Prespecifying Interim Enrichment Decision Rules

Why an interim rule must be operational

An adaptive enrichment design can change which population is enrolled after interim data become available. That flexibility is useful only if the change follows a rule written before the data are examined. A rule such as “continue in the subgroup if the interim results look better” leaves open which result, which analysis, and how much better. Different analysts could reasonably make different choices from the same data.

A prespecified decision rule instead names the interim analysis time, the populations to evaluate, the efficacy and futility criteria, the safety criteria, and the outcome for every combination that can occur. It also states what “continue” means: for example, stop further enrollment in the biomarker-negative stratum while continuing the trial in biomarker-positive participants. Patients already enrolled are generally followed according to the protocol; narrowing eligibility need not remove them from follow-up.

For illustration, let \(A\) denote the all-comer population and \(M+\) the biomarker-positive population. Let \(\theta_g\) be the treatment effect in population \(g\), oriented so that positive values favor treatment. The populations overlap: \(M+\) is contained within \(A\). The estimands, endpoint, treatment contrast, and analysis set for each population must be defined in the protocol. This tutorial uses an estimated treatment effect \(\hat{\theta}_g\), its standard error \(SE_g\), and the standardized statistic \(Z_g=\hat{\theta}_g/SE_g\). The direction of benefit and the method for estimating \(SE_g\) must match the endpoint and analysis model.

Fix the timing and the criteria first

The analysis time should be determined by a rule that does not depend on whether the emerging results look favorable. It can be based on a prespecified information fraction, a target number of mature primary-endpoint observations, or another objective milestone. The protocol should specify the data cutoff, minimum endpoint maturity, the populations to be analyzed, and how delayed or missing biomarker measurements are handled. If the intended milestone is not met, the rule should say whether to wait for more data or proceed under a prespecified fallback.

The efficacy criterion and futility criterion answer different questions. An interim efficacy boundary may support an early efficacy stop if the boundary and inferential procedure were calibrated for the design. A futility rule identifies a population that is not worth continuing under the prespecified criterion; a nonbinding futility rule allows the sponsor or monitoring committee not to stop, but that discretion and its consequences should be described. Neither criterion should be improvised after seeing the results.

For a simple illustration, define a futility flag \(F_g\) when \(Z_g\leq 0.20\), and an interim efficacy boundary \(c_E=2.80\). These numbers are hypothetical and design-specific. The efficacy boundary is assumed to have been calibrated under the planned analysis and monitoring scheme; it is not a universal cutoff. A safety rule can be expressed in the units of a harm measure. For example, let \(\Delta_g\) be the treatment-minus-control risk difference for a specified adverse event, where positive values mean more harm on treatment. With harm margin \(\delta=0.08\), define a safety concern if the one-sided 95% upper confidence limit for \(\Delta_g\) exceeds \(\delta\). In an actual design, the confidence-limit method, event definition, and margin must be justified and prespecified.

Constructing a complete decision table

A useful table follows a fixed order. First apply safety rules to each candidate population. Next identify any efficacy-boundary crossings that permit an efficacy stop under the prespecified inferential plan. For the remaining safe populations, apply the futility rule. Finally, select the broadest population that remains eligible, or stop if no candidate remains. The priority order matters: without it, two individually reasonable criteria can point to incompatible actions.

One illustrative rule is: stop for safety in any population that crosses its safety boundary; if a safe population crosses the calibrated efficacy boundary, stop for efficacy according to the inferential plan; otherwise continue in \(A\) if it is safe and not futile; if not, continue in \(M+\) if it is safe and not futile; if neither population qualifies, stop for futility or safety. If \(A\) and \(M+\) both qualify, retain \(A\), the broader population. This is one possible algorithm, not a default recommendation.

This table uses a broad-population-first tie-break rule. It does not say that an all-comer effect is larger than a subgroup effect, nor does it treat the correlated statistics from \(A\) and \(M+\) as independent. The hierarchy is an operational choice: when both populations remain acceptable, preserve broader eligibility. Another design could prioritize a different population, but it must state that choice and assess its operating characteristics before the interim data are observed.

Worked examples

Why the ordering and calibration matter

The basic logic can be written as a sequence of eligibility checks. Let \(S_g=1\) mean population \(g\) is eligible under the safety rule, and let \(C_g=1\) mean it passes the continuation criterion. For the example above, \(S_g=1\) when the safety upper limit does not exceed \(\delta\), and \(C_g=1\) when \(Z_g>0.20\), provided no efficacy stop applies. The safe continuation set is

If \(\mathcal{G}\) is empty, stop. If it contains \(A\), continue in \(A\). If it contains \(M+\) but not \(A\), enrich to \(M+\). This derivation shows why the rule must state the tie-break: when both are in \(\mathcal{G}\), the set alone does not determine the action. It also shows why safety comes before population selection: a statistically promising population cannot be chosen for continued enrollment if it has failed the specified safety gate.

Thresholds must be assessed as part of the design, not merely selected for convenience. At a minimum, evaluate the probability of each possible action under plausible treatment effects, biomarker prevalence, endpoint variability, and safety-event rates. Check the chance of stopping too early, continuing an ineffective population, or selecting a narrower group when effects are similar. Because \(M+\) is part of \(A\), their estimates and decision flags are dependent; design evaluation should preserve that dependence. Any confirmatory claims following selection also require an inferential plan that accounts for the adaptive design.

Diagnostics, alternatives, and limitations

At implementation, verify that the scheduled analysis actually satisfies the prespecified timing condition and that each population has the required endpoint and biomarker data. Check that the treatment-effect direction, standard-error calculation, safety-event coding, and confidence-limit method match the protocol. A table that cannot classify an unusual case—such as an unavailable subgroup estimate, a late endpoint, or a safety estimate based on very few events—is incomplete. Specify whether to delay, stop, or use a defined fallback; do not invent a response after seeing the results.

The broad-population-first rule is one alternative to a rule that always favors \(M+\) when its estimate is more favorable. The former preserves generalizability when the broad population remains acceptable; the latter may focus enrollment but can discard evidence and participants from the broader group. A fixed all-comer design avoids interim population switching but cannot adapt eligibility to emerging subgroup evidence. An informal committee judgment is more flexible, but less reproducible unless the permitted discretion and its documentation are clearly bounded. These choices have different operational and inferential consequences.

Common failure modes include setting thresholds after reviewing interim effects, using an efficacy boundary that has not been calibrated for the design, treating a continuation flag as proof of benefit, and allowing safety criteria to be overridden by a favorable efficacy statistic without an explicit rule. Another pitfall is to state “enrich if the subgroup looks better.” That is not a decision rule: it gives no threshold, safety condition, timing, tie-break, or action when both populations look promising. Replace it with an ordered algorithm that classifies every possible result pattern and has been evaluated before use.