Skip to main content
SpringerLink
Log in

Survival Analysis

  • Chapter
Bayesian Nonparametric Data Analysis

Part of the book series: Springer Series in Statistics ((SSS))

  • 7129 Accesses

Abstract

Inference for event time data is one of the most traditional applications of nonparametric Bayesian inference. For survival data, especially in biomedical applications, it is natural to focus on inference for detailed features of the survival function rather than only summaries like mean and variance. We extensively discuss semi- and nonparametric Bayesian methods for survival regression. Inference for such data has been traditionally dominated by the proportional hazards model. We review in detail nonparametric Bayesian alternatives which we introduce as natural generalizations of a parametric accelerated failure time model. We conclude with a discussion of three case studies.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aalen OO (1980) A model for nonparametric regression analysis of counting processes. In: Lecture notes in statistics, vol 2. Springer, New York, pp 1–25

    Google Scholar 

  • Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10(4):1100–1120

    Article  MATH  MathSciNet  Google Scholar 

  • Arjas E, Gasbarra D (1994) Nonparametric Bayesian inference from right censored survival data, using the Gibbs sampler. Stat Sin 4:505–524

    MATH  MathSciNet  Google Scholar 

  • Banerjee S, Dey DK (2005) Semi-parametric proportional odds models for spatially correlated survival data. Lifetime Data Anal 11:175–191

    Article  MATH  MathSciNet  Google Scholar 

  • Banerjee S, Carlin BP, Gelfand AE (2004) Hierarchical modeling and analysis for spatial data. Chapman & Hall/CRC, Boca Raton

    MATH  Google Scholar 

  • Beadle G, Harris J, Silver B, Botnick L, Hellman S (1984) Cosmetic results following primary radiation therapy for early breast cancer. Cancer 54:2911–2918

    Article  Google Scholar 

  • Belitz C, Brezger A, Kneib T, Lang S (2009) BayesX—software for Bayesian inference in structured additive regression models. Version 2.00. Available from http://www.stat.uni-muenchen.de/~bayesx

  • Carlin BP, Hodges JS (1999) Hierarchical proportional hazards regression models for highly stratified data. Biometrics 55:1162–1170

    Article  MATH  Google Scholar 

  • Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Graph Stat 8(1):69–92

    MathSciNet  Google Scholar 

  • Chen Y, Hanson T, Zhang J (2014) Accelerated hazards model based on parametric families generalized with Bernstein polynomials. Biometrics 70:192–201

    Article  MATH  MathSciNet  Google Scholar 

  • Chen YQ, Wang MC (2000) Analysis of accelerated hazards models. J Am Stat Assoc 95:608–618

    Article  MATH  Google Scholar 

  • Christensen R, Johnson WO (1988) Modeling accelerated failure time with a Dirichlet process. Biometrika 75:693–704

    Article  MATH  MathSciNet  Google Scholar 

  • Chung Y, Dunson DB (2009) Nonparametric Bayes conditional distribution modeling with variable selection. J Am Stat Assoc 104:1646–1660

    Article  MATH  MathSciNet  Google Scholar 

  • Cox DR (1972) Regression models and life-tables (with discussion). J R Stat Soc Ser B 34:187–220

    MATH  Google Scholar 

  • Cox DR (1975) Partial likelihood. Biometrika 62(2):269–276

    Article  MATH  MathSciNet  Google Scholar 

  • Damien P, Laud P, Smith A (1996) Implementation of Bayesian nonparametric inference using beta processes. Scand J Stat 23:27–36

    MATH  MathSciNet  Google Scholar 

  • De Iorio M, Johnson WO, Müller P, Rosner GL (2009) Bayesian nonparametric non-proportional hazards survival modelling. Biometrics 65:762–771

    Article  MATH  MathSciNet  Google Scholar 

  • Doksum K (1974) Tailfree and neutral random probabilities and their posterior distributions. Ann Probab 2:183–201

    Article  MATH  MathSciNet  Google Scholar 

  • Dykstra RL, Laud P (1981) A Bayesian nonparametric approach to reliability. Ann Stat 9:356–367

    Article  MATH  MathSciNet  Google Scholar 

  • Ferguson TS (1973) A Bayesian analysis of some nonparametric problems. Ann Stat 1:209–230

    Article  MATH  MathSciNet  Google Scholar 

  • Ferguson TS (1974) Prior distribution on the spaces of probability measures. Ann Stat 2:615–629

    Article  MATH  MathSciNet  Google Scholar 

  • Ferguson TS, Klass MJ (1972) A representation of independent increment processes without Gaussian components. Ann Math Stat 43:1634–1643

    Article  MATH  MathSciNet  Google Scholar 

  • Ferguson TS, Phadia EG (1979) Bayesian nonparametric estimation based on censored data. Ann Stat 7:163–186

    Article  MATH  MathSciNet  Google Scholar 

  • Gamerman D (1991) Dynamic Bayesian models for survival data. Appl Stat 40:63–79

    Article  MATH  Google Scholar 

  • Geisser S, Eddy W (1979) A predictive approach to model selection. J Am Stat Assoc 74:153–160

    Article  MATH  MathSciNet  Google Scholar 

  • Gelfand AE, Mallick BK (1995) Bayesian analysis of proportional hazards models built from monotone functions. Biometrics 51:843–852

    Article  MATH  Google Scholar 

  • Gray RJ (1994) A Bayesian analysis of institutional effects in a multicenter cancer clinical trial. Biometrics 50:244–253

    Article  MATH  Google Scholar 

  • Hanson T, Johnson WO (2002) Modeling regression error with a mixture of Polya trees. J Am Stat Assoc 97:1020–1033

    Article  MATH  MathSciNet  Google Scholar 

  • Hanson T, Johnson WO (2004) A Bayesian semiparametric AFT model for interval-censored data. J Comput Graph Stat 13:341–361

    Article  MathSciNet  Google Scholar 

  • Hanson T, Kottas A, Branscum A (2008) Modelling stochastic order in the analysis of receiver operating characteristic data: Bayesian nonparametric approaches. J R Stat Soc Ser C 57:207–225

    Article  MATH  MathSciNet  Google Scholar 

  • Hanson TE (2006a) Inference for mixtures of finite Polya tree models. J Am Stat Assoc 101(476):1548–1565

    Article  MATH  MathSciNet  Google Scholar 

  • Hanson TE (2006b) Modeling censored lifetime data using a mixture of gammas baseline. Bayesian Anal 1:575–594

    Article  MathSciNet  Google Scholar 

  • Hanson TE, Yang M (2007) Bayesian semiparametric proportional odds models. Biometrics 63(1):88–95

    Article  MATH  MathSciNet  Google Scholar 

  • Hanson TE, Branscum A, Johnson WO (2005) Bayesian nonparametric modeling and data analysis: an introduction. In: Dey D, Rao C (eds) Bayesian thinking: modeling and computation. Handbook of statistics, vol 25. Elsevier, Amsterdam, pp 245–278

    Chapter  Google Scholar 

  • Hanson TE, Branscum A, Johnson WO (2011) Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches. Lifetime Data Anal 17:3–28

    Article  MathSciNet  Google Scholar 

  • Hennerfeind A, Brezger A, Fahrmeir L (2006) Geoadditive survival models. J Am Stat Assoc 101(475):1065–1075

    Article  MATH  MathSciNet  Google Scholar 

  • Hjort NL (1990) Nonparametric Bayes estimators based on beta processes in models for life history data. Ann Stat 18:1259–1294

    Article  MATH  MathSciNet  Google Scholar 

  • Hutton JL, Monaghan PF (2002) Choice of parametric accelerated life and proportional hazards models for survival data: asymptotic results. Lifetime Data Anal 8:375–393

    Article  MATH  MathSciNet  Google Scholar 

  • Ibrahim JG, Chen MH, Sinha D (2001) Bayesian survival analysis. Springer, New York

    Book  MATH  Google Scholar 

  • Jara A, Hanson TE (2011) A class of mixtures of dependent tailfree processes. Biometrika 98:553–566

    Article  MATH  MathSciNet  Google Scholar 

  • Jara A, Lesaffre E, De Iorio M, Quintana FA (2010) Bayesian semiparametric inference for multivariate doubly-interval-censored data. Ann Appl Stat 4:2126–2149

    Article  MATH  MathSciNet  Google Scholar 

  • Jara A, Hanson TE, Quintana FA, Müller P, Rosner GL (2011) DPpackage: Bayesian semi- and nonparametric modeling in R. J Stat Softw 40(5):1–30

    Google Scholar 

  • Johnson WO, Christensen R (1989) Nonparametric Bayesian analysis of the accelerated failure time model. Stat Probab Lett 8:179–184

    Article  MATH  MathSciNet  Google Scholar 

  • Kalbfleisch JD (1978) Nonparametric Bayesian analysis of survival time data. J R Stat Soc Ser B Methodol 40:214–221

    MATH  MathSciNet  Google Scholar 

  • Kay R, Kinnersley N (2002) On the use of the accelerated failure time model as an alternative to the proportional hazards model in the treatment of time to event data: a case study in influenza. Drug Inform J 36:571–579

    Google Scholar 

  • Kneib T, Fahrmeir L (2007) A mixed model approach for geoadditive hazard regression. Scand J Stat 34(1):207–228

    Article  MATH  MathSciNet  Google Scholar 

  • Komárek A, Lesaffre E, Legrand C (2007) Baseline and treatment effect heterogeneity for survival times between centers using a random effects accelerated failure time model with flexible error distribution. Stat Med 26:5457–5472

    Article  MathSciNet  Google Scholar 

  • Kottas A, Gelfand AE (2001) Bayesian semiparametric median regression modeling. J Am Stat Assoc 96:1458–1468

    Article  MATH  MathSciNet  Google Scholar 

  • Kuo L, Mallick B (1997) Bayesian semiparametric inference for the accelerated failure-time model. Can J Stat 25:457–472

    Article  MATH  Google Scholar 

  • Li L, Hanson T, Zhang J (2015) Spatial extended hazard model with application to prostate cancer survival. Biometrics (in press)

    Google Scholar 

  • Maksymiuk AW, Jett JR, Earle JD, Su JQ, Diegert FA, Mailliard JA, Kardinal CG, Krook JE, Veeder MH, Wiesenfeld M, Tschetter, LK, Levitt R (1994) Sequencing and schedule effects of cisplatin plus etoposide in small cell lung cancer results of a north central cancer treatment group randomized clinical trial. J Clin Oncol 12:70–76

    Google Scholar 

  • Müller P, Erkanli A, West M (1996) Bayesian curve fitting using multivariate normal mixtures. Biometrika 83:67–79

    Article  MATH  MathSciNet  Google Scholar 

  • Nieto-Barajas L, Walker SG (2002) Markov beta and gamma processes for modelling hazard rates. Scand J Stat 29:413–424

    Article  MATH  MathSciNet  Google Scholar 

  • Orbe J, Ferreira E, Núñez Antón V (2002) Comparing proportional hazards and accelerated failure time models for survival analysis. Stat Med 21(22):3493–3510

    Article  Google Scholar 

  • Portnoy S (2003) Censored regression quantiles. J Am Stat Assoc 98:1001–1012

    Article  MATH  MathSciNet  Google Scholar 

  • Reid N (1994) A conversation with Sir David Cox. Stat Sci 9:439–455

    Article  MATH  Google Scholar 

  • Rosner GL (2005) Bayesian monitoring of clinical trials with failure-time endpoints. Biometrics 61:239–245

    Article  MathSciNet  Google Scholar 

  • Ryan T, Woodall W (2005) The most-cited statistical papers. J Appl Stat 32(5):461–474

    Article  MATH  MathSciNet  Google Scholar 

  • Sinha D, Dey DK (1997) Semiparametric Bayesian analysis of survival data. J Am Stat Assoc 92:1195–1212

    Article  MATH  MathSciNet  Google Scholar 

  • Sinha D, McHenry MB, Lipsitz SR, Ghosh M (2009) Empirical Bayes estimation for additive hazards regression models. Biometrika 96(3):545–558

    Article  MATH  MathSciNet  Google Scholar 

  • Somoza JL (1980) Illustrative analysis: infant and child mortality in Colombia. World Fertility Survey Scientific Reports 10

    Google Scholar 

  • Susarla V, Van Ryzin J (1976) Nonparametric Bayesian estimation of survival curves from incomplete observations. J Am Stat Assoc 71:897–902

    Article  MATH  Google Scholar 

  • Therneau TM, Grambsch PM (2000) Modeling survival data: extending the Cox model. Springer, New York

    Book  Google Scholar 

  • Walker SG, Damien P (2000) Representation of Lévy processes without Gaussian components. Biometrika 87:447–483

    Article  MathSciNet  Google Scholar 

  • Walker SG, Mallick BK (1997) Hierarchical generalized linear models and frailty models with Bayesian nonparametric mixing. J R Stat Soc Ser B 59:845–860

    Article  MATH  MathSciNet  Google Scholar 

  • Walker SG, Mallick BK (1999) A Bayesian semiparametric accelerated failure time model. Biometrics 55(2):477–483

    Article  MATH  MathSciNet  Google Scholar 

  • Walker SG, Damien P, Laud PW, Smith AFM (1999) Bayesian nonparametric inference for random distributions and related functions (with discussion). J R Stat Soc B 61:485–527

    Article  MATH  MathSciNet  Google Scholar 

  • Wang L, Dunson DB (2011) Semiparametric Bayes’ proportional odds models for current status data with underreporting. Biometrics 67(3):1111–1118

    Article  MATH  MathSciNet  Google Scholar 

  • Yang S (1999) Censored median regression using weighted empirical survival and hazard functions. J Am Stat Assoc 94:137–145

    Article  MATH  Google Scholar 

  • Yin G, Ibrahim JG (2005) A class of Bayesian shared gamma frailty models with multivariate failure time data. Biometrics 61:208–216

    Article  MATH  MathSciNet  Google Scholar 

  • Ying Z, Jung SH, Wei LJ (1995) Survival analysis with median regression models. J Am Stat Assoc 90:178–184

    Article  MATH  MathSciNet  Google Scholar 

  • Zellner A (1983) Applications of Bayesian analysis in econometrics. Statistician 32:23–34

    Article  Google Scholar 

  • Zhang M, Davidian M (2008) “Smooth” semiparametric regression analysis for arbitrarily censored time-to-event data. Biometrics 64(2):567–576

    Article  MATH  MathSciNet  Google Scholar 

  • Zhao L, Hanson TE, Carlin BP (2009) Mixtures of Polya trees for flexible spatial frailty survival modelling. Biometrika 96(2):263–276

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, USA

    Peter Müller

  2. Departamento de Estadística, Pontificia Universidad Católica, Santiago, Chile

    Fernando Andrés Quintana & Alejandro Jara

  3. Department of Statistics, University of South Carolina, Columbia, SC, USA

    Tim Hanson

Authors
  1. Peter Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fernando Andrés Quintana
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alejandro Jara
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tim Hanson
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Müller, P., Quintana, F.A., Jara, A., Hanson, T. (2015). Survival Analysis. In: Bayesian Nonparametric Data Analysis. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-18968-0_6

Download citation

Publish with us

Policies and ethics

Search

Navigation