Calculate predicted values after validation of logistic model I have a simple logistic model, and I internally validated it using rms::validate in R. The estimated overoptimism for the intercept is -0.015 and for the slope is 0.05. How do I calculate the corrected predicted values? I know I have to multiply the coefficients by 0.95, but what should I do with the intercept? I guess I have to subtract -0.015, but I am not really sure.

First of all, note that recalibration of predictions is better than not recalibrating them, but you don't get a way to validate the recalibrated model.  In your particular case where the calibration intercept and slope are -0.015 and 0.95 most statisticians would conclude that the model is OK as-is.
But all that depends on the calibration being linear.  Show us the smooth estimated entire calibration curve and some indexes of miscalibration.  One of my favorite indexes is the 0.9 quantile of absolute prediction error.
Iif you proceed with a linear recalibration then you would multiply $X\hat{\beta}$ by 0.95 and subtract 0.015.
See Ewout Steyerberg's book Clinical Prediction Models for much more on this subject.
