Why Recalibrate a Score?
A polygenic risk score (PRS) can rank individuals reasonably well in a new population yet give probabilities that are systematically too high or too low. The target population may have a different outcome prevalence, or the relationship between the score and outcome may differ in strength. Recalibration uses outcomes measured in the target population to adjust the score’s predicted risks. It does not, by itself, change the genetic variants or their weights that formed the score.
For a binary outcome, let \(Y_i=1\) indicate that individual \(i\) has the outcome by a specified time horizon and \(Y_i=0\) otherwise. Let \(S_i\) be that individual’s fixed PRS, and let \(p_i^{(0)}\) be the probability supplied by the original prediction model. When the original model supplies probabilities, define its linear predictor as \(\eta_i=\operatorname{logit}(p_i^{(0)})\), where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). The recalibration model relates target-population outcomes to this fixed linear predictor.
The model is fitted to target-population data with known outcomes. It assumes that the outcome is coded consistently, the time horizon is the same as the one for which risks are being reported, and the target data represent the population where predictions will be used. The logistic relationship is an approximation that must be assessed; two coefficients cannot repair every form of model misspecification. As with ordinary logistic regression, the likelihood-based fit also needs enough outcome and non-outcome observations to estimate its coefficients reliably and should not be undermined by complete or near-complete separation.
Two Recalibration Models
An intercept-only update holds the source scale fixed at one and estimates \(a\):
This model changes the baseline while preserving the relative spacing of predictions on the log-odds scale. It is appropriate when the source predictor’s scale is considered adequate but the target population has a different baseline risk. A slope-and-intercept update estimates both coefficients:
Here \(b\) can compress or expand the source linear predictor. With this parameterization, ideal calibration corresponds to \(a=0\) and \(b=1\). A fitted slope below one means that, conditional on this model, the source predictions are too spread out on the log-odds scale for the target outcomes; a slope above one means they are not spread out enough. The intercept and slope are estimated together in the two-parameter model, so the fitted intercept is not generally the same as the intercept-only correction.
If the available input is a raw score \(S_i\), rather than a source probability or logit, one may fit \(\operatorname{logit}\{P(Y_i=1\mid S_i)\}=a+bS_i\). In that model, \(b\) is per unit of the score, so its numerical value depends on the score’s units and scaling. When recalibrating an existing probability model, using its linear predictor \(\eta_i\) makes the reference scale explicit.
Worked Example: Deriving an Intercept-Only Correction
Suppose, hypothetically, that a source model gives every member of a target validation sample a risk of \(0.40\) for a binary outcome by the specified horizon. In \(400\) representative target individuals, \(20\) have the outcome, so the observed target prevalence is \(20/400=0.05\). The predictions are constant in this example, so an intercept-only logistic update can match the target prevalence exactly.
The source logit is \(\eta=\operatorname{logit}(0.40)=\log(0.40/0.60)\approx-0.4055\). The updated intercept must make the fitted probability equal \(0.05\), so the intercept correction is
The updated logit is \(-2.5390-0.4055\approx-2.9444\), and \(\operatorname{expit}(-2.9444)=0.0500\), where \(\operatorname{expit}(x)=1/\{1+\exp(-x)\}\). Thus the calibrated risk is \(5.00\%\), matching the observed target prevalence in this constant-prediction example. This is not a general rule to subtract the logit of the average predicted risk from the logit of prevalence: with varying \(\eta_i\), the intercept must account for all individual predictions.
How the Coefficients Are Estimated
For independent target observations with binary outcomes, logistic recalibration estimates \(a\) and, when included, \(b\) by maximizing the Bernoulli log-likelihood. Write \(q_i=\operatorname{expit}(a+b\eta_i)\). Apart from terms that do not involve the coefficients, the log-likelihood is
The fitted coefficients make the likelihood stationary. For example, the score equation for \(a\) is \(\sum_i(y_i-q_i)=0\); for \(b\), it is \(\sum_i\eta_i(y_i-q_i)=0\). These equations express the fitted model’s balance between observed outcomes and predicted probabilities, with the slope equation also weighting that balance by the source linear predictor.
For an intercept-only update, set \(b=1\) and use \(q_i=\operatorname{expit}(a+\eta_i)\). Differentiating the log-likelihood with respect to \(a\) gives
At an interior maximum, this derivative is zero, so \(\sum_i y_i=\sum_i\operatorname{expit}(a+\eta_i)\): the total fitted number of outcomes equals the observed number. If all source probabilities equal a common value \(p_0\), then all \(\eta_i=\operatorname{logit}(p_0)\). Writing the observed prevalence as \(\hat p\), the equation reduces to \(\hat p=\operatorname{expit}\{a+\operatorname{logit}(p_0)\}\). Taking logits yields \(a=\operatorname{logit}(\hat p)-\operatorname{logit}(p_0)\), the correction used in the preceding example. When predictions vary, the equation is instead solved numerically using each person’s \(\eta_i\).
Worked Example: Estimating a Calibration Slope From Two Score Groups
Consider hypothetical target data with \(100\) individuals at source logit \(\eta=-2\), of whom \(10\) have the outcome, and \(100\) individuals at \(\eta=0\), of whom \(40\) have the outcome. Fit a logistic recalibration model \(a+b\eta\). With two distinct predictor values and two parameters, the fitted probabilities reproduce the two observed group proportions.
At \(\eta=0\), the observed proportion is \(40/100=0.40\), so \(a=\operatorname{logit}(0.40)=\log(0.40/0.60)\approx-0.4055\). At \(\eta=-2\), the observed proportion is \(10/100=0.10\), so \(a-2b=\operatorname{logit}(0.10)=\log(0.10/0.90)\approx-2.1972\). Therefore,
The fitted model is approximately \(\operatorname{logit}\{P(Y=1\mid\eta)\}=-0.4055+0.8959\eta\). It predicts \(0.40\) at \(\eta=0\) and \(0.10\) at \(\eta=-2\), matching the group proportions. The source probabilities at those logits were \(\operatorname{expit}(0)=0.5000\) and \(\operatorname{expit}(-2)\approx0.1192\). In this hypothetical sample, both source predictions exceed their corresponding observed proportions, and the fitted slope is below one. This small grouped example illustrates the coefficients; it does not establish that the same calibration relationship holds throughout a real target population.
Applying the Recalibrated Model
Worked Example: Updating an Individual Risk
Suppose, hypothetically, that target outcome data have produced recalibration estimates \(a=-0.2000\) and \(b=0.8000\). An individual’s fixed source-model risk is \(0.2000\). First convert that risk to its source logit:
Apply the fitted recalibration model, retaining full precision in the intermediate calculation:
The updated probability is \(\operatorname{expit}(-1.309035)\approx0.2127\). The recalibrated risk is therefore about \(21.27\%\) by the same outcome horizon, rather than the source model’s \(20.00\%\). These hypothetical coefficients alter the mapping from prediction to risk; they do not change the individual’s PRS or any variant weight.
Diagnostics, Alternatives, and Limits
Recalibration is useful when a score is to be used in a population for which local outcomes are available. It can correct systematic risk-scale errors without rebuilding the score. But its performance must be checked on data not used to estimate the recalibration coefficients, or by an appropriate resampling strategy when the target sample is limited. Reuse of the fitting data can make calibration appear better than it will be for future individuals.
Inspect calibration across the prediction range, not only the overall mean. A suitable calibration plot compares observed outcome frequencies with predicted risks in groups or by a smooth curve, while accounting for uncertainty in the observed frequencies. If discrepancies remain curved or differ across score ranges, a two-parameter logistic mapping may be inadequate. Also check for sparse outcome counts, influential observations, unstable coefficients, and separation. In a small target cohort, a flexible-looking recalibration may have high variance even when its in-sample fit is good.
The main options answer different questions:
| Approach | What changes? | When it may be suitable | Main caution |
|---|---|---|---|
| No update | Nothing; retain source risks. | Target and source risks are already adequately calibrated. | A risk-scale shift can persist in the target population. |
| Intercept-only recalibration | Baseline log odds; source scale fixed. | Baseline risk differs, but the source predictor scale is considered suitable. | Cannot fix a mis-scaled predictor or more complex calibration shape. |
| Intercept-and-slope recalibration | Baseline and source log-odds scale. | Target outcomes support estimating both parameters. | More parameters can be unstable with limited outcome information. |
| Refit variant weights | The score’s genetic effects, as well as potentially its predictions. | The goal is to estimate or revise genetic associations using suitable data. | This is model development, not merely recalibration, and needs its own design and validation. |
Re-estimating variant weights is not the same as recalibrating. Recalibration treats the supplied score or source linear predictor as fixed and estimates a small number of mapping parameters. Refitting weights changes how genetic variants contribute to the score and generally requires genotype data, outcome data, and a model-development strategy. A two-coefficient recalibration cannot be interpreted as evidence that particular variants have changed effects.
The target outcomes must also support the intended interpretation. If participants were sampled in a way that changes the observed case fraction, that fraction may not represent population prevalence, so a simple prevalence-based intercept correction can be inappropriate without accounting for the sampling design. Outcome definitions, follow-up, and ascertainment should match the intended use. Recalibration cannot compensate for a different outcome horizon, unmeasured shifts in who enters the target population, or all differences in predictor-outcome relationships.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A source model predicts \(0.25\) for every person, while a representative target sample has outcome prevalence \(0.10\). What intercept-only correction is required?
- A fitted recalibration model is \(\operatorname{logit}(P(Y=1\mid\eta))=0.3+1.2\eta\). What risk does it assign when \(\eta=-1\), approximately?
- In a recalibration model with \(b=0.7\), what does the slope indicate about the source predictor’s spread on the log-odds scale?
- Does an intercept-only recalibration change the PRS variant weights? Explain the difference between its role and refitting weights.
Solutions. (1) The correction is \(\operatorname{logit}(0.10)-\operatorname{logit}(0.25)\approx-2.1972-(-1.0986)=-1.0986\). Adding this to the source logit \(-1.0986\) gives \(-2.1972\), whose expit is \(0.1000\). (2) The recalibrated logit is \(0.3+1.2(-1)=-0.9\); hence the risk is \(\operatorname{expit}(-0.9)\approx0.2891\). (3) A slope below one compresses the source linear predictor: in the fitted calibration relationship, its variation is too extreme on the log-odds scale relative to the target outcomes. (4) No. Intercept-only recalibration changes the baseline log odds for predictions while holding the score and its variant weights fixed. Refitting weights changes how variants contribute to the score and is a separate model-development task.