Abstract
Variance of parameter estimate in Cox’s proportional hazards model is based on asymptotic variance. When sample size is small, variance can be estimated by bootstrap method. However, if censoring rate in a survival data set is high, bootstrap method may fail to work properly. This is because bootstrap samples may be even more heavily censored due to repeated sampling of the censored observations. This paper proposes a random weighting method for variance estimation and confidence interval estimation for proportional hazards model. This method, unlike the bootstrap method, does not lead to more severe censoring than the original sample does. Its large sample properties are studied and the consistency and asymptotic normality are proved under mild conditions. Simulation studies show that the random weighting method is not as sensitive to heavy censoring as bootstrap method is and can produce good variance estimates or confidence intervals.
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References
Cox D R. Regression models and lifetables (with discussion). J Roy Statist Soc Ser B, 34: 187–220 (1972)
Cox D R. Partial likelihood. Biometrika, 62: 269–276 (1975)
Ansersen P K, Gill R D. Cox’s regression model for counting processes: a large sample study. Ann Statist, 10: 1100–1120 (1982)
Fleming T R, Harrington D P. Counting Processes and Survival Analysis. New York: John Wiley & Sons, Inc., 1991
Efron B. Censored data and the bootstrap. J Amer Statist Assoc, 76: 312–319 (1981)
Davison A C, Hinkley D V. Bootstrap Methods and Their Application. Cambridge: Cambridge University Press, 1997
James L F. A study of a class of weighted bootstraps for censored data. Ann Statist, 25: 1595–1621 (1997)
Efron B. Nonparametric standard errors and confidence intervals (with discussion). Canad J Statist, 9: 139–319 (1981)
Hjort N L. Bootstrapping Cox’s regression model. Technical report NSF-241, Department of Statistics, Stanford University, 1985
Zheng Z. Random weighting method (In Chinese). Acta Math Appl Sin, 10: 247–253 (1987)
Rao C R, Zhao L C. Approximation to the distribution of M-estimaties in linear models by randomly weighted bootsrap. Sankhyā, 54: 323–331 (1992)
Rubin D B. The Bayesian bootstrap. Ann Statist, 9: 130–134 (1981)
Weng C S. On a second-order asymptotic property of the Bayesian bootstrap. Ann Statist, 17: 705–710 (1989)
Kim Y, Lee J. Bayesian bootstrap for proportional hazards models. Ann Statist, 31: 1905–1922 (2003)
Ortega J M, Rheinboldt W G. Iterative Solutions of Nonlinear Equations in Several Variable. New York: Academic Press, 1970
van der Vaart A W, Wellner J A. Weak Convergence and Empirical Processes. New York: Springer-Verlag, 1996
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 10471136, 10671189), PhD Program Foundation of Ministry of Education of China and Foundations from the Chinese Academy of Sciences
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Cui, W., Li, K., Yang, Y. et al. Random weighting method for Cox’s proportional hazards model. Sci. China Ser. A-Math. 51, 1843–1854 (2008). https://doi.org/10.1007/s11425-007-0164-7
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DOI: https://doi.org/10.1007/s11425-007-0164-7