Abstract
Proportional hazards (PH) regression is a standard methodology for analyzing survival and time-to-event data. The proportional hazards assumption of PH regression, however, is not always appropriate. In addition, PH regression focuses mainly on hazard ratios and thus does not offer many insights into underlying determinants of survival. These limitations have led statistical researchers to explore alternative methodologies. Threshold regression (TR) is one of these alternative methodologies (see Lee and Whitmore, Stat Sci 21:501–513, 2006, for a review). The connection between PH regression and TR has been examined in previous published work but the investigations have been limited in scope. In this article, we study the connections between these two regression methodologies in greater depth and show that PH regression is, for most purposes, a special case of TR. We show two methods of construction by which TR models can yield PH functions for survival times, one based on altering the TR time scale and the other based on varying the TR boundary. We discuss how to estimate the TR time scale and boundary, with or without the PH assumption. A case demonstration is used to highlight the greater understanding of scientific foundations that TR can offer in comparison to PH regression. Finally, we discuss the potential benefits of positioning PH regression within the first-hitting-time context of TR regression.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aalen OO, Gjessing HK (2001) Understanding the shape of the hazard rate: a process point of view. Stat Sci 16: 1–22
Aalen OO, Borgan O, Gjessing HK (2008) Survival and event history analysis: a process point of view (statistics for biology and health). Springer, New York
Balka J, Desmond AF, McNicholas PD (2009) Review and implementation of cure models based on first hitting times for Wiener processes. Lifetime Data Anal 15: 147–176
Beran R (1981) Nonparametric regression with randomly censored survival data. Technical Report. University of California, Berkeley, CA
Cox DR (1972) Regression models and life tables (with discussion). J R Stat Soc Ser B 34: 187–230
Cox DR, Oakes D (1984) Analysis of survival data. Chapman and Hall, London
Duchesne T, Lawless J (2000) Alternative time scales and failure time models. Lifetime Data Anal 6: 157–179
Duchesne T, Rosenthal JS (2003) On the collapsibility of lifetime regression models. Adv Appl Prob 35: 755–772
Klein JP, Moeschberger ML (2003) Survival analysis: techniques for censored and truncated data, 2nd edn. Springer-Verlag, New York
Kordonsky KB, Gertsbakh I (1997) Multiple time scales and the lifetime coefficient of variation: engineering applications. Lifetime Data Anal 3: 139–156
Lee M-LT (2009) Personal research webpage for TR and other software. http://sph.umd.edu/epib/faculty/mltlee/index.html
Lee M-LT, Whitmore GA (2006) Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary. Stat Sci 21: 501–513
Lee M-LT, Whitmore GA, Chang M (2008) A threshold regression mixture model for assessing treatment efficacy in a multiple myeloma clinical trial. J Biopharm Stat 18: 1136–1149
Lee M-LT, Whitmore GA, Laden F, Hart JE, Garshick E (2009) A case–control study relating railroad worker mortality to diesel exhaust exposure using a threshold regression model. J Stat Plan Infer 139: 1633–1642
Lee M-LT, Whitmore GA, Rosner B (2009b) Benefits of threshold regression: a case-study comparison with Cox proportional hazards regression (submitted, under revision)
Lee M-LT, Whitmore GA, Rosner B (2009c) Threshold regression for survival data with time-varying covariates. Stat Med (accepted, in press)
Nahman NS Jr, Middendorf DF, Bay WH, McElligott R, Powell S, Anderson J (1992) Modification of the percutaneous approach to peritoneal dialysis catheter placement under peritoneoscopic visualization: Clinical results in 78 patients. J Am J Nephrol 3: 103–107
Oakes D (1995) Multiple time scales in survival analysis. Lifetime Data Anal 1: 7–18
Pennell ML, Whitmore GA, Lee M-LT (2009) Bayesian random effects threshold regression with application to survival data with nonproportional hazards. Biostatistics (accepted, in press)
Ross SM (1996) Stochastic processes, 2nd edn. Wiley, New York
Tong X, He X, Sun J, Lee M-LT (2008) Joint analysis of current status and marker data: an extension of a bivariate threshold model. Int J Biostat 4(1):Article 21
Whitmore GA (1984) Barrier estimation using first passage time data from Brownian motion. Working paper. McGill University, Montreal
Whitmore GA (1986) First passage time models for duration data—regression structures and competing risks. Statistician 35: 207–219
Whitmore GA, Su Y (2007) Modeling low birth weights using threshold regression: results for U.S. birth data. Lifetime Data Anal 13: 161–190
Yu Z, Tu W, Lee M-LT (2009) A semiparametric threshold regression analysis of sexually transmitted infections in adolescent women. Stat Med (accepted, in press)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lee, ML.T., Whitmore, G.A. Proportional hazards and threshold regression: their theoretical and practical connections. Lifetime Data Anal 16, 196–214 (2010). https://doi.org/10.1007/s10985-009-9138-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10985-009-9138-0