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Abstract
Many models have been proposed that relate failure times and stochastic time-varying covariates. In some of these models, failure occurs when a particular observable marker crosses a threshold level. We are interested in the more difficult, and often more realistic, situation where failure is not related deterministically to an observable marker. In this case, joint models for marker evolution and failure tend to lead to complicated calculations for characteristics such as the marginal distribution of failure time or the joint distribution of failure time and marker value at failure. This paper presents a model based on a bivariate Wiener process in which one component represents the marker and the second, which is latent (unobservable), determines the failure time. In particular, failure occurs when the latent component crosses a threshold level. The model yields reasonably simple expressions for the characteristics mentioned above and is easy to fit to commonly occurring data that involve the marker value at the censoring time for surviving cases and the marker value and failure time for failing cases. Parametric and predictive inference are discussed, as well as model checking. An extension of the model permits the construction of a composite marker from several candidate markers that may be available. The methodology is demonstrated by a simulated example and a case application.
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References
Raj S. Chhikara and J. Leroy Folks,The Inverse Gaussian Distribution: Theory, Methodology and Applications, Marcel Dekker: New York, 1989.
D. R. Cox and H. D. Miller,The Theory of Stochastic Processes, Chapman and Hall: London, 1965.
K. Doksum and A. Hoyland, “Models for variable-stress accelerated testing experiments based on Wiener processes and the inverse Gaussian distribution,”Technometrics vol. 34 pp. 74–82, 1992.
J. W. Hogan and N. Laird, “Mixture models for the joint distribution of repeated measures and event times,”Statistics in Medicine vol. 16 pp. 239–257, 1997.
N. P. Jewell and J. D. Kalbfleisch, “Marker Models in Survival Analysis and Applications to Issues Associated with AIDS,” inAIDS Epidemiology: Methodological Issues, N. Jewell, K. Dietz and V. Farewell, eds., Birkhausen: Boston, 1992.
N. P. Jewell and J. D. Kalbfleisch, “Marker processes in survival analysis,”Lifetime Data Analysis vol. 2 pp. 15–29, 1996.
Jin Lu, “A Reliability Model Based on Degradation and Lifetime Data,” Ph.D. thesis, McGill University, Montreal, Canada, 1995.
C. J. Lu and W. Q. Meeker, “Using degradation measures to estimate time-to-failure distribution,”Technometrics vol. 35 pp. 161–174, 1993.
NAG,Library Manual Mark 13, Numerical Algorithms Group: Oxford, 1988.
S. Self and Y. Pawitan, “Modeling aMarker of Disease Progression and Onset of Disease,” inAIDS Epidemiology: Methodological Issues, N. Jewell, K. Dietz and V. Farewell, eds., Birkhausen: Boston, 1992.
M. Shi, J. M. G. Taylor and A. Munoz, “Models for residual time to AIDS,”Lifetime Data Analysis vol. 2 pp. 31–49, 1996.
N. D. Singpurwalla, “Survival in dynamic environments,”Statistical Science vol. 10 pp. 86–103, 1995.
G. A. Whitmore, “An inverse Gaussian model for labour turnover,”Journal of the Royal Statistical Society Series A, vol. 142 pp. 468–478, 1979.
G. A. Whitmore, “Estimating degradation by a Wiener diffusion process subject to measurement error,”Lifetime Data Analysis vol. 1 pp. 307–319, 1995.
G. A. Whitmore and Fred Schenkelberg, “Modelling accellerated degradation data using Wiener diffusion with a time scale transformation,”Lifetime Data Analysis vol. 3 pp. 1–19, 1997.
M. S. Wolfshon and A. A. Tsiatis, “Ajoint model for survival and longitudinal data measured with error,”Biometrics vol. 53 pp. 330–339, 1997.
A. I. Yashin and K. G. Manton, “Effects of unobserved and partially observed covariate processes on system failure: a review of models and estimation strategies,”Statistical Science vol. 12 pp. 20–34, 1997.
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McGill University, Montreal, Canada
G. A. Whitmore
University of Surrey, Guildford, U.K.
M. J. Crowder
University of Waterloo, Waterloo, Canada
J. F. Lawless
- G. A. Whitmore
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- M. J. Crowder
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- J. F. Lawless
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Whitmore, G.A., Crowder, M.J. & Lawless, J.F. Failure Inference From a Marker Process Based on a Bivariate Wiener Model.Lifetime Data Anal4, 229–251 (1998). https://doi.org/10.1023/A:1009617814586
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