Architectures For Adaptive Population Enrichment

Why Trial Architecture Determines the Population You Can Speak About

An adaptive trial may change which participants it recruits as evidence accumulates. That change can improve efficiency when treatment benefit appears concentrated in a biomarker-defined group. But the recruitment architecture also determines which populations remain represented in the data. A precise estimate in biomarker-positive participants is not automatically evidence of benefit in everyone who might receive the treatment.

This tutorial compares three architectures: an all-comer trial that later enriches recruitment, an enriched-only trial, and a multi-cohort design that recruits distinct biomarker groups. The central question is not simply which design enrolls fewer participants. It is whether the design preserves information for the population-level claim that matters.

Definition: Let \(B=+\) denote the biomarker-positive subgroup and \(B=-\) the biomarker-negative subgroup. Let \(\pi=P(B=+)\) be the prevalence of the positive subgroup in the target population. For a continuous outcome on which larger values are better, define \(\Delta_{+}\) and \(\Delta_{-}\) as the treatment-minus-control mean differences within the two subgroups. The all-comer treatment effect for that target population is \(\Delta_{\mathrm{all}}=\pi\Delta_{+}+(1-\pi)\Delta_{-}\), provided the two subgroups are exhaustive and their prevalence is the same in the treatment and control populations.

This weighted mean-difference identity concerns a specified target population: changing its prevalence \(\pi\) can change the all-comer effect even if both subgroup effects stay the same. It also relies on measuring the same outcome at the same follow-up time in both subgroups and defining subgroup membership using a baseline characteristic. For randomized comparisons within subgroups, the effects have a causal treatment interpretation under the usual consistency and follow-up assumptions; generalizing from trial participants to a target population additionally requires that the relevant subgroup effects transport to that population.

Three Architectures and Their Evidence

Architecture	Who is enrolled?	Broader-population evidence
All-comer-to-enriched	Initially all eligible participants; later recruitment may be limited to a selected subgroup.	Can retain direct all-comer evidence from the initial stage. Later subgroup data may contribute to a standardized all-comer estimate if subgroup effects and target prevalence are appropriately estimated.
Enriched-only	Only the selected subgroup, commonly biomarker-positive participants.	Directly informs the enrolled subgroup. It does not, by itself, estimate effects in excluded groups or the all-comer effect.
Multi-cohort	Distinct cohorts, such as positive and negative groups, are recruited, often concurrently.	Can estimate subgroup effects and a target-prevalence-weighted all-comer effect if the cohorts are sufficiently informative and the weighting and inference are prespecified.

An all-comer-to-enriched design begins with both biomarker groups represented under randomization, then directs later recruitment toward the subgroup of interest. It therefore has a possible bridge to broader claims: the initial all-comer stage supplies direct observations from both groups, while later positive-subgroup participants can add information about \(\Delta_{+}\). The strength of that bridge depends on the amount and quality of information retained for the negative subgroup and on valid inference after the adaptation.

An enriched-only design is different at its foundation. It can be efficient for estimating benefit in its enrolled population, but it has no randomized treatment comparison in the excluded subgroup. Broader benefit can only be inferred by adding assumptions or evidence not supplied by the enriched-only randomized comparison. A multi-cohort design deliberately retains separate subgroup evidence; unlike an all-comer sample, however, its allocation across cohorts may not match population prevalence. Standardization may therefore be essential.

Why Prevalence Weighting Works

The all-comer mean under treatment is a mixture of the subgroup means, with mixture weights given by the target population’s subgroup prevalence. The same is true under control. Subtracting those two mixture means gives the weighted effect. This short derivation also shows why simply pooling cohorts in their trial proportions can target the wrong population.

\begin{aligned} \mu_{T,\mathrm{all}} &= \pi\mu_{T,+}+(1-\pi)\mu_{T,-},\\ \mu_{C,\mathrm{all}} &= \pi\mu_{C,+}+(1-\pi)\mu_{C,-},\\ \Delta_{\mathrm{all}} &=\mu_{T,\mathrm{all}}-\mu_{C,\mathrm{all}}\\ &=\pi(\mu_{T,+}-\mu_{C,+})+(1-\pi)(\mu_{T,-}-\mu_{C,-})\\ &=\pi\Delta_{+}+(1-\pi)\Delta_{-}. \end{aligned}

Suppose the subgroup effect estimates \(\widehat{\Delta}_{+}\) and \(\widehat{\Delta}_{-}\) are independent, and \(\pi\) is treated as fixed and known. The standardized estimator is \(\widehat{\Delta}_{\mathrm{all}}=\pi\widehat{\Delta}_{+}+(1-\pi)\widehat{\Delta}_{-}\). The variance of a weighted sum of independent estimates is the sum of their weighted variances, so its standard error is:

SE(\widehat{\Delta}_{\mathrm{all}}) = \sqrt{\pi^2 SE(\widehat{\Delta}_{+})^2+ (1-\pi)^2 SE(\widehat{\Delta}_{-})^2}.

This formula is conditional on the stated assumptions. If estimates share participants or otherwise have nonzero covariance, the variance also includes \(2\pi(1-\pi)\operatorname{Cov}(\widehat{\Delta}_{+},\widehat{\Delta}_{-})\). If prevalence is estimated rather than fixed, uncertainty in that estimate may also matter. In an adaptive trial, selection and repeated looks can further affect inference; the simple standard error alone does not establish that a test or interval has its advertised operating characteristics.

Worked Example: Updating an All-Comer-to-Enriched Estimate

Consider a hypothetical trial with a continuous outcome measured in points, with larger values better. In stage 1, both biomarker groups are enrolled and randomized. Each arm has 20 positive participants and 30 negative participants. The common within-arm standard deviation is 10 points. The observed treatment-minus-control differences are 6 points in the positive subgroup and 1 point in the negative subgroup. Recruitment then continues only in the positive subgroup, adding 30 participants per arm; the stage 2 positive-subgroup difference is 5 points. Suppose the target prevalence is \(\pi=0.40\).

The combined positive-subgroup estimate weights the two stage-specific differences by their equal per-arm sample sizes:

\widehat{\Delta}_{+} =\frac{20(6)+30(5)}{20+30} =\frac{270}{50} =5.40\text{ points}.

For independent treatment and control samples with common standard deviation 10 points, the standard error of a difference in means is \(\sqrt{10^2/n_T+10^2/n_C}\). The combined positive subgroup has 50 participants per arm; the negative subgroup, observed only in stage 1, has 30 per arm. Thus:

SE(\widehat{\Delta}_{+})=\sqrt{\frac{100}{50}+\frac{100}{50}}=2.000\text{ points}, \qquad SE(\widehat{\Delta}_{-})=\sqrt{\frac{100}{30}+\frac{100}{30}} =\sqrt{6.6667}\approx2.582\text{ points}.

Standardizing to the target prevalence gives:

\widehat{\Delta}_{\mathrm{all}} =0.40(5.40)+0.60(1.00) =2.16+0.60 =2.760\text{ points}.

Assuming the subgroup estimates are independent and treating prevalence as fixed, the standard error is:

SE(\widehat{\Delta}_{\mathrm{all}}) =\sqrt{(0.40)^2(2.000)^2+(0.60)^2(2.582)^2} =\sqrt{0.6400+2.400} =\sqrt{3.040} \approx1.744\text{ points}.

The architecture retains observations from both groups and provides a route to an all-comer estimate. The calculation does not, on its own, guarantee valid confirmatory inference: if the decision to enrich depended on interim results, the analysis must account for that adaptation. The initial all-comer data also provide a direct stage 1 estimate of the broad effect: \(0.40(6)+0.60(1)=3.00\) points. That estimate and the standardized estimate using later positive participants are distinct analyses, with different information and inferential requirements.

What Enriched-Only Data Cannot Determine

Worked Example: Two Broad Effects Compatible With the Same Enriched Result

Suppose a hypothetical enriched-only randomized trial estimates a positive-subgroup treatment effect of 8 points. The positive subgroup makes up 25% of the target population. There are no randomized negative-subgroup participants in the trial, so \(\Delta_{-}\) is unobserved.

If the negative-subgroup effect were \(-4\) points, the target-population effect would be:

\Delta_{\mathrm{all}}=0.25(8)+0.75(-4)=2-3=-1\text{ point}.

If instead the negative-subgroup effect were \(4\) points, the same positive-subgroup result would imply:

\Delta_{\mathrm{all}}=0.25(8)+0.75(4)=2+3=5\text{ points}.

Both possibilities agree with the enriched trial’s observed subgroup result, yet one implies an unfavorable all-comer mean difference and the other a favorable one. The trial directly supports an estimate for the positive subgroup; without information or defensible assumptions about the negative subgroup, it cannot resolve the broader effect.

A claim that the treatment works in the positive subgroup is not the same claim as a claim that it benefits the broader population. Nor does a positive overall mean difference necessarily show benefit in every subgroup. The estimand and intended population should be explicit before choosing an architecture; the introductory material on population-specific estimands treats that distinction in more detail.

Multi-Cohort Designs: Retaining Both Groups Deliberately

Worked Example: Standardizing Unequally Representative Cohorts

In a hypothetical multi-cohort trial, positive and negative participants are randomized within their respective cohorts. Each cohort has 40 participants per arm, and the common within-arm standard deviation is 12 points. The observed treatment effects are 5 points in the positive cohort and 2 points in the negative cohort. The target population is 30% positive, so \(\pi=0.30\).

The target-population effect is:

\widehat{\Delta}_{\mathrm{all}} =0.30(5)+0.70(2) =1.50+1.40 =2.900\text{ points}.

For either subgroup, the standard error is \(\sqrt{12^2/40+12^2/40}=\sqrt{7.2}\approx2.683\) points. Assuming the subgroup estimates are independent, the standard error of the standardized estimate is:

SE(\widehat{\Delta}_{\mathrm{all}}) =\sqrt{(0.30)^2(7.2)+(0.70)^2(7.2)} =\sqrt{0.648+3.528} =\sqrt{4.176} \approx2.044\text{ points}.

The trial enrolled equal numbers in the two cohorts, so an unweighted pooled contrast would be \((5+2)/2=3.50\) points. That is not the target-population effect here: the target population is 30% positive, not 50% positive. Cohort recruitment proportions are design choices; use target-population weights when the scientific estimand calls for them.

Conditions, Diagnostics, and Limitations

Retaining broader-population evidence requires more than collecting a few participants from each subgroup. Check that the design and analysis meet the following conditions:

Both groups are defined and measured reliably. Assay error, missing biomarker results, or changes in assay procedures can distort subgroup membership and the resulting effects.
Each needed treatment contrast is represented. For subgroup-specific randomized effects, verify that both treatment and control observations exist in each subgroup used in the standardization.
The target prevalence matches the claim. State whether \(\pi\) is estimated from the trial or taken from an external target population, and assess how uncertainty or population differences affect the result.
Outcomes are comparable. Differences in follow-up, measurement, missingness, or calendar-time care across cohorts can make subgroup contrasts difficult to interpret together.
The inference reflects adaptation and selection. If recruitment or analysis choices depend on interim evidence, ordinary fixed-design calculations may not control error or provide valid coverage. Use an analysis plan designed for the actual adaptation.

An all-comer-to-enriched design can be vulnerable to an underinformative initial stage: the negative subgroup may be too small to support its effect estimate, making any standardized broad estimate highly uncertain. Enriched-only recruitment avoids spending resources on excluded groups but forfeits direct randomized evidence about them. Multi-cohort recruitment preserves comparisons by subgroup, but may require substantial enrollment if one subgroup is rare or if precise estimates are needed in each group.

A common mistake is to reason, “The positive subgroup benefited, so the all-comer population benefited.” The correction is to specify the target population and account for every subgroup’s effect and prevalence. Another mistake is to treat the trial’s pooled contrast as automatically all-comer evidence. If cohort proportions differ from target prevalence, the pooled contrast generally targets the trial’s composition instead. Finally, retaining both cohorts does not by itself solve multiplicity or selection concerns: a confirmatory claim still needs an analysis that matches the adaptive design.

Key takeaway: Enriched-only designs directly support claims about the enrolled subgroup. All-comer-to-enriched and multi-cohort designs can retain a route to broader-population evidence, but only when both subgroup effects are adequately informed, target-population weighting is appropriate, and inference accounts for the design and adaptation.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A target population is 20% biomarker-positive. If the subgroup effects are 7 points in positive participants and \(-1\) point in negative participants, what is the all-comer effect?
Why can an enriched-only randomized trial not, by itself, establish the treatment effect in the full target population?
Independent subgroup effect estimates have standard errors of 2 and 4 points. If \(\pi=0.50\), calculate the standard error of the prevalence-weighted estimate, treating prevalence as fixed.
A multi-cohort trial recruits equal numbers from the positive and negative groups. The effects are 6 and 0 points, respectively, while the target population is 20% positive. Compare the unweighted pooled effect with the target-standardized effect.

Solutions. 1. \(\Delta_{\mathrm{all}}=0.20(7)+0.80(-1)=1.40-0.80=0.60\) point. 2. The trial does not randomize or observe the treatment contrast in excluded participants, so their effect is unknown from that trial; different excluded-group effects could produce different all-comer effects. 3. \(SE=\sqrt{(0.50)^2(2^2)+(0.50)^2(4^2)}=\sqrt{1+4}=\sqrt{5}\approx2.236\) points. 4. The unweighted contrast is \((6+0)/2=3.00\) points, reflecting equal cohort weights. The target-standardized effect is \(0.20(6)+0.80(0)=1.20\) points.