Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach

Kipkirui Jephtah; Tonui Benard Cheruiyot

doi:doi:10.11648/j.bsi.20261102.12

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Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach

Kipkirui Jephtah^*

, Tonui Benard Cheruiyot

Published in Biomedical Statistics and Informatics (Volume 11, Issue 2)

Received: 5 August 2025 Accepted: 27 November 2025 Published: 4 September 2026

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Abstract

Background: Analysis of time-to-event data, generally called survival analysis, arise in many fields of study. Conventional methods of analyzing this type of data are historically well established and continue to be applied. The methods rely on the assumptions that subjects on follow-up can only fail from a well-defined single type of event, and that, conditioned on subject covariates, censoring time and the impending time-of-failure are independent. In real-world settings these assumptions are rarely fulfilled: subjects are typically exposed to multiple events that act concurrently to impede or modify the probability of failure from the event of interest. Events “compete” with each other, so the eventual failure of a subject can only be attributed to the first-occurring event – a competing risks scenario in which both the time to the first occurring event and the type of event that occurs at that time are of interest. Objective: To provide you the reader with a friendly and sufficient introduction to the theory underpinning the sub-distribution hazard model, based on the familiar or rather “traditional” description of survival data. This model describes the absolute risk of a subject experiencing an outcome while adjusting for right censoring, subject covariates, and naturally existing competing outcomes. Conclusions: In the competing risks framework, data may be modelled either through the cause-specific hazard model or the sub-distribution hazard model. Although both strategies are founded on the proportional hazards assumption, they differ in how risk sets are constituted and thus in what they measure, in their context of application, and in how the resulting hazard ratios are interpreted. For research questions where the sub-distribution hazard model is appropriate, the theoretical framework presented in this paper applies.

Published in	Biomedical Statistics and Informatics (Volume 11, Issue 2)
DOI	10.11648/j.bsi.20261102.12
Page(s)	60-64
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Competing Risks, Survival Analysis, Censoring, Cumulative Incidence Function, Sub-distribution Hazard, Cause-specific Hazard

References

[1]	Turkson, A. J., Ayiah-Mensah, F. and Nimoh, V. (2021). Handling Censoring and Censored Data in Survival Analysis: A Standalone Systematic Literature Review. International Journal of Mathematics and Mathematical Sciences, 2021, 1-16. https://doi.org/10.1155/2021/9307475
[2]	Moeschberger, M. L. and Klein, J. P. (1995). Statistical Methods for Dependent Competing Risks. Lifetime Data Analysis, 1, 195-204 https://doi.org/10.1007/BF00985770
[3]	Gooley, T. A., Leisenring, W., Crowley, J. and Storer, B. E. (1999). Estimation of Failure Probabilities in the Presence of Competing Risks: New Representations of Old Estimators. Statistics in Medicine, 18(6), 695-706. https://doi.org/10.1002/(SICI)1097-0258(19990330)18:6 textless695::AID-SIM60textgreater3.0.CO;2-O
[4]	Cox, D. R. (1959). The Analysis of Exponentially Distributed Life-Times with Two Types of Failure. Journal of the Royal Statistical Society: Series B, 21(2), 411-421. https://doi.org/10.1111/j.2517-6161.1959.tb00349.x
[5]	Elandt-Johnson, R. C. and Johnson, N. L. (1980). Survival Models and Data Analysis, Vol. 110. John Wiley & Sons, New York.
[6]	Gail, M. (1975). A Review and Critique of Some Models Used in Competing Risk Analysis. Biometrics, 31(1), 209-222. https://doi.org/10.2307/2529721
[7]	Prentice, R. L., Kalbfleisch, J. D., Peterson Jr., A. V., Flournoy, N., Farewell, V. T. and Breslow, N. E. (1978). The Analysis of Failure Times in the Presence of Competing Risks. Biometrics, 34(4), 541-554. https://doi.org/10.2307/2530374
[8]	Fine, J. P. and Gray, R. J. (1999). A Proportional Hazards Model for the Subdistribution of a Competing Risk. Journal of the American Statistical Association, 94(446), 496-509. https://doi.org/10.1080/01621459.1999.10474144
[9]	Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B, 34(2), 187-202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
[10]	Andersen, P. K., Borgan, {O}., Gill, R. D. and Keiding, N. (2012). Statistical Models Based on Counting Processes. Springer Science & Business Media, New York. https://doi.org/10.1007/978-1-4612-4348-9
[11]	Collett, D. (2015). Modelling Survival Data in Medical Research (3rd ed.). CRC Press. https://doi.org/10.1201/9781003282525
[12]	Bajorunaite, R. and Klein, J. P. (2008). Comparison of Failure Probabilities in the Presence of Competing Risks. Journal of Statistical Computation and Simulation, 78(10), 951-966.
[13]	Pintilie, M. (2006). Competing Risks: A Practical Perspective. John Wiley & Sons, Chichester. https://doi.org/10.1002/9780470870709
[14]	Donoghoe, M. W. and Gebski, V. (2017). The Importance of Censoring in Competing Risks Analysis of the Subdistribution Hazard. BMC Medical Research Methodology, 17, 52. https://doi.org/10.1186/s12874-017-0327-3
[15]	Willems, S. J. and Fiocco, M. (2014). Inverse Probability Censoring Weights for Routine Outcome Monitoring Data. Technical Report, Universiteit Leiden, Leiden, The Netherlands. https://hdl.handle.net/1887/3597243
[16]	Austin, P. C. and Fine, J. P. (2017). Practical Recommendations for Reporting Fine-Gray Model Analyses for Competing Risk Data. Statistics in Medicine, 36(27), 4391-4400. https://doi.org/10.1002/sim.7501
[17]	Mao, L. and Lin, D. Y. (2017). Efficient Estimation of Semiparametric Transformation Models for the Cumulative Incidence of Competing Risks. Journal of the Royal Statistical Society: Series B, 79(2), 573-587. https://doi.org/10.1111/rssb.12177

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APA Style

Jephtah, K., Cheruiyot, T. B. (2026). Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach. Biomedical Statistics and Informatics, 11(2), 60-64. https://doi.org/10.11648/j.bsi.20261102.12

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ACS Style

Jephtah, K.; Cheruiyot, T. B. Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach. Biomed. Stat. Inform. 2026, 11(2), 60-64. doi: 10.11648/j.bsi.20261102.12

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AMA Style

Jephtah K, Cheruiyot TB. Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach. Biomed Stat Inform. 2026;11(2):60-64. doi: 10.11648/j.bsi.20261102.12

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@article{10.11648/j.bsi.20261102.12,
  author = {Kipkirui Jephtah and Tonui Benard Cheruiyot},
  title = {Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach},
  journal = {Biomedical Statistics and Informatics},
  volume = {11},
  number = {2},
  pages = {60-64},
  doi = {10.11648/j.bsi.20261102.12},
  url = {https://doi.org/10.11648/j.bsi.20261102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.bsi.20261102.12},
  abstract = {Background: Analysis of time-to-event data, generally called survival analysis, arise in many fields of study. Conventional methods of analyzing this type of data are historically well established and continue to be applied. The methods rely on the assumptions that subjects on follow-up can only fail from a well-defined single type of event, and that, conditioned on subject covariates, censoring time and the impending time-of-failure are independent. In real-world settings these assumptions are rarely fulfilled: subjects are typically exposed to multiple events that act concurrently to impede or modify the probability of failure from the event of interest. Events “compete” with each other, so the eventual failure of a subject can only be attributed to the first-occurring event – a competing risks scenario in which both the time to the first occurring event and the type of event that occurs at that time are of interest. Objective: To provide you the reader with a friendly and sufficient introduction to the theory underpinning the sub-distribution hazard model, based on the familiar or rather “traditional” description of survival data. This model describes the absolute risk of a subject experiencing an outcome while adjusting for right censoring, subject covariates, and naturally existing competing outcomes. Conclusions: In the competing risks framework, data may be modelled either through the cause-specific hazard model or the sub-distribution hazard model. Although both strategies are founded on the proportional hazards assumption, they differ in how risk sets are constituted and thus in what they measure, in their context of application, and in how the resulting hazard ratios are interpreted. For research questions where the sub-distribution hazard model is appropriate, the theoretical framework presented in this paper applies.},
 year = {2026}
}

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TY  - JOUR
T1  - Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach
AU  - Kipkirui Jephtah
AU  - Tonui Benard Cheruiyot
Y1  - 2026/09/04
PY  - 2026
N1  - https://doi.org/10.11648/j.bsi.20261102.12
DO  - 10.11648/j.bsi.20261102.12
T2  - Biomedical Statistics and Informatics
JF  - Biomedical Statistics and Informatics
JO  - Biomedical Statistics and Informatics
SP  - 60
EP  - 64
PB  - Science Publishing Group
SN  - 2578-8728
UR  - https://doi.org/10.11648/j.bsi.20261102.12
AB  - Background: Analysis of time-to-event data, generally called survival analysis, arise in many fields of study. Conventional methods of analyzing this type of data are historically well established and continue to be applied. The methods rely on the assumptions that subjects on follow-up can only fail from a well-defined single type of event, and that, conditioned on subject covariates, censoring time and the impending time-of-failure are independent. In real-world settings these assumptions are rarely fulfilled: subjects are typically exposed to multiple events that act concurrently to impede or modify the probability of failure from the event of interest. Events “compete” with each other, so the eventual failure of a subject can only be attributed to the first-occurring event – a competing risks scenario in which both the time to the first occurring event and the type of event that occurs at that time are of interest. Objective: To provide you the reader with a friendly and sufficient introduction to the theory underpinning the sub-distribution hazard model, based on the familiar or rather “traditional” description of survival data. This model describes the absolute risk of a subject experiencing an outcome while adjusting for right censoring, subject covariates, and naturally existing competing outcomes. Conclusions: In the competing risks framework, data may be modelled either through the cause-specific hazard model or the sub-distribution hazard model. Although both strategies are founded on the proportional hazards assumption, they differ in how risk sets are constituted and thus in what they measure, in their context of application, and in how the resulting hazard ratios are interpreted. For research questions where the sub-distribution hazard model is appropriate, the theoretical framework presented in this paper applies.
VL  - 11
IS  - 2
ER  -

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Author Information

Kipkirui Jephtah

Department of Mathematics, Actuarial Science & Statistics, University of Kabianga, Kericho, Kenya

Contact Email

http://orcid.org/0009-0005-2597-1342
Tonui Benard Cheruiyot

Department of Mathematics, Actuarial Science & Statistics, University of Kabianga, Kericho, Kenya

Contact Email

http://orcid.org/0009-0009-1630-3827

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APA Style

Jephtah, K., Cheruiyot, T. B. (2026). Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach. Biomedical Statistics and Informatics, 11(2), 60-64. https://doi.org/10.11648/j.bsi.20261102.12

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ACS Style

Jephtah, K.; Cheruiyot, T. B. Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach. Biomed. Stat. Inform. 2026, 11(2), 60-64. doi: 10.11648/j.bsi.20261102.12

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AMA Style

Jephtah K, Cheruiyot TB. Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach. Biomed Stat Inform. 2026;11(2):60-64. doi: 10.11648/j.bsi.20261102.12

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@article{10.11648/j.bsi.20261102.12,
  author = {Kipkirui Jephtah and Tonui Benard Cheruiyot},
  title = {Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach},
  journal = {Biomedical Statistics and Informatics},
  volume = {11},
  number = {2},
  pages = {60-64},
  doi = {10.11648/j.bsi.20261102.12},
  url = {https://doi.org/10.11648/j.bsi.20261102.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.bsi.20261102.12},
  abstract = {Background: Analysis of time-to-event data, generally called survival analysis, arise in many fields of study. Conventional methods of analyzing this type of data are historically well established and continue to be applied. The methods rely on the assumptions that subjects on follow-up can only fail from a well-defined single type of event, and that, conditioned on subject covariates, censoring time and the impending time-of-failure are independent. In real-world settings these assumptions are rarely fulfilled: subjects are typically exposed to multiple events that act concurrently to impede or modify the probability of failure from the event of interest. Events “compete” with each other, so the eventual failure of a subject can only be attributed to the first-occurring event – a competing risks scenario in which both the time to the first occurring event and the type of event that occurs at that time are of interest. Objective: To provide you the reader with a friendly and sufficient introduction to the theory underpinning the sub-distribution hazard model, based on the familiar or rather “traditional” description of survival data. This model describes the absolute risk of a subject experiencing an outcome while adjusting for right censoring, subject covariates, and naturally existing competing outcomes. Conclusions: In the competing risks framework, data may be modelled either through the cause-specific hazard model or the sub-distribution hazard model. Although both strategies are founded on the proportional hazards assumption, they differ in how risk sets are constituted and thus in what they measure, in their context of application, and in how the resulting hazard ratios are interpreted. For research questions where the sub-distribution hazard model is appropriate, the theoretical framework presented in this paper applies.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Modelling Time-to-Failure in Presence of Competing Risks, the Sub-distribution Hazard Approach
AU  - Kipkirui Jephtah
AU  - Tonui Benard Cheruiyot
Y1  - 2026/09/04
PY  - 2026
N1  - https://doi.org/10.11648/j.bsi.20261102.12
DO  - 10.11648/j.bsi.20261102.12
T2  - Biomedical Statistics and Informatics
JF  - Biomedical Statistics and Informatics
JO  - Biomedical Statistics and Informatics
SP  - 60
EP  - 64
PB  - Science Publishing Group
SN  - 2578-8728
UR  - https://doi.org/10.11648/j.bsi.20261102.12
AB  - Background: Analysis of time-to-event data, generally called survival analysis, arise in many fields of study. Conventional methods of analyzing this type of data are historically well established and continue to be applied. The methods rely on the assumptions that subjects on follow-up can only fail from a well-defined single type of event, and that, conditioned on subject covariates, censoring time and the impending time-of-failure are independent. In real-world settings these assumptions are rarely fulfilled: subjects are typically exposed to multiple events that act concurrently to impede or modify the probability of failure from the event of interest. Events “compete” with each other, so the eventual failure of a subject can only be attributed to the first-occurring event – a competing risks scenario in which both the time to the first occurring event and the type of event that occurs at that time are of interest. Objective: To provide you the reader with a friendly and sufficient introduction to the theory underpinning the sub-distribution hazard model, based on the familiar or rather “traditional” description of survival data. This model describes the absolute risk of a subject experiencing an outcome while adjusting for right censoring, subject covariates, and naturally existing competing outcomes. Conclusions: In the competing risks framework, data may be modelled either through the cause-specific hazard model or the sub-distribution hazard model. Although both strategies are founded on the proportional hazards assumption, they differ in how risk sets are constituted and thus in what they measure, in their context of application, and in how the resulting hazard ratios are interpreted. For research questions where the sub-distribution hazard model is appropriate, the theoretical framework presented in this paper applies.
VL  - 11
IS  - 2
ER  -

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