Weighted Population Estimands Under Adaptive Enrichment

Why Enrollment Mix and Target Population Must Be Separated

Adaptive enrichment can change which baseline subgroups contribute participants as a trial proceeds. That may be appropriate for recruitment or trial efficiency, but it also changes the subgroup proportions among enrolled participants. If treatment effects differ across subgroups, a pooled treatment contrast can then change even when each subgroup’s treatment effect has not changed.

The solution begins by defining the population the overall effect is meant to describe. The earlier material on standardizing effects to a fixed population and fixed baseline subgroup estimands establishes the relevant ingredients: subgroup-specific effects and a prespecified target population. Here, the focus is how to combine those effects when enrollment proportions change.

Definition: Let $G$ denote membership in one of $K$ mutually exclusive, exhaustive baseline subgroups. For subgroup $g$, let $\tau_g$ be the treatment effect under the specified treatment conditions, outcome summary, and intercurrent-event strategy. Let $q_g$ be the proportion of subgroup $g$ in the fixed target population, where $q_g \geq 0$ and $\sum_{g=1}^{K} q_g=1$. The weighted population estimand is $$ \tau_{\mathrm{target}}=\sum_{g=1}^{K}q_g\tau_g. $$ The weights $q_g$ describe the target population; they are not defined by the numbers enrolled in the adapted trial.

For a mean-difference estimand, one possible subgroup effect is $\tau_g=E[Y^1-Y^0\mid G=g]$, where $Y^1$ and $Y^0$ are the outcomes that would be observed under the two specified treatment conditions. The target effect is then in the same units as the outcome, such as points on a symptom scale. Other prespecified subgroup contrasts can be combined in the same way only when their definitions and scales are compatible.

Deriving the Weighted Effect

The weighted formula follows by applying the law of total expectation separately to the two treatment conditions. Suppose the target population is partitioned by $G$, and $q_g=P(G=g)$ in that population. The target-population mean under treatment condition 1 is

E[Y^1]=\sum_{g=1}^{K}P(G=g)E[Y^1\mid G=g] =\sum_{g=1}^{K}q_gE[Y^1\mid G=g].

The same identity holds under treatment condition 0. Subtracting the second mean from the first gives

\begin{aligned} E[Y^1]-E[Y^0] &=\sum_{g=1}^{K}q_gE[Y^1\mid G=g] -\sum_{g=1}^{K}q_gE[Y^0\mid G=g]\\ &=\sum_{g=1}^{K}q_g\left\{E[Y^1\mid G=g]-E[Y^0\mid G=g]\right\}\\ &=\sum_{g=1}^{K}q_g\tau_g. \end{aligned}

This is a derivation of the estimand, not a claim that every trial’s raw pooled contrast estimates it. A pooled contrast implicitly uses the subgroup proportions represented in its treatment and control groups. Those proportions can differ from the target weights, and may even differ between arms.

Conditions for Interpreting the Weighted Combination

The definition is meaningful only when its components refer to a coherent target and compatible effects. Before estimating it, check the following conditions.

The subgroups form a partition. Every member of the target population belongs to one, and only one, subgroup. The weights are nonnegative and sum to one.
The target weights are specified independently of the adapted enrollment mix. They may come from the target population’s known subgroup proportions or another justified, prespecified target distribution. Trial enrollment proportions are not a substitute merely because they are easy to calculate.
Each $\tau_g$ uses the same treatment contrast and outcome definition. The treatment conditions, outcome summary, and strategy for intercurrent events must be aligned across subgroups. The contrast may vary in magnitude by subgroup; its definition must remain comparable.
Subgroup effects are identifiable from relevant data or justified assumptions. In a randomized trial, randomization within the relevant subgroup supports comparisons of treatment arms there, subject to the trial’s assumptions for outcome observation and analysis. If a subgroup with positive target weight is never enrolled or randomized to one treatment condition, its effect is not identified from those trial data alone.
The observed participants represent the subgroup effect being targeted, or transport is justified. If enrichment selects a narrow subset within a broad subgroup, the subgroup-specific effect among enrollees need not equal the target subgroup effect. The subgroup definition, sampling process, and any required assumptions must support that interpretation.

In particular, a fixed-weight formula preserves the target only if the subgroup effects being inserted are the effects for the intended target subgroups. Fixed weights cannot repair a mismatch between enrolled participants and the target population within a subgroup, nor can they supply an unobserved subgroup effect.

Estimating the Target Effect

For a mean difference, estimate each subgroup’s effect using the treatment-arm mean difference within that subgroup, then combine the estimates using the fixed target weights. Write $\hat{\tau}_g$ for the estimated effect in subgroup $g$. The direct standardized estimate is

\hat{\tau}_{\mathrm{target}}=\sum_{g=1}^{K}q_g\hat{\tau}_g.

With randomized treatment assignment within subgroups and adequate observations in each treatment arm, the subgroup contrasts can estimate their corresponding effects. The weights in this calculation remain $q_g$, even if the adaptive trial enrolls a very different proportion of each subgroup. This calculation defines a point estimate; it does not by itself establish that a confidence interval or test accounts for every consequence of adaptation.

Worked Example: Combining Two Prespecified Subgroup Effects

Suppose a hypothetical trial targets a population that is 60% subgroup A and 40% subgroup B. On a continuous outcome measured in points, the estimated treatment effect is 4 points in subgroup A and 1 point in subgroup B. Assume both contrasts use the same outcome definition and treatment conditions, and that each subgroup estimate is supported by randomized comparisons in that subgroup.

The fixed target weights are $q_A=0.60$ and $q_B=0.40$. Therefore,

\hat{\tau}_{\mathrm{target}} =(0.60)(4\text{ points})+(0.40)(1\text{ point}) =2.4\text{ points}+0.4\text{ points} =2.8\text{ points}.

The estimated target-population treatment effect is 2.8 points. The trial’s enrollment proportions do not enter this calculation: the 60% and 40% weights describe the specified target population.

What Changes When Enrichment Changes the Mix?

If subgroup treatment effects are heterogeneous, changing the mix changes an enrollment-weighted average. That is a different estimand unless the enrolled mix happens to match the target weights. The distinction is easiest to see when both treatment arms have the same subgroup proportions: their pooled mean difference then weights subgroup effects by that shared enrollment mix. If the proportions differ between arms, even that simple interpretation of the pooled contrast may fail.

Worked Example: Adaptive Enrollment Changes the Pooled Contrast

Consider a hypothetical two-stage trial. Its target population is 50% subgroup A and 50% subgroup B. After an adaptation, 20% of enrolled participants are from A and 80% are from B. Within each subgroup, the trial has equal numbers in the two treatment arms. The subgroup means on an outcome measured in points are:

Subgroup	Treatment mean	Control mean	Estimated effect
A	8 points	4 points	4 points
B	6 points	5 points	1 point

The subgroup contrasts are 8 − 4 = 4 points in A and 6 − 5 = 1 point in B. Applying the target weights gives

\hat{\tau}_{\mathrm{target}} =(0.50)(4)+(0.50)(1)=2.5\text{ points}.

By contrast, the pooled treatment mean under the enrolled mix is $(0.20)(8)+(0.80)(6)=6.4$ points, and the pooled control mean is $(0.20)(4)+(0.80)(5)=4.8$ points. Their difference is $6.4-4.8=1.6$ points. That 1.6-point contrast answers a question about the enrolled mix, not the prespecified 50:50 target. The subgroup effects are unchanged; the difference arises from using different weights.

Worked Example: Unequal Subgroup Representation Across Arms

In another hypothetical trial, the fixed target population is 70% subgroup A and 30% subgroup B. The observed arm means and sample sizes are:

Subgroup	Treatment: n and mean	Control: n and mean
A	40 participants; 10 points	20 participants; 6 points
B	10 participants; 7 points	30 participants; 5 points

Assuming the within-subgroup treatment comparisons support the specified subgroup effects, the estimates are $\hat{\tau}_A=10-6=4$ points and $\hat{\tau}_B=7-5=2$ points. The fixed-weight estimate is

\hat{\tau}_{\mathrm{target}} =(0.70)(4)+(0.30)(2)=2.8+0.6=3.4\text{ points}.

The raw pooled means instead combine different subgroup proportions in the two arms. The treatment mean is $\{(40)(10)+(10)(7)\}/50=470/50=9.4$ points. The control mean is $\{(20)(6)+(30)(5)\}/50=270/50=5.4$ points. Their difference is 4.0 points, not 3.4 points. The raw pooled contrast is affected both by the changed subgroup mix and by its imbalance between arms. Standardizing the subgroup effects to the target weights directly answers the stated target-population question.

Comparison, Diagnostics, and Limitations

The appropriate summary depends on the question being asked. The alternatives below are not interchangeable when effects vary across subgroups.

Approach	Weights used	Question answered	Main caution
Fixed-weight standardization	Prespecified target weights $q_g$	What is the effect in the fixed target population?	Requires support for estimating each target subgroup effect.
Enrollment-weighted pooled contrast	Subgroup proportions among enrolled participants, if common across arms	What contrast corresponds to the enrolled mix?	Changing enrollment can change the answer even if subgroup effects are stable.
Separate subgroup effects	No single combined set of weights	How does the effect differ across subgroups?	Does not provide one overall target-population effect.

Useful diagnostics include tabulating, by subgroup and trial stage, the number randomized to each treatment condition and the observed subgroup proportions in the trial and target population. Also check whether subgroup treatment contrasts and outcome definitions remain comparable across stages. Large differences between a raw pooled contrast and a fixed-weight estimate are not automatically evidence of a calculation error; they can reveal that the two summaries answer different questions.

The method has important limits. A target weight does not create information: if an outcome contrast is unavailable for a subgroup with positive weight, the weighted target effect cannot be estimated from the observed subgroup effects alone. Nor does a coarse subgroup definition guarantee that changing enrollment within that subgroup leaves its effect representative of the target. Finally, the weighted estimate is not a claim that all participants share the same treatment effect. It is an average for the specified target population.

Common Mistake

Common mistake: “The adaptation enrolled 80% subgroup B, so subgroup B should receive 80% of the weight in the overall estimate.” That reasoning silently changes the target population to the adapted trial’s enrollment mix. If the intended target remains 50% A and 50% B, use those prespecified weights. Use enrollment proportions only when the enrolled mix itself is the explicitly intended target and the subgroup effects represent that mix.

Check Your Understanding

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

A target population is 30% subgroup A and 70% subgroup B. The treatment effects are 5 and 2 points, respectively. What is the weighted target effect?
If the treatment effect is 3 points in both subgroups, does changing the target weights change the weighted effect? Show why.
A subgroup with a positive target weight has no randomized participants in the control arm. Can its treatment effect be estimated from the trial’s observed within-subgroup contrast? Explain.
Why can an enrollment-weighted pooled contrast differ from a fixed-weight target estimate after enrichment?
What additional concern arises if enrichment selects a nonrepresentative subset within a broad baseline subgroup?

Solutions. (1) $(0.30)(5)+(0.70)(2)=1.5+1.4=2.9$ points. (2) No: for any weights summing to one, $\sum_g q_g(3)=3\sum_g q_g=3$ points. (3) No. Without control observations in that subgroup, the within-subgroup treatment contrast is not identified from the trial’s randomized comparison alone; further data or justified assumptions are needed. (4) The pooled contrast reflects the enrolled subgroup mix (and can also reflect different mixes between arms), while the fixed-weight estimate uses the target population’s prespecified weights. (5) The effect among enrolled participants may not represent the target subgroup effect, so fixed weights alone do not remove that selection mismatch.

Approach	Weights used	Question answered	Main caution
Fixed-weight standardization	Prespecified target weights \(q_g\)	What is the effect in the fixed target population?	Requires support for estimating each target subgroup effect.
Enrollment-weighted pooled contrast	Subgroup proportions among enrolled participants, if common across arms	What contrast corresponds to the enrolled mix?	Changing enrollment can change the answer even if subgroup effects are stable.
Separate subgroup effects	No single combined set of weights	How does the effect differ across subgroups?	Does not provide one overall target-population effect.