Firth's logistic regression with rare events: accurate effect estimates and predictions?
Open accessAbstract
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.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Firth's penalized logistic regression is widely recommended for binary outcomes in small samples because it adds a Jeffreys-prior penalty to the likelihood function, yielding stable parameter estimates and valid confidence intervals.
“This approach is widely recommended for binary outcomes in small samples [60] because it adds a Jeffreys-prior penalty to the likelihood function, yielding stable parameter estimates and valid confidence intervals [61,62], and was implemented via the logistf package.”
From Wildfire Ash Alters Freshwater Crustacean Behavior: Disentangling the Contributions of Suspended Macroparticles and Ash‐Induced Alkalinization (2026) · cites this paper as [60]
The probability of responding in Aegla jarai was analyzed using Firth's penalized logistic regression, a bias-reduction method suited to rare events.
“Probability of responding analyzed using Firth's penalized logistic regression”
From Wildfire Ash Alters Freshwater Crustacean Behavior: Disentangling the Contributions of Suspended Macroparticles and Ash‐Induced Alkalinization (2026) · cites this paper as [60]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
R: A Language and Environment for Statistical Computing
R Core Team (2000) · doi:10.32614/r.manuals · Cited by 355,706 · see citing works
Assessing the Performance of Prediction Models: A Framework for Traditional and Novel Measures
Steyerberg, E. W., Vickers, A. J., Cook, N. R., Gerds, T., Gonen, M., Obuchowski, N., Pencina, M. J., & Kattan, M. W. (2010) · Epidemiology · doi:10.1097/ede.0b013e3181c30fb2 · Cited by 5,170 · see citing works
Bias reduction of maximum likelihood estimates
FIRTH, D. (1993) · Biometrika · doi:10.1093/biomet/80.1.27 · Cited by 5,035 · see citing works
Data Analysis Using Regression and Multilevel/Hierarchical Models
Gelman, A., & Su, Y.-S. (2007) · CRAN: Contributed Packages · doi:10.32614/cran.package.arm · Cited by 4,629 · see citing works
Logistic Regression in Rare Events Data
King, G., & Zeng, L. (2001) · Political Analysis · doi:10.1093/oxfordjournals.pan.a004868 · Cited by 4,316 · cited 3× · see citing works
“[3], penalization by log-F(1, 1) [4] or Cauchy priors [5], King and Zeng’s ‘approximate Bayesian’, and ‘approximate unbiased’ method [6] and ridge regression.”
MethodA solution to the problem of separation in logistic regression
Heinze, G., & Schemper, M. (2002) · Statistics in Medicine · doi:10.1002/sim.1047 · Cited by 2,711 · see citing works
Self-citationA weakly informative default prior distribution for logistic and other regression models
Gelman, A., Jakulin, A., Pittau, M. G., & Su, Y.-S. (2008) · The Annals of Applied Statistics · doi:10.1214/08-aoas191 · Cited by 2,115 · see citing works
Ridge Estimators in Logistic Regression
le Cessie, S., & van Houwelingen, H. C. (1992) · Journal of the Royal Statistical Society Series C (Applied Statistics) · doi:10.2307/2347628 · Cited by 1,689 · see citing works
Improvements on Cross-Validation: The .632+ Bootstrap Method
Efron, B., & Tibshirani, R. J. (1997) · Journal of the American Statistical Association · doi:10.1080/01621459.1997.10474007 · Cited by 1,638 · see citing works
A WEAKLY INFORMATIVE DEFAULT PRIOR DISTRIBUTION FOR LOGISTIC AND OTHER REGRESSION MODELS
Gelman, A., Jakulin, A., & Grazia, M. (2009) · Cited by 1,352 · see citing works
Computation of Multivariate Normal and t Probabilities
Genz, A. C., & Bretz, F. (2009) · Lecture notes in statistics · doi:10.1007/978-3-642-01689-9 · Cited by 884 · see citing works
Improvements on Cross-Validation: The .632+ Bootstrap Method
Efron, B., & Tibshirani, R. J. (1997) · Journal of the American Statistical Association · doi:10.2307/2965703 · Cited by 723 · see citing works
rms: Regression Modeling Strategies
Harrell, F. E. (2009) · doi:10.32614/cran.package.rms · Cited by 477 · see citing works
Greenland, S., & Mansournia, M. A. (2015) · Statistics in Medicine · doi:10.1002/sim.6537 · Cited by 237 · see citing works
Penalized likelihood in Cox regression
Verweij, P. J. M., & van Houwelingen, H. C. (1994) · Statistics in Medicine · doi:10.1002/sim.4780132307 · Cited by 211 · see citing works
logistf: Firth's Bias-Reduced Logistic Regression
Heinze, G., Ploner, M., Jiricka, L., & Steiner, G. (2003) · doi:10.32614/cran.package.logistf · Cited by 129 · see citing works
Self-citationFIRTHLOGIT: Stata module to calculate bias reduction in logistic regression
Coveney, J. (2008) · RePEc: Research Papers in Economics · Cited by 83 · see citing works
Separation-Resistant and Bias-Reduced Logistic Regression: STATISTICA Macro
Fijorek, K., & Sokoowski, A. (2012) · Journal of Statistical Software · doi:10.18637/jss.v047.c02 · Cited by 17 · see citing works
Penalised logistic regression and dynamic prediction for discrete-time recurrent event data
Elgmati, E. A., Fiaccone, R. L., Henderson, R., & Matthews, J. N. S. (2015) · Lifetime Data Analysis · doi:10.1007/s10985-015-9321-4 · Cited by 15 · see citing works
Greenland, S. (2010) · The American Statistician · doi:10.1198/tast.2010.10006 · Cited by 13 · see citing works
Flagged first, then by citations
Research works printed but not linked: 1
- [11][11] Binder H, Sauerbrei W, Royston P. Multivariable Model-Building with Continuous Covariates: 1. Performance Measures and Simulation Design. Technical Report FDM-Preprint 105 , University of Freiburg, Germany 2011. -
Cited by
Sorted by citations
Van Calster, B., van Smeden, M., De Cock, B., & Steyerberg, E. W. (2020) · Statistical Methods in Medical Research · doi:10.1177/0962280220921415 · Cited by 120 · see citing works
Ju, K., Lin, L., Chu, H., Cheng, L.-L., & Xu, C. (2020) · BMC Medical Research Methodology · doi:10.1186/s12874-020-01035-6 · Cited by 61 · see citing works
Pereira‐Silva, P. B., Pacheco, D. R., Kirsch, G., Araujo, R. M. G., Oscar, B. V. C., Matias, W. G., Santos, S., & Figueiredo, B. R. S. (2026) · Environmental Toxicology · doi:10.1002/tox.70178
- Authors
- Puhr, R., Heinze, G., Nold, M., Lusa, L., & Geroldinger, A.
- Affiliations
- UNSW Sydney (AU)
- Medical University of Vienna (AT)
- Jena University Hospital (DE)
- University of Ljubljana (SI)
- Corresponding author
- Geroldinger, Angelika
- Venue
- Statistics in Medicine
- Publisher
- Wiley
- Published
- 2017-03-12
- Volume
- 36
- Issue
- 14
- Pages
- 2302-2317
- ISSN
- 0277-6715 (ISSN-L)
- 1097-0258
- OpenAlex
- W2593185514
- Cited by
- 483
- References
- 20
- Citations by year
- 2026: 159 · 2025: 77 · 2024: 56 · 2023: 48 · 2022: 45 · 2021: 44 · 2020: 32 · 2019: 13 · 2018: 5 · 2017: 3
- FWCI
- 10.27
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Statistics in Medicine (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Austrian Science Fund
- Javna Agencija za Raziskovalno Dejavnost RS
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Keywords
- rare events, binary outcomes, predicted probabilities, bias correction, data augmentation, penalty method, effect estimation, prediction accuracy, vascular closure devices, minimally invasive cardiac surgery
- MeSH terms
- Computer Simulation, Cardiac Surgical Procedures, Humans, Probability, Models, Statistical, Bias, Likelihood Functions, Logistic Models, Sample Size, Minimally Invasive Surgical Procedures, Biostatistics, Vascular Closure Devices
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
