Generalizing Enriched-Trial Effects To Target Populations

Why an Enriched-Trial Effect May Not Be the Target Effect

An enriched trial deliberately recruits more participants from a biomarker stratum of interest, or may enroll only that stratum. Randomization can support a credible treatment comparison among the participants enrolled. But the average treatment effect in that enrolled sample need not equal the average effect in the population where treatment would be used: the prevalence of biomarker strata may differ, and treatment effects may differ across them.

To generalize a trial result, define the target population first. The target might be all eligible patients in a health system, all patients meeting a clinical indication, or a specified future population. The question addressed here is how to combine treatment effects within biomarker strata using the target population’s stratum proportions. This is a standardization problem, not a license to extrapolate an unobserved stratum effect.

Definition: Let \(B\) denote a baseline biomarker stratum, with \(B=1\) indicating biomarker-positive and \(B=0\) biomarker-negative. Let \(Y(1)\) and \(Y(0)\) be the outcomes that a person would have under treatment and control, respectively. On the mean-difference scale, the target-population average treatment effect is \(\Delta_T=E_T[Y(1)-Y(0)]\). The target prevalence is \(q_T=P_T(B=1)\), and the stratum-specific target effects are \(\Delta_{1,T}=E_T[Y(1)-Y(0)\mid B=1]\) and \(\Delta_{0,T}=E_T[Y(1)-Y(0)\mid B=0]\).

Standardizing Stratum-Specific Effects

For a binary biomarker, the law of total expectation gives the target effect as a weighted average of the two conditional effects. The weights are the proportions of biomarker-positive and biomarker-negative people in the target population, not necessarily the proportions in the trial.

\Delta_T = q_T\Delta_{1,T}+(1-q_T)\Delta_{0,T}.

If a trial estimates both stratum-specific effects, and those effects can be transported to the target population, an estimator is \(\widehat{\Delta}_T=\hat q_T\widehat{\Delta}_1+(1-\hat q_T)\widehat{\Delta}_0\). Here \(\hat q_T\) estimates target prevalence and \(\widehat{\Delta}_b\) estimates the effect in stratum \(b\). The estimated effects must use the same outcome definition, follow-up time, and treatment contrast.

Key identification condition: For each stratum used in the standardization, the treatment effect among trial participants in that stratum must equal the corresponding target-population effect: \(\ E[Y(1)-Y(0)\mid B=b,S=1]=E_T[Y(1)-Y(0)\mid B=b]\), where \(S=1\) denotes trial participation. This is conditional effect transportability. It is an assumption about how effects relate across trial and target populations; randomization within the trial does not establish it.

A short derivation shows why both the weights and the conditional effects matter. Partition the target population by biomarker status and apply the law of total expectation:

\begin{aligned} \Delta_T &=E_T[Y(1)-Y(0)]\\ &=P_T(B=1)E_T[Y(1)-Y(0)\mid B=1]\\ &\quad+P_T(B=0)E_T[Y(1)-Y(0)\mid B=0]\\ &=q_T\Delta_{1,T}+(1-q_T)\Delta_{0,T}. \end{aligned}

This identity is exact for the mean treatment-effect difference. To estimate its terms from trial data, further assumptions are required. In particular, randomization identifies a causal treatment contrast within a trial stratum, while transportability connects that contrast to the target stratum.

Assumptions That Support Transport

The following conditions clarify what the calculation does and does not establish. They should be considered in relation to a specific trial and a precisely described target population.

Well-defined treatment and outcome. Treatment and control correspond to the interventions of interest, and \(Y\) is measured comparably in the trial and target setting. The outcome scale and assessment time are the same.
Consistency and no interference. Each participant’s observed outcome under the received treatment equals the corresponding potential outcome, and one participant’s treatment does not change another participant’s outcome in a way relevant to the estimand.
Valid within-trial comparisons. Treatment is randomized, or otherwise exchangeable, within the biomarker strata used. Each relevant stratum has a positive probability of assignment to each treatment arm, and missing outcomes or protocol deviations do not invalidate the chosen estimand and analysis.
Conditional effect transportability. Within each stratum, the trial effect applies to the target population. If effect modifiers other than \(B\) differ between trial and target, this condition may fail even when biomarker prevalence is known.
Stratum coverage. Every target stratum with nonzero prevalence has a usable effect estimate or a defensible source for that estimate. A trial containing only biomarker-positive participants cannot, by itself, estimate the biomarker-negative effect.
Valid target prevalence. \(q_T\) is known or estimated from data representative of the defined target population, with the same biomarker definition. A trial’s enriched prevalence is not a substitute for target prevalence.

For a richer set of effect modifiers, the same logic extends by standardizing over joint covariate strata, but sparse or absent trial strata can make the required effects unsupported. The binary-marker formula is useful precisely because it makes the support and weighting requirements visible.

Worked Example: Reweighting Effects to the Target Biomarker Mix

Suppose a hypothetical randomized trial estimates a mean treatment benefit of 8.0 days in biomarker-positive participants and 3.0 days in biomarker-negative participants. Assume these effects are transportable to the defined target population. In the target population, 35% are biomarker-positive. What is the target average treatment effect?

The target prevalence is \(q_T=0.35\), so the negative-stratum proportion is \(1-q_T=0.65\). Standardize using the target mix:

\begin{aligned} \widehat{\Delta}_T &=q_T\widehat{\Delta}_1+(1-q_T)\widehat{\Delta}_0\\ &=(0.35)(8.0\text{ days})+(0.65)(3.0\text{ days})\\ &=2.80\text{ days}+1.95\text{ days}\\ &=4.75\text{ days}. \end{aligned}

Under the stated assumptions, the estimated average treatment benefit in the target population is 4.75 days. If the enriched trial sample were 80% positive, weighting these same effects by its sample composition would give \((0.80)(8.0)+(0.20)(3.0)=7.00\) days. That is the effect for the trial’s mixture under these assumptions, not the target effect.

What If the Trial Enrolled Only One Stratum?

An enriched trial restricted to biomarker-positive participants can estimate the positive-stratum effect, subject to valid randomization and other trial assumptions. It does not identify the negative-stratum effect merely because target prevalence is available. If \(q_T<1\), the target formula still contains \(\Delta_{0,T}\); without information about that quantity, there is no unique target average effect.

Worked Example: Identifying What Is Missing

A hypothetical biomarker-positive-only trial estimates a 10-percentage-point increase in response probability. The target population is 30% biomarker-positive. First determine whether the trial and prevalence alone identify the target risk difference. Then suppose an independent, relevant randomized data source supports a 2-percentage-point risk difference among biomarker-negative target patients, and assume it is transportable.

With only the enriched trial and \(q_T=0.30\), the target effect would be \(\Delta_T=(0.30)(10)+(0.70)\Delta_{0,T}\), in percentage points. Since \(\Delta_{0,T}\) is unknown, the target effect is not identified.

With the additional negative-stratum estimate and its transportability assumption, calculate:

\begin{aligned} \widehat{\Delta}_T &=(0.30)(10\text{ percentage points}) +(0.70)(2\text{ percentage points})\\ &=3.00+1.40\\ &=4.40\text{ percentage points}. \end{aligned}

Thus, the combined evidence supports an estimated target risk difference of 4.40 percentage points, conditional on the stated assumptions. The external estimate is not made valid simply by being available: its population, endpoint, treatment contrast, and follow-up must be sufficiently comparable, or its own transport assumptions must be justified.

Uncertainty in the Standardized Estimate

A point estimate alone does not describe its precision. If target prevalence \(q_T\) is treated as known and the two stratum-effect estimates have variances \(V_1\) and \(V_0\), then the variance of the weighted estimate is \(q_T^2V_1+(1-q_T)^2V_0+2q_T(1-q_T)\operatorname{Cov}(\widehat{\Delta}_1,\widehat{\Delta}_0)\). The covariance term should not be dropped without justification. If estimates come from independent samples or independent randomized strata, zero covariance may be reasonable; otherwise it must be estimated or accounted for.

If prevalence is estimated rather than known, its uncertainty also contributes. For independent estimates, a first-order approximation adds \((\Delta_1-\Delta_0)^2\operatorname{Var}(\hat q_T)\) to the variance expression. This term is larger when stratum effects are more different. If the prevalence and effect estimates are statistically dependent, additional covariance terms are needed. A confidence interval then reflects sampling uncertainty under the model and assumptions; it does not quantify uncertainty from an incorrect transportability assumption.

Worked Example: A Confidence Interval for a Standardized Effect

Suppose a hypothetical analysis has \(q_T=0.40\), a positive-stratum effect estimate of 6.0 percentage points with \(SE=1.5\) percentage points, and a negative-stratum estimate of 1.0 percentage point with \(SE=1.0\) percentage point. Assume the estimates are independent, target prevalence is known, and a normal approximation is adequate.

First standardize the effects:

\widehat{\Delta}_T=(0.40)(6.0)+(0.60)(1.0)=2.40+0.60=3.00\text{ percentage points}.

Independence makes the covariance zero. The variance and standard error are:

\begin{aligned} \widehat{\operatorname{Var}}(\widehat{\Delta}_T) &=(0.40)^2(1.5)^2+(0.60)^2(1.0)^2\\ &=(0.16)(2.25)+(0.36)(1.00)\\ &=0.36+0.36=0.72,\\ SE(\widehat{\Delta}_T)&=\sqrt{0.72}=0.8485\text{ percentage points}. \end{aligned}

A 95% normal-approximation confidence interval is \(3.00\pm1.96(0.8485)\), or \(3.00\pm1.663\), giving approximately 1.34 to 4.66 percentage points. Under the model and assumptions, this procedure has 95% long-run coverage for the standardized target effect. It does not mean there is a 95% probability that this fixed effect lies in this particular interval.

Choice of Effect Scale and Alternative Approaches

The weighted-average formula applies directly to mean differences, including risk differences for a binary outcome when effects are expressed as differences in response probabilities. It does not generally apply by taking a prevalence-weighted average of stratum-specific risk ratios or odds ratios. For a relative measure, first obtain or estimate the relevant target stratum-specific outcome means under each treatment, average those means using target prevalence, and then form the target-level ratio or odds ratio. The resulting marginal relative effect need not equal an average of the conditional relative effects.

Approach	What it estimates	Main requirement or limitation
Target standardization	A target-average effect assembled from stratum-specific effects and target prevalence	Requires target weights and supported, transportable effects for all relevant strata
Unweighted pooled trial contrast	An effect for the trial’s enrolled mixture	Matches the target only if the trial mix and target mix yield the same average effect
Direct all-comer randomized trial	A target-like average effect if trial participants represent the target population	Can be less efficient for a rare stratum; still needs generalizability from trial participants to the target
Model-based standardization	A target-average effect predicted from an outcome or effect model	Can accommodate measured effect modifiers but depends on model specification and covariate support

Standardization is therefore not inherently superior to an all-comer design or a correctly specified model. It makes the target composition explicit and can use evidence from more than one source. Its advantage is transparency about which stratum effects and weights determine the answer; its limitation is that unsupported effects and unmeasured differences cannot be repaired by arithmetic.

Diagnostics, Failure Modes, and a Common Mistake

Before reporting a transported estimate, check whether the analysis is being driven by extrapolation or fragile inputs. Useful diagnostics include:

Compare the target and trial biomarker proportions and verify that the target prevalence refers to the same eligibility definition and time period.
Confirm that each target stratum has evidence for its effect. A missing stratum estimate should appear as missing information, not be silently replaced by zero or by the other stratum’s effect.
Review sample sizes, standard errors, and outcome completeness within each stratum. A very imprecise stratum estimate can dominate uncertainty even if its target weight is modest.
Examine whether known effect modifiers differ between trial and target within biomarker strata. If they do, consider whether suitable data support finer standardization; if not, acknowledge the transportability limitation.
Check that all component effects use the same treatment contrast, outcome scale, and follow-up. Combining a risk difference at one time point with a hazard ratio or a different follow-up does not implement the stated estimand.

Common mistake: “The trial enrolled 80% biomarker-positive patients, so the trial effect is the effect for a target population that is 35% positive.” This confuses the trial’s composition with the target’s composition. If stratum effects differ, the pooled trial contrast weights them by the trial mix. The correction is to estimate stratum-specific effects, apply the target weights, and state the transportability and coverage assumptions. If the trial measured only one stratum, the other effect remains unidentified unless supported by additional valid evidence or an explicit, defensible assumption.

Key takeaway: A target-population effect is a target-weighted combination of stratum-specific effects on the appropriate scale. The weights must describe the target, the effects must apply to its strata, and every stratum with positive target prevalence must have adequate support.

Check Your Understanding

Answer each question using the definitions and assumptions above. Brief solutions follow the questions.

A target population is 20% biomarker-positive. The transportable mean benefits are 5 days in positive patients and 2 days in negative patients. Calculate the target mean benefit.
A biomarker-positive-only trial estimates a risk difference, and the target prevalence is known to be 50%. Is the target risk difference identified from those facts alone? Explain.
Why might an unweighted pooled effect from an enriched trial differ from a standardized target effect, even when both stratum-specific estimates are unbiased?
For two independent stratum-effect estimates, target prevalence is known and fixed. Write the variance of the standardized estimate.
Why is it generally incorrect to average stratum-specific risk ratios using target prevalence as weights?

Solutions. (1) \((0.20)(5)+(0.80)(2)=1.00+1.60=2.60\) days. (2) No. The negative-stratum effect is absent, and it receives positive target weight, so the target effect cannot be calculated without additional evidence or an assumption. (3) The pooled estimate uses the trial’s stratum proportions, whereas the standardized estimate uses target proportions; the two averages differ when the mixes differ and stratum effects are heterogeneous. (4) With \(V_1=\operatorname{Var}(\widehat{\Delta}_1)\) and \(V_0=\operatorname{Var}(\widehat{\Delta}_0)\), the variance is \(q_T^2V_1+(1-q_T)^2V_0\). (5) A risk ratio is a ratio of outcome risks, not an additive effect; the marginal ratio must be formed from target-weighted risks under each treatment, and need not equal a weighted average of conditional ratios.