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Research article
Julius S Ngwa
Boston University
Corresponding Author
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Background: Statistical methods for modeling longitudinal and time-to-event data has received much attention in medical research and is becoming increasingly useful. In clinical studies, such as cancer and AIDS, longitudinal biomarkers are used to monitor disease progression and to predict survival. These longitudinal measures are often missing at failure times and may be prone to measurement errors. More importantly, time-dependent survival models that include the raw longitudinal measurements may lead to biased results. In previous studies these two types of data are frequently analyzed separately where a mixed effects model is used for the longitudinal data and a survival model is applied to the event outcome.
Methods: In this paper we compare joint maximum likelihood methods, a two-step approach and a time dependent covariate method that link longitudinal data to survival data with emphasis on using longitudinal measures to predict survival. We apply a Bayesian semi-parametric joint method and maximum likelihood joint method that maximizes the joint likelihood of the time-to-event and longitudinal measures. We also implement the Two-Step approach, which estimates random effects separately, and a classic Time Dependent Covariate Model. We use simulation studies to assess bias, accuracy, and coverage probabilities for the estimates of the link parameter that connects the longitudinal measures to survival times.
Results: Simulation results demonstrate that the Two-Step approach performed best at estimating the link parameter when variability in the longitudinal measure is low but is somewhat biased downwards when the variability is high. Bayesian semi-parametric and maximum likelihood joint methods yield higher link parameter estimates with low and high variability in the longitudinal measure. The Time Dependent Covariate method resulted in consistent underestimation of the link parameter. We illustrate these methods using data from the Framingham Heart Study in which lipid measurements and Myocardial Infarction data were collected over a period of 26 years.
Conclusions: Traditional methods for modeling longitudinal and survival data, such as the time dependent covariate method, that use the observed longitudinal data, tend to provide downwardly biased estimates. The two-step approach and joint models provide better estimates, although a comparison of these methods may depend on the underlying residual variance.
Keywords
Joint longitudinal and Survival Model, Cox Model, Two-step Approach, Mixed Effect Modeling, Time Dependent Covariate Models, Weibull Distribution, Residual Variance
Figure 1
The full text of this article is available to read as a PDF.
This is a list of supplementary files associated with this preprint. Click to download.
- Tables.docx
Tables 1-6
- SupplementalFigures.docx
S1: Type I Errors for Link (Exponential and Weibull Distribution, N = 100) S2: Estimates and Confidence Intervals for Link (Weibull Distribution, N = 100) S3: Estimates and Confidence Intervals for Age (Weibull Distribution, N = 100) S4: Estimates and Confidence Intervals for Sex (Weibull Distribution, N = 100) S5: Bayesian Semi-Parametric Joint Modeling Exact Prior Distributions
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View the published article at BMC Medical Research Methodology.
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Posted 19 Jan, 2021
No community comments so far
Submission checks complete
On 12 Jan, 2021
Editorial decision: Accept
On 11 Jan, 2021
No comments provided
Editorial decision: Minor revision
On 04 Jan, 2021
Review #1 received
Received 27 Dec, 2020
Reviewers invited
Invitations sent on 09 Dec, 2020
Reviewer #1 agreed
On 09 Dec, 2020
Editor assigned
On 08 Dec, 2020
Submission checks complete
On 08 Dec, 2020
Editor invited
On 08 Dec, 2020
No comments provided
Editorial decision: Major revision
On 28 Oct, 2020
Review #2 received
Received 28 Sep, 2020
Review #1 received
Received 25 Aug, 2020
Reviewer #2 agreed
On 03 Aug, 2020
Reviewer #1 agreed
On 28 Jul, 2020
Editor assigned
On 15 Jun, 2020
Reviewers invited
Invitations sent on 15 Jun, 2020
Submission checks complete
On 10 Jun, 2020
Editor invited
On 10 Jun, 2020
Metrics
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See the published version of this preprint at BMC Medical Research Methodology.
This is a preprint, a preliminary version of a manuscript that has not completed peer review at a journal. Research Square does not conduct peer review prior to posting preprints. The posting of a preprint on this server should not be interpreted as an endorsement of its validity or suitability for dissemination as established information or for guiding clinical practice.
Learn more about In Review
Research article
Julius S Ngwa
Boston University
Corresponding Author
Badges
Prescreen
This manuscript passed our Prescreen assessment
Complete author information
Appropriate declaration statements
Potential risks to human health
Prescreen
Peer Review Timeline
CURRENT STATUS: Published
View the published article at BMC Medical Research Methodology.
Version 3
Posted 19 Jan, 2021
No community comments so far
Submission checks complete
On 12 Jan, 2021
Editorial decision: Accept
On 11 Jan, 2021
No comments provided
Editorial decision: Minor revision
On 04 Jan, 2021
Review #1 received
Received 27 Dec, 2020
Reviewers invited
Invitations sent on 09 Dec, 2020
Reviewer #1 agreed
On 09 Dec, 2020
Editor assigned
On 08 Dec, 2020
Submission checks complete
On 08 Dec, 2020
Editor invited
On 08 Dec, 2020
No comments provided
Editorial decision: Major revision
On 28 Oct, 2020
Review #2 received
Received 28 Sep, 2020
Review #1 received
Received 25 Aug, 2020
Reviewer #2 agreed
On 03 Aug, 2020
Reviewer #1 agreed
On 28 Jul, 2020
Editor assigned
On 15 Jun, 2020
Reviewers invited
Invitations sent on 15 Jun, 2020
Submission checks complete
On 10 Jun, 2020
Editor invited
On 10 Jun, 2020
Metrics
Comments: 0
PDF Downloads: ...
HTML Views: ...
License:
This work is licensed under a CC BY 4.0 License. Read Full License
View the published article at BMC Medical Research Methodology.
Background: Statistical methods for modeling longitudinal and time-to-event data has received much attention in medical research and is becoming increasingly useful. In clinical studies, such as cancer and AIDS, longitudinal biomarkers are used to monitor disease progression and to predict survival. These longitudinal measures are often missing at failure times and may be prone to measurement errors. More importantly, time-dependent survival models that include the raw longitudinal measurements may lead to biased results. In previous studies these two types of data are frequently analyzed separately where a mixed effects model is used for the longitudinal data and a survival model is applied to the event outcome.
Methods: In this paper we compare joint maximum likelihood methods, a two-step approach and a time dependent covariate method that link longitudinal data to survival data with emphasis on using longitudinal measures to predict survival. We apply a Bayesian semi-parametric joint method and maximum likelihood joint method that maximizes the joint likelihood of the time-to-event and longitudinal measures. We also implement the Two-Step approach, which estimates random effects separately, and a classic Time Dependent Covariate Model. We use simulation studies to assess bias, accuracy, and coverage probabilities for the estimates of the link parameter that connects the longitudinal measures to survival times.
Results: Simulation results demonstrate that the Two-Step approach performed best at estimating the link parameter when variability in the longitudinal measure is low but is somewhat biased downwards when the variability is high. Bayesian semi-parametric and maximum likelihood joint methods yield higher link parameter estimates with low and high variability in the longitudinal measure. The Time Dependent Covariate method resulted in consistent underestimation of the link parameter. We illustrate these methods using data from the Framingham Heart Study in which lipid measurements and Myocardial Infarction data were collected over a period of 26 years.
Conclusions: Traditional methods for modeling longitudinal and survival data, such as the time dependent covariate method, that use the observed longitudinal data, tend to provide downwardly biased estimates. The two-step approach and joint models provide better estimates, although a comparison of these methods may depend on the underlying residual variance.
Figure 1
The full text of this article is available to read as a PDF.
This is a list of supplementary files associated with this preprint. Click to download.
- Tables.docx
Tables 1-6
- SupplementalFigures.docx
S1: Type I Errors for Link (Exponential and Weibull Distribution, N = 100) S2: Estimates and Confidence Intervals for Link (Weibull Distribution, N = 100) S3: Estimates and Confidence Intervals for Age (Weibull Distribution, N = 100) S4: Estimates and Confidence Intervals for Sex (Weibull Distribution, N = 100) S5: Bayesian Semi-Parametric Joint Modeling Exact Prior Distributions
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