Transportability of Surrogate Relationships Across Populations

Why Transportability Needs to Be Assessed

A biomarker may predict clinical outcomes well in one population without providing equally accurate treatment-benefit predictions in another. The new population may have a different baseline risk, a different mix of patients who modify the biomarker-outcome relationship, or different disease mechanisms and treatment patterns. Transportability is the question of whether evidence about the surrogate can be used to predict clinical outcomes in that target population, and what adjustments are needed.

A crucial distinction is between association and causal prediction. An association between a post-treatment biomarker and outcome does not, on its own, show that changing the biomarker would change the outcome. Nor does a biomarker-outcome relationship automatically capture treatment effects that operate through other pathways. Earlier material on potential-outcome estimands for biomarker mediation and the surrogate paradox establishes why these distinctions matter. Here the focus is whether a relationship validated in a source population can be applied to a different target population.

Definition: A surrogate-outcome relationship is transportable from a source population to a target population when the relationship, together with the relevant target-population information, identifies accurate predictions of clinical outcomes under the treatment strategies of interest. Transport may require recalibration or standardization; it does not necessarily mean that all marginal risks or treatment effects are identical across populations.

What Must Remain Comparable?

Let \(Z=S\) indicate the source population and \(Z=T\) the target population. Let \(A\) denote treatment, \(S\) the biomarker measured at an appropriate time, \(Y\) the clinical outcome by a specified follow-up time, and \(X\) measured baseline characteristics that may modify biomarker-outcome relationships or treatment effects. For a binary clinical event, a useful target is the event risk under each treatment strategy and their difference:

\mu_T(a)=P_T\{Y(a)=1\}, \qquad \Delta_T=\mu_T(1)-\mu_T(0).

A negative \(\Delta_T\) indicates a lower event risk under treatment 1 than under treatment 0. To use the surrogate to predict these risks, one needs more than a source-population correlation between \(S\) and \(Y\). One needs a defensible outcome-response model and the target distribution of the biomarker under each treatment.

For example, write \(m_z(a,s,x)=E_z\{Y(a,s)\mid X=x\}\) for the expected outcome if treatment were set to \(a\) and the biomarker to \(s\), conditional on \(X=x\), in population \(z\). If the source response model is valid in the target conditional on the included variables, then it can be combined with the target distribution of \(S\) and \(X\):

\mu_T(a)=\int m_S(a,s,x)\, f_T(s\mid a,x)\, f_T(x)\, ds\, dx.

For discrete variables, the integral is replaced by a sum. This standardization averages the source conditional response over the target population's distribution of baseline characteristics and treatment-specific biomarker values. It is not enough to insert the target's average biomarker value into a source model if the relationship is nonlinear or if important modifiers are distributed differently.

Transport assumptions: The outcome and biomarker are defined and measured comparably, including the time horizon. Conditional on the variables in the model, the source outcome-response relationship applies in the target. The target has adequate overlap in relevant \(X\), treatment, and biomarker values with the evidence supporting the source model. Treatment strategies are sufficiently well specified to have comparable meaning. For a causal interpretation involving \(S\), the required mediator-outcome exchangeability assumptions must also hold; a predictive association alone does not establish them.

Why These Assumptions Matter

The transport formula follows by averaging conditional outcome risks over the target distribution. Under consistency and the relevant exchangeability assumptions, the risk under treatment \(a\) can be written as the conditional risk at each \(s,x\), averaged over the biomarker distribution induced by \(a\) and the target distribution of \(X\). If the conditional risk function is invariant across populations, the source function can supply that conditional risk in the target. Thus:

\begin{aligned} P_T\{Y(a)=1\} &=\int P_T\{Y(a)=1\mid S(a)=s,X=x\}\, f_T(s\mid a,x)f_T(x)\,ds\,dx\\ &=\int m_S(a,s,x)\, f_T(s\mid a,x)f_T(x)\,ds\,dx, \end{aligned}

The second equality is the transport step: it replaces the target conditional outcome response with the source response model. It is justified only if that conditional response is comparable after conditioning on the variables included. If an unmeasured disease mechanism modifies the response, or a treatment-specific pathway differs between populations, the equality need not hold. Randomization of treatment in the source trial does not by itself guarantee that a post-treatment biomarker-outcome relationship is causal or transportable.

In practice, a validated relationship might be a model for outcome risk conditional on the biomarker, a model for how treatment effects on the biomarker predict treatment effects on the clinical outcome across trials, or a mechanistic model. These are not interchangeable. A trial-level relationship can predict an average treatment effect in a new trial only to the extent that the new trial is represented by the validation evidence and relevant trial-level modifiers are addressed. Individual-level prediction also requires appropriate individual-level information. The earlier material on meta-analytic validation and trial-level \(R^2\) addresses those distinct validation questions.

Worked Example: Same Biomarker Relationship, Different Baseline Risk

Suppose a hypothetical source study supports a model in which, at a fixed covariate profile, the odds of a clinical event are multiplied by 1.5 for each one-unit increase in a biomarker \(S\). The model is calibrated to a source risk of 10% at \(S=0\). A target population has a risk of 30% at \(S=0\). Predict the risk associated with a one-unit decrease in \(S\) in each population, assuming the conditional odds relationship transports.

For the source, the odds at \(S=0\) are \(0.10/(1-0.10)=0.1111\). A one-unit decrease divides the odds by 1.5:

\text{odds}_{S=-1}=\frac{0.1111}{1.5}=0.07407,\qquad p_{S=-1}=\frac{0.07407}{1+0.07407}=0.06897.

The source risk difference is \(0.06897-0.1000=-0.03103\), a decrease of about 3.10 percentage points. In the target, the odds at \(S=0\) are \(0.30/0.70=0.4286\). Applying the same odds multiplier:

\text{odds}_{S=-1}=\frac{0.4286}{1.5}=0.2857,\qquad p_{S=-1}=\frac{0.2857}{1+0.2857}=0.2222.

The target risk difference is \(0.2222-0.3000=-0.07778\), a decrease of about 7.78 percentage points. The conditional odds relationship was held fixed, but the absolute risk change differed because baseline risk differed. This calculation is a causal treatment-benefit prediction only if the model represents the relevant causal response to biomarker change and the other transport assumptions hold. If it is merely an observational association, the result is a model-based risk comparison, not evidence that lowering the biomarker causes the risk reduction.

Standardizing for Effect Modifiers

A population shift in the distribution of effect modifiers can change the average surrogate-based prediction even when the conditional relationships are stable. The appropriate response is to estimate or specify the relationship within modifier strata and then average using the target population's modifier distribution. This is different from assuming that the source population's average relationship applies unchanged.

Worked Example: A Different Mix of Effect Modifiers

In a hypothetical setting, a one-unit treatment-induced decrease in the biomarker corresponds to a 2-percentage-point reduction in event risk among patients in a low-risk mechanism stratum and a 6-percentage-point reduction in the high-risk mechanism stratum. Assume these stratum-specific reductions are valid in both populations and apply over the biomarker change being considered. The source population is 25% high-risk; the target population is 75% high-risk. Calculate the average predicted reductions.

The source average reduction is \(0.75(0.02)+0.25(0.06)=0.015+0.015=0.030\), or 3.00 percentage points. The target average is \(0.25(0.02)+0.75(0.06)=0.005+0.045=0.050\), or 5.00 percentage points. The target prediction is therefore 2.00 percentage points larger, not because the stratum-specific relationship changed, but because the target contains more people in the stratum with the larger reduction.

This standardization is justified only if both strata are represented well enough to estimate their relationships, the target stratum proportions are known or estimable, and the specified effects apply to the target treatment and biomarker changes. If the target contains a stratum absent from the source evidence, the stratum-specific prediction is extrapolation rather than transport supported by overlap.

When the Relationship Does Not Transport

Baseline-risk differences can sometimes be handled by recalibration, and measured differences in modifier distributions can sometimes be handled by standardization. Neither adjustment repairs a genuinely different conditional response. For instance, the same biomarker may lie on the principal treatment pathway in the source population but only one of several pathways in the target. Or treatment patterns may differ so that the same biomarker change is accompanied by a different direct treatment effect on the clinical outcome.

Worked Example: A Target-Specific Treatment Pathway

Consider a hypothetical source population in which a treatment reduces event risk by 8 percentage points through the biomarker pathway, with no additional treatment pathway relevant to the outcome. A source-based surrogate prediction for the target is therefore an 8-percentage-point reduction. In the target, suppose the treatment produces the same biomarker change and the same 8-percentage-point reduction through that pathway, but also increases event risk by 3 percentage points through a separate pathway.

The target's net event-risk change is the sum of the pathway changes: \(-0.08+0.03=-0.05\). Thus the net treatment effect is a 5-percentage-point risk reduction, not the 8-percentage-point reduction predicted from the surrogate alone. The surrogate still reflects one beneficial pathway, but it does not capture the target-specific harmful pathway. The prediction fails because the source conditional treatment-outcome relationship does not represent the target, not because the arithmetic of standardization is wrong.

Diagnostics, Alternatives, and Limitations

Transport assessment is strongest when it checks the assumptions directly where data permit. Compare the source and target on baseline risk, disease severity, biomarker distributions, relevant effect modifiers, outcome definition, follow-up time, and treatment regimen. Examine whether the target's covariate and biomarker values fall within the source evidence; a fitted model can return a number outside that range, but the number may depend heavily on unsupported extrapolation. Check calibration in the source and, where target outcomes are available, assess calibration in the target. Investigate clinically plausible interactions rather than treating a good pooled fit as proof that no subgroup relationship differs.

A practical sequence is:

Define the prediction target.
Specify the target population, treatment strategies, clinical outcome, follow-up time, and effect scale, such as an absolute risk difference.

Identify plausible modifiers.
Use biological and clinical knowledge to name measured factors that may alter the biomarker-outcome or treatment-outcome relationship.

Assess overlap and comparability.
Check whether source evidence covers the target's relevant covariate, biomarker, and treatment values, and whether measurement and follow-up are aligned.

Transport and qualify the result.
Recalibrate baseline risk or standardize over target covariates when justified; state where the prediction relies on assumptions or extrapolation.

The main alternatives are to recalibrate a source risk model using target outcome data, standardize a model with target covariate distributions, fit a hierarchical model that allows population-specific relationships, or obtain direct randomized clinical-outcome evidence in the target population. Recalibration can correct a baseline-risk mismatch but cannot establish that the biomarker causally mediates treatment benefit. Hierarchical models can represent between-population variation, but they still need enough informative populations and can shrink toward a misleading common relationship if the populations differ in unmodeled ways. A target outcome trial is more direct but may be costly or infeasible.

No diagnostic can prove transportability when target outcomes are unavailable. Good source fit, high trial-level association, or agreement with a mechanistic expectation is supportive evidence, not a guarantee. Predictions become especially fragile when the target has poor overlap, a different endpoint or time horizon, treatment regimens not represented in the source evidence, or plausible unmeasured modifiers. A surrogate-based prediction should therefore report its assumptions and uncertainty, and should not be presented as equivalent to observed clinical benefit.

Common Pitfall

A common error is to carry a source population's average biomarker-outcome slope or absolute risk reduction directly into the target and call the result transportable. This confuses a marginal relationship, which depends on the population's baseline risk and modifier mix, with a conditional relationship that might be stable. The correction is to specify the effect scale, assess conditional comparability, and recalculate target risks using target baseline risk and modifier distributions. If the conditional relationship itself may differ, a numerical adjustment alone is not enough; target-population evidence or a more cautious conclusion is needed.

Key takeaway: Transporting a surrogate relationship is a conditional prediction problem. Baseline risk and measured modifier distributions can sometimes be handled by recalibration and standardization, but different mechanisms, direct treatment pathways, or unsupported extrapolation can invalidate the prediction.

Check Your Understanding

Answer each question, then compare with the solutions below.

A source and target population share the same conditional biomarker-outcome odds ratio, but the target has higher baseline risk. Must their absolute risk reductions be equal?
Why might the target population's average predicted benefit differ when stratum-specific biomarker relationships are stable?
What does lack of overlap between source and target biomarker values imply for transport?
Why is a post-treatment biomarker-outcome association not, by itself, proof that treatment benefit can be predicted causally from the biomarker?

Solutions. (1) No. A common odds ratio can imply different absolute risk changes at different baseline risks, as the worked calculation illustrates. (2) The target may have a different distribution of effect modifiers; averaging the stratum-specific predictions over target proportions can change the overall estimate. (3) The conditional relationship must be extrapolated into values unsupported by the source data, so the prediction depends on unverified model assumptions. (4) Confounding of the biomarker-outcome relationship or treatment effects through pathways not captured by the biomarker may remain; causal prediction requires additional assumptions and a suitable causal model.