External Data and Transport Assumptions

What Can an External Outcome Dataset Tell Us?

Suppose a study records an outcome for some participants but not others. A separate dataset may contain that outcome for a different population, perhaps with a much higher completion rate. It is tempting to use the external outcome distribution as a substitute for the outcomes missing in the study. That substitution is justified only if the external data speak to the target population’s full outcome distribution, not merely to the external population’s own outcomes.

The distinction matters because the outcome distribution among target-study respondents may differ from the distribution among nonrespondents. External data can help determine the latter only through assumptions linking the external and target populations, together with the known response pattern and observed outcomes in the target study.

The central transport assumption is that, after conditioning on relevant covariates, the outcome distribution is the same in the target and external populations:

for every outcome set \(A\) and covariate value \(x\) relevant to the target population. This is a distributional assumption, not merely equality of means. For a binary outcome it implies equality of conditional event probabilities; for a continuous outcome it concerns the whole conditional distribution.

The equation alone is not enough. The external data must estimate the external conditional distribution in a way that represents the stated external population—for example, through a valid sampling design or appropriate weights. The external outcome must be measured on a scale and at a time comparable to the target outcome. The covariates used for transport must have comparable definitions, and they must be sufficiently rich to account for outcome-relevant differences between populations. Finally, covariate overlap is required: target covariate values must occur in the external data, so the conditional outcome distribution needed for the target is supported by external observations.

From Transport to the Missing-Outcome Distribution

For a fixed covariate value \(x\), write \(r(x)=P(R=1\mid X=x,S=1)\), \(F_1(y\mid x)=P(Y\leq y\mid R=1,X=x,S=1)\), and \(F_0(y\mid x)=P(Y\leq y\mid R=0,X=x,S=1)\). Let \(F_T(y\mid x)=P(Y\leq y\mid X=x,S=1)\) be the full target-population outcome distribution. Partitioning target members into respondents and nonrespondents gives

If the transport assumption holds, the external data identify or estimate \(F_T(y\mid x)\) through the external conditional distribution. The target study supplies \(r(x)\) and \(F_1(y\mid x)\). When \(r(x)<1\), rearranging the mixture equation gives

This is a population identity under the stated assumptions, not a recipe that guarantees every estimated distribution will be valid. For example, the right-hand side must be nondecreasing in \(y\), bounded between zero and one, and have the properties of a distribution function. If it does not, the assumed transported distribution and target respondent data are incompatible at the population level, or their estimates are too noisy to satisfy the identity.

For a binary outcome, let \(p_T(x)=P(Y=1\mid X=x,S=1)\), \(p_1(x)=P(Y=1\mid R=1,X=x,S=1)\), and \(p_0(x)=P(Y=1\mid R=0,X=x,S=1)\). Then

A necessary feasibility condition is \(r(x)p_1(x)\leq p_T(x)\leq r(x)p_1(x)+1-r(x)\). These inequalities say that the total target event probability cannot be smaller than the event mass already contributed by respondents, or larger than that mass plus the maximum possible event mass among nonrespondents.

Why Covariate-Specific Transport Matters

When the target and external populations have different covariate distributions, comparing their unadjusted outcome prevalences can be misleading even if the conditional transport assumption holds. The target population’s overall distribution is formed by averaging conditional outcome distributions over the target covariate distribution:

for discrete \(X\), with an integral replacing the sum for continuous covariates. The external data contribute the conditional outcome distributions; the target study contributes the target covariate distribution. Applying external conditional estimates to the external covariate mix instead answers a question about the external population, not the target.

Assumptions, Diagnostics, and Failure Modes

The calculation is informative only to the extent that its assumptions are credible. In practice, the key checks are conceptual as well as numerical:

• Population comparability: after conditioning on \(X\), no remaining population difference changes the outcome distribution. If, for example, an unmeasured severity factor affects outcomes and differs between populations, equality conditional on the recorded covariates may fail.
• Overlap: each target covariate profile requiring transport must be represented in the external data. Extrapolation into unsupported profiles is not identification from the external outcomes.
• Measurement comparability: outcome definitions, instruments, eligibility rules, and measurement times must support the same interpretation. A one-year outcome in one dataset is not automatically comparable with a six-month outcome in another.
• Data-source validity: the external sample’s outcome and covariates must support inference about its stated source population. Differential outcome recording or unaccounted sampling can undermine the transported distribution.
• Mixture compatibility: transported probabilities must satisfy the feasibility inequalities in each stratum. For a full outcome distribution, the reconstructed \(F_0\) must be a valid distribution function.

A useful distinction is whether the external data identify the target full outcome distribution or only contribute a restriction. If the transport assumptions and data are accepted and the relevant target covariate strata have support, the mixture identity can identify the nonrespondent distribution when the response rate is below one. If transport is only partly credible, or the external data yield a range rather than a single target margin, the same identity can instead translate that range into bounds on missing outcomes. The strength of the conclusion should match the strength of the assumptions.

External outcome data differ from a refreshment sample: the latter samples from the target population under its own design and timing assumptions, whereas external data require a link between distinct populations. They also differ from a validation sample, which can directly measure outcomes missing in a subset of the target study under its validation design. Neither source automatically removes assumptions. When the transport link is not defensible, the earlier material on bounds and restrictions, or sensitivity analysis, is preferable to presenting a transported point estimate as established by the observed data.

Common Mistake and Practical Interpretation

A common mistake is to say, “The external dataset has complete outcomes, so it tells us the missing outcomes in the study.” Completeness addresses missingness in the external dataset; it does not establish population comparability. A correct statement is conditional: if the external outcome distribution transports to the target conditional on adequately measured covariates, the external data identify the target outcome margin in supported strata; combining that margin with the target response mixture then identifies the missing-outcome distribution.

Even when the assumptions are scientifically plausible, estimates can be unstable when target response rates are close to one, because the denominator \(1-r(x)\) is small and small discrepancies in the target margin or respondent estimate are amplified. Sparse external strata also yield uncertain conditional estimates. Sampling uncertainty should therefore be carried through the standardization and mixture calculation; a point estimate that satisfies feasibility is not proof that transport holds.