title: Firth's logistic regression with rare events: accurate effect estimates and predictions?
abstract: Firth's 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 one-half 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's 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's penalization as an iterative data augmentation procedure. Our suggested modification consists in introducing an indicator variable that distinguishes between original and pseudo-observations in the augmented data. In a comprehensive simulation study, these approaches are compared with other attempts to improve predictions based on Firth's penalization and to other published penalization strategies intended for routine use. For instance, we consider a recently suggested compromise between maximum likelihood and Firth's logistic regression. Simulation results are scrutinized with regard to prediction and effect estimation. We find that both our suggested methods do not only give unbiased predicted probabilities but also improve the accuracy conditional on explanatory variables compared with Firth's penalization. While one method results in effect estimates identical to those of Firth's penalization, the other introduces some bias, but this is compensated by a decrease in the mean squared error. Finally, all methods considered are illustrated and compared for a study on arterial closure devices in minimally invasive cardiac surgery. Copyright © 2017 John Wiley & Sons, Ltd.
authors.affiliation: The Kirby Institute, University of New South Wales, Sydney, Australia.
authors.forename: Rainer
authors.initials: R
authors.lastname: Puhr
authors.affiliation: Center for Medical Statistics, Informatics and Intelligent Systems, Medical University of Vienna, Vienna, Austria.
authors.forename: Georg
authors.initials: G
authors.lastname: Heinze
authors.affiliation: Institute of Medical Statistics, Computer Sciences and Documentation, University Hospital Jena, Jena, Germany.
authors.forename: Mariana
authors.initials: M
authors.lastname: Nold
authors.affiliation: Institute for Biostatistics and Medical Informatics, Faculty of Medicine, University of Ljubljana, Ljubljana, Slovenia.
authors.forename: Lara
authors.initials: L
authors.lastname: Lusa
authors.affiliation: Center for Medical Statistics, Informatics and Intelligent Systems, Medical University of Vienna, Vienna, Austria.
authors.forename: Angelika
authors.identifier: 0000-0002-4659-4911
authors.initials: A
authors.lastname: Geroldinger
mesh_terms: D015982:Bias; D056808:Biostatistics; D006348:Cardiac Surgical Procedures / Q000009:adverse effects / Q000295:instrumentation / Q000706:statistics & numerical data; D003198:Computer Simulation; D006801:Humans; D016013:Likelihood Functions; D016015:Logistic Models; D019060:Minimally Invasive Surgical Procedures / Q000009:adverse effects / Q000295:instrumentation / Q000706:statistics & numerical data; D015233:Models, Statistical; D011336:Probability; D018401:Sample Size; D065506:Vascular Closure Devices / Q000009:adverse effects / Q000706:statistics & numerical data
keywords: Jeffreys prior; bias reduction; data augmentation; penalized likelihood; sparse data