Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards

Michael Richard Crager; Jerome Victor Braun

doi:doi:10.11648/j.ajam.20221002.15

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Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards

Michael Richard Crager, Jerome Victor Braun

Published in American Journal of Applied Mathematics (Volume 10, Issue 2)

Received: 4 April 2022 Accepted: 19 April 2022 Published: 28 April 2022

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Abstract

It is often important to account for the effects of a competing risk when estimating the risk of a particular event of interest by estimating its absolute risk. Available methodology for interval estimation of the absolute risk using the proportional regression of cause-specific hazards (CSH) has been limited to situations with time-invariant covariates and a single random censoring mechanism, without accommodation of cohort sampling study designs. Here we derive asymptotic pointwise confidence intervals in closed form for the absolute risk of an event at a specified time (the value of the cumulative incidence function) in the presence of competing risks using proportional CSH regression, accommodating external time-dependent covariates, cohort sampling study designs and multiple censoring mechanisms. Different covariates may be used for the event of interest and the various competing risks. Consistent with the definition of absolute risk, the CSH method produces absolute risk estimates that are less than or approximately equal to corresponding “conditional” risk estimates that do not account for competing risks. An example shows that this property is not necessarily shared by methods based on subdistribution hazard regression. Simulation studies indicate that the CSH method confidence intervals computed on the log cumulative hazard or the risk scale have coverage probabilities that approximate the nominal level for small and moderate samples, provided that the number of events per covariate is at least 10 and, when using cohort sampling, the ratio of patients without events to patients with events is at least 2:1.

Published in	American Journal of Applied Mathematics (Volume 10, Issue 2)
DOI	10.11648/j.ajam.20221002.15
Page(s)	59-85
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), 2024. Published by Science Publishing Group

Keywords

Absolute Risk, Cause-Specific Hazards, Cohort Sampling, Competing Risks, Cumulative Incidence Function, Interval Estimation, Time-Dependent Covariates

References

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APA Style

Michael Richard Crager, Jerome Victor Braun. (2022). Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards. American Journal of Applied Mathematics, 10(2), 59-85. https://doi.org/10.11648/j.ajam.20221002.15

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ACS Style

Michael Richard Crager; Jerome Victor Braun. Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards. Am. J. Appl. Math. 2022, 10(2), 59-85. doi: 10.11648/j.ajam.20221002.15

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AMA Style

Michael Richard Crager, Jerome Victor Braun. Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards. Am J Appl Math. 2022;10(2):59-85. doi: 10.11648/j.ajam.20221002.15

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@article{10.11648/j.ajam.20221002.15,
  author = {Michael Richard Crager and Jerome Victor Braun},
  title = {Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards},
  journal = {American Journal of Applied Mathematics},
  volume = {10},
  number = {2},
  pages = {59-85},
  doi = {10.11648/j.ajam.20221002.15},
  url = {https://doi.org/10.11648/j.ajam.20221002.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221002.15},
  abstract = {It is often important to account for the effects of a competing risk when estimating the risk of a particular event of interest by estimating its absolute risk. Available methodology for interval estimation of the absolute risk using the proportional regression of cause-specific hazards (CSH) has been limited to situations with time-invariant covariates and a single random censoring mechanism, without accommodation of cohort sampling study designs. Here we derive asymptotic pointwise confidence intervals in closed form for the absolute risk of an event at a specified time (the value of the cumulative incidence function) in the presence of competing risks using proportional CSH regression, accommodating external time-dependent covariates, cohort sampling study designs and multiple censoring mechanisms. Different covariates may be used for the event of interest and the various competing risks. Consistent with the definition of absolute risk, the CSH method produces absolute risk estimates that are less than or approximately equal to corresponding “conditional” risk estimates that do not account for competing risks. An example shows that this property is not necessarily shared by methods based on subdistribution hazard regression. Simulation studies indicate that the CSH method confidence intervals computed on the log cumulative hazard or the risk scale have coverage probabilities that approximate the nominal level for small and moderate samples, provided that the number of events per covariate is at least 10 and, when using cohort sampling, the ratio of patients without events to patients with events is at least 2:1.},
 year = {2022}
}

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TY  - JOUR
T1  - Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards
AU  - Michael Richard Crager
AU  - Jerome Victor Braun
Y1  - 2022/04/28
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221002.15
DO  - 10.11648/j.ajam.20221002.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 59
EP  - 85
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20221002.15
AB  - It is often important to account for the effects of a competing risk when estimating the risk of a particular event of interest by estimating its absolute risk. Available methodology for interval estimation of the absolute risk using the proportional regression of cause-specific hazards (CSH) has been limited to situations with time-invariant covariates and a single random censoring mechanism, without accommodation of cohort sampling study designs. Here we derive asymptotic pointwise confidence intervals in closed form for the absolute risk of an event at a specified time (the value of the cumulative incidence function) in the presence of competing risks using proportional CSH regression, accommodating external time-dependent covariates, cohort sampling study designs and multiple censoring mechanisms. Different covariates may be used for the event of interest and the various competing risks. Consistent with the definition of absolute risk, the CSH method produces absolute risk estimates that are less than or approximately equal to corresponding “conditional” risk estimates that do not account for competing risks. An example shows that this property is not necessarily shared by methods based on subdistribution hazard regression. Simulation studies indicate that the CSH method confidence intervals computed on the log cumulative hazard or the risk scale have coverage probabilities that approximate the nominal level for small and moderate samples, provided that the number of events per covariate is at least 10 and, when using cohort sampling, the ratio of patients without events to patients with events is at least 2:1.
VL  - 10
IS  - 2
ER  -

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Author Information

Michael Richard Crager

Department of Biostatistics, Exact Sciences Corporation, Redwood City, USA
Jerome Victor Braun

Department of Biostatistics, Exact Sciences Corporation, Redwood City, USA

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APA Style

Michael Richard Crager, Jerome Victor Braun. (2022). Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards. American Journal of Applied Mathematics, 10(2), 59-85. https://doi.org/10.11648/j.ajam.20221002.15

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ACS Style

Michael Richard Crager; Jerome Victor Braun. Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards. Am. J. Appl. Math. 2022, 10(2), 59-85. doi: 10.11648/j.ajam.20221002.15

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AMA Style

Michael Richard Crager, Jerome Victor Braun. Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards. Am J Appl Math. 2022;10(2):59-85. doi: 10.11648/j.ajam.20221002.15

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@article{10.11648/j.ajam.20221002.15,
  author = {Michael Richard Crager and Jerome Victor Braun},
  title = {Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards},
  journal = {American Journal of Applied Mathematics},
  volume = {10},
  number = {2},
  pages = {59-85},
  doi = {10.11648/j.ajam.20221002.15},
  url = {https://doi.org/10.11648/j.ajam.20221002.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20221002.15},
  abstract = {It is often important to account for the effects of a competing risk when estimating the risk of a particular event of interest by estimating its absolute risk. Available methodology for interval estimation of the absolute risk using the proportional regression of cause-specific hazards (CSH) has been limited to situations with time-invariant covariates and a single random censoring mechanism, without accommodation of cohort sampling study designs. Here we derive asymptotic pointwise confidence intervals in closed form for the absolute risk of an event at a specified time (the value of the cumulative incidence function) in the presence of competing risks using proportional CSH regression, accommodating external time-dependent covariates, cohort sampling study designs and multiple censoring mechanisms. Different covariates may be used for the event of interest and the various competing risks. Consistent with the definition of absolute risk, the CSH method produces absolute risk estimates that are less than or approximately equal to corresponding “conditional” risk estimates that do not account for competing risks. An example shows that this property is not necessarily shared by methods based on subdistribution hazard regression. Simulation studies indicate that the CSH method confidence intervals computed on the log cumulative hazard or the risk scale have coverage probabilities that approximate the nominal level for small and moderate samples, provided that the number of events per covariate is at least 10 and, when using cohort sampling, the ratio of patients without events to patients with events is at least 2:1.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Interval Estimation of the Absolute Risk of an Event with Competing Risks Using Proportional Regression of Cause-Specific Hazards
AU  - Michael Richard Crager
AU  - Jerome Victor Braun
Y1  - 2022/04/28
PY  - 2022
N1  - https://doi.org/10.11648/j.ajam.20221002.15
DO  - 10.11648/j.ajam.20221002.15
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 59
EP  - 85
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20221002.15
AB  - It is often important to account for the effects of a competing risk when estimating the risk of a particular event of interest by estimating its absolute risk. Available methodology for interval estimation of the absolute risk using the proportional regression of cause-specific hazards (CSH) has been limited to situations with time-invariant covariates and a single random censoring mechanism, without accommodation of cohort sampling study designs. Here we derive asymptotic pointwise confidence intervals in closed form for the absolute risk of an event at a specified time (the value of the cumulative incidence function) in the presence of competing risks using proportional CSH regression, accommodating external time-dependent covariates, cohort sampling study designs and multiple censoring mechanisms. Different covariates may be used for the event of interest and the various competing risks. Consistent with the definition of absolute risk, the CSH method produces absolute risk estimates that are less than or approximately equal to corresponding “conditional” risk estimates that do not account for competing risks. An example shows that this property is not necessarily shared by methods based on subdistribution hazard regression. Simulation studies indicate that the CSH method confidence intervals computed on the log cumulative hazard or the risk scale have coverage probabilities that approximate the nominal level for small and moderate samples, provided that the number of events per covariate is at least 10 and, when using cohort sampling, the ratio of patients without events to patients with events is at least 2:1.
VL  - 10
IS  - 2
ER  -

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