Taiwanese Journal of Mathematics

A Survey of Threshold Regression for Time-to-event Analysis and Applications

Mei-Ling Ting Lee

Full-text: Open access

Abstract

In analyzing time-to-event data, proportional hazards (PH) regression is an ubiquitous model used in many fields. PH regression, however, requires a strong assumption that is not always appropriate. Threshold regression (TR) is one of the alternative models. A first-hitting-time (FHT) survival model postulates a health status process for a patient that gradually declines until the patient dies when the health level first reaches a critical threshold. In this article, we review the development of threshold regression models and their applications.

Article information

Source
Taiwanese J. Math., Volume 23, Number 2 (2019), 293-305.

Dates
Received: 7 May 2018
Accepted: 20 December 2018
First available in Project Euclid: 25 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1548406820

Digital Object Identifier
doi:10.11650/tjm/190107

Mathematical Reviews number (MathSciNet)
MR3936001

Subjects
Primary: 92B15: General biostatistics [See also 62P10] 62P10: Applications to biology and medical sciences

Keywords
boundary Brownian motion first hitting time inverse Gaussian distribution running time stochastic process survival analysis Wiener process

Citation

Lee, Mei-Ling Ting. A Survey of Threshold Regression for Time-to-event Analysis and Applications. Taiwanese J. Math. 23 (2019), no. 2, 293--305. doi:10.11650/tjm/190107. https://projecteuclid.org/euclid.twjm/1548406820


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References

  • O. O. Aalen, Ø. Borgan and H. K. Gjessing, Survival and Event History Analysis: A Process Point of View, Statistics for Biology and Health, Springer, New York, 2008.
  • O. O. Aalen and H. K. Gjessing, Understanding the shape of the hazard rate: a process point of view, Statist. Sci. 16, (2001), no. 1, 1–22.
  • S. D. Aaron, T. Ramsay, K. Vandemheen and G. A. Whitmore, A threshold regression model for recurrent exacerbations in chronic obstructive pulmonary disease, J. Clin. Epidemiol. 63 (2010), no. 12, 1324–1331.
  • S. D. Aaron, A. L. Stephenson, D. W. Cameron and G. A. Whitmore, A statistical model to predict one-year risk of death in patients with cystic fibrosis, J. Clin. Epidemiol. 68 (2015), no. 11, 1336–1345.
  • J. Balka, A. F. Desmond and P. D. McNicholas, Review and implementation of cure models based on first hitting times for Wiener processes, Lifetime Data Anal. 15 (2009), no. 2, 147–176.
  • D. R. Cox and H. D. Miller, The Theory of Stochastic Processes, John Wiley & Sons, New York, 1965.
  • D. R. Cox and D. Oakes, Analysis of Survival Data, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1984.
  • X. He, G. A. Whitmore, G. Y. Loo, M. C. Hochberg and M.-L. T. Lee, A model for time to fracture with a shock stream superimposed on progressive degradation: the study of osteoporotic fractures, Stat. Med. 34 (2015), no. 4, 652–663.
  • M.-L. T. Lee, M. Chang and G. A. Whitmore, A threshold regression mixture model for assessing treatment efficacy in a multiple myeloma clinical trial, J. Biopharm. Statist. 18 (2008), no. 6, 1136–1149.
  • M.-L. T. Lee, V. DeGruttola and D. Schoenfeld, A model for markers and latent health status, J. R. Stat. Soc. Ser. B Stat. Methodol. 62 (2000), no. 4, 747–762.
  • M.-L. T. Lee and G. A. Whitmore, Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary, Statist. Sci. 21 (2006), no. 4, 501–513.
  • ––––, Proportional hazards and threshold regression: their theoretical and practical connections, Lifetime Data Anal. 16 (2010), no. 2, 196–214.
  • ––––, Practical applications of a family of shock-degradation failure models, in: Statistical Modeling for Degradation Data, 211–229, ICSA Book Ser. Stat., Springer, Singapore, 2017.
  • M.-L. T. Lee, G. A. Whitmore, F. Laden, J. E. Hart and E. Garshick, Assessing lung cancer risk in railroad workers using a first hitting time regression model, Environmetrics 15 (2004), no. 5, 501–512.
  • ––––, A case-control study relating railroad worker mortality to diesel exhaust exposure using a threshold regression model, J. Statist. Plann. Inference 139 (2009), no. 5, 1633–1642.
  • M.-L. T. Lee, G. A. Whitmore and B. A. Rosner, Threshold regression for survival data with time-varying covariates, Stat. Med. 29 (2010), no. 7-8, 896–905.
  • J. Li and M.-L. T. Lee, Analysis of failure time using threshold regression with semi-parametric varying coefficients, Stat. Neerl. 65 (2011), no. 2, 164–182.
  • C. M. Mulatya, A. C. McLain, B. Cai, J. W. Hardin and P. S. Albert, Estimating time to event characteristics via longitudinal threshold regression models–an application to cervical dilation progression, Stat. Med. 35 (2016), no. 24, 4368–4379.
  • M. L. Pennell, G. A. Whitmore and M.-L. T. Lee, Bayesian random-effects threshold regression with application to survival data with nonproportional hazards, Biostatistics 11 (2009), no. 1, 111–126.
  • D. Stogiannis, C. Caroni, C. E. Anagnostopoulos and I. K. Toumpoulis, Comparing first hitting time and proportional hazards regression models, J. Appl. Stat. 38 (2011), no. 7, 1483–1492.
  • C. L. William and C. Law, Threshold regression and first hitting time models, Research & Reviews: J. Stat. Math. Sci. 1 (2015), no. 1, 38–48.
  • G. A. Whitmore, A regression method for censored inverse-Gaussian data, Canad. J. Statist. 11 (1983), no. 4, 305–315.
  • G. A. Whitmore, M. J. Crowder and J. F. Lawless, Failure inference from a marker process based on a bivariate Wiener model, Lifetime Data Anal. 4 (1998), no. 3, 229–251.
  • G. A. Whitmore, T. Ramsay and S. D. Aaron, Recurrent first hitting times in Wiener diffusion under several observation schemes, Lifetime Data Analysis 18 (2012), no. 2, 157–176.
  • T. Xiao, G. A. Whitmore, X. He and M.-L. T. Lee, Threg: a new command to implement threshold regression model in STATA, The STATA Journal 12 (2012), 257–283.
  • ––––, The R package threg to implement threshold regression model, J. Stat. Softw. 66 (2015), no. 8, 16 pp.