Firth's logistic regression with rare events: accurate effect estimates AND predictions?

TL;DR

This paper addresses bias in predicted probabilities from Firth's logistic regression for small samples with rare events, proposing modifications to achieve unbiased predictions and evaluating them through simulations and real data.

Contribution

It introduces two simple modifications to Firth-type logistic regression that correct bias in predicted probabilities, enhancing its accuracy for small sample and rare event scenarios.

Findings

Modified methods reduce bias in predicted probabilities.

Simulation studies show improved prediction accuracy.

Application to medical data demonstrates practical benefits.

Abstract

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.