Why an Eligibility Amendment Changes the Question
An eligibility amendment changes more than a screening rule: it can change which patients are enrolled and, therefore, the population described by an estimated treatment effect. If a trial initially enrolls patients with a narrow range of disease severity and later broadens that range, the effect estimated after the amendment need not describe the same population as the earlier effect. The difference may reflect treatment-effect variation across severity levels, rather than a change in treatment performance for comparable patients.
The aim is not to make the enrolled groups appear identical at any cost. It is to state which population an effect is intended to describe, then estimate effects for that population using data that support the comparison. The earlier material on choosing a time-specific treatment-effect estimand and standardizing effects to a common target population provides useful context; here the focus is how an eligibility rule defines and limits the populations available for those analyses.
The criteria may become broader, narrower, or simply different. If the new criteria are a strict expansion of the old criteria, the old eligible population is contained within the new one, and the old population is their intersection. If the rules are not nested—for example, one version excludes a comorbidity while the other excludes a different condition—the intersection is the patients meeting both rules. Eligibility should be determined using baseline information available before randomization. Restricting on a characteristic affected by treatment or measured after treatment starts can introduce bias rather than create a valid baseline subgroup.
A randomized comparison within each period identifies an effect for that period’s randomized, eligible population, subject to the trial’s usual assumptions. It does not automatically identify the effect in patients who would have been eligible under a different rule. To compare periods meaningfully, define a common target and make clear what assumptions connect the observed patients to it.
Two Coherent Strategies
There are two principal ways to align the target populations. Restriction retains only patients satisfying the chosen common eligibility definition. Standardization estimates treatment effects across relevant baseline strata and averages those effects using a specified target distribution. Restriction is direct when the target is the common eligible subset; standardization can target a broader or otherwise defined population, but requires stronger assumptions about transporting stratum-specific effects.
The formula makes the target explicit. The weights \(q(x)\) come from the target population—not separately from each period’s enrolled population. If the aim is the effect among patients satisfying the common criteria, \(Q\) can be that common population. If \(Q\) includes patients not represented in one period, the analysis needs adequate information to estimate their stratum-specific effects and a defensible transportability assumption. In either case, treatment and control outcomes must be available across the target’s relevant strata.
A short derivation explains the weighting. Partition the target population into mutually exclusive strata defined by \(X\). Within each stratum, the expected treatment contrast is \(\Delta_t(x)=E[Y(1)-Y(0)\mid X=x,t]\). The law of total expectation gives
Thus, changing the target weights changes the average effect when effects differ across strata, even if every stratum-specific effect remains unchanged. Under randomization, the treatment and control means within a baseline stratum estimate the corresponding potential-outcome means for that period, assuming treatment assignment is followed and outcomes are adequately observed. To apply those conditional means to a target population, one additionally assumes that, conditional on the modeled \(X\), the treatment-specific potential-outcome means in the observed eligible patients apply to the target patients. Treatment assignment must have positive probability in each target stratum, and the target strata must be represented in the relevant period.
Worked Example: A Broader Eligibility Rule Changes the Enrolled Mix
Suppose a hypothetical trial broadens eligibility between two periods. The treatment effect varies by baseline severity, but the stratum-specific risks remain the same in both periods. In each period, severity is recorded before randomization, and patients are randomized to treatment or control within the eligible population. The observed risks are:
| Severity stratum | Treatment risk | Control risk | Period 0 proportion | Period 1 proportion |
|---|---|---|---|---|
| Lower | 0.10 | 0.15 | 0.80 | 0.30 |
| Higher | 0.25 | 0.45 | 0.20 | 0.70 |
Assume the target is a common population with 60% lower severity and 40% higher severity. The stratum-specific treatment risk differences are \(\Delta(\text{lower})=0.10-0.15=-0.05\) and \(\Delta(\text{higher})=0.25-0.45=-0.20\). A negative risk difference means the event risk is lower under treatment.
In period 0, the crude risks are \(0.80(0.10)+0.20(0.25)=0.13\) for treatment and \(0.80(0.15)+0.20(0.45)=0.21\) for control. The crude risk difference is \(0.13-0.21=-0.08\). In period 1, they are \(0.30(0.10)+0.70(0.25)=0.205\) and \(0.30(0.15)+0.70(0.45)=0.36\), respectively. The crude difference is \(0.205-0.36=-0.155\). These crude differences are not directly comparable: the periods have different severity mixes.
For the specified common target, standardize both effects using weights 0.60 and 0.40:
The standardized risk difference is \(-0.11\) in either period under the stated assumptions. In this hypothetical example, the crude difference changes because eligibility changes the severity composition, while the common-target treatment effect does not. The estimate is an 11-percentage-point reduction in event risk for the specified target population; it is not a claim that the effect must be the same in other populations.
Restriction to the Common Eligible Population
Restriction is often the most transparent option when a clinically interpretable common population can be defined and is represented in both periods. Apply the intersection of the two eligibility rules, or another explicitly justified common rule, to patients in each period. Then estimate each period’s randomized treatment contrast among those who meet that rule. Because the restriction uses baseline eligibility information, it preserves the randomized comparison within the retained subgroup in expectation. It does, however, reduce the sample size and changes the estimand: the result applies to the restricted population, not automatically to everyone eligible under either protocol version.
Worked Example: Restricting to a Common Eligibility Definition
In a hypothetical platform trial, the original rule requires a biomarker value of at least 40 units. The amended rule lowers the threshold to 25 units. Assume all other criteria remain unchanged. The original eligible patients therefore satisfy both rules, while the amended period also enrolls patients with values from 25 up to, but not including, 40. The team chooses the original rule as the common target and analyzes only patients with biomarker values of at least 40 in each period.
Among patients meeting that common rule, the earlier period has 12 events among 100 treatment patients and 20 among 100 control patients. Its risk difference is
The later period’s common-eligibility subgroup has 15 events among 120 treatment patients and 24 among 120 control patients. Its risk difference is
These estimates describe treatment-control risk differences among patients meeting the original biomarker threshold in each period. Their numerical similarity does not prove that the effects are identical; sampling uncertainty and other assumptions still matter. Crucially, the analysis does not combine the later period’s newly eligible patients with the common subgroup and then compare that mixture with the earlier period’s narrower population.
Restriction is not always sufficient. If criteria changed in non-nested ways, the intersection may be small or clinically unrepresentative. If some target strata occur in only one period, restriction may leave too little information for a useful period comparison. Standardization can retain more information when there is adequate overlap and credible modeling, but it cannot recover an effect for a target stratum with no relevant data merely by adjusting a regression model.
Assumptions, Diagnostics, and Failure Modes
A defensible analysis should state its target, its eligibility rule, and the conditions under which the observed data identify the target effect. At minimum, consider the following:
- Well-defined baseline eligibility. The original and amended rules can be applied consistently using information determined before treatment assignment. Any ambiguity in the criteria should be resolved before defining the analysis population.
- Valid within-period treatment comparisons. Randomization is preserved within the analyzed subgroup, and treatment assignment, outcome ascertainment, and follow-up support estimation of the stated effect.
- Positivity and representation. Every target stratum has relevant patients in each period being compared, and both randomized treatment groups are represented within those strata. For standardization, a target stratum absent from the data has no directly estimated conditional effect.
- Appropriate transport across selected populations. If the target is not simply the observed restricted subgroup, conditional treatment-specific outcome means must transport to the target given the measured covariates used for standardization. Unmeasured effect modifiers can make this assumption implausible.
- Comparable outcome definition. The outcome is defined on a basis that supports the intended comparison. A common eligibility population alone does not ensure that other aspects of measurement or follow-up are comparable.
Useful diagnostics start with a rule-by-rule account of who is included and excluded. Tabulate the number and baseline characteristics of patients satisfying the old rule, the amended rule, and their intersection. Check whether the target strata are represented by treatment arm and period, and inspect whether estimated effects depend heavily on a few sparse strata or extreme model-based extrapolations. These checks complement the earlier material on assessing covariate overlap and trimming when historical controls lack overlap; here they should be applied specifically to the populations induced by the eligibility amendment.
A common failure is to report the pre-amendment estimate for the old population and the post-amendment estimate for the new population as though they estimated the same quantity. Another is to “adjust for eligibility version” in a regression and assume that a model term alone creates comparability. Adjustment cannot identify an unobserved stratum’s treatment effect without additional assumptions. Likewise, a nonsignificant interaction between treatment and period does not establish that the target populations are comparable; it may simply be imprecise.
Choosing an Analysis and Stating Its Limits
The main alternatives answer different questions. Restriction estimates an effect for a shared, explicitly defined subgroup and is easiest to explain when that subgroup is clinically meaningful and sufficiently large. Standardization estimates effects for a declared target distribution and can account for different covariate mixes, but depends on adequate representation and transport assumptions. Reporting period-specific effects without forcing a single pooled comparison is appropriate when a common target cannot be supported; the estimates should then be labeled as effects for their respective eligibility-defined populations.
| Approach | Target | Main strength | Main limitation |
|---|---|---|---|
| Restriction | Patients meeting the chosen common eligibility rule | Directly aligns eligibility definitions | Can reduce precision and narrow clinical relevance |
| Standardization | A specified distribution of baseline characteristics | Allows a common target despite different enrolled mixes | Needs data support and transportability assumptions |
| Separate period-specific estimates | Each period’s own eligible population | Avoids unsupported pooling | Does not by itself provide a common-population comparison |
No method removes all uncertainty. Restriction trades breadth for direct comparability; standardization trades fewer exclusions for stronger modeling and transport assumptions. If the amendment changes which patients are eligible in a way that is not captured by measured baseline characteristics, or if the common target is poorly represented, the appropriate conclusion may be that a single cross-period effect for that target is not reliably identified.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- An amended rule expands eligibility to include a new baseline severity stratum. Why might the crude treatment risk difference change even if every severity-specific treatment effect stays constant?
- Two criteria sets are non-nested. What population does their intersection define, and what is one limitation of restricting to it?
- A target consists of 25% mild and 75% severe patients. The treatment risk differences are \(-0.04\) for mild and \(-0.16\) for severe patients. Calculate the standardized target risk difference.
- One period contains no patients in a target severity stratum. Can standardization alone identify the treatment effect in that stratum? Explain.
Solutions. (1) The crude effect averages severity-specific effects using the enrolled severity proportions; expansion changes those proportions, so the average can change despite stable stratum-specific effects. (2) The intersection consists of patients satisfying both criteria sets. It can be small or unrepresentative, reducing precision and limiting the population to which the result applies. (3) \(0.25(-0.04)+0.75(-0.16)=-0.01-0.12=-0.13\), a 13-percentage-point reduction in risk for that target. (4) No. The missing stratum has no observed data from that period to estimate its conditional treatment effect; identifying it would require additional, explicit assumptions or external information.