Why Define a Population by Both Potential Statuses?
An intercurrent event can occur after treatment assignment and can differ between treatment groups. A comparison restricted to participants who did not experience the event may therefore compare groups selected in different ways. A principal-stratum estimand instead defines its target population by asking what each participant’s event status would be under both treatment assignments. This makes the target group a counterfactual population, not simply a subset that can be picked out from the observed data.
For example, suppose the condition of interest is avoiding rescue medication by the study endpoint. Some participants would avoid rescue under either assignment; others would use it under one or both assignments. The principal-stratum question is about the first group. It is distinct from asking for the effect among people observed not to use rescue, because each participant is observed under only one assignment.
The condition must be stated precisely, including its timing and operational definition. “No intercurrent event” is not sufficient if more than one event or definition is possible. The estimand must also specify the outcome and treatment contrast. In the definition above, \(Y(a)\) is the outcome under assignment \(a\), with its handling of intercurrent events made explicit. If the outcome is defined only for some participants under a chosen strategy, the estimand requires a suitable outcome definition before this contrast can be used.
The target population is not “those who avoided the event.” It is those who would avoid it under both assignments. That distinction is central: \(S(0)\) and \(S(1)\) are both potential statuses, but only \(S(A)\) is observed for any one participant. This estimand is a causal contrast in a latent, counterfactual subgroup.
Assumptions and What Randomization Does Not Reveal
The potential-status definition presumes well-defined treatment assignments and a well-defined event condition under each assignment. For causal interpretation, the usual framework also requires consistency: the observed status and outcome equal the potential status and outcome under the assignment actually received. Random assignment supports comparisons between treatment groups for quantities observed under each assignment. As in other potential-outcome analyses, the interpretation also presumes no interference between participants.
These conditions do not make principal-stratum membership observable. Randomization can reveal, in expectation, the marginal distribution of \(S(0)\) from the control arm and of \(S(1)\) from the treatment arm. It does not reveal their joint distribution for the same people. Nor does it, by itself, reveal the mean of \(Y(1)\) or \(Y(0)\) specifically among those with \(S(0)=S(1)=1\).
Worked Example: Bounding the Size of the Target Population
Hypothetical data. In a randomized trial, 70 of 100 participants assigned to control avoid rescue medication by the endpoint, compared with 80 of 100 assigned to treatment. Suppose these rates are treated as the population marginal probabilities \(P\{S(0)=1\}=0.70\) and \(P\{S(1)=1\}=0.80\). What can be concluded about the proportion who would avoid rescue under both assignments?
Let \(q=P\{S(0)=1,S(1)=1\}\). The probability of the union of the two status events cannot exceed 1, so $$ P\{S(0)=1\}+P\{S(1)=1\}-q\leq 1. $$ Therefore \(q\geq 0.70+0.80-1=0.50\). The joint probability cannot exceed either marginal probability, so \(q\leq\min(0.70,0.80)=0.70\). Thus $$ 0.50\leq P\{S(0)=1,S(1)=1\}\leq 0.70. $$
The observed arm-specific rates are compatible with a target group comprising anywhere from 50% to 70% of the population. They do not determine its exact size. These are population bounds; in an actual sample, uncertainty in the estimated arm-specific rates would also need to be considered. Even knowing the target group’s exact size would not, by itself, identify its treatment-effect mean.
Why an Observed-Status Comparison Can Be Different
The observed group with \(S=1\) in the control arm consists of participants with \(S(0)=1\). The observed group with \(S=1\) in the treatment arm consists of participants with \(S(1)=1\). Unless the two groups coincide, their outcome means do not estimate the same principal-stratum contrast. The following hypothetical population makes the difference explicit.
Worked Example: The Selected Comparison Is Not the Principal Effect
Hypothetical population. There are 100 participants. Forty are in the always-avoids stratum \((S(0),S(1))=(1,1)\), 20 would avoid rescue only under treatment \((0,1)\), and 40 would use rescue under either assignment \((0,0)\). No participant is in stratum \((1,0)\). Among the 40 always-avoiders, the mean outcome is 10 units under control and 14 units under treatment. Among the 20 treatment-only avoiders, the mean under treatment is 20 units.
The principal-stratum estimand is the contrast among the 40 always-avoiders: $$ \Delta_{11}=14-10=4\text{ outcome units}. $$ Under control, the participants observed to avoid rescue are exactly the 40 always-avoiders, with mean outcome 10 units. Under treatment, those observed to avoid rescue include both the 40 always-avoiders and the 20 treatment-only avoiders. Their observed-status mean is $$ \frac{40(14)+20(20)}{60} =\frac{560+400}{60} =16\text{ outcome units}. $$
The observed-status comparison is \(16-10=6\) outcome units, not the principal-stratum contrast of 4 units. The discrepancy arises because the treated observed-status group includes participants who would not have avoided rescue under control. A comparison within observed status therefore cannot simply be relabelled as an effect in the always-avoids stratum.
Identification Requires Additional Assumptions
A principal-stratum estimand is well-defined even when it is not identified from observed data. Identification means that the distribution of observed data, together with stated assumptions, determines the estimand. Randomization alone generally leaves the joint distribution of \(S(0)\) and \(S(1)\), and the stratum-specific outcome means, unknown.
One possible additional assumption is monotonicity. If \(S=1\) means avoiding rescue and treatment is assumed never to make someone less likely to avoid rescue, monotonicity states \(S(1)\geq S(0)\) for every participant. This rules out the stratum \((1,0)\). It does not rule out \((0,1)\), so treated participants who avoid rescue can still be a mixture of always-avoiders and treatment-only avoiders.
Another possible assumption is conditional principal ignorability. Let \(X\) be measured baseline covariates. Under monotonicity, one may assume that among participants with \(S(1)=1\), the treatment potential outcome has the same conditional mean for the always-avoiders and the treatment-only avoiders, given \(X\). In notation, this mean restriction is $$ E\{Y(1)\mid S(0)=1,S(1)=1,X\} = E\{Y(1)\mid S(0)=0,S(1)=1,X\}, $$ where both sides are defined in covariate strata where the relevant groups occur. This is a substantive, generally untestable assumption; randomization does not guarantee it. Positivity is also needed: covariate values in the target group must have suitable representation among the observed groups used to estimate the conditional means.
Worked Example: Standardizing Under Explicit Assumptions
Hypothetical data and assumptions. Suppose \(S=1\) means avoiding rescue, monotonicity \(S(1)\geq S(0)\) is assumed, and conditional principal ignorability holds for the treatment outcome given baseline severity \(X\). Among control participants with \(S=1\), there are 30 with low severity and 10 with high severity. Their mean control outcomes are 8 and 12 units, respectively. Among treated participants with \(S=1\), the mean treatment outcomes are 11 units for low severity and 15 units for high severity. Assume each relevant covariate group is represented.
Under monotonicity, a control participant with \(S(0)=1\) must also have \(S(1)=1\); the control participants with \(S(0)=1\) therefore belong to the always-avoids stratum. Their baseline-severity proportions are \(30/40=0.75\) low and \(10/40=0.25\) high. The mean control outcome in that stratum is $$ \widehat{E}\{Y(0)\mid G=(1,1)\} =0.75(8)+0.25(12)=6+3=9\text{ units}. $$
The treated participants with \(S(1)=1\) may include treatment-only avoiders. Conditional principal ignorability is what permits their treatment outcome means, within severity groups, to stand in for the always-avoiders’ conditional means. Standardize those means to the severity distribution of the target group: $$ \widehat{E}\{Y(1)\mid G=(1,1)\} =0.75(11)+0.25(15)=8.25+3.75=12\text{ units}. $$ The resulting estimated contrast is \(12-9=3\) outcome units. This calculation illustrates identification only under the stated assumptions; the observed data do not verify conditional principal ignorability.
Diagnostics, Alternatives, and Limitations
Diagnostics can check aspects of the analysis, but they cannot establish membership in a latent principal stratum. The operational definition and timing of the event should be checked against trial procedures and recorded data. Arm-specific event rates and baseline covariate distributions can be examined for data errors or patterns that challenge the analysis. Under monotonicity, the implied direction of the arm-specific status rates should also be plausible: if the population rate of \(S=1\) is lower under treatment, the assumption \(S(1)\geq S(0)\) is contradicted. Sampling variation must be considered when rates are estimated from finite samples. Conversely, rates in the expected direction do not prove monotonicity at the individual level.
The principal-stratum approach answers a different question from the main alternatives already used for intercurrent events. A treatment-policy estimand includes outcomes regardless of the event according to its specified treatment-policy question. A hypothetical estimand targets outcomes under a specified hypothetical scenario. A while-on-treatment estimand focuses on outcomes during the specified treatment period. A principal-stratum estimand instead targets people defined by their potential event status under both assignments. None is universally preferable; the choice depends on the clinical question.
The main limitations are substantive as well as statistical. The target group is counterfactual and usually cannot be identified participant by participant. Its prevalence and treatment effect may depend heavily on untestable assumptions, and an estimated contrast can be imprecise when the stratum is small. A result for this group also does not automatically describe all trial participants or people in routine care. Sensitivity analyses can show how conclusions change under alternative assumptions, but they do not turn an assumption-dependent estimate into a directly observed fact.
- Wrong reasoning: “The participants who avoided rescue in each arm are the same kind of people, so their observed mean difference estimates the effect among those who would avoid rescue under either assignment.”
- Correction: Observed avoidance reveals \(S(A)\), not both \(S(0)\) and \(S(1)\). The two observed groups can have different principal-stratum compositions, so equality with the \((1,1)\) contrast requires additional justification.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Let \(S=1\) denote meeting the specified event condition. Write the principal stratum for participants who would meet the condition under control but not under treatment.
- In a hypothetical population, \(P\{S(0)=1\}=0.60\) and \(P\{S(1)=1\}=0.70\). Give the possible range for \(P\{S(0)=1,S(1)=1\}\).
- Why does randomization not, by itself, identify the mean treatment outcome among participants with \(S(0)=S(1)=1\)?
- If \(S=1\) means avoiding rescue and monotonicity \(S(1)\geq S(0)\) holds, which principal stratum is ruled out? Does this make all treated participants with \(S(1)=1\) always-avoiders?
Solutions. (1) The stratum is \((S(0),S(1))=(1,0)\). (2) The lower bound is \(\max(0,0.60+0.70-1)=0.30\), and the upper bound is \(\min(0.60,0.70)=0.60\), so the range is 0.30 to 0.60. (3) Each participant reveals status and outcome under only the assigned treatment. Randomization identifies marginal distributions by assignment, not the joint pair of statuses or the treated outcome mean within the latent \((1,1)\) group. (4) Monotonicity rules out \((1,0)\). No: treated participants with \(S(1)=1\) may be either always-avoiders \((1,1)\) or treatment-only avoiders \((0,1)\).