  Firth-type logistic regression has become a standard approach for the
analysis of binary outcomes with small samples. Whereas it reduces the bias in
maximum likelihood estimates of coefficients, bias towards 1/2 is introduced in
the predicted probabilities. The stronger the imbalance of the outcome, the
more severe is the bias in the predicted probabilities. We propose two simple
modifications of Firth-type logistic regression resulting in unbiased predicted
probabilities. The first corrects the predicted probabilities by a post-hoc
adjustment of the intercept. The other is based on an alternative formulation
of Firth-types estimation as an iterative data augmentation procedure. Our
suggested modification consists in introducing an indicator variable which
distinguishes between original and pseudo observations in the augmented data.
In a comprehensive simulation study these approaches are compared to other
attempts to improve predictions based on Firth-type penalization and to other
published penalization strategies intended for routine use. For instance, we
consider a recently suggested compromise between maximum likelihood and
Firth-type logistic regression. Simulation results are scrutinized both with
regard to prediction and regression coefficients. Finally, the methods
considered are illustrated and compared for a study on arterial closure devices
in minimally invasive cardiac surgery.
