Bayesian Analysis

Flexible Bayesian Survival Modeling with Semiparametric Time-Dependent and Shape-Restricted Covariate Effects

Thomas A. Murray, Brian P. Hobbs, Daniel J. Sargent, and Bradley P. Carlin

Full-text: Open access

Abstract

Presently, there are few options with available software to perform a fully Bayesian analysis of time-to-event data wherein the hazard is estimated semi- or non-parametrically. One option is the piecewise exponential model, which requires an often unrealistic assumption that the hazard is piecewise constant over time. The primary aim of this paper is to construct a tractable semiparametric alternative to the piecewise exponential model that assumes the hazard is continuous, and to provide modifiable, user-friendly software that allows the use of these methods in a variety of settings. To accomplish this aim, we use a novel model formulation for the log-hazard based on a low-rank thin plate linear spline that readily facilitates adjustment for covariates with time-dependent and proportional hazards effects, possibly subject to shape restrictions. We investigate the performance of our model choices via simulation. We then analyze colorectal cancer data from a clinical trial comparing the effectiveness of two novel treatment regimes relative to the standard of care for overall survival. We estimate a time-dependent hazard ratio for each novel regime relative to the standard of care while adjusting for the effect of aspartate transaminase, a biomarker of liver function, that is subject to a non-decreasing shape restriction.

Article information

Source
Bayesian Anal. Volume 11, Number 2 (2016), 381-402.

Dates
First available in Project Euclid: 14 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.ba/1431607819

Digital Object Identifier
doi:10.1214/15-BA954

Mathematical Reviews number (MathSciNet)
MR3471995

Zentralblatt MATH identifier
1357.62281

Subjects
Primary: 62F15: Bayesian inference 62N86: Fuzziness, and survival analysis and censored data
Secondary: 62F30: Inference under constraints

Keywords
Bayesian methods survival analysis semiparametric methods penalized splines shape-restricted effects time-dependent effects colorectal cancer

Citation

Murray, Thomas A.; Hobbs, Brian P.; Sargent, Daniel J.; Carlin, Bradley P. Flexible Bayesian Survival Modeling with Semiparametric Time-Dependent and Shape-Restricted Covariate Effects. Bayesian Anal. 11 (2016), no. 2, 381--402. doi:10.1214/15-BA954. https://projecteuclid.org/euclid.ba/1431607819


Export citation

References

  • Bender, R., Augustin, T., and Blettner, M. (2005). “Generating Survival Times to Simulate Cox Proportional Hazards Models.” Statistics in Medicine, 24(11): 1713–1723.
  • Brezger, A. and Steiner, W. (2004). “Monotonic regression Bayesian P-splines.” Journal of Business and Economic Statistics, 26(1): 90–104.
  • Cai, T., Hyndman, R., and Wand, M. (2002). “Mixed model-based hazard estimation.” Journal of Computational and Graphical Statistics, 11(4): 784–798.
  • Cox, D. R. (1975). “Partial likelihood.” Biometrika, 62(2): 269–276.
  • Crainiceanu, C. M., Ruppert, D., and Wand, M. P. (2005). “Bayesian Analysis for Penalized Spline Regression Using WinBUGS.” Journal of Statistical Software, 14(14): 1–24.
  • Dunson, D. B. (2005). “Bayesian semiparametric isotonic regression for count data.” Journal of the American Statistical Association, 100(470): 618–627.
  • Fahrmeir, L. and Hennerfeind, A. (2003). “Nonparametric Bayesian hazard rate models based on penalized splines.” Discussion paper // Sonderforschungsbereich 386 der Ludwig-Maximilians-Universität München 361.
  • Fahrmeir, L. and Lang, S. (2001). “Bayesian inference for generalized additive mixed models based on Markov random field priors.” Journal of the Royal Statistical Society: Series C (Applied Statistics), 50(2): 201–220.
  • Gamerman, D. (1991). “Dynamic Bayesian Models for Survival Data.” Applied Statistics, 40(1): 63–79.
  • Gelfand, A. E. and Mallick, B. K. (1995). “Bayesian analysis of proportional hazards models built from monotone functions.” Biometrics, 51: 843–852.
  • Gelman, A. (2006). “Prior distributions for variance parameters in hierarchical models.” Bayesian Analysis, 1: 1–19.
  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2014). Bayesian Data Analysis, 3rd Edition. Boca-Raton, FL: Chapman & Hall/CRC Press.
  • Goldberg, R. M., Sargent, D. J., Morton, R. F., Fuchs, C. S., Ramanathan, R. K., Williamson, S. K., Findlay, B. P., Pitot, H. C., and Alberts, S. R. (2004). “A randomized controlled trial of fluorouracil plus leucovorin, irinotecan, and oxaliplatin combinations in patients with previously untreated metastatic colorectal cancer.” Journal of Clinical Oncology, 22(1): 23–30.
  • Hennerfeind, A., Brezger, A., and Fahrmeir, L. (2006). “Geoadditive Survival Models.” Journal of the American Statistical Association, 101(475): 1065–1075.
  • Henschel, V., Engel, J., Hölzel, D., and Mansmann, U. (2009). “A semiparametric Bayesian proportional hazards model for interval censored data and frailty effects.” BMC Medical Research Methodology, 9(1): 1–15.
  • Ibrahim, J. G., Chen, M.-H., and Sinha, D. (2001). Bayesian Survival Analysis. New York: Springer.
  • Klein, J. P. and Moeschberger, M. L. (2003). Survival Analysis: Techniques for Censored and Truncated Data. New York, NY: Springer.
  • Lin, X., Cai, B., Wang, L., and Zhang, Z. (2014). “A Bayesian proportional hazards model for general interval-censored data.” Lifetime Data Analysis, 1–21. doi: http://dx.doi.org/10.1007/s10985-014-9305-9
  • Lunn, D., Spiegelhalter, D., Thomas, A., and Best, N. (2009). “The BUGS project: Evolution, critique and future directions.” Statistics in Medicine, 28(25): 3049–3067.
  • Müller, P. and Mitra, R. (2013). “Bayesian nonparametric inference: why and how.” Bayesian Analysis, 8(2): 269–302.
  • Plummer, M. (2003). “JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling.” http://mcmc-jags.sourceforge.net/.
  • Ruppert, D. (2002). “Selecting the number of knots for penalized splines.” Journal of Computational and Graphical Statistics, 11(1): 735–757.
  • Ruppert, D., Wand, M., and Carroll, R. (2003). Semiparametric Regression. New York: Cambridge University Press.
  • Sharef, E., Strawderman, R. L., Ruppert, D., Cowen, M., and Halasyamani, L. (2010). “Bayesian adaptive B-spline estimation in proportional hazards frailty models.” Electronic Journal of Statistics, 4: 606–642.
  • Shively, T. S. and Sager, T. W. (2009). “A Bayesian approach to non-parametric monotone function estimation.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(1): 159–175.
  • Shively, T. S., Walker, S. G., and Damian, P. (2011). “Nonparametric function estimation subject to monotonicity, convexity and other shape constraints.” Journal of Econometrics, 161: 166–181.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and Van Der Linde, A. (2002). “Bayesian measures of model complexity and fit.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4): 583–639.