Transporting Exposure-Response Relationships

Why Transport the Exposure-Response Relationship?

A source population and a target population can have different exposure distributions even when the drug’s exposure-response relationship is the same in both. In that case, the target population’s expected outcome can differ because its members experience different exposures—not necessarily because the response to a given exposure differs. Conversely, a shared exposure distribution does not guarantee a shared response relationship.

Transporting an exposure-response relationship makes these possibilities explicit. The analysis uses a model for outcomes conditional on exposure, evaluates that model over the target population’s exposure distribution, and examines whether key model features appear compatible across populations. The result is a prediction under stated assumptions, not proof that exposure itself causes the outcome.

Definition: Let $S=0$ identify the source population and $S=1$ the target population. Let $E$ denote exposure, $C$ relevant covariates, and $Y$ the outcome. If the source conditional mean is $m_0(e,c)=E(Y\mid E=e,C=c,S=0)$, the transported target mean is $$ \mu_{1\leftarrow 0}=\int m_0(e,c)\,dF_1(e,c), $$ where $F_1$ is the target population’s joint distribution of exposure and covariates. This prediction applies the source response function to target values of $E$ and $C$.

For a binary outcome, $m_0(e,c)$ is a probability; for a continuous outcome, it is an expected value in the outcome’s units. The integral is an average over the target distribution, not simply the source model evaluated at the target mean exposure. Unless the response function is linear on the outcome scale, those calculations generally differ.

Assumptions and a Standardization Derivation

A transported prediction requires a well-defined outcome, exposure metric, time window, and target population. The exposure must be measured comparably in both populations, and the source model must adequately describe the outcome across the exposure and covariate values represented in the target. In particular, the target should not require extrapolation far beyond the source data without additional justification.

For the prediction to be valid, the conditional response relationship must also be transportable after accounting for relevant covariates: $E(Y\mid E,C,S=1)$ must be adequately represented by the source relationship $m_0(E,C)$. This is a substantive assumption, not something established by fitting a source model alone. If exposure is affected by prognosis, adherence, or other factors that also affect the outcome, a conditional association need not be a causal exposure effect. A causal interpretation requires additional assumptions about confounding and the exposure intervention being considered.

The standardization formula follows from averaging conditional means. For discrete exposure and covariate values, partition the target population into groups indexed by $e,c$. The mean outcome is the sum of each group’s conditional mean multiplied by its target-population proportion:

E_1(Y)=\sum_{e,c}E(Y\mid E=e,C=c,S=1)P_1(E=e,C=c).

If the conditional relationship transports, replace each target conditional mean by the source model prediction $m_0(e,c)$. The resulting weighted sum is precisely the discrete form of $\mu_{1\leftarrow 0}$. For continuous variables, the sum becomes an integral. This derivation also shows why the target distribution matters: each conditional prediction receives weight according to how common its exposure and covariate values are in the target.

Worked Example: Predicting Target Efficacy Under a Shifted Exposure Distribution

Suppose a hypothetical source study estimates the probability of clinical response using the logistic model $\operatorname{logit}\{P(Y=1\mid E)\}=-2+1.2E$, where $E$ is exposure in a specified concentration unit. Assume, for this calculation, that the model is valid in the target and there are no additional covariates. In the source, half the population has $E=0$ and half has $E=1$. In the target, 25% has $E=1$ and 75% has $E=2$. Find each population’s model-standardized response probability.

The conditional probability is $p(e)=1/[1+\exp\{-(-2+1.2e)\}]$. Thus $p(0)=1/[1+\exp(2)]=0.1192$, $p(1)=1/[1+\exp(0.8)]=0.3100$, and $p(2)=1/[1+\exp(-0.4)]=0.5987$.

The source standardized probability is

\mu_0=0.50(0.1192)+0.50(0.3100)=0.2146.

The target prediction under the same conditional model is

\mu_{1\leftarrow 0}=0.25(0.3100)+0.75(0.5987) =0.07750+0.4490=0.5265.

The predicted response probability is 21.46% in the source exposure distribution and 52.65% in the target exposure distribution. This difference arises from averaging the same model over different exposure distributions; it does not demonstrate a population-specific response slope or establish that changing exposure would cause the predicted change.

Assessing Whether the Relationship Transports

A practical starting point is to fit a model that allows selected features to differ by population. For a continuous outcome, one simple specification is $E(Y\mid E,S)=a+bE+\gamma S+\delta ES$. Here $b$ is the source slope, $\delta$ is the target-minus-source slope difference, and $\gamma$ is the difference in conditional mean at $E=0$. For a binary outcome, the same linear predictor can be used in a logistic model, with slopes and intercepts interpreted on the log-odds scale. More flexible exposure functions may be needed when the relationship is nonlinear.

The reference exposure matters: the intercept difference is the population contrast at $E=0$. If zero is outside the observed range or clinically uninformative, center exposure at a meaningful value $E_0$, for example by fitting $E-E_0$. The population coefficient then concerns the conditional outcome at $E_0$. This is a model-based comparison at a specified exposure, not a general statement that all aspects of baseline risk are equal or different.

For the slope, test $H_0:\delta=0$ against an appropriate alternative. A Wald statistic is $z=\hat{\delta}/SE(\hat{\delta})$ when the estimate is approximately normal; a likelihood-ratio comparison of nested models is another common approach. The test depends on adequate model form, comparable exposure measurement, and sufficient data across populations and exposure ranges. A large p-value does not prove equal slopes. If practical equivalence matters, define a scientifically acceptable difference and assess whether the interval for $\delta$ is sufficiently narrow relative to that margin.

Worked Example: Testing Slope and Reference-Exposure Differences

In a hypothetical analysis of a binary endpoint, a logistic model includes population, exposure, and their interaction. The estimated target-minus-source slope difference is $\hat{\delta}=-0.25$ log-odds per exposure unit, with $SE(\hat{\delta})=0.30$. The estimated target-minus-source intercept difference at the chosen reference exposure is $\hat{\gamma}=-0.30$ log-odds, with $SE(\hat{\gamma})=0.25$. Assume the model is correctly specified and these estimates have approximately normal sampling distributions.

For the slope difference, $z=-0.25/0.30=-0.833$. The two-sided p-value is approximately 0.405, and a 95% Wald confidence interval is

-0.25\pm1.96(0.30)=[-0.838,\ 0.338] \quad\text{log-odds per exposure unit}.

At the 0.05 level, these data do not provide evidence against equal slopes. However, the interval includes meaningfully negative and positive differences, so the result does not establish that the slopes are practically equivalent.

For the reference-exposure difference, $z=-0.30/0.25=-1.20$, giving a two-sided p-value of approximately 0.230. Its 95% interval is

-0.30\pm1.96(0.25)=[-0.790,\ 0.190] \quad\text{log-odds at the reference exposure}.

This analysis does not provide strong evidence of an intercept difference either, but it is not evidence that the reference-exposure outcomes are identical. Both conclusions depend on the model and the specified reference exposure. A joint test of the population terms would require their covariance, which is not provided here.

Residual Variability and Model Checks

For a continuous outcome, transport can fail even when the fitted conditional means and slopes are similar: outcomes may be more variable around the fitted relationship in one population. A model can allow separate residual variances, for example $\epsilon\mid S=s\sim N(0,\sigma_s^2)$, while retaining a specified mean model. Pooling residual variability assumes $\sigma_0^2=\sigma_1^2$. The normality and independence assumptions matter for a classical variance-ratio test; residual plots and robust alternatives can be important when those assumptions are doubtful.

Worked Example: Comparing Residual Standard Deviations

Consider a hypothetical continuous endpoint analyzed with the same linear exposure-response form in two independent populations. Each population has 30 observations and two fitted mean-model parameters, an intercept and a slope. The residual standard deviations are $s_0=1.00$ outcome unit in the source and $s_1=1.50$ outcome units in the target. Assume independent observations, a correctly specified mean model, and normally distributed residuals within each population.

The residual degrees of freedom are $30-2=28$ in each population. Under equal residual variances, the ratio of residual sample variances follows an $F$ distribution with 28 and 28 degrees of freedom. The observed ratio, taking the larger variance over the smaller, is

F=\frac{s_1^2}{s_0^2} =\frac{(1.50)^2}{(1.00)^2} =2.25,\qquad df=(28,28).

The two-sided p-value is approximately 0.04. At $\alpha=0.05$, this gives evidence against equal residual variances under the stated assumptions. The target residual standard deviation is 1.50 outcome units, compared with 1.00 in the source. This suggests that a common residual variance may be unsuitable for these data; it does not by itself identify why variability differs or establish that the difference will recur in a new study.

Diagnostics should examine the response curve and the data supporting it, not just individual coefficient tests. Plot outcomes and fitted values against exposure by population; look for curvature, systematic residual patterns, unequal spread, influential observations, and calibration differences. Inspect exposure ranges and covariate distributions for gaps: a model prediction at exposure values absent from the source is extrapolation, even if the target average exposure appears plausible. For binary outcomes, check calibration across exposure ranges and whether sparse outcomes or separation make estimates unstable.

Choosing an Approach and Interpreting Limitations

Standardization is useful when a defensible conditional response model is available and the question concerns an outcome averaged over a specified target distribution. It uses the whole target distribution and can accommodate nonlinear response curves. Replacing the distribution with its mean is generally not equivalent: for a nonlinear function $m$, $E\{m(E)\}$ need not equal $m\{E(E)\}$. Exposure matching, discussed in the earlier material on regional dose bridging, instead focuses on aligning exposure distributions; it does not, by itself, test whether conditional response relationships are shared.

A shared model with population interactions directly assesses selected differences, such as slope or reference-exposure outcome. A population-specific model is more flexible but may be imprecise when target data are sparse. These approaches are not substitutes for checking overlap and model fit. A lack of statistically significant interactions can reflect low information rather than true transportability; a pooled relationship should not be assumed merely because a test is nonsignificant.

Transported predictions have uncertainty from at least two sources: estimation of the response-model parameters and estimation of the target exposure distribution. Confidence intervals or simulation-based uncertainty summaries should propagate both when both are estimated. A confidence interval for a population mean is not a prediction interval for an individual outcome; individual prediction also includes residual variability. Predictions may be especially fragile when exposure is measured differently, response is nonlinear, the target has little source-population support, or important effect modifiers are unmeasured.

Specify the target quantity.
Name the outcome, exposure definition, time point, target population, and whether the aim is a population mean or an individual prediction.

Fit and check the conditional model.
Assess functional form, exposure support, calibration, and residual behavior; allow population differences that are scientifically plausible.

Standardize and quantify uncertainty.
Average fitted outcomes over the target distribution, and propagate uncertainty in both model parameters and target-distribution estimates.

Common pitfall: “The slope interaction was not statistically significant, so the exposure-response relationship is the same and the transported prediction is valid.” The test only says the data did not show a sufficiently clear difference under the fitted model. It may be imprecise, misspecified, or unsupported in parts of the target exposure range. Examine the confidence interval, overlap, calibration, and model assumptions before treating the relationship as transportable.

Check Your Understanding

Answer each question, then compare with the solutions below.

Why is a target marginal prediction generally not obtained by evaluating a nonlinear response model at the target mean exposure?
A target exposure distribution is supported mainly above the source study’s observed range. What issue does this raise for a transported prediction?
A slope-difference test has $p=0.62$. Does this establish that the slopes are equivalent? Explain.
What does a population intercept difference represent when exposure is centered at $E_0$?
Name two uncertainty sources that should be considered when standardizing a fitted model to an estimated target distribution.

Solutions. (1) The average of a nonlinear function generally differs from the function evaluated at the average: $E\{m(E)\}\ne m\{E(E)\}$. (2) The model must extrapolate beyond source support, so its predictions rely strongly on assumptions not directly checked by source data. (3) No. A large p-value is not proof of equivalence; the interval may still allow important slope differences. (4) It is the conditional outcome contrast between populations at exposure $E_0$, on the model’s scale. (5) Uncertainty in the fitted response-model parameters and uncertainty in the target exposure distribution.