Why Overlap Is a Requirement for Bridging
A source population supplies evidence about a treatment effect, while a target population is the population to which that effect is intended to apply. Even when the estimand is aligned and relevant treatment-effect modifiers have been identified, a further question remains: does the source contain evidence for the covariate profiles found in the target? If important target profiles are absent or rare in the source, the data provide little or no direct support for extrapolating to them.
Let \(X\) denote the measured covariates relevant to the bridging analysis, \(S=1\) denote membership in the source population, and \(S=0\) denote membership in the target population. Let \(f_S(x)\) and \(f_T(x)\) denote the covariate densities in the source and target, respectively. For discrete covariates, these densities are probability masses. The condition is about the joint covariate profile \(X\), not only one covariate at a time.
This condition is distinct from exchangeability. Exchangeability concerns whether, conditional on the relevant covariates, the treatment effect can be related across populations. Positivity concerns whether the source contains observations at those covariate values in the first place. Neither condition guarantees the other. A bridging analysis may have a plausible exchangeability argument and still lack support for some target profiles.
Density and Propensity-Score Diagnostics
A first diagnostic is to compare source and target covariate distributions. Overlay density plots for continuous variables and compare proportions or counts for categorical variables. Strongly separated distributions, target mass beyond the source range, or thin source tails are warnings. A marginal plot is useful for locating potential problems in one variable, but it cannot establish joint overlap: source and target may each include low and high values of two covariates while having very different combinations of those values.
A second diagnostic summarizes the covariates through the probability of source membership. In a pooled sample of source and target participants, define the source-membership propensity score \(e(x)=P(S=1\mid X=x)\). If the pooled sample contains a source fraction \(\pi=P(S=1)\), Bayes' rule gives:
When both populations have positive density at \(x\), the score lies strictly between 0 and 1. A score near 0 indicates that the profile is much more characteristic of the target than the source; a score near 1 indicates the reverse. Scores at or near the boundaries therefore help flag limited overlap. These scores depend on the source-to-target sampling fraction in the pooled data, so the numerical values are not universal properties of the populations.
The connection between the score and support follows directly by taking its odds:
If \(f_T(x)>0\), then \(e(x)=0\) exactly when \(f_S(x)=0\). Thus, a target profile absent from the source has a source-membership score of zero in the population. In finite data, however, an estimated score is a model-based diagnostic: a flexible or poorly specified model can hide or exaggerate separation. Inspect the covariate distributions as well as the score distribution, and examine profiles that are scientifically important.
Worked Example: A Thin Source Tail
Suppose a hypothetical pooled bridging data set has 100 source participants and 100 target participants. Among people aged 70–74 years, there are 2 source participants and 20 target participants. Treat the age band as a discrete covariate profile for this calculation.
The estimated source-membership proportion in this band is \(2/(2+20)=0.0909\). The corresponding target proportion in the same pooled band is \(20/22=0.909\). The estimated source-to-target count ratio is \(2/20=0.100\), indicating that the band is much less represented in the source than in the target. The source is not empty in this band, so this is not a demonstrated structural positivity failure. It is a practical-overlap warning: two source participants provide little direct information for characterizing the source response in this age range.
This calculation is descriptive, not a test of whether transportability holds. The result also depends on how the pooled data were sampled. The appropriate conclusion is that the age band deserves scrutiny and may require more evidence, not that an automatic score cutoff proves the analysis invalid.
Worked Example: Marginal Overlap Can Hide a Missing Joint Profile
Consider two binary covariates in a hypothetical study: \(X_1\) indicates renal impairment and \(X_2\) indicates use of a concomitant medicine. The source contains 100 people: 50 with profile \((0,0)\), 30 with \((0,1)\), 20 with \((1,0)\), and none with \((1,1)\). The target also contains 100 people: 30, 20, 20, and 30 people in those respective profiles.
Both populations include people with and without renal impairment: the source has 20 with impairment, and the target has 50. Both also include people with and without the concomitant medicine: the source has 30 medicine users, and the target has 50. Marginal comparisons therefore show representation of each category in each population.
But the joint profile \((1,1)\) occurs in 30 target participants and no source participants. Its target probability is \(30/100=0.300\), while its source probability is \(0/100=0\). Thus \(f_T(x)>0\) and \(f_S(x)=0\) for that profile, violating positivity. In the pooled data, the empirical source-membership score for that cell is \(0/(0+30)=0\). Separate plots for the two covariates would not reveal this joint gap; a cross-tabulation or joint diagnostic does.
If the treatment effect could differ for people with both characteristics, the source data do not directly identify how the source effect behaves in that group. A model that predicts an effect there is extrapolating beyond observed source support; a smooth prediction does not remove the lack of evidence.
What to Do When Overlap Is Limited
The response should reflect both the severity of the gap and the scientific question. Before changing the analysis, verify that covariates are defined and measured comparably in both populations, that the plotted samples represent the intended populations, and that the diagnostic includes the relevant joint profiles. Then identify which target participants are poorly supported and how much of the target they represent.
| Response | What it does | Main implication |
|---|---|---|
| Restriction | Defines a target subgroup with covariate profiles supported by the source. | The conclusion applies to this restricted target, not automatically to the original target. |
| Trimming | Excludes profiles with inadequate overlap according to a stated rule. | Changes the population represented by the analysis; the excluded fraction and rule must be reported. |
| New evidence | Obtains relevant evidence in unsupported or thinly represented profiles. | May improve support, but evidence must be suitable for the same estimand and population question. |
Restriction is appropriate when a scientifically meaningful target subgroup can be specified and the decision can be limited to that subgroup. Trimming is an analytic exclusion rule, not a way to make unsupported target individuals supported. A threshold such as \(0.1<e(x)<0.9\) is sometimes used as a practical screen, but no universal cutoff defines acceptable overlap. The chosen rule should be justified in context, and results should state who was excluded and what target population remains.
When the unsupported group matters to the decision, new evidence may be necessary. This could mean additional source-like participants with the relevant profiles or another appropriate evidence source that addresses the same clinical question. A mechanistic rationale may help explain why effects could be similar, but it does not create observed source support. The need for additional evidence should be evaluated against the consequences of making a decision for the unsupported group.
Worked Example: Quantifying the Consequence of Restriction
Return to the hypothetical joint-profile example. Suppose the target contains 30 people with profile \((1,1)\), which is absent from the source, and 70 people in the other profiles, all represented in the source. A proposed analysis restricts its target to profiles represented in the source.
The restricted target covers \(70/100=0.700\), or 70.0%, of the original target. It excludes \(30/100=0.300\), or 30.0%. The result may address the 70 supported target participants if the other bridging assumptions hold, but it does not establish an effect for the excluded 30%. If those people are important to the intended decision, restriction alone is not a complete answer; the analysis needs a justified basis for extrapolation or additional evidence relevant to their profile.
This conclusion depends on the stated counts and rule. If the excluded profiles were instead a small, clinically irrelevant portion of the intended target, a restricted conclusion might be sufficient for a narrower decision. The fraction excluded and the scientific relevance of that fraction should both be reported.
Derivation: Why the Score Flags Support
The propensity-score diagnostic does not replace direct covariate assessment. Its usefulness follows from the relationship between the two population densities. Starting with the definition of conditional probability, the probability of source membership given a profile is the joint probability of source membership and that profile divided by the probability of the profile in the pooled population:
Substituting \(P(S=1)=\pi\) and \(P(S=0)=1-\pi\) yields the score formula above. If \(f_T(x)>0\), its denominator is positive. Therefore \(e(x)=0\) if and only if the numerator \(\pi f_S(x)\) is zero; when both populations are sampled, \(\pi>0\), so this occurs if and only if \(f_S(x)=0\). The corresponding odds expression also shows why a very low score indicates that source density is small relative to target density, subject to the pooled sampling fraction.
Diagnostics, Alternatives, and Limitations
Use several views of the data rather than relying on a single summary. Plot source and target distributions for continuous covariates, inspect category counts, compare clinically important combinations, and examine the distribution of estimated source-membership scores by population. A model that predicts source membership almost perfectly is a warning of separation; a model that predicts little separation does not prove adequate support if it omits important interactions or nonlinearities.
Density plots are intuitive and can show where tails diverge, but overlapping marginal curves can conceal a joint gap. Propensity scores compress many covariates into one diagnostic, but the result depends on model specification and can obscure which variables drive separation. Direct tables or plots of important joint profiles can be more interpretable, although the number of combinations grows quickly with many covariates. These approaches are complementary rather than interchangeable.
Common failure modes include empty or very sparse source cells, extreme score estimates, separation between populations, and inconsistent covariate measurement. Small samples can produce an empty observed cell even when the population probability is not literally zero; that is still a serious practical limitation. Conversely, observing one or two source participants in a target-relevant profile does not guarantee useful precision. Diagnostics cannot prove positivity in the underlying populations, and overlap alone does not verify exchangeability, estimand alignment, correct outcome modeling, or adequate measurement.
The central limitation is that no diagnostic can manufacture information for unsupported profiles. Restriction and trimming make the question narrower; new evidence can address the missing profiles; and model-based extrapolation requires explicit scientific justification beyond the overlap diagnostic itself. A credible report therefore states the support problem, its size and clinical relevance, the response, and the population to which the resulting conclusion applies.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a pooled sample, a target-relevant profile has \(f_T(x)>0\) and \(f_S(x)=0\). What is the population source-membership score \(e(x)\), and what condition does this violate?
- A target subgroup contains 18% of the target population and is absent from the source. If analysis is restricted to the supported remainder, what fraction of the original target is covered?
- Why can separate marginal plots of two binary covariates fail to reveal a positivity problem?
- A fitted score is near 0.05 for some target participants. What does this suggest, and why is it not by itself proof of structural positivity failure?
Solutions. (1) The score is \(e(x)=0\), since the numerator \(\pi f_S(x)\) is zero while the target contributes positive density to the denominator. This violates positivity because a target-supported profile has no source support. (2) The supported remainder is \(100\%-18\%=82\%\), so restriction covers 82% of the original target and excludes 18%. (3) Each covariate category can occur in both populations while a particular combination of categories occurs only in the target. Joint tables or suitable joint diagnostics are needed. (4) A score near 0 suggests that the profile is much more characteristic of the target than the source, so practical overlap may be weak. The estimate depends on the fitted model and finite sample; it does not establish that the true source density is exactly zero. Inspect the underlying profiles, counts, and model specification before concluding that structural positivity fails.