Transport Weights for External-Control Populations

Why Transport Weights Are Needed

An external control cohort may differ from a trial population in characteristics that modify the outcome or treatment effect. If those characteristics are measured, weighting can make the external cohort’s covariate distribution resemble the trial population’s distribution. The resulting weighted outcome summary then targets the trial population, rather than the original external cohort, provided the necessary assumptions hold.

The goal is not simply to make two data sets look alike. A weight must represent how much each external participant stands in for participants in the target trial population. In particular, weighting measured effect modifiers cannot correct differences in unmeasured prognostic factors, incompatible outcome measurement, or calendar-time changes that are not adequately represented in the weighting variables.

Definition: Let \(X\) denote measured baseline covariates, \(f_T(x)\) their density in the target trial population, and \(f_E(x)\) their density in the external cohort. The transport weight for an external participant with covariates \(X=x\) is proportional to the density ratio \(f_T(x)/f_E(x)\). It upweights covariate profiles more common in the target population and downweights profiles less common there.

For discrete covariates, densities in this definition are probabilities. In practice, the ratio is generally estimated from observed data. A convenient scale is to normalize the estimated weights so that their sum equals the number of external participants. Multiplying all weights by the same positive constant does not change a weighted mean or risk, but the scale matters when weights are used to construct a likelihood.

Assumptions and the Weighting Identity

A weighted external-control outcome can represent the target population only under substantive and statistical conditions. The covariates used in weighting should include the measured factors that are relevant to outcome prognosis or modify the effect being transported. Outcomes and covariates must be defined comparably across data sources, and the external cohort must represent the source population described by its covariate distribution. Most importantly, target-population covariate profiles must be represented in the external cohort: if \(f_T(x)>0\), the external density \(f_E(x)\) must also be positive in the relevant region. This is the support or positivity requirement.

For transporting a control outcome, a further assumption is that, conditional on the measured covariates, the relevant control outcome distribution is transportable from the external cohort to the target trial population. This is sometimes expressed as conditional exchangeability across populations. It is not guaranteed by balancing \(X\): important unmeasured differences can violate it. If the purpose is to transport a treatment effect, the analogous conditional relationship for treatment effects must also be credible.

The rationale for the density ratio follows directly from integration. For any integrable outcome function \(g(X)\), weighting the external covariate distribution by \(f_T(X)/f_E(X)\) recovers its target-population expectation, when the support condition holds.

\begin{aligned} E_E\left[\frac{f_T(X)}{f_E(X)}g(X)\right] &= \int \frac{f_T(x)}{f_E(x)}g(x)f_E(x)\,dx\\ &= \int g(x)f_T(x)\,dx\\ &= E_T[g(X)]. \end{aligned}

Taking \(g(X)=E(Y\mid X,\text{control},E)\) shows why weighting can recover the target control mean if that conditional outcome relationship is transportable. The identity itself is mathematical; the transportability and measurement assumptions are scientific claims that must be assessed separately.

Estimating Weights From Data

There are several ways to estimate the ratio. With a small number of categorical effect modifiers, direct standardization or stratum-specific ratios may be transparent. With multiple covariates, one common approach is to pool target and external participants and model population membership. Let \(S=1\) identify a target trial participant and \(S=0\) an external participant. If the pooled data contain \(n_T\) target participants and \(n_E\) external participants, then

\frac{f_T(x)}{f_E(x)} = \frac{P(S=1\mid X=x)}{P(S=0\mid X=x)} \frac{n_E}{n_T}.

Thus, the estimated odds of target membership are multiplied by \(n_E/n_T\) to estimate the density ratio. The factor corrects for the sampling proportions in the pooled data. A logistic regression is one possible membership model; flexible models can also be used, but their predictions still require support checks and balance assessment. Estimated weights should be examined in the external cohort, since that is the data being reweighted.

Worked Example: Reweighting a Binary Effect Modifier

Consider a hypothetical target trial with 500 participants: 200 are marker-positive and 300 are marker-negative. A separate hypothetical external control cohort has 1,000 participants: 300 are marker-positive and 700 are marker-negative. Suppose marker status is a measured effect modifier and both marker groups are represented in both populations.

The target and external proportions are 0.40 and 0.30 for marker-positive participants, and 0.60 and 0.70 for marker-negative participants. The raw density-ratio weights are therefore

w_+ = \frac{0.40}{0.30}=1.333,\qquad w_- = \frac{0.60}{0.70}=0.8571.

The external cohort’s weighted marker-positive count is \(300(1.333)=400\), and its weighted marker-negative count is \(700(0.8571)=600\). The total weight is 1,000, so the normalized weighted proportions are \(400/1{,}000=0.40\) and \(600/1{,}000=0.60\), matching the target distribution.

Suppose, hypothetically, that there are 60 control events among the 300 marker-positive participants and 70 among the 700 marker-negative participants. The two observed event rates are \(60/300=0.20\) and \(70/700=0.10\). The weighted event total is \(60(1.333)+70(0.8571)=80+60=140\), giving a weighted control risk of \(140/1{,}000=0.140\). Equivalently, standardizing the two stratum-specific rates to the target proportions gives \(0.40(0.20)+0.60(0.10)=0.140\). This is a target-population estimate only if the conditional control outcome relationship is transportable.

Using a Membership Model

A membership model is useful when the covariates are numerous or continuous. The fitted probability \(e(x)=P(S=1\mid X=x)\) yields an estimated raw transport weight \(\widehat w(x)=\{e(x)/(1-e(x))\}(n_E/n_T)\). Predictions near zero or one can generate very small or very large weights, respectively. This can reveal weak support, but a smooth model does not create information for covariate profiles absent from the external cohort.

Worked Example: Converting Membership Probabilities to Weights

Suppose a hypothetical pooled data set contains \(n_T=500\) trial participants and \(n_E=1{,}000\) external participants. A fitted membership model gives a target-membership probability of 0.20 for one external participant’s covariate profile. The estimated raw weight is

\widehat w(x)=\frac{0.20}{1-0.20}\frac{1{,}000}{500} =0.25(2)=0.500.

For a second profile with fitted probability 0.60, the weight is \(\{0.60/0.40\}(2)=3.000\). The second profile is estimated to be more common in the target population relative to the external cohort, so its external observations receive more weight. These values are relative representation weights, not probabilities and not measures of how much causal information an individual supplies.

In this example, the model and sample-size factor are specified for illustration; in an analysis they would be estimated from the actual pooled covariates. Before interpreting a weighted outcome, check whether the weighted external covariates reproduce the target covariate distribution. A fitted model can have good predictive performance for membership yet still yield inadequate balance for important effect modifiers.

Support, Stability, and Weight Modification

Weight diagnostics should address both the covariate distributions and the weights themselves. Compare target and weighted-external covariate summaries, including the effect modifiers used in the weight model; inspect the range, quantiles, and concentration of the weights; and identify target profiles with little or no external representation. These checks complement the earlier material on positivity and covariate overlap, balance after adjustment, and effective sample size. A high maximum weight is a warning, not by itself proof of failure; the covariate profile producing it and its influence on the analysis matter.

A common stability summary is the effective sample size (ESS) of the external weights:

ESS=\frac{\left(\sum_{i=1}^{n_E}w_i\right)^2} {\sum_{i=1}^{n_E}w_i^2}.

It equals \(n_E\) when all weights are equal and decreases as weight concentration increases. It is a diagnostic of concentration, not a replacement for the actual sample size or a guarantee that the weighted estimate is unbiased.

Worked Example: Weight Concentration and Capping

In a hypothetical external cohort of 1,000 participants, suppose 900 have weight 0.500 and 100 have weight 5.500. The weights sum to \(900(0.500)+100(5.500)=450+550=1{,}000\), so their mean is 1. The ESS is

ESS=\frac{1{,}000^2}{900(0.500^2)+100(5.500^2)} =\frac{1{,}000{,}000}{225+3{,}025} =307.7.

Thus, although 1,000 participants were observed, the concentration of weights corresponds to an ESS of about 308 for this diagnostic. Now cap weights at 2.000 without rescaling. Their sum becomes \(900(0.500)+100(2.000)=650\), and the ESS becomes \(650^2/\{900(0.500^2)+100(2.000^2)\}=422{,}500/625=676.0\). Capping has made the weights less concentrated, but it has also altered their relative representation of covariate profiles.

To see the potential consequence, suppose the 900 lower-weight participants have an outcome rate of 0.10 and the 100 higher-weight participants have a rate of 0.40. The uncapped weighted rate is \(\{900(0.500)(0.10)+100(5.500)(0.40)\}/1{,}000=0.265\). After capping, it is \(\{900(0.500)(0.10)+100(2.000)(0.40)\}/650=0.1923\). The difference is not a mere numerical stabilization: it reflects a changed weighting of the groups. The capping rule and its effect on balance and the target estimand must therefore be reported and justified.

Trimming excludes participants outside a chosen range, whereas capping limits the largest weights. Both can reduce variance or sensitivity to a few observations, but can introduce bias for the original target population. Renormalizing all weights by a common constant changes neither a weighted mean nor the ESS; it does not undo the change in relative weights caused by capping or trimming.

Choosing an Approach and Interpreting Results

Direct stratification is transparent when there are few categorical covariates and adequate observations in every relevant stratum. A parametric membership model can be more compact, but depends on its functional form and specification. Flexible membership models can capture nonlinearities and interactions, yet still require empirical checks of support and balance. Calibration approaches instead select weights to match specified target covariate summaries; matching those summaries does not ensure balance on omitted features or correct transport of outcomes.

Transport weights also have a different role from Bayesian borrowing. They adjust the representation of observed external participants to target a covariate population; they do not establish that external and trial outcomes are exchangeable, and they are not automatically a prior discount factor. If weighted observations enter a Bayesian analysis through a power likelihood such as \(\prod_i p(y_i\mid\theta)^{w_i}\), the weights alter the likelihood contribution. This is a modeling choice, not an automatic consequence of density-ratio weighting. Treating estimated weights as fixed also omits their estimation uncertainty; a joint model or another justified uncertainty procedure may be needed when that uncertainty is material.

Key takeaway: A transport weight estimates how an external covariate profile should contribute to a target-population summary. Credible transport requires measured relevant effect modifiers, adequate support, acceptable balance, stable weights, and a defensible conditional outcome-transport assumption. No weighting diagnostic can verify all of these scientific conditions.

Common Mistakes and Failure Modes

Confusing a large weight with a large outcome effect. A weight describes relative representation, not the participant’s prognosis or treatment response.
Assuming balance proves transportability. Balance concerns measured covariates; it does not rule out unmeasured population differences or outcome measurement discrepancies.
Ignoring the scale factor in membership odds. The target-to-external odds must be multiplied by \(n_E/n_T\) for the pooled sampling setup described here.
Capping weights and claiming the original target is unchanged. Capping changes relative contributions and may change the effective target; show its impact on balance and estimates.
Reading ESS as a literal number of independent participants. ESS summarizes weight concentration and does not account for every source of uncertainty or dependence.

Check Your Understanding

Answer each question, then compare with the solutions below.

A target population has 25% of participants in a covariate stratum, while the external cohort has 50%. What is the raw transport weight for an external participant in that stratum?
In a pooled membership model, \(n_T=400\), \(n_E=800\), and an external participant has fitted target-membership probability 0.75. Calculate the estimated raw weight.
Why does adequate weighted balance on measured effect modifiers not by itself establish that a weighted external control risk is valid for the trial population?
What happens to a weighted mean if all weights are multiplied by the same positive constant? Does this operation repair a lack of covariate support?

Solutions. (1) The ratio is \(0.25/0.50=0.500\), so that profile is downweighted. (2) The weight is \(\{0.75/(1-0.75)\}(800/400)=3(2)=6.000\). (3) Balance verifies measured covariate representation, not conditional outcome transportability; unmeasured differences, incompatible measurement, or time-related changes may remain. (4) The common factor cancels between the weighted numerator and denominator of a weighted mean, so the mean is unchanged. It cannot create external observations in unsupported target profiles or repair positivity failure.