Transporting Absolute Risk Across Populations

Why Transport Requires More Than Changing the Baseline

An absolute-risk estimate is tied to the population and time horizon in which it is interpreted. A genetic score can rank people similarly in two populations while their absolute risks differ because the underlying incidence differs. Conversely, matching an overall incidence does not guarantee that the score has the same predictive relationship with the outcome in both populations. Transport therefore asks two distinct questions: does the genetic relative effect carry over, and what baseline risk applies in the target population?

The distinction matters in clinical use. If only baseline incidence changes and the genetic relative effect is stable, it may be reasonable to update the model's baseline while retaining its genetic coefficient. If the relative effect changes, a baseline adjustment alone can make the average predicted risk look appropriate while leaving risk estimates systematically wrong across score levels.

A useful working model for a binary outcome at a fixed horizon is

Here \(S\) is on the same scale in source and target populations, and \(C\) is a vector of clinical predictors. The intercept \(\alpha_S\) sets the model's baseline on the log-odds scale; \(\beta_S\) and \(\gamma_S\) describe conditional associations on that scale. A basic transport strategy assumes the conditional slopes transfer, while allowing the intercept to change:

This is an assumption, not a consequence of observing different population incidences. It can fail if, for example, genetic score effects differ because the score captures different underlying variants, the outcome definition or follow-up differs, or predictor effects vary with environmental or clinical context. A score's distribution may also differ between populations; even with stable conditional effects, that difference changes the average risk.

Updating Risk When Relative Effects Are Assumed Stable

For fixed clinical predictors and a specified reference score, a relative effect can be expressed as an odds ratio. If the target-population risk at score \(S=0\) is \(q_T\), and the odds ratio per one-unit increase in \(S\) is \(R\), then the risk at score \(S=s\) is

The odds ratio multiplies odds, not risk. Thus, substituting a target baseline risk into this formula is appropriate only when the odds ratio is a valid conditional relative effect for the target population and the other predictors are held at the same values.

In practice, a baseline risk for a single reference profile may not be available. Instead, a representative target sample may provide an overall event incidence, and the target distribution of predictors can be used to recalibrate the model intercept. Let \(\pi_T\) be the target incidence over the same horizon among people at risk at its start. Retaining the source slopes, choose \(\alpha_T\) to satisfy

The expectation is over the target population's distribution of \(S\) and \(C\). Consequently, a changed predictor distribution is part of the recalibration: the same conditional model can imply different overall incidence when applied to different mixtures of predictor values. In data, the expectation is often approximated by averaging predictions across a representative target sample, or by a suitably weighted sample if the sampling design requires weights.

Why an Intercept Can Be Solved—and What It Does

For fixed slopes and target predictor values, define \(f(a)=E_T[\operatorname{expit}(a+\beta_S S+\gamma_S^{\mathsf T}C)]\). When predictors are finite, \(f\) is continuous and strictly increasing: each expit term increases with \(a\), with derivative \(\operatorname{expit}(u)\{1-\operatorname{expit}(u)\}>0\). As \(a\) tends to negative infinity, \(f(a)\) tends to 0; as \(a\) tends to positive infinity, it tends to 1. Therefore, for any target incidence \(0<\pi_T<1\), there is a unique intercept that makes the average predicted risk equal \(\pi_T\). It can be found numerically.

This argument establishes only a match in the target average. It does not establish that people with a particular score or clinical profile have accurate predictions. Intercept recalibration shifts predictions on the log-odds scale; it cannot repair an incorrect genetic slope, an incorrect clinical slope, or a relationship whose form differs in the target population.

Target-Population Calibration Is a Necessary Check

After recalibration, predictions should be assessed in target-population data that are appropriate for the intended population and horizon. The observed outcome must be ascertained comparably, and the validation sample should represent the target population or be analyzed with appropriate weights. Useful checks include comparing average predicted risk with observed incidence, examining observed outcomes across ranges of predicted risk, and checking whether errors vary with score or clinical predictors. An overall match alone is insufficient.

Assumptions and Common Pitfalls

A transport analysis should make its assumptions explicit. In particular, check that the predicted outcome has the same definition and time horizon, and that the target incidence refers to the same eligible population and starting point. Confirm that the score is constructed and scaled comparably; standardizing the score differently in the target population changes the meaning of a one-unit coefficient. The target sample must also provide a credible estimate of incidence and predictor distribution. A case-control sample, for example, does not generally reveal population incidence from its sampled case fraction alone.

Most importantly, assess whether retaining the source conditional effects is defensible. Genetic effects estimated through a score can vary across settings, including when the score's relationship to underlying genetic variants differs or when relevant interactions differ. A changed score distribution by itself can change marginal risks without changing conditional effects; it should not be mistaken for proof that the genetic slope has changed. Conversely, a correct target average does not prove the genetic slope is stable.

• Wrong reasoning: “The target population has higher incidence, so multiply every source risk by the ratio of incidences.” Correction: Risks are bounded by 1, and relative odds or risks do not generally translate through simple multiplication. Use a model-appropriate recalibration, such as a target intercept fitted to target incidence and predictor distribution.
• Wrong reasoning: “The recalibrated predictions average to the target incidence, so the model transports.” Correction: The average can match while predictions are wrong within score or clinical subgroups. Check target calibration across predicted risks and relevant predictors.
• Wrong reasoning: “A changed target incidence proves that the genetic effect changed.” Correction: Incidence can change because of baseline risk or predictor distributions even when conditional genetic effects are stable. Evaluate baseline and relative effects as separate questions.