Assessing the goodness of fit of logistic regression models in large samples: a modification of the Hosmer-Lemeshow test
Abstract
Evaluating the goodness of fit of logistic regression models is crucial to ensure the accuracy of the estimated probabilities. Unfortunately, such evaluation is problematic in large samples. Because the power of traditional goodness of fit tests increases with the sample size, practically irrelevant discrepancies between estimated and true probabilities are increasingly likely to cause the rejection of the hypothesis of perfect fit in larger and larger samples. This phenomenon has been widely documented for popular goodness of fit tests, such as the Hosmer-Lemeshow test. To address this limitation, we propose a modification of the Hosmer-Lemeshow approach. By standardizing the noncentrality parameter that characterizes the alternative distribution of the Hosmer-Lemeshow statistic, we introduce a parameter that measures the goodness of fit of a model but does not depend on the sample size. We provide the methodology to estimate this parameter and construct confidence intervals for it. Finally, we propose a formal statistical test to rigorously assess whether the fit of a model, albeit not perfect, is acceptable for practical purposes. The proposed method is compared in a simulation study with a competing modification of the Hosmer-Lemeshow test, based on repeated subsampling. We provide a step-by-step illustration of our method using a model for postneonatal mortality developed in a large cohort of more than 300 000 observations.
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The proposed summary measures for calibration plots (ECE, ECI, ICI) cannot inform on the direction of miscalibration.
“Similar to statistical tests such as the Hosmer-Lemeshow test, the proposed summary measures cannot inform on the direction of miscalibration.”
From Evaluation of performance measures in predictive artificial intelligence models to support medical decisions: overview and guidance (2025) · cites this paper as [61]
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- Authors
- Nattino, G., Pennell, M. L., & Lemeshow, S.
- Affiliations
- The Ohio State University (US)
- Corresponding author
- Nattino, Giovanni
- Venue
- Biometrics
- Publisher
- Oxford University Press (OUP)
- Published
- 2020-04-06
- Volume
- 76
- Issue
- 2
- Pages
- 549-560
- ISSN
- 0006-341X (ISSN-L)
- 1541-0420
- OpenAlex
- W3009925463
- Cited by
- 229
- References
- 27
- Citations by year
- 2026: 29 · 2025: 48 · 2024: 67 · 2023: 27 · 2022: 35 · 2021: 21 · 2020: 2
- FWCI
- 14.98
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Biometrics (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Keywords
- goodness of fit, logistic regression, Hosmer-Lemeshow test, large samples, noncentrality parameter, confidence intervals, model fit assessment, postneonatal mortality
- MeSH terms
- Biometry, Computer Simulation, Female, Humans, Infant, Infant Mortality, Male, Monte Carlo Method, Ohio, Probability, Models, Statistical, Confidence Intervals, Chi-Square Distribution, Linear Models, Logistic Models, Sample Size
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