Recalibrating Score Slopes Across Ancestries

Why Recalibrate a Score?

A polygenic score can rank people usefully yet map to different average outcome values in different target populations. One reason is a difference in scale: a one-unit difference in the score may correspond to a different difference in the outcome across groups. A difference in baseline outcome levels can also shift the mapping. An ancestry-specific calibration regression estimates these mappings for a fixed score.

This regression describes how the score relates to the outcome in a specified target group and on a specified outcome scale. It does not, by itself, identify why the relationship differs. In particular, a slope difference is not proof of a biological difference in genetic effects: score variance, tagging, population composition, measurement, and other features of the validation setting may matter.

What the Intercept and Slope Mean

The intercept \(\alpha_g\) is the modelled mean outcome when \(S=0\). Its interpretation therefore depends on how the score is coded and centred. If zero is outside the range of observed scores, the intercept may be an extrapolation rather than a useful description of a typical individual. The slope \(\beta_g\) is the modelled change in mean outcome for a one-unit increase in score, in the outcome’s units per score unit. A positive slope means higher scores are associated with higher modelled outcomes in that group; it does not establish that changing a person’s score would cause an outcome change.

An important distinction is whether the regression uses a raw score or an existing prediction. If \(S\) is an arbitrary-scale genetic score, the fitted slope is the outcome change per score unit; there is no general reason it should equal one. If a prediction \(\hat{Y}_{\mathrm{ref}}\) is already on the outcome scale, its calibration can instead be assessed by regressing observed outcomes on that prediction. Under ideal linear calibration, the intercept is zero and the slope is one: on average, the observed outcome matches the predicted value across the prediction range.

Thus, a difference between ancestry-specific slopes is evidence that the score-to-outcome mapping has a different scale in the groups, subject to uncertainty and model assumptions. Calling it “scale miscalibration” is most direct when comparing predictions that are intended to be on the same outcome scale. For a raw score, it is safer to say that the estimated mapping slopes differ.

Fitting Separate Regressions

For each group, ordinary least squares estimates the intercept and slope by minimizing the sum of squared residuals. With one predictor and an intercept, the estimates can be written using the group’s sample means and standard deviations:

Here \(r_g\) is the sample correlation between score and outcome in group \(g\), while \(s_{S,g}\) and \(s_{Y,g}\) are their sample standard deviations. The formula makes a scale issue visible: even with the same correlation, slopes can differ if the score standard deviations differ. Conversely, equal slopes do not mean equal correlations or equal predictive performance.

These regressions should use a comparable definition of ancestry group, the same score construction and allele coding, and a common outcome definition and scale. If scores are standardized separately within groups, a one-unit score increase means one group-specific standard deviation; the resulting slopes answer a different question than slopes for a common raw score. Any covariate adjustment also changes the interpretation: an unadjusted slope describes the group-level linear association, whereas an adjusted slope describes the association conditional on the included covariates.

Using a Calibration Regression to Diagnose an Existing Prediction

When a score has already been turned into an outcome prediction, the calibration regression provides a direct comparison with the intended scale. For group \(g\), write the prediction as \(\hat{Y}_{\mathrm{ref}}\), then fit:

If this prediction is calibrated in the linear sense, \(A_g=0\) and \(B_g=1\). A fitted \(B_g<1\) indicates that observed outcomes vary less across the prediction range than the predictions do; \(B_g>1\) indicates greater variation. A nonzero \(A_g\) indicates a systematic shift on the outcome scale, interpreted together with the slope. These are population features estimated from validation data, not statements that each individual’s outcome should equal their prediction.

Why the OLS Formula Has This Form

The least-squares slope is the value of \(b\) that minimizes the sum of squared residuals after choosing an intercept \(a\). For a fixed \(b\), differentiating the residual sum of squares with respect to \(a\) gives an intercept that makes the residuals sum to zero, namely \(a=\bar{Y}_g-b\bar{S}_g\). Substituting this expression into the residual sum of squares leaves a function of \(b\) based on centred scores and outcomes. Setting its derivative to zero gives:

Solving this equation gives the centred-product formula for \(\hat{\beta}_g\). Replacing the centred-product sum by the sample covariance expression yields \(r_gs_{Y,g}/s_{S,g}\). This derivation explains the estimate, but it does not guarantee that a linear regression is a suitable model for a particular validation data set.

Assumptions and Scope of Interpretation

The regression’s mean mapping is useful when the conditional mean outcome is reasonably represented by a straight line over the score range being studied. The score must vary within each group; if every person has the same score, the slope cannot be estimated. Observations should also be independent for the usual individual-level regression analysis, or the analysis must account for dependence such as relatedness or clustered sampling. For conventional standard errors and tests, the usual model additionally assumes an appropriate residual variance structure and, for exact small-sample inference, normally distributed errors. These inferential assumptions are distinct from the calculation of the least-squares point estimates.

Inspect group-specific scatterplots and residual patterns rather than assuming the model is adequate because it is easy to fit. Strong curvature, influential observations, or substantially different residual spread can make a single linear slope incomplete or make conventional uncertainty estimates unreliable. A slope comparison also requires attention to the score scale: if one group’s score is standardized within group and another’s is not, the numerical slopes are not directly comparable as changes per common score unit.

If a fitted slope or intercept is to be used for prediction, estimate it on calibration data and assess performance on data not used to estimate that mapping. Estimating and evaluating the mapping on the same observations generally makes its apparent fit optimistic. The target group and outcome scale should be defined before interpreting the estimates, and uncertainty in each estimate should be reported when making substantive comparisons.

Recalibration Is Not Variant-Weight Re-estimation

A fixed polygenic score has the form \(S=\sum_j w_jG_j\), where \(G_j\) denotes the coded genotype at variant \(j\) and \(w_j\) is its score weight. An ancestry-specific affine recalibration applies an intercept and a common multiplier to the already computed score:

This estimates a new mapping from the fixed score to the outcome. It does not estimate a new relative contribution for each variant. Algebraically, the expression can be written as \(\hat{\alpha}_g+\sum_j(\hat{\beta}_gw_j)G_j\), so every weight is multiplied by the same factor. That algebraic rewriting should not be confused with re-estimating variant effects: the relative weight pattern is unchanged. By contrast, fine-tuning or refitting can change variant-specific weights differently and therefore change the score itself.

Common Mistake

Wrong reasoning: “The target-ancestry regression slope is \(0.8\), so the polygenic score must be reduced by 20%, and a slope below one always means the model is overpredicting.”

Correction: A slope of \(0.8\) for \(Y\) regressed on a raw score means that the fitted mean outcome changes by \(0.8\) outcome units per score unit in that group. It is not inherently a 20% correction: the score may not be in outcome units, and the slope’s reference value is not automatically one. The “below one means predictions vary too widely” interpretation applies when regressing observed outcomes on predictions already expressed on the intended outcome scale. Always state which variable is the regression predictor and identify its units.