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Lifetime Data Analysis

, Volume 23, Issue 3, pp 495–515 | Cite as

Generalized accelerated failure time spatial frailty model for arbitrarily censored data

  • Haiming Zhou
  • Timothy Hanson
  • Jiajia Zhang
Article

Abstract

Flexible incorporation of both geographical patterning and risk effects in cancer survival models is becoming increasingly important, due in part to the recent availability of large cancer registries. Most spatial survival models stochastically order survival curves from different subpopulations. However, it is common for survival curves from two subpopulations to cross in epidemiological cancer studies and thus interpretable standard survival models can not be used without some modification. Common fixes are the inclusion of time-varying regression effects in the proportional hazards model or fully nonparametric modeling, either of which destroys any easy interpretability from the fitted model. To address this issue, we develop a generalized accelerated failure time model which allows stratification on continuous or categorical covariates, as well as providing per-variable tests for whether stratification is necessary via novel approximate Bayes factors. The model is interpretable in terms of how median survival changes and is able to capture crossing survival curves in the presence of spatial correlation. A detailed Markov chain Monte Carlo algorithm is presented for posterior inference and a freely available function frailtyGAFT is provided to fit the model in the R package spBayesSurv. We apply our approach to a subset of the prostate cancer data gathered for Louisiana by the surveillance, epidemiology, and end results program of the National Cancer Institute.

Keywords

Interval-censored data Heteroscedastic survival Linear dependent tailfree process Spatial data Stratified AFT model 

Notes

Acknowledgments

This work was supported by NCI grant 5R03CA176739. The authors would like to thank the editor, the associate editor, and the two referees for their valuable comments, which led to great improvements to the paper.

Supplementary material

10985_2016_9361_MOESM1_ESM.pdf (1.1 mb)
Supplementary material 1 (pdf 1122 KB)

References

  1. Banerjee S, Carlin BP (2003) Semiparametric spatio-temporal frailty modeling. Environmetrics 14(5):523–535CrossRefGoogle Scholar
  2. Banerjee S, Dey DK (2005) Semiparametric proportional odds models for spatially correlated survival data. Lifetime Data Anal 11(2):175–191MathSciNetCrossRefMATHGoogle Scholar
  3. Banerjee S, Wall MM, Carlin BP (2003) Frailty modeling for spatially correlated survival data, with application to infant mortality in Minnesota. Biostatistics 4(1):123–142CrossRefMATHGoogle Scholar
  4. Besag J (1974) Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc 36(2):192–236MathSciNetMATHGoogle Scholar
  5. Bouliotis G, Billingham L (2011) Crossing survival curves: alternatives to the log-rank test. Trials 12(Suppl 1):A137CrossRefGoogle Scholar
  6. Chiou SH, Kang S, Yan J (2015) Semiparametric accelerated failure time modeling for clustered failure times from stratified sampling. J Am Stat Assoc 110:621–629MathSciNetCrossRefGoogle Scholar
  7. Christensen R, Johnson W (1988) Modeling accelerated failure time with a Dirichlet process. Biometrika 75(4):693–704MathSciNetCrossRefMATHGoogle Scholar
  8. Cox DR (1975) Partial likelihood. Biometrika 62(2):269–276MathSciNetCrossRefMATHGoogle Scholar
  9. De Iorio M, Johnson WO, Müller P, Rosner GL (2009) Bayesian nonparametric nonproportional hazards survival modeling. Biometrics 65(3):762–771MathSciNetCrossRefMATHGoogle Scholar
  10. Dickey JM (1971) The weighted likelihood ratio, linear hypotheses on normal location parameters. Ann Math Stat 42(1):204–223MathSciNetCrossRefMATHGoogle Scholar
  11. Gamerman D (1997) Sampling from the posterior distribution in generalized linear mixed models. Stat Comput 7(1):57–68CrossRefGoogle Scholar
  12. Geisser S, Eddy WF (1979) A predictive approach to model selection. J Am Stat Assoc 74(365):153–160MathSciNetCrossRefMATHGoogle Scholar
  13. Gelfand AE, Dey DK (1994) Bayesian model choice: asymptotics and exact calculations. J R Stat Soc 56(3):501–514MathSciNetMATHGoogle Scholar
  14. Gelfand AE, Vounatsou P (2003) Proper multivariate conditional autoregressive models for spatial data analysis. Biostatistics 4(1):11–15CrossRefMATHGoogle Scholar
  15. Haario H, Saksman E, Tamminen J (2001) An adaptive Metropolis algorithm. Bernoulli 7(2):223–242MathSciNetCrossRefMATHGoogle Scholar
  16. Hanson T, Johnson WO (2002) Modeling regression error with a mixture of Polya trees. J Am Stat Assoc 97(460):1020–1033MathSciNetCrossRefMATHGoogle Scholar
  17. Hanson T, Johnson WO (2004) A Bayesian semiparametric AFT model for interval-censored data. J Comput Gr Stat 13(2):341–361MathSciNetCrossRefGoogle Scholar
  18. 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 57(2):207–225CrossRefMATHGoogle Scholar
  19. Hanson TE (2006) Inference for mixtures of finite Polya tree models. J Am Stat Assoc 101(476):1548–1565MathSciNetCrossRefMATHGoogle Scholar
  20. Hanson TE, Jara A (2013) Surviving fully Bayesian nonparametricregression models. In: Bayesian theory and applications. Oxford University Press, Oxford, pp 592–615Google Scholar
  21. Hanson TE, Jara A, Zhao L et al (2012) A Bayesian semiparametric temporally-stratified proportional hazards model with spatial frailties. Bayesian Anal 7(1):147–188MathSciNetCrossRefMATHGoogle Scholar
  22. Henderson R, Shimakura S, Gorst D (2002) Modeling spatial variation in leukemia survival data. J Am Stat Assoc 97(460):965–972MathSciNetCrossRefMATHGoogle Scholar
  23. Hennerfeind A, Brezger A, Fahrmeir L (2006) Geoadditive survival models. J Am Stat Assoc 101(475):1065–1075MathSciNetCrossRefMATHGoogle Scholar
  24. Jara A, Hanson TE (2011) A class of mixtures of dependent tailfree processes. Biometrika 98(3):553–566MathSciNetCrossRefMATHGoogle Scholar
  25. Koenker R (2008) Censored quantile regression redux. J Stat Softw 27(6):1–25CrossRefGoogle Scholar
  26. Kottas A, Gelfand AE (2001) Bayesian semiparametric median regression modeling. J Am Stat Assoc 96(456):1458–1468CrossRefMATHGoogle Scholar
  27. Kuo L, Mallick B (1997) Bayesian semiparametric inference for the accelerated failure-time model. Can J Stat 25(4):457–472CrossRefMATHGoogle Scholar
  28. Li Y, Ryan L (2002) Modeling spatial survival data using semiparametric frailty models. Biometrics 58(2):287–297MathSciNetCrossRefMATHGoogle Scholar
  29. Logan BR, Klein JP, Zhang M-J (2008) Comparing treatments in the presence of crossing survival curves: an application to bone marrow transplantation. Biometrics 64(3):733–740MathSciNetCrossRefMATHGoogle Scholar
  30. Neal RM (2003) Slice sampling. Ann Stat 31(3):705–767MathSciNetCrossRefMATHGoogle Scholar
  31. Pang L, Lu W, Wang HJ (2015) Local Buckley-James estimation for heteroscedastic accelerated failure time model. Stat Sin 25(3):863–877MathSciNetMATHGoogle Scholar
  32. Portnoy S (2003) Censored regression quantiles. J Am Stat Assoc 98(464):1001–1012MathSciNetCrossRefMATHGoogle Scholar
  33. Raftery AE (1996) Hypothesis testing and model selection via posterior simulation. In: Markov Chain Monte Carlo in practice. Springer, New York, pp 163–187Google Scholar
  34. Robert C, Casella G (2005) Monte Carlo statistical methods. Springer, New YorkMATHGoogle Scholar
  35. Verdinelli I, Wasserman L (1995) Computing Bayes factors using a generalization of the Savage-Dickey density ratio. J Am Stat Assoc 90(430):614–618MathSciNetCrossRefMATHGoogle Scholar
  36. Walker SG, Mallick BK (1999) A Bayesian semiparametric accelerated failure time model. Biometrics 55(2):477–483MathSciNetCrossRefMATHGoogle Scholar
  37. Wang S, Zhang J, Lawson AB (2012) A Bayesian normal mixture accelerated failure time spatial model and its application to prostate cancer. Stat Methods Med Res. doi: 10.1177/0962280212466189
  38. Zellner A (1983) Applications of Bayesian analysis in econometrics. Statistician 32(1/2):23–34CrossRefGoogle Scholar
  39. Zhang J, Lawson AB (2011) Bayesian parametric accelerated failure time spatial model and its application to prostate cancer. J Appl Stat 38(3):591–603MathSciNetCrossRefGoogle Scholar
  40. Zhao L, Hanson TE (2011) Spatially dependent Polya tree modeling for survival data. Biometrics 67(2):391–403MathSciNetCrossRefMATHGoogle Scholar
  41. Zhao L, Hanson TE, Carlin BP (2009) Mixtures of Polya trees for flexible spatial frailty survival modelling. Biometrika 96(2):263–276MathSciNetCrossRefMATHGoogle Scholar
  42. Zhou H, Hanson T, Jara A, Zhang J (2015) Modeling county level breast cancer survival data using a covariate-adjusted frailty proportional hazards model. Ann Appl Stat 9(1):43–68MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Division of StatisticsNorthern Illinois UniversityDeKalbUSA
  2. 2.Department of StatisticsUniversity of South CarolinaColumbiaUSA
  3. 3.Department of Epidemiology and BiostatisticsUniversity of South CarolinaColumbiaUSA

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