Recalibrating Scores to Local Baseline Incidence

Why Recalibrate a Local Baseline?

A genetic score can preserve useful information about relative risk while producing absolute risks that are too high or too low in a particular setting. For example, the score’s relative effects may be used in a region where the event rate differs from the population in which the risk model was developed. If the target population has a different baseline incidence, applying the original absolute-risk model unchanged can misstate the number of people expected to experience the event.

A baseline-only recalibration changes the model’s overall risk level while keeping its specified relative effects fixed. The update is meaningful only when the local incidence refers to the same outcome definition and time horizon as the model, and when it represents the population to which predictions will be applied. Matching one overall event rate does not, by itself, establish that risks are accurate for every person or subgroup.

Definition: Recalibrating to local baseline incidence means updating an absolute-risk model’s intercept or baseline hazard using local outcome information, while holding the model’s relative-effect coefficients fixed. In an intercept-only update, each person’s linear predictor retains the same score and covariate contributions; a common intercept change shifts the predicted risks. In a proportional-hazards model, the baseline hazard changes while the relative hazard coefficients remain fixed.

Intercept Updating for a Fixed-Horizon Risk Model

Suppose a model predicts whether an event will occur by a specified horizon, such as five years. Write the original logistic risk model for person \(i\) as \(p_i=\operatorname{expit}(\alpha+\eta_i)\), where \(\operatorname{expit}(u)=1/(1+\exp(-u))\), \(\alpha\) is the intercept, and \(\eta_i\) is the fixed linear predictor. The linear predictor can include a genetic score and other specified predictors. In this update, their coefficients are not re-estimated.

For local individuals with outcomes \(Y_i\), where \(Y_i=1\) indicates an event by the horizon and \(Y_i=0\) otherwise, the observed local incidence is \(\hat{p}_{\mathrm{local}}=\sum_i Y_i/n\). The updated intercept \(\hat{\alpha}_{\mathrm{local}}\) is chosen so that the mean predicted risk in the local sample equals this incidence:

\frac{1}{n}\sum_{i=1}^{n}\operatorname{expit}(\hat{\alpha}_{\mathrm{local}}+\eta_i) = \frac{1}{n}\sum_{i=1}^{n}Y_i.

This is generally not the same as adding the difference between the logit of the local incidence and the old intercept. The local incidence averages risks across people with different \(\eta_i\); the logit of that average is not generally the intercept of the model. Instead, the updated intercept can be estimated by fitting a logistic regression with \(\eta_i\) as a fixed offset and estimating only the intercept.

The local sample must represent the population for which the updated risks will be used, or the calculation must account for the sampling design. The outcome must be observed consistently through the specified horizon. If the sample is selected according to case status, its raw event proportion does not estimate population incidence; appropriate sampling corrections or external incidence information are needed. Likewise, different age distributions or outcome definitions can make a crude local rate inappropriate for the intended predictions.

Worked Example: Updating a Logistic Intercept

Hypothetical data. A model predicts a three-year event risk. In a representative local sample of 400 people, 200 have fixed linear predictor \(\eta=0\), and 200 have \(\eta=\ln(3)\). There are 50 events in the first group and 100 in the second, for 150 events overall. The original intercept is \(\alpha=0\), and both groups’ score effects are to remain fixed.

The observed local incidence is \(150/400=0.375\). The updated intercept solves

200\operatorname{expit}(\alpha_{\mathrm{local}}) + 200\operatorname{expit}(\alpha_{\mathrm{local}}+\ln 3) =150.

Set \(\alpha_{\mathrm{local}}=-\ln(3)\), which is approximately \(-1.099\). The predicted risks are then

\operatorname{expit}(-\ln 3)=\frac{1}{4}=0.25, \qquad \operatorname{expit}(-\ln 3+\ln 3)=\operatorname{expit}(0)=0.50.

The updated model predicts \(200(0.25)+200(0.50)=50+100=150\) events, or \(150/400=0.375\). Thus it matches the observed local incidence while retaining the difference in the two groups’ linear predictors. With the original intercept, the risks would have been \(0.50\) and \(0.75\), for a mean predicted risk of \(0.625\), too high for this local sample.

The result is a local baseline update for this three-year outcome in a representative sample. It does not show that the model is accurate for every subgroup or that its score effects are correct in other populations.

Why the Intercept Solution Is Well-Defined

For fixed \(\eta_i\), the average predicted risk is a continuous, strictly increasing function of the intercept. Its derivative is

\frac{d}{d\alpha} \left[ \frac{1}{n}\sum_{i=1}^{n}\operatorname{expit}(\alpha+\eta_i) \right] = \frac{1}{n}\sum_{i=1}^{n} \operatorname{expit}(\alpha+\eta_i) \left[1-\operatorname{expit}(\alpha+\eta_i)\right] >0.

For finite \(\eta_i\), the average tends to 0 as \(\alpha\) tends to negative infinity and to 1 as \(\alpha\) tends to positive infinity. Therefore, for a local incidence strictly between 0 and 1, there is a unique intercept matching that incidence. In practice, a numerical root-finding algorithm or an intercept-only logistic fit with \(\eta_i\) as an offset finds it.

The same result follows from the likelihood. Under independent Bernoulli outcomes with the fixed predictors and logistic model, the log-likelihood as a function of the intercept is

\ell(\alpha)= \sum_{i=1}^{n} \left\{ Y_i(\alpha+\eta_i) - \log\left[1+\exp(\alpha+\eta_i)\right] \right\}.

Its first derivative is \(\sum_i[Y_i-\operatorname{expit}(\alpha+\eta_i)]\), so the likelihood equation sets the total predicted event count equal to the observed count. Its second derivative is \(-\sum_i p_i(1-p_i)<0\) when the fitted probabilities are strictly between 0 and 1, making the log-likelihood concave. This gives a unique finite maximum when the data contain both events and non-events. If every person has the same outcome, a finite logistic intercept estimate does not exist; external information or a suitable alternative estimation approach is needed.

Updating a Baseline Hazard

For a time-to-event model with proportional hazards, write the hazard for person \(i\) at time \(t\) as \(h_i(t)=h_0(t)\exp(\eta_i)\). Here \(h_0(t)\) is the baseline hazard and \(\eta_i\) contains the fixed relative-effect contributions, including any genetic score effect. If there is no competing event, the risk of the event by time \(T\) is \(1-\exp[-H_0(T)\exp(\eta_i)]\), where \(H_0(T)=\int_0^T h_0(t)\,dt\) is the baseline cumulative hazard.

A local update can estimate the baseline hazard from local follow-up data while treating the relative-effect coefficients as fixed. For example, with right-censored data and distinct event times, the Breslow estimate of the baseline cumulative hazard at time \(t\) is

\widehat{H}_{0,\mathrm{local}}(t) = \sum_{t_j\leq t} \frac{d_j} {\sum_{i\in R(t_j)}\exp(\eta_i)}.

Here \(t_j\) is an observed event time, \(d_j\) is the number of events at that time, and \(R(t_j)\) is the set of people still at risk just before it. The resulting local baseline estimate reflects the event experience and risk-set composition in the local data, conditional on the fixed relative effects. It requires suitable follow-up and censoring assumptions; in particular, censoring must be handled by the survival analysis method rather than treating people lost to follow-up as event-free through the horizon.

Sometimes a simpler model assumes that the local baseline hazard is a constant multiple \(c\) of the original one: \(h_{0,\mathrm{local}}(t)=c h_{0,\mathrm{original}}(t)\), with \(c>0\). Then \(H_{0,\mathrm{local}}(T)=cH_{0,\mathrm{original}}(T)\). This is a restricted adjustment: it changes the overall hazard level but preserves the original baseline hazard’s shape over time. A single multiplier will not correct a local event pattern whose difference from the original varies substantially over time.

Worked Example: Scaling a Baseline Cumulative Hazard

Hypothetical data. Consider a two-year proportional-hazards model with original baseline cumulative hazard \(H_{0,\mathrm{original}}(2)=0.10\). In the target population, half the people have fixed \(\eta=0\) and half have fixed \(\eta=\ln(2)\). Suppose local follow-up indicates a two-year event incidence of approximately \(0.2555\), and assume a constant baseline-hazard multiplier is appropriate over this horizon.

Under multiplier \(c\), the two predicted risks are \(1-\exp(-0.10c)\) and \(1-\exp(-0.20c)\). The multiplier must satisfy

\frac{ [1-\exp(-0.10c)]+[1-\exp(-0.20c)] }{2} =0.2555.

At \(c=2\), the risks are \(1-\exp(-0.20)\approx0.1813\) and \(1-\exp(-0.40)\approx0.3297\). Their average is \((0.1813+0.3297)/2\approx0.2555\), so \(c\approx2.000\) matches the local incidence to the stated precision. The updated baseline cumulative hazard at two years is \(2(0.10)=0.20\); the relative hazard ratio between the groups remains \(\exp(\ln 2)=2\).

This calculation is appropriate only under the stated proportional-hazards model and constant-scaling assumption. With censoring or a varying baseline hazard, local survival data should be used in an analysis that accounts for event times and risk sets. If competing events are relevant, the event probability must be calculated using a competing-risks model rather than the single-event survival formula above.

When a Baseline Update Is Not Enough

An intercept or baseline-hazard update has limited flexibility. It can correct a shared baseline level under its assumptions, but it cannot repair a changed relationship between the score and the outcome. Nor can matching the population average guarantee that risks agree within score groups, age groups, or other clinically important strata. A useful check is to compare observed and predicted outcomes across relevant subgroups after the update, using data that represent the intended population and time horizon.

Worked Example: Matching the Overall Rate but Missing Subgroup Risks

Hypothetical data. In a target population with two equally sized score groups, a fixed model predicts risks of \(0.10\) in the lower-score group and \(0.30\) in the higher-score group. Local observed risks are \(0.05\) and \(0.35\), respectively.

Both the mean predicted risk and the mean observed risk are \(0.20\): \((0.10+0.30)/2=0.20\) and \((0.05+0.35)/2=0.20\). An intercept-only update therefore has no need to shift the overall mean, yet it leaves a 0.05 overprediction in the lower-score group and a 0.05 underprediction in the higher-score group.

A common intercept shift adds the same amount to both groups’ log-odds, so it cannot change the difference between their predicted log-odds. The model’s predicted log-odds difference is \(\operatorname{logit}(0.30)-\operatorname{logit}(0.10)\approx1.350\). The observed difference is \(\operatorname{logit}(0.35)-\operatorname{logit}(0.05)\approx2.325\). Since these differences are not equal, no single intercept can make both group risks match their observed values. The local overall incidence is matched, but the model’s relative risk pattern is not.

This pattern calls for investigation beyond a baseline-only adjustment, such as evaluating whether relative effects differ in the target setting. It does not, by itself, establish why they differ. A baseline recalibration should not be presented as a complete correction when subgroup discrepancies remain.

Common Mistakes and Practical Checks

Adding a logit-rate difference without checking the predictor distribution. The logit of the local average risk is generally not the model intercept. Solve the average-risk equation with the local predictors instead.
Using an incompatible incidence. A rate for a different time horizon, age range, case definition, or target population is not automatically the right baseline for the model. Align these quantities or explicitly model the differences.
Equating a matched average with accurate individual risks. A model can reproduce the overall event rate while overpredicting some groups and underpredicting others. Examine relevant strata as well as the aggregate.
Changing relative effects unintentionally. A baseline-only update keeps the specified score and covariate coefficients fixed. If they are re-estimated, the procedure is no longer just baseline recalibration.
Ignoring the sampling and follow-up design. Case-control sampling does not give population incidence from the raw case fraction, and censoring means observed event fractions may not estimate horizon risk. Use methods appropriate to the design.

Key takeaway: Recalibrate the intercept or baseline hazard only with local incidence information aligned to the model’s population, outcome, and time horizon. A successful baseline update matches the overall risk level under its assumptions; it does not establish that the model’s relative genetic effects or subgroup predictions are correct.

Check Your Understanding

Answer these questions by referring to the model assumptions and calculations in this tutorial.

Why is the updated intercept generally not equal to the logit of the local event proportion?
A logistic model has fixed individual linear predictors. What equation defines an intercept that matches the local event incidence?
In a proportional-hazards model, what does multiplying the baseline hazard by a constant leave unchanged?
Why can matching an overall incidence fail to correct risks in two score groups?
Name two ways that local outcome information can be incompatible with the target of an absolute-risk model.