Source: Braz. J. Probab. Stat.
Volume 26, Number 3
Long-term survival models have historically been considered for analyzing time-to-event data with long-term survivors fraction. However, situations in which a fraction (1 − p) of systems is subject to failure from independent competing causes of failure, while the remaining proportion p is cured or has not presented the event of interest during the time period of the study, have not been fully considered in the literature. In order to accommodate such situations, we present in this paper a new long-term survival model. Maximum likelihood estimation procedure is discussed as well as interval estimation and hypothesis tests. A real dataset illustrates the methodology.
Box-Steffensmeier, J. M. and Jones, B. S. (2004). Event History Modeling: A Guide for Social Scientists. Cambridge/New York: Cambridge Univ. Press.
Chan, V. V. and Meeker, W. Q. (1998). A competing-risk limited failure population model for product failure times. Technical report, Dept. Statistics, Iowa State Univ., Ames, IA.
Chan, V. V. and Meeker, W. Q. (1999). A Failure-time model for infant-mortality and wearout failure modes. IEEE Transactions on Reliability 48, 377–387.
Chao, E. C. (1998). Gibbs sampling for long-term survival data with competing risks. Biometrics 54, 350–366.
Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. London: Chapman & Hall.
Mathematical Reviews (MathSciNet): MR370837
David, H. A. and Moeschberger, M. L. (1978). The Theory of Competing Risks. Griffin’s Statistical Monograph Series 39. New York: Macmillan.
Mathematical Reviews (MathSciNet): MR592960
Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Applications. Cambridge/New York: Cambridge Univ. Press.
Ghitany, M. E. and Maller, R. A. (1992). Asymptotic results for exponential mixture models with long-term survivors. Statistics 23, 321–336.
Gordon, S. C. (2002). Stochastic dependence in competing risks. American Journal of Political Science 46, 200–217.
Hall, P. P. (1986). On the bootstrap and confidence intervals. The Annals of Statistics 14, 1431–1452.
Mathematical Reviews (MathSciNet): MR868310
Ihaka, R. R. and Gentleman, R. R. (1996). R: A language for data analysis and graphics. Journal of Computational and Graphical Statistics 5, 299–314.
Jasra, A. A., Holmes, C. C. and Stephens, D. A. (2005). Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Statistical Science 20, 50–67.
Jeng, S. L. and Meeker, W. Q. (1999). Comparisons of approximate confidence interval procedures for type I censored data. Scandinavian Journal of Statistics 20, 1–15.
Lagakos, S. W., Sommer, C. J. and Zelen, M. M. (1978). Semi-Markov models for partially censored data. Biometrika 65, 311–317.
Mathematical Reviews (MathSciNet): MR513932
Larson, M. G. and Dinse, G. E. (1985). A mixture model for the regression analysis of competing risks data. Journal of the Royal Statistical Society, Ser. C: Applied Statistics 34, 201–211.
Mathematical Reviews (MathSciNet): MR827668
Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data. New York: Wiley.
Mathematical Reviews (MathSciNet): MR640866
Louzada-Neto, F. F. (1998). Hazard models for lifetime data. Ph.D. thesis, Oxford Univ.
Louzada-Neto, F. F. (1999). Polyhazard regression models for lifetime data. Biometrics 55, 1281–1285.
Maller, R. A. and Zhou, X. X. (1996). Survival Analysis With Long-Term Survivors. Chichester: Wiley.
Peng, Y. Y., Dear, K. B. G. and Denham, J. W. (1998). A generalized F mixture model for cure rate estimation. Statistics in Medicine 17, 813–830.
SAS (2010). The NLP Procedure, SAS/OR® 9.22 User’s Guide: Mathematical Programming. Cary, NC: SAS Institute Inc.
Yamaguchi, K. K. (1992). Accelerated failure-time regression models with a regression model of surviving fraction: An application to the analysis of permanent employment in Japan. Journal of the American Statistical Association 87, 284–292.