A solution to the problem of separation in logistic regression
Abstract
The phenomenon of separation or monotone likelihood is observed in the fitting process of a logistic model if the likelihood converges while at least one parameter estimate diverges to ± infinity. Separation primarily occurs in small samples with several unbalanced and highly predictive risk factors. A procedure by Firth originally developed to reduce the bias of maximum likelihood estimates is shown to provide an ideal solution to separation. It produces finite parameter estimates by means of penalized maximum likelihood estimation. Corresponding Wald tests and confidence intervals are available but it is shown that penalized likelihood ratio tests and profile penalized likelihood confidence intervals are often preferable. The clear advantage of the procedure over previous options of analysis is impressively demonstrated by the statistical analysis of two cancer studies. Copyright © 2002 John Wiley & Sons, Ltd.
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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 [62]
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 [62]
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Self-citationAutoantibodies against type I IFNs in patients with critical influenza pneumonia
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- Authors
- Heinze, G., & Schemper, M.
- Affiliations
- University of Vienna (AT)
- Corresponding author
- Heinze, Georg
- Venue
- Statistics in Medicine
- Publisher
- Wiley
- Published
- 2002-07-26
- Volume
- 21
- Issue
- 16
- Pages
- 2409-2419
- ISSN
- 0277-6715 (ISSN-L)
- 1097-0258
- OpenAlex
- W2013416277
- Cited by
- 2,711
- References
- 36
- Citations by year
- 2026: 770 · 2025: 160 · 2024: 159 · 2023: 147 · 2022: 142 · 2021: 193 · 2020: 151 · 2019: 142 · 2018: 129 · 2017: 104 · 2016: 112 · 2015: 100 · 2014: 82 · 2013: 73 · 2012: 49
- FWCI
- 7.98
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Statistics in Medicine (publishedVersion)
- PubMed (publishedVersion)
- 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
- penalized likelihood, Wald test, likelihood ratio test, small sample bias
- MeSH terms
- Breast Neoplasms, Computer Simulation, Female, Humans, Lung Neoplasms, Monte Carlo Method, Neovascularization, Pathologic, Radiotherapy, Smoking, Likelihood Functions, Logistic Models, Case-Control Studies, Endometrial Neoplasms
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