×

Nonparametric Bayesian survival analysis using mixtures of Weibull distributions. (English) Zbl 1079.62095

Summary: Bayesian nonparametric methods have been applied to survival analysis problems since the emergence of the area of Bayesian nonparametrics. However, the use of the flexible class of Dirichlet process mixture models has been rather limited in this context. This is, arguably, to a large extent due to the standard way of fitting such models that precludes full posterior inference for many functionals of interest in survival analysis applications. To overcome this difficulty, we provide a computational approach to obtain the posterior distribution of general functionals of a Dirichlet process mixture. We model the survival distribution employing a flexible Dirichlet process mixture, with a Weibull kernel, that yields rich inference for several important functionals. In the process, a method for hazard function estimation emerges. Methods for simulation-based model fitting, in the presence of censoring, and for prior specification are provided. We illustrate the modeling approach with simulated and real data.

MSC:

62N02 Estimation in survival analysis and censored data
62F15 Bayesian inference
62P10 Applications of statistics to biology and medical sciences; meta analysis
62G05 Nonparametric estimation
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Antoniadis, A.; Grégoire, G.; Nason, G., Density and hazard rate estimation for right-censored data by using wavelet methods, J. roy. statist. soc. ser. B, 61, 63-84, (1999) · Zbl 0915.62020
[2] Antoniak, C.E., Mixtures of Dirichlet processes with applications to nonparametric problems, Ann. statist., 2, 1152-1174, (1974) · Zbl 0335.60034
[3] Arjas, E.; Gasbarra, D., Nonparametric Bayesian inference from right censored survival data, using the Gibbs sampler, Statist. sinica, 4, 505-524, (1994) · Zbl 0823.62030
[4] Blackwell, D.; MacQueen, J.B., Ferguson distributions via Pólya urn schemes, Ann. statist., 1, 353-355, (1973) · Zbl 0276.62010
[5] Brunner, L.J., Bayesian nonparametric methods for data from a unimodal density, Statist. probab. lett., 14, 195-199, (1992) · Zbl 0806.62038
[6] Brunner, L.J.; Lo, A.Y., Bayes methods for a symmetric unimodal density and its mode, Ann. statist., 17, 1550-1566, (1989) · Zbl 0697.62003
[7] Bush, C.A.; MacEachern, S.N., A semiparametric Bayesian model for randomised block designs, Biometrika, 83, 275-285, (1996) · Zbl 0864.62052
[8] Damien, P.; Walker, S., A Bayesian non-parametric comparison of two treatments, Scand. J. statist., 29, 51-56, (2002) · Zbl 1017.62108
[9] Damien, P.; Laud, P.W.; Smith, A.F.M., Implementation of Bayesian non-parametric inference based on beta processes, Scand. J. statist., 23, 27-36, (1996) · Zbl 0888.62034
[10] Damien, P.; Wakefield, J.; Walker, S., Gibbs sampling for Bayesian non-conjugate and hierarchical models by using auxiliary variables, J. roy. statist. soc. ser. B, 61, 331-344, (1999) · Zbl 0913.62028
[11] Doss, H., Bayesian nonparametric estimation for incomplete data via successive substitution sampling, Ann. statist., 22, 1763-1786, (1994) · Zbl 0824.62027
[12] Dykstra, R.L.; Laud, P., A Bayesian nonparametric approach to reliability, Ann. statist., 9, 356-367, (1981) · Zbl 0469.62077
[13] Escobar, M.D., Estimating normal means with a Dirichlet process prior, J. amer. statist. assoc., 89, 268-277, (1994) · Zbl 0791.62039
[14] Escobar, M.D.; West, M., Bayesian density estimation and inference using mixtures, J. amer. statist. assoc., 90, 577-588, (1995) · Zbl 0826.62021
[15] Ferguson, T.S., A Bayesian analysis of some nonparametric problems, Ann. statist., 1, 209-230, (1973) · Zbl 0255.62037
[16] Ferguson, T.S., Prior distributions on spaces of probability measures, Ann. statist., 2, 615-629, (1974) · Zbl 0286.62008
[17] Ferguson, T.S., Bayesian density estimation by mixtures of normal distributions, (), 287-302
[18] Ferguson, T.S.; Phadia, E.G., Bayesian nonparametric estimation based on censored data, Ann. statist., 7, 163-186, (1979) · Zbl 0401.62031
[19] Gelfand, A.E.; Kottas, A., A computational approach for full nonparametric Bayesian inference under Dirichlet process mixture models, J. comput. graph. statist., 11, 289-305, (2002)
[20] Gelfand, A.E.; Mukhopadhyay, S., On nonparametric Bayesian inference for the distribution of a random sample, Canad. J. statist., 23, 411-420, (1995) · Zbl 0858.62028
[21] Haupt, G., Mansmann, U., 1995. Survcart: S and C code for classification and regression trees analysis with survival data (available from Statlib shar archive at http://lib.stat.cmu.edu/S/survcart).
[22] Hjort, N.L., Nonparametric Bayes estimators based on beta processes in models for life history data, Ann. statist., 18, 1259-1294, (1990) · Zbl 0711.62033
[23] Ibrahim, J.G.; Chen, M-H.; Sinha, D., Bayesian survival analysis, (2001), Springer New York · Zbl 0978.62091
[24] Kalbfleisch, J.D., Non-parametric Bayesian analysis of survival time data, J. roy. statist. soc. ser. B, 40, 214-221, (1978) · Zbl 0387.62030
[25] Kottas, A.; Gelfand, A.E., Bayesian semiparametric Median regression modeling, J. amer. statist. assoc., 96, 1458-1468, (2001) · Zbl 1051.62038
[26] Kuo, L.; Mallick, B., Bayesian semiparametric inference for the accelerated failure-time model, Canad. J. statist., 25, 457-472, (1997) · Zbl 0894.62033
[27] Lavine, M., Some aspects of polya tree distributions for statistical modelling, Ann. statist., 20, 1222-1235, (1992) · Zbl 0765.62005
[28] Lawless, J.F., Statistical models and methods for lifetime data, (1982), Wiley New York · Zbl 0541.62081
[29] Lo, A.Y., On a class of Bayesian nonparametric estimatesi. density estimates, Ann. statist., 12, 351-357, (1984) · Zbl 0557.62036
[30] MacEachern, S.N.; Müller, P., Estimating mixture of Dirichlet process models, J. comput. graph. statist., 7, 223-238, (1998)
[31] Merrick, J.R.W.; Soyer, R.; Mazzuchi, T.A., A Bayesian semiparametric analysis of the reliability and maintenance of machine tools, Technometrics, 45, 58-69, (2003)
[32] Muliere, P.; Walker, S., A Bayesian non-parametric approach to survival analysis using polya trees, Scand. J. statist., 24, 331-340, (1997) · Zbl 0888.62031
[33] Müller, P.; Erkanli, A.; West, M., Bayesian curve Fitting using multivariate normal mixtures, Biometrika, 83, 67-79, (1996) · Zbl 0865.62029
[34] Neal, R.M., Markov chain sampling methods for Dirichlet process mixture models, J. comput. graph. statist., 9, 249-265, (2000)
[35] Nieto-Barajas, L.E.; Walker, S.G., Markov beta and gamma processes for modelling hazard rates, Scand. J. statist., 29, 413-424, (2002) · Zbl 1036.62105
[36] Sethuraman, J., A constructive definition of Dirichlet priors, Statist. sinica, 4, 639-650, (1994) · Zbl 0823.62007
[37] Sethuraman, J.; Tiwari, R.C., Convergence of Dirichlet measures and the interpretation of their parameter, (), 305-315
[38] Susarla, V.; Van Ryzin, J., Nonparametric Bayesian estimation of survival curves from incomplete observations, J. amer. statist. assoc., 71, 897-902, (1976) · Zbl 0344.62036
[39] Walker, S.; Damien, P., A full Bayesian non-parametric analysis involving a neutral to the right process, Scand. J. statist., 25, 669-680, (1998) · Zbl 0927.62036
[40] Walker, S.; Muliere, P., Beta-stacy processes and a generalization of the Pólya-urn scheme, Ann. statist., 25, 1762-1780, (1997) · Zbl 0928.62067
[41] West, M.; Müller, P.; Escobar, M.D., Hierarchical priors and mixture models, with application in regression and density estimation, (), 363-386 · Zbl 0842.62001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.