, not a consequence of randomization.

For example, under monotonicity \(S(1)\geq S(0)\), control participants with \(S=1\) are always-status participants. In the treatment arm, status-positive participants are a mixture. If principal ignorability holds for \(Y(1)\) conditional on \(X\), then their conditional outcome mean can be used to estimate the treatment outcome mean for always-status participants at each \(X\). That conditional mean must then be averaged over the covariate distribution of the always-status group, not automatically over all randomized participants. Identification therefore depends on the monotonicity and principal-ignorability assumptions, correct measurement of \(X\), and sufficient data to estimate the needed conditional means and weights.

Other restrictions may sometimes be used, but each has its own substantive meaning and consequences. Sensitivity analyses can show how conclusions vary over plausible departures from an identifying assumption. The crucial discipline is to label which parts are observed-data estimates and which depend on assumptions about unobserved statuses or outcomes.

Diagnostics, Comparisons, and Limitations

The defining cross-world quantities cannot generally be checked directly in the data. Diagnostics can still expose problems with a proposed analysis. Check that the event status and its timing are defined consistently in both arms; compare observed status rates and their uncertainty; examine whether a monotonicity claim is clinically credible; and assess whether the covariates used for principal ignorability were measured before treatment and capture plausible predictors of both status and outcome. Sparse status-positive groups, covariate imbalance after conditioning on observed status, and strong dependence of outcomes on predictors of the event status can make estimates unstable or the assumptions implausible. Similar observed status rates do not prove that individual statuses agree across treatments.

A principal-stratum effect differs from the main alternatives. A treatment-policy estimand compares outcomes according to randomized treatment regardless of an intercurrent event, when the outcome remains defined. A hypothetical estimand targets outcomes under a specified hypothetical handling of an event. A principal-stratum estimand instead targets people classified by their potential event statuses under the treatment options; it does not, by definition, remove the event or imagine that it did not occur. These estimands answer different questions, so they need not have the same numerical value.

The approach is especially interpretable when the target group has a clear substantive meaning, such as people who would survive under either treatment. Its limitations are substantial: the group is latent, identifying assumptions are often untestable, and the target may be difficult to communicate to decision-makers. The estimand also does not directly describe the treatment effect for all randomized participants. A carefully specified principal-stratum result should therefore be reported with its assumptions and alongside relevant observable summaries, not as though it were an ordinary randomized comparison among a known baseline subgroup.