Why the Target Population Is Part of the Effect
An enrichment design may recruit or retain patients based on a biomarker, but a treatment effect is not defined by the treatment comparison alone. It also depends on which population the comparison describes, what outcome is measured, when it is measured, and which effect scale is used. A treatment can have one average effect among marker-positive patients, another among marker-negative patients, and a different average effect in a population containing both groups.
The key distinction is between a subgroup-specific effect and an all-comer effect. The all-comer effect is not a universal number attached to a treatment: for an additive effect scale, it depends on the proportion of marker-positive and marker-negative patients in the target population. Thus, an effect estimated in an enriched sample need not equal the effect for the population from which that sample was drawn.
These are estimands: precisely defined quantities the trial aims to learn about. They are distinct from estimates computed from observed data. A complete estimand specifies the target population, treatment conditions, outcome variable, summary measure, and time point or follow-up period. For example, “the mean difference in symptom score at week 12 between treatment and control among marker-positive adults eligible for the trial” is more precise than “the effect in marker-positive patients.” The marker-positive definition and the outcome scale must also be operationally clear.
For a binary outcome, a common additive estimand is the risk difference. If \(Y(a)\) is an indicator of an event under treatment condition \(a\), then \(\tau_+\) and \(\tau_-\) are differences in event probabilities within the respective marker groups. When the event is undesirable, a negative risk difference means a lower event risk under treatment. The effect scale matters: a difference in means, risk difference, risk ratio, and odds ratio are different estimands.
How Prevalence Determines the All-Comer Effect
Let \(\pi=P(M=1)\) be the prevalence of marker-positive status in the target population. The two marker categories are assumed to be exhaustive and mutually exclusive for this calculation. The law of total expectation gives the all-comer effect on an additive scale:
This derivation uses the same target population for all three estimands: the two subgroup effects are conditional averages within its marker groups, and \(\pi\) is its marker-positive proportion. It does not require the subgroup effects to be equal. If they are equal, the weighted average equals that common effect for every prevalence; if they differ, changing the population mix generally changes the all-comer effect.
The same weighting applies to an all-comer risk difference because risks are probabilities and the difference of the two population risks is additive. Specifically, if \(\mu_{a,+}=E[Y(a)\mid M=1]\) and \(\mu_{a,-}=E[Y(a)\mid M=0]\), then the all-comer mean under condition \(a\) is \(\pi\mu_{a,+}+(1-\pi)\mu_{a,-}\). Subtracting the control mean from the treatment mean yields the weighted difference above.
Worked Example: A Continuous Outcome in Two Marker Groups
Suppose a hypothetical trial targets a population in which 30% of patients are marker-positive. The outcome is a symptom score at week 12, measured in points, with higher scores indicating greater improvement. The estimated treatment-minus-control mean difference is 5.0 points among marker-positive patients and 1.0 point among marker-negative patients. Calculate the all-comer mean difference for this target population.
The required inputs are \(\pi=0.30\), \(\tau_+=5.0\) points, and \(\tau_-=1.0\) point. The additive-scale formula gives:
For a target population with this 30% prevalence, the average treatment-minus-control difference is 2.20 points. It is neither the unweighted average of the subgroup effects nor the marker-positive effect alone. This calculation assumes the subgroup effects apply to the corresponding groups in the target population and that the stated prevalence describes that population.
Worked Example: All-Comer Risk Difference From Subgroup Risks
Consider a hypothetical binary outcome of experiencing an adverse event by day 90. In marker-positive patients, the event risk is 0.30 under control and 0.18 under treatment. In marker-negative patients, the corresponding risks are 0.10 and 0.08. Suppose the target population is 40% marker-positive. Calculate the subgroup risk differences and the all-comer risk difference.
Define each risk difference as treatment risk minus control risk, so a negative value indicates fewer adverse events with treatment. The subgroup differences are:
The prevalence-weighted all-comer difference is:
The all-comer risk difference is \(-0.060\), or 6.0 percentage points fewer adverse events under treatment than control for this target population. As a cross-check, the all-comer risks are \(0.40(0.18)+0.60(0.08)=0.120\) under treatment and \(0.40(0.30)+0.60(0.10)=0.180\) under control. Their difference is \(0.120-0.180=-0.060\), in agreement with the weighted subgroup differences.
Assumptions for Interpreting the Estimand Causally
The estimand is defined by the target population and outcome, whether or not a trial can estimate it without bias. A causal interpretation of an observed treatment contrast requires conditions linking the observed data to the potential outcomes. In a randomized trial, the main requirements include clearly specified treatment conditions, treatment assignment independent of potential outcomes within the relevant marker groups, and a positive probability of receiving each treatment condition in each group. Consistency means that a patient’s observed outcome under the treatment received equals the potential outcome for that treatment condition. The analysis also needs an appropriate approach to missing outcomes; randomization alone does not resolve bias from outcome data that are missing in a treatment-dependent or outcome-dependent way.
Marker status must refer to the intended baseline classification. If marker status is measured after treatment, conditioning on it may define a different and potentially biased comparison. For an all-comer estimate, the prevalence used in the weighting must refer to the intended target population, not automatically to the trial’s recruited sample. If the trial is enriched, its sample marker proportion may differ substantially from the prevalence among all eligible patients or another population named in the estimand.
An adaptive decision to focus recruitment on a subgroup does not, by itself, change the scientific question originally posed. The target population and estimand should be stated separately from the adaptation rule. The design and subsequent inference must support estimation of that target; selection and multiplicity concerns are separate from the definition and weighting of these population-specific effects.
Worked Example: Reweighting an Enriched Sample to a Target Population
Suppose a hypothetical enriched trial has 75% marker-positive participants, but the intended all-comer target population has only 25% marker-positive patients. Assume the mean treatment effects are 8.0 outcome units among marker-positive patients and 2.0 units among marker-negative patients. Calculate the additive effect for the trial’s observed mix and for the target population.
Using the trial mix gives:
For the target population, use its prevalence, \(\pi=0.25\), rather than the trial’s 0.75:
The 6.5-unit result describes a population with the trial’s stated marker mix, while 3.5 units is the standardized effect for the intended target mix, assuming the subgroup effects transport to that target. The change is not a contradiction: the populations assign different weights to the two subgroup effects.
Choosing the Right Scale and Checking the Result
The prevalence-weighted formula applies directly to effects that are differences of means or risks. A relative effect, such as a risk ratio, is generally not the prevalence-weighted average of the subgroup risk ratios. For example, the all-comer risk ratio must be formed from the all-comer treatment and control risks, each calculated using the target prevalence, and then divided. This distinction is important because an estimate on one scale cannot be relabelled as an estimate on another scale.
| Estimand | Population and summary | How prevalence enters |
|---|---|---|
| Marker-positive effect | Average treatment contrast among \(M=1\) | No mixture weighting within this subgroup |
| Marker-negative effect | Average treatment contrast among \(M=0\) | No mixture weighting within this subgroup |
| All-comer additive effect | Average difference in the specified target population | \(\pi\tau_+ +(1-\pi)\tau_-\) |
| All-comer relative effect | Relative contrast of target-population outcomes | Combine group-specific outcomes within each arm first; do not generally average relative effects |
Several checks help detect a mismatch between the estimand and calculation:
- Check the target. State whether the all-comer population is the recruited sample, all eligible patients, or another defined group. Use that population’s prevalence.
- Check the weights. The weights must be nonnegative and sum to one: \(\pi+(1-\pi)=1\). If \(\pi=0\), the all-comer effect reduces to \(\tau_-\); if \(\pi=1\), it reduces to \(\tau_+\).
- Check the scale and units. Both subgroup effects must use the same outcome, time point, and additive scale before they are combined.
- Check the marker groups. The two groups must cover the target population without overlap. If some patients have unknown or intermediate status, the two-group formula does not describe all-comers unless those patients are addressed explicitly.
- Check transportability. Applying trial subgroup effects to a different target assumes those effects are applicable to that target’s corresponding marker groups. Reweighting alone cannot establish this.
A common mistake is to report the marker-positive treatment effect as the all-comer treatment effect because the trial enrolled mostly marker-positive patients. That reasoning confuses the effect in a subgroup with the effect in a population mixture. The correction is to define the target population, obtain or specify its marker prevalence, and combine the subgroup effects on the appropriate scale. Another mistake is to use the sample prevalence automatically; that is appropriate only if the sample mix is itself the intended target mix.
Check Your Understanding
Try each question, then compare with the concise solutions below.
- A target population is 20% marker-positive. The mean treatment effects are 4.0 units in marker-positive patients and 1.0 unit in marker-negative patients. What is the all-comer mean effect?
- Why should an enriched trial’s marker prevalence not automatically be used to estimate an all-comer effect for all eligible patients?
- For an adverse event, the marker-positive risk difference is \(-0.10\), the marker-negative risk difference is \(0.02\), and the target prevalence is 0.50. Calculate and interpret the all-comer risk difference.
- Is the all-comer risk ratio generally the prevalence-weighted average of the two subgroup risk ratios? Briefly explain.
Solutions. (1) \(0.20(4.0)+0.80(1.0)=0.80+0.80=1.60\) units. (2) Enrichment can make the recruited sample’s marker mix differ from the target population’s mix; use the target prevalence for the target estimand. (3) \(0.50(-0.10)+0.50(0.02)=-0.05+0.01=-0.04\), so the target-population event risk is 4 percentage points lower under treatment than control. (4) No. Calculate the prevalence-weighted treatment and control risks separately, then take their ratio; ratios do not generally combine by simple prevalence weighting.