Should we still use pointwise confidence intervals for the Kaplan–Meier estimator?
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Szilárd Nemes
Abstract
To represent uncertainty in survival analysis, the Kaplan–Meier estimator is frequently presented, along with pointwise confidence intervals, though rarely with confidence bands. Using theoretical reasoning and simulations, this paper investigates the coverage properties and interpretation of pointwise confidence intervals. In addition, we examine the technicalities and potential pitfalls of confidence bands. As both the sample size and the number of events increase, the simultaneous coverage of pointwise confidence intervals decreases. The portion of the survival curve covered by interconnected pointwise confidence intervals corresponds to the nominal coverage of pointwise confidence intervals. Contrary to expectations, most confidence bands cover only a portion of the survival curve. The coverage of confidence bands built for the entire curve may be below the nominal coverage. Despite their limitations, pointwise confidence intervals remain practical and easy to use for illustrating uncertainty in survival estimates. They do not cover a continuous portion of the survival curve but, on average, cover a portion equal to the pointwise nominal confidence level.
Acknowledgments
I would like to thank the two anonymous reviewers for their careful reading of the manuscript and for their insightful comments and suggestions, which have significantly improved the clarity and quality of this work.
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved its submission.
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Use of Large Language Models, AI and Machine Learning Tools: Grammatical and language corrections in this document were performed using a combination of ChatGPT (OpenAI) and Microsoft Word. These tools were used to improve clarity and readability; the author remains solely responsible for the final content.
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Conflict of interest: The author states no conflict of interest. Employee of AstraZeneca PLC.
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Research funding: None declared.
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Data availability: Publicly available data is used and referenced.
References
1. Bie, O, Borgan, Ø, Liestøl, K. Confidence intervals and confidence bands for the cumulative hazard rate function and their small sample properties. Scand J Stat 1987:221–33.Search in Google Scholar
2. Borgan, Ø, Liestøl, K. A note on confidence intervals and bands for the survival function based on transformations. Scand J Stat 1990:35–41.Search in Google Scholar
3. Sachs, MC, Brand, A, Gabriel, EE. Confidence bands in survival analysis. Br J Cancer 2022;127:1636–41. https://doi.org/10.1038/s41416-022-01920-5.Search in Google Scholar PubMed PubMed Central
4. Moertel, CG, Fleming, TR, Macdonald, JS, Haller, DG, Laurie, JA, Tangen, CM, et al.. Fluorouracil plus levamisole as effective adjuvant therapy after resection of stage III colon carcinoma: a final report. Ann Intern Med 1995;122:321–6. https://doi.org/10.7326/0003-4819-122-5-199503010-00001.Search in Google Scholar PubMed
5. Kaplan, EL, Meier, P. Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457–81. https://doi.org/10.1080/01621459.1958.10501452.Search in Google Scholar
6. Breslow, N, Crowley, J. A large sample study of the life table and product limit estimates under random censorship. Ann Stat 1974:437–53. https://doi.org/10.1214/aos/1176342705.Search in Google Scholar
7. Meier, P. Estimation of a distribution function from incomplete observations. J Appl Probab 1975;12:67–87. https://doi.org/10.1017/s0021900200047574.Search in Google Scholar
8. Nemes, S. Asymptotic relative efficiency of parametric and nonparametric survival estimators. Stats 2023;6:1147–59. https://doi.org/10.3390/stats6040072.Search in Google Scholar
9. Gillespie, MJ, Fisher, L. Confidence bands for the Kaplan–Meier survival curve estimate. Ann Stat 1979:920–4. https://doi.org/10.1214/aos/1176344742.Search in Google Scholar
10. Hall, WJ, Wellner, JA. Confidence bands for a survival curve from censored data. Biometrika 1980;67:133–43. https://doi.org/10.1093/biomet/67.1.133.Search in Google Scholar
11. Nair, VN. Confidence bands for survival functions with censored data: a comparative study. Technometrics 1984;26:265–75. https://doi.org/10.2307/1267553.Search in Google Scholar
12. Zheng, G, Gastwirth, JL. On the Fisher information in randomly censored data. Stat Probab Lett 2001;52:421–6. https://doi.org/10.1016/s0167-7152(01)00025-6.Search in Google Scholar
13. Nemes, S, Gustavsson, A, Taib, Z. Variance inflation due to censoring in survival probability estimates. Statistica 2020;80:395–412.Search in Google Scholar
14. Hollander, M, McKeague, IW, McKeague, IW. Likelihood ratio-based confidence bands for survival functions. J Am Stat Assoc 1997;92:215–26. https://doi.org/10.2307/2291466.Search in Google Scholar
15. Chen, T, Tracy, S, Uno, H. OptBand: optimization-based confidence bands for functions to characterize time-to-event distributions. Lifetime Data Anal 2021;27:481–98. https://doi.org/10.1007/s10985-021-09522-8.Search in Google Scholar PubMed PubMed Central
16. Andersen, PK, Borgan, O, Gill, RD, Keiding, N. Statistical models based on counting processes. New York, NY: Springer Science and Business Media; 2012.Search in Google Scholar
17. Laurie, JA, Moertel, CG, Fleming, TR, Wieand, HS, Leigh, JE, Rubin, J, et al.. Surgical adjuvant therapy of large-bowel carcinoma: an evaluation of levamisole and the combination of levamisole and fluorouracil. The North Central Cancer Treatment Group and the Mayo Clinic. J Clin Oncol 1989;7:1447–56. https://doi.org/10.1200/jco.1989.7.10.1447.Search in Google Scholar PubMed
18. Simpson, AL, Peoples, J, Creasy, JM, Fichtinger, G, Gangai, N, Keshavamurthy, KN, et al.. Preoperative CT and survival data for patients undergoing resection of colorectal liver metastases. Sci Data 2024;11:172. https://doi.org/10.1038/s41597-024-02981-2.Search in Google Scholar PubMed PubMed Central
19. Hassen, HY, Teka, MA, Beksisa, J, Feyisa, JD. Data on histological characteristics, survival patterns and determinants of mortality among colorectal, esophageal and prostate cancer patients in Ethiopia. Data Brief 2021;38:107279. https://doi.org/10.1016/j.dib.2021.107279.Search in Google Scholar PubMed PubMed Central
20. Nitsche, U, Weber, C, Kaufmann, B, von Figura, G, Assfalg, V, Miller, G, et al.. Survival data on timing of resection of liver metastases in colorectal cancer patients. Data Brief 2020;31:105973. https://doi.org/10.1016/j.dib.2020.105973.Search in Google Scholar PubMed PubMed Central
21. Wu, ZY, Han, HH, Kim, HJ, Lee, J, Chung, IY, Kim, J, et al.. Data on distant metastasis and survival after locoregional recurrence following nipple-sparing mastectomy and immediate breast reconstruction. Data Brief 2021;35:106837. https://doi.org/10.1016/j.dib.2021.106837.Search in Google Scholar PubMed PubMed Central
22. Howe, MS, Keys, W, Richards, D. Long-term (10-year) dental implant survival: a systematic review and sensitivity meta-analysis. J Dent 2019;84:9–21. https://doi.org/10.1016/j.jdent.2019.03.008.Search in Google Scholar PubMed
23. Pakarinen, O, Ponkilainen, V, Varnum, C, Pedersen, AB, Overgaard, S, Kärrholm, J, et al.. Choice of bearings influences the implant survival of total hip arthroplasty in patients who have osteoarthritis aged 55 years or older: results of 158,044 patients from the Nordic arthroplasty register association from 2005 to 2017. J Arthroplasty 2025;40:437–42. https://doi.org/10.1016/j.arth.2024.08.015.Search in Google Scholar PubMed
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Articles in the same Issue
- Research Articles
- Definition, identification, and estimation of the direct and indirect number needed to treat
- Sample size determination for external validation of risk models with binary outcomes using the area under the ROC curve
- Analysis of the drug resistance level of malaria disease: a fractional-order model
- Extending the scope of the capture-recapture experiment: a multilevel approach with random effects to provide reliable estimates at national level
- Discrete-time compartmental models with partially observed data: a comparison among frequentist and Bayesian approaches for addressing likelihood intractability
- Sensitivity analysis for unmeasured confounding for a joint effect with an application to survey data
- Investigating the association between school substance programs and student substance use: accounting for informative cluster size
- The quantiles of extreme differences matrix for evaluating discriminant validity
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- Causal mediation analysis for difference-in-difference design and panel data
- What if dependent causes of death were independent?
- Bot invasion: protecting the integrity of online surveys against spamming
- A study of a stochastic model and extinction phenomenon of meningitis epidemic
- Understanding the impact of media and latency in information response on the disease propagation: a mathematical model and analysis
- Time-varying reproductive number estimation for practical application in structured populations
- Perspective
- Should we still use pointwise confidence intervals for the Kaplan–Meier estimator?
- Leveraging data from multiple sources in epidemiologic research: transportability, dynamic borrowing, external controls, and beyond
- Regression calibration for time-to-event outcomes: mitigating bias due to measurement error in real-world endpoints
