Why the Patient Population Is Part of the Effect
A treatment effect estimated in a clinical trial describes a particular population under a particular endpoint definition and follow-up. Applying that result to a different group of patients is not simply a matter of removing confounding. The new group may contain different proportions of patients for whom the treatment works better or worse. Consequently, even when the trial’s within-group effects apply, its average effect need not equal the average effect in the new population.
Transporting an effect means using evidence from one population to estimate a treatment-effect quantity for a separately specified target population. The target must be named: for example, all patients meeting the trial’s eligibility criteria in a defined health system, rather than “patients in practice” in general. The target definition determines which patients and which distribution of patient characteristics the answer concerns.
A different target population can have a different value of \(\Delta_T\). Likewise, a risk difference, risk ratio, and odds ratio are different effect measures and need not transport in the same way. A report should state the population, outcome, follow-up, treatment contrast, and effect scale before interpreting an estimate.
Confounding Adjustment Is Not Transport
Confounding adjustment addresses a comparison of treatments within a population when treatment assignment is associated with prognosis. In a randomized trial, randomization identifies the treatment contrast in the trial population, subject to adherence to the assigned-treatment estimand and appropriate outcome ascertainment. Transport addresses a different question: whether that contrast, or its relevant conditional components, applies to another population and what average it implies there.
The distinction matters even if all treatment groups have been perfectly balanced on measured covariates. Balance within the trial does not make the trial participants representative of the target population. Conversely, a population difference is not automatically a confounder: it threatens transport of an average effect when it changes the distribution of characteristics that modify the treatment effect, or when outcome relations differ across populations in ways relevant to the effect.
For example, disease severity may affect both treatment choice and outcome in a nonrandomized comparison, making it a confounder for that comparison. It may also modify the treatment effect. Adjustment for severity can address treatment confounding in the source data, while transporting an effect still requires accounting for how severity is distributed in the target population and whether the conditional effects apply there. One variable can play both roles; the questions remain distinct.
Identification Conditions for Transport
Write \(m_a(x)=E\{Y(a)\mid X=x,S=1\}\) for the trial’s conditional mean outcome under treatment \(a\). If the target distribution of \(X\) is known, one can average the trial’s conditional effects over that distribution. This identifies the target effect under assumptions, not by virtue of standardization alone.
Conditional mean transportability says that, within measured \(X\) strata, the average potential outcome under each treatment is the same in the trial and target populations. It permits the distribution of \(X\) to differ. It does not require identical populations, but it can fail if an unmeasured characteristic modifies the effect and its distribution differs by trial participation. For an average treatment effect, this mean condition is sufficient; equality of the full outcome distributions is stronger than necessary.
The trial’s randomization supports identification of \(m_a(x)\) only for covariate patterns represented in the trial and with treatment assignment possible. It does not establish conditional transportability. Nor does a large trial compensate for absent target-relevant strata: extrapolating to such strata requires additional, untestable modeling assumptions.
Standardizing Conditional Effects to the Target
Under the conditions above, the target effect is the target-population average of the conditional trial effects. For a discrete \(X\), this is a sum; for continuous \(X\), it is an integral. The formula makes explicit why different target populations can have different average effects even when the same conditional effects apply.
A short justification follows from iterated expectation. First condition each potential outcome on \(X\) in the target, then use conditional mean transportability to replace the target conditional means with trial conditional means:
In a Bayesian analysis, the same calculation can be applied to each posterior draw of the conditional outcome means (and, where modeled, the target covariate distribution). The resulting draws describe posterior uncertainty about \(\Delta_T\), conditional on the model and transport assumptions. This does not make those assumptions empirically guaranteed: uncertainty from an unmeasured effect modifier is not automatically captured unless it is explicitly modeled.
Worked Example: Transporting a Risk Difference to a Target Population
Suppose a hypothetical randomized trial compares a new treatment with control for a binary outcome by 12 weeks. Baseline renal impairment is measured in both the trial and a defined target population. The estimated risks are as follows.
| Renal impairment | Control risk | Treatment risk | Risk difference |
|---|---|---|---|
| No | 0.10 | 0.06 | −0.04 |
| Yes | 0.30 | 0.24 | −0.06 |
Assume treatment was randomized within the trial, the endpoint and follow-up are harmonized, these strata are represented in both populations, and conditional mean transportability is credible. Renal impairment is an effect modifier on the risk-difference scale because the conditional differences are not equal.
In the trial, 20% of participants had renal impairment. Its trial-population average risk difference is $$ (0.80)(-0.04)+(0.20)(-0.06)=-0.032-0.012=-0.044. $$ That is a 4.4-percentage-point lower risk under treatment in the trial population. Suppose 60% of the target population has renal impairment. The transported target effect is $$ (0.40)(-0.04)+(0.60)(-0.06)=-0.016-0.036=-0.052. $$ Under the stated assumptions, treatment is associated with a 5.2-percentage-point lower 12-week risk in the target population. The target effect is more negative because the target contains a larger share of the stratum with the larger absolute risk reduction. This calculation changes the population average; it does not assert that the treatment has a different effect within either renal-impairment stratum.
The Target Definition Changes the Answer
“Transport to patients outside the trial” is too vague to define an estimand. Two plausible targets may have different covariate distributions and therefore different average effects. The analysis should specify the eligibility rules, care setting, calendar period, and other features that define the population, while distinguishing those features from the patient characteristics used in the outcome model.
Worked Example: Two Targets, Two Average Effects
In a hypothetical trial, a binary severity indicator modifies the 6-month risk difference. The conditional treatment effects are −0.02 for patients with lower severity and −0.10 for patients with higher severity. Assume the required transportability and overlap conditions hold for both target populations.
Target A is all eligible patients in a regional health system, 30% of whom have higher severity. Its transported effect is $$ (0.70)(-0.02)+(0.30)(-0.10)=-0.014-0.030=-0.044. $$ The estimated risk difference is −4.4 percentage points.
Target B is the subset of eligible patients referred to specialist care, 70% of whom have higher severity. Its transported effect is $$ (0.30)(-0.02)+(0.70)(-0.10)=-0.006-0.070=-0.076. $$ The estimated risk difference is −7.6 percentage points. Both results use the same conditional effects; they differ because the populations have different severity distributions. It would be misleading to call either answer “the” effect outside the trial without naming the target.
Diagnostics, Limitations, and Alternatives
Observed data cannot prove conditional mean transportability, because potential outcomes under the unassigned treatment are not jointly observed and target outcomes may be unavailable. Diagnostics can nevertheless reveal serious problems. Compare trial and target covariate distributions, paying particular attention to plausible effect modifiers and to combinations of covariates, not just marginal averages. Check whether target strata have trial support. Earlier material on positivity and covariate overlap and on balance after adjustment is relevant to these checks.
- Limited overlap: If a target subgroup is absent or very rare in the trial, its conditional treatment effect is weakly identified or unidentified from trial data. A precise-looking model-based estimate can conceal extrapolation.
- Unmeasured effect modification: If a prognostic treatment modifier differs between populations but is omitted, averaging over measured characteristics may yield the wrong target effect.
- Endpoint or treatment mismatch: Different outcome ascertainment, follow-up, treatment versions, or care pathways can make conditional outcome means incomparable. Harmonized labels alone do not ensure identical measurement or clinical meaning.
- Scale dependence: Effect modification is scale-specific. Equal conditional risk ratios need not imply equal risk differences, and equal conditional odds ratios do not imply equal marginal odds ratios. Define the effect scale before deciding whether effects vary.
- Model dependence: Conditional outcome models may be misspecified, particularly in sparse strata. Examine calibration where outcome data allow it and report uncertainty and sensitivity analyses rather than relying on a single extrapolated estimate.
The main alternative is to report the trial-population effect, which avoids claiming that it represents a different population but may not answer the clinical decision question. A subgroup effect describes a conditional contrast for a subgroup; it is not automatically the average effect in a target population. A meta-analysis combines evidence across studies, but does not by itself identify an effect for a specified target unless the population and transport assumptions are addressed. A target-standardized outcome model and a transport-weighting approach can target the same estimand under suitable conditions, but differ in modeling and diagnostic demands; the choice of estimator does not remove the need to define the target or defend the assumptions.
Transport is most defensible when the target is explicit, relevant effect modifiers are measured consistently, the trial covers target covariate patterns, and the conditional effects are clinically plausible across populations. When these conditions are doubtful, report which assumption is uncertain and how conclusions change under plausible alternatives. The estimate is conditional on the specified target and assumptions, not a universal treatment effect.
Worked Example: An Omitted Modifier Can Mislead
Consider a hypothetical binary biomarker that is measured in the target population but omitted from the transport model. In the trial, 10% of patients are biomarker-positive; in the target, 70% are positive. Suppose the conditional risk difference is −0.10 among biomarker-positive patients and 0 among biomarker-negative patients. Assume these conditional effects apply in both populations and overlap is adequate.
The trial average is $$ (0.10)(-0.10)+(0.90)(0)=-0.010, $$ or a 1-percentage-point reduction. The true target average, accounting for the modifier, is $$ (0.70)(-0.10)+(0.30)(0)=-0.070, $$ or a 7-percentage-point reduction. Transporting the trial average unchanged would understate the absolute risk reduction by 6 percentage points. The issue is not treatment confounding: even a randomized trial can have a different average effect from the target because the modifier’s prevalence differs. If the biomarker is unobserved or its interaction with treatment is not credibly estimated, the target effect cannot be recovered from these numbers alone.
Common Mistake
Wrong reasoning: “The external patients were adjusted to have the same measured covariates as trial patients, so the treatment effect has been transported.” This confuses covariate balance for a treatment comparison with defining the population over which an effect is averaged. It may also confuse adjusting treatment confounding with accounting for effect modification.
Correction: State the target population and effect scale first. Then identify plausible modifiers, assess whether their conditional effects are estimable in the trial, and average those conditional effects over the target’s covariate distribution under explicit transportability and overlap assumptions. Balance alone establishes none of these assumptions.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A risk difference is −0.03 for one subgroup and −0.09 for another. What additional information is needed to calculate the target average risk difference?
- In the trial, 25% of patients are in a subgroup with a risk difference of −0.08; the remaining patients have a risk difference of −0.02. The target subgroup prevalence is 50%. Calculate the target effect.
- Why does randomization in the trial not, by itself, establish that the trial effect applies to a target population?
- A target includes a covariate pattern absent from the trial. Which condition is in difficulty, and what does this imply for estimation?
Solutions. (1) The target prevalence of each subgroup, plus the definition of the target, treatment contrast, outcome, follow-up, and effect scale. The subgroup effects must also be applicable to the target under the transport assumptions. (2) The target average is \((0.50)(-0.08)+(0.50)(-0.02)=-0.04-0.01=-0.05\), a 5-percentage-point lower risk under treatment. (3) Randomization identifies treatment contrasts within the trial; it does not establish equality of conditional potential-outcome means across trial and target populations or equal modifier distributions. (4) Covariate overlap, or positivity for trial representation of target covariate patterns, fails. The relevant conditional effect is not identified directly by the trial in that pattern, so an estimate requires extrapolation or additional evidence and assumptions.