Bayesian regularization for flexible baseline hazard functions in Cox survival models

Fully Bayesian methods for Cox models specify a model for the baseline hazard function. Parametric approaches generally provide monotone estimations. Semi-parametric choices allow for more flexible patterns but they can suffer from overfitting and instability. Regularization methods through prior distributions with correlated structures usually give reasonable answers to these types of situations. We discuss Bayesian regularization for Cox survival models defined via flexible baseline hazards specified by a mixture of piecewise constant functions and by a cubic B-spline function. For those "semi-parametric" proposals, different prior scenarios ranging from prior independence to particular correlated structures are discussed in a real study with microvirulence data and in an extensive simulation scenario that includes different data sample and time axis partition sizes in order to capture risk variations. The posterior distribution of the parameters was approximated using Markov chain Monte Carlo methods. Model selection was performed in accordance with the deviance information criteria and the log pseudo-marginal likelihood. The results obtained reveal that, in general, Cox models present great robustness in covariate effects and survival estimates independent of the baseline hazard specification. In relation to the "semi-parametric" baseline hazard specification, the B-splines hazard function is less dependent on the regularization process than the piecewise specification because it demands a smaller time axis partition to estimate a similar behavior of the risk.

[1]  J G Ibrahim,et al.  On Bayesian Inference for Proportional Hazards Models Using Noninformative Priors , 2000, Lifetime data analysis.

[2]  D. Ruppert,et al.  Bayesian adaptive B-spline estimation in proportional hazards frailty models , 2010 .

[3]  S. Geer,et al.  Regularization in statistics , 2006 .

[4]  S. Geisser,et al.  A Predictive Approach to Model Selection , 1979 .

[5]  Timothy Hanson,et al.  Bayesian Semiparametric Proportional Odds Models , 2007, Biometrics.

[6]  A. Branscum,et al.  Bayesian Nonparametric Meta‐Analysis Using Polya Tree Mixture Models , 2008, Biometrics.

[7]  T. Kneib,et al.  BayesX: Analyzing Bayesian Structural Additive Regression Models , 2005 .

[8]  Nan M. Laird,et al.  Covariance Analysis of Censored Survival Data Using Log-Linear Analysis Techniques , 1981 .

[9]  D K Dey,et al.  A Weibull Regression Model with Gamma Frailties for Multivariate Survival Data , 1997, Lifetime data analysis.

[10]  Axel Benner,et al.  High‐Dimensional Cox Models: The Choice of Penalty as Part of the Model Building Process , 2010, Biometrical journal. Biometrische Zeitschrift.

[11]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[12]  J. Ramsay.  Monotone Regression Splines in Action , 1988 .

[13]  J. Bernardo.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[14]  A. Tsiatis.  A Large Sample Study of Cox's Regression Model , 1981 .

[15]  Carmen Armero,et al.  S. Typhimurium virulence changes caused by exposure to different non-thermal preservation treatments using C. elegans. , 2017, International journal of food microbiology.

[16]  A. Gelman.  Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper) , 2004 .

[17]  Adrian F. M. Smith,et al.  Bayesian Inference for Generalized Linear and Proportional Hazards Models Via Gibbs Sampling , 1993 .

[18]  Rob J Hyndman,et al.  Mixed Model-Based Hazard Estimation , 2002 .

[19]  Ralf Bender,et al.  Generating survival times to simulate Cox proportional hazards models , 2005, Statistics in medicine.

[20]  S. Lang,et al.  Bayesian P-Splines , 2004 .

[21]  T. Holford.  The analysis of rates and of survivorship using log-linear models. , 1980, Biometrics.

[22]  Dai Feng,et al.  Bayesian random threshold estimation in a Cox proportional hazards cure model , 2014, Statistics in medicine.

[23]  Michael J Crowther,et al.  Simulating biologically plausible complex survival data , 2013, Statistics in medicine.

[24]  Bradley P Carlin,et al.  Flexible Bayesian survival modeling with semiparametric time-dependent and shape-restricted covariate effects. , 2016, Bayesian analysis.

[25]  N. Breslow.  Covariance analysis of censored survival data. , 1974, Biometrics.

[26]  Kyu Ha Lee,et al.  Hierarchical Models for Semicompeting Risks Data With Application to Quality of End-of-Life Care for Pancreatic Cancer , 2015, Journal of the American Statistical Association.

[27]  J. Kalbfleisch,et al.  Marginal likelihoods based on Cox's regression and life model , 1973 .

[28]  R. Gill,et al.  Cox's regression model for counting processes: a large sample study : (preprint) , 1982 .

[29]  Ciprian M. Crainiceanu,et al.  Bayesian Analysis for Penalized Spline Regression Using WinBUGS , 2005 .

[30]  Ulrich Mansmann,et al.  A semiparametric Bayesian proportional hazards model for interval censored data with frailty effects , 2009, BMC medical research methodology.

[31]  P. Austin.  Generating survival times to simulate Cox proportional hazards models with time-varying covariates , 2012, Statistics in medicine.

[32]  J. Ibrahim,et al.  Bayesian local influence for survival models , 2011, Lifetime data analysis.

[33]  F. Girosi,et al.  From regularization to radial, tensor and additive splines , 1993, Proceedings of 1993 International Conference on Neural Networks (IJCNN-93-Nagoya, Japan).

[34]  James O. Berger,et al.  An overview of robust Bayesian analysis , 1994 .