Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T04:12:46.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Frailty models based on Lévy processes

Part of: Survival analysis and censored data Probability theory on algebraic and topological structures Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Håkon K. Gjessing ,
Odd O. Aalen  and
Nils Lid Hjort
Håkon K. Gjessing*
Affiliation:
University of Oslo and Norwegian Institute of Public Health
Odd O. Aalen*
Affiliation:
University of Oslo
Nils Lid Hjort*
Affiliation:
University of Oslo
*
∗ Postal address: Norwegian Institute of Public Health, PO Box 4404, Nydalen, N-0403 Oslo, Norway. Email address: [email protected]
∗∗ Postal address: Section of Medical Statistics, University of Oslo, PO Box 1122, Blindern, N-0317 Oslo, Norway.
∗∗∗ Postal address: Department of Mathematics, University of Oslo, PO Box 1053, Blindern, N-0316 Oslo, Norway.
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalizing the standard frailty models of survival analysis, we propose to model frailty as a weighted Lévy process. Hence, the frailty of an individual is not a fixed quantity, but develops over time. Formulae for the population hazard and survival functions are derived. The power variance function Lévy process is a prominent example. In many cases, notably for compound Poisson processes, quasi-stationary distributions of survivors may arise. Quasi-stationarity implies limiting population hazard rates that are constant, in spite of the continual increase of the individual hazards. A brief discussion is given of the biological relevance of this finding.

Keywords

Frailtyquasi-stationarycompound Poisson processcumulative damage processLévy processhazard processpower variance function distribution

MSC classification

Primary: 62N01: Censored data models
Secondary: 60G51: Processes with independent increments; L'evy processes 60B10: Convergence of probability measures
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 532 - 550
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aalen, O. O. (1992). Modelling heterogeneity in survival analysis by the compound Poisson distribution. Ann. Appl. Prob. 2, 951–972.Google Scholar
[2] Aalen, O. O. (1994). Effects of frailty in survival analysis. Statist. Meth. Med. Res. 3, 227–243.Google Scholar
[3] Aalen, O. O. and Gjessing, H. K. (2001). Understanding the shape of the hazard rate: a process point of view. Statist. Sci. 16, 1–22.CrossRefGoogle Scholar
[4] Aalen, O. O. and Hjort, N. L. (2002). Frailty models that yield proportional hazards. Statist. Prob. Lett. 58, 335–342.Google Scholar
[5] Aalen, O. O. and Tretli, S. (1999). Analyzing incidence of testis cancer by means of a frailty model. Cancer Causes Control 10, 285–292.Google Scholar
[6] Andersen, P. K., Borgan, Ø., Gill, R. and Keiding, N. (1994). Statistical Models Based on Counting Processes. Springer, New York.Google Scholar
[7] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Statist. Soc. B 63, 167–241.Google Scholar
[8] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[9] Clarke, C., Lumsden, C. J. and McInnes, R. R. (2001). Inherited neurodegenerative diseases: the one-hit model of neurodegeneration. Human Molecular Genetics 10, 2269–2275.CrossRefGoogle ScholarPubMed
[10] Clarke, C. et al., (2000). A one-hit model of cell death in inherited neuronal degenerations. Nature 406, 195–199.Google Scholar
[11] Hjort, N. L. (2003). Topics in non-parametric Bayesian statistics (with discussion). In Highly Structured Stochastic Systems, eds Green, P. J., Hjort, N. L. and Richardson, S., Oxford University Press, pp. 455–478.Google Scholar
[12] Hougaard, P. (1986). Survival models for heterogeneous populations derived from stable distributions. Biometrika 73, 387–396. (Correction: 75 (1998), 395.).Google Scholar
[13] Hougaard, P. (2000). Analysis of Multivariate Survival Data. Springer, New York.Google Scholar
[14] Kebir, Y. (1991). On hazard rate processes. Naval Res. Logistics 38, 865–876.Google Scholar
[15] Lebedev, N. N. (1972). Special Functions and Their Applications. Dover Publications, New York.Google Scholar
[16] Lo, C.-C. et al., (2002). Dynamics of sleep–wake transitions during sleep. Europhys. Lett. 57, 625–631.Google Scholar
[17] Singpurwalla, N. D. (1995). Survival in dynamic environments. Statist. Sci. 10, 86–103.Google Scholar
[18] Slate, E. H. and Turnbull, B. W. (2000). Statistical models for longitudinal biomarkers of disease onset. Statist. Med. 19, 617–637.Google Scholar
[19] Triarhou, L. C. (1998). Rate of neuronal fallout in a transsynaptic cerebellar model. Brain Res. Bull. 47, 219–222.Google Scholar
[20] Yashin, A. I. and Manton, K. G. (1997). Effects of unobserved and partially observed covariate processes on system failure: a review of models and estimation strategies. Statist. Sci. 12, 20–34.Google Scholar