Two-Stage Designs That Narrow Eligibility

Why Narrow Eligibility in Stages?

A trial may begin with broad eligibility because the treatment could benefit several biomarker groups, or because the size of a treatment-effect difference between groups is uncertain. Once interim biomarker-specific evidence is available, investigators may decide that later recruitment should focus on a narrower population. The design challenge is not just deciding who may enroll later. It is preserving a credible randomized treatment comparison while making the transition.

A two-stage design addresses this challenge by randomizing an initial broad population, then changing the eligibility criteria for subsequent enrollment. The first stage can provide randomized evidence in multiple biomarker groups. The second stage adds randomized evidence only in the group or groups that remain eligible. The initial participants do not lose their randomized status when eligibility changes, but how their data are used in a final analysis depends on the estimand, the design, and the adaptation.

Let \(S=1\) denote membership in a biomarker-defined subgroup and \(S=0\) denote those outside it. The stage-1 population is broad, \(S\in\{0,1\}\); if narrowing selects \(S=1\), the stage-2 population is restricted to \(S=1\). Biomarker status should be determined using the same qualified assay and definition in both stages, before treatment assignment. The two-stage structure describes when eligibility changes; it does not by itself specify the evidence threshold or decision rule for changing it.

Designing the Two Stages

A useful design starts by fixing the target population and treatment contrast. For example, the primary target might be all biomarker-positive participants randomized in either stage, with a treatment-versus-control contrast on a specified outcome and follow-up time. The protocol should also clarify whether broader-population effects remain of interest, or whether they are exploratory after narrowing. A change in who may enter the trial does not automatically change the scientific question or make every pooled analysis appropriate.

At stage 1, biomarker status is measured for all participants to whom the planned subgroup assessment applies. Randomization within biomarker strata is often operationally helpful: within \(S=1\) and \(S=0\), participants are separately assigned to the randomized arms. This supports treatment balance within each group, particularly when the subgroup is not common. If randomization is not stratified by biomarker, treatment assignment can still be randomized, but chance imbalance within a small subgroup may be more likely.

At the interim transition, the trial uses the available biomarker-specific evidence to decide which population remains eligible for new enrollment. This decision should not break the randomized comparison. In stage 2, eligible participants continue to be randomized between treatment and control, with concurrent control participants enrolled during the same period. Continuing only the experimental treatment would leave no randomized stage-2 comparison and would make treatment differences difficult to separate from calendar-time changes in care, recruitment, or outcome measurement.

Before combining stage-specific information, investigators need to consider whether the treatment, comparator, endpoint definition, follow-up, assay, and conduct are sufficiently consistent across stages. Randomization supports causal comparisons within the randomized groups, subject to the usual trial assumptions, including appropriate handling of missing outcomes and adherence to the specified analysis. Combining results across stages additionally requires a clear target estimand and a defensible account of any stage differences.

Worked Examples of the Design

Why the Combination Needs Justification

The basic rationale for combining stages can be stated directly. Let \(\hat{\Delta}_1\) and \(\hat{\Delta}_2\) be unbiased randomized estimates of the same target treatment effect \(\Delta\) in the selected biomarker population, with independent stage data and variances \(V_1\) and \(V_2\). The inverse-variance weighted estimate is

Because each stage estimate has expectation \(\Delta\), the weighted estimate also has expectation \(\Delta\): the weights are fixed and sum to one after normalization. Independence gives its variance as

These properties rely on the assumptions just stated. In particular, if treatment effects differ between stages, the weighted estimate targets a precision-weighted average of stage-specific effects, not necessarily one common effect. If stage 1 informs which subgroup is selected, its estimate is also involved in the selection process; ordinary fixed-design inference may then fail to have its advertised operating characteristics. A final inferential plan must address the adaptation and the claims being made rather than treating the stages as if they had been planned independently.

Diagnostics, Alternatives, and Limitations

Several practical checks help reveal whether the design is functioning as intended. Review recruitment, assay success, marker prevalence, and the number of mature outcomes by stage and biomarker group. Confirm that randomization was implemented within the intended groups, that the treatment and control arms remained concurrent, and that follow-up and outcome measurement were comparable across stages. Large shifts in subgroup composition, missing biomarker results, or outcome availability can undermine the planned comparison even when the written design is sound.

The main alternatives serve different goals. Fixed enrichment enrolls only the biomarker-defined population from the start; it can be efficient when the target is already well established, but provides less direct randomized evidence in the excluded group. An all-comer trial that continues broad enrollment preserves a broad-population comparison, but may spend resources on groups whose treatment effects are less promising. A two-stage narrowing design begins broad and adapts later, retaining stage-1 information while directing later recruitment. Its cost is added operational complexity and a need to handle data-dependent selection carefully.

Narrowing also has limits. It can reduce the ability to make strong claims about biomarker-negative participants, particularly if stage 1 was small or subgroup outcomes were immature at interim review. A biomarker result that arrives too late for eligibility decisions can delay recruitment or create inconsistent procedures. Treatment-effect differences across stages, changes in standard care, or assay drift can make simple pooling inappropriate. Finally, subgroup selection and multiple population claims can affect error rates; the trial needs an analysis strategy that supports the intended claims.