Recalibration by regressing on intercept only with offset

Question

Consider a risk prediction model developed on one population. When transporting this model to another population, the calibration can be off. There are several strategies for recalibration. One is to fit a new logistic regression model with an intercept only and an offset term for the linear predictor (coefficients from developed model times new data). The resulting linear predictor is considered the recalibrated risk (Steyerberg, Clinical Predictions Models, Chapter 20). What is the exact logic of this approach? I can imagine if there is only a difference in prevalence, then the intercept will be something like the average difference across scores.

Specifically, what is the exact interpretation of the linear predictor resulting from the logistic regression of an intercept only with offset and why?

Answer 1

If you read:

Validation and updating of predictive logistic regression models: a study on sample size and shrinkage (pdf)

The same author explains in a little bit of detail. The recalibration approach is to define i linear predictor, $Z$ as $Z = \tilde{\alpha}_{model} + \tilde{\beta}_1 X_1 + ... + \tilde{\beta}_p X_p$ and to use this linear predictor in a regression model: $\hat{Y}_{cal} = \hat{\alpha}_{overall} + \hat{\beta}_{overall}$. If no recalibration is required, $\hat{\alpha}_{overall} = 0 $ and $\hat{\beta}_{overall}$ = 1. If they significantly differ from this case then there is a need to recalibrate the model.

Recalibrated regression coefficients can be calculated as $\hat{\beta}_{cal} = \hat{\beta}_{overall} \tilde{\beta}_i$. One could say that the recalibrated model borrows the relative effects (ratios) of the regression coefficients from the model in the training set.

The case which you describe in your question arises when $\hat{\beta}_{overall} = 1$, i.e. $\hat{\alpha}_{overall} | \hat{\beta}_{overall}=1$ where the slope is set to unity by entering $Z$ as an offset term to the model. Predicted risks are on average understimated if $\hat{\alpha}_{overall} | \hat{\beta}_{overall}=1>0$ and overestimated if $\hat{\alpha}_{overall} | \hat{\beta}_{overall}=1 < 0$.

Updating of the intercept and fixing $\hat{\beta}_{overall}$ to one is a simple recalibration method that intends to correct calibration at the macro scale, i.e. to make the average predicted probability equal to the observed overall event rate.