Bayesian proportional hazards model with time‐varying regression coefficients: a penalized Poisson regression approach

One can fruitfully approach survival problems without covariates in an actuarial way. In narrow time bins, the number of people at risk is counted together with the number of events. The relationship between time and probability of an event can then be estimated with a parametric or semi‐parametric model. The number of events observed in each bin is described using a Poisson distribution with the log mean specified using a flexible penalized B‐splines model with a large number of equidistant knots. Regression on pertinent covariates can easily be performed using the same log‐linear model, leading to the classical proportional hazard model. We propose to extend that model by allowing the regression coefficients to vary in a smooth way with time. Penalized B‐splines models will be proposed for each of these coefficients. We show how the regression parameters and the penalty weights can be estimated efficiently using Bayesian inference tools based on the Metropolis‐adjusted Langevin algorithm. Copyright © 2005 John Wiley & Sons, Ltd.

[1]  Edmund Taylor Whittaker.  On a New Method of Graduation , 1922, Proceedings of the Edinburgh Mathematical Society.

[2]  Stuart A. Klugman,et al.  Graduation: The Revision of Estimates , 1985 .

[3]  B. Efron.  Logistic Regression, Survival Analysis, and the Kaplan-Meier Curve , 1988 .

[4]  P. J. Verweij,et al.  Cross-validation in survival analysis. , 1993, Statistics in medicine.

[5]  H C van Houwelingen,et al.  Time-dependent effects of fixed covariates in Cox regression. , 1995, Biometrics.

[6]  R. Tweedie,et al.  Exponential convergence of Langevin distributions and their discrete approximations , 1996 .

[7]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[8]  Paul H. C. Eilers,et al.  Flexible smoothing with B-splines and penalties , 1996 .

[9]  Dani Gamerman,et al.  Sampling from the posterior distribution in generalized linear mixed models , 1997, Stat. Comput..

[10]  J. Rosenthal,et al.  Optimal scaling of discrete approximations to Langevin diffusions , 1998 .

[11]  Ludwig Fahrmeir,et al.  Recent Advances in Semiparametric Bayesian Function Estimation , 1999 .

[12]  P. Eilers,et al.  Non-proportional hazards models in survival analysis , 2000 .

[13]  C. Biller.  Adaptive Bayesian Regression Splines in Semiparametric Generalized Linear Models , 2000 .

[14]  Scott M. Berry,et al.  Bayesian Smoothing and Regression Splines for Measurement Error Problems , 2002 .

[15]  Christopher Holmes,et al.  Bayesian Methods for Nonlinear Classification and Regressing , 2002 .

[16]  P. Eilers.  A perfect smoother. , 2003, Analytical chemistry.

[17]  L. Fahrmeir,et al.  PENALIZED STRUCTURED ADDITIVE REGRESSION FOR SPACE-TIME DATA: A BAYESIAN PERSPECTIVE , 2004 .

[18]  Refik Soyer,et al.  Bayesian Methods for Nonlinear Classification and Regression , 2004, Technometrics.

[19]  G. Tutz,et al.  Flexible modelling of discrete failure time including time‐varying smooth effects , 2004, Statistics in medicine.

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

[21]  Göran Kauermann,et al.  Penalized spline smoothing in multivariable survival models with varying coefficients , 2005, Comput. Stat. Data Anal..

[22]  L. Fahrmeir,et al.  Geoadditive Survival Models , 2006 .

[23]  Andreas Brezger,et al.  Generalized structured additive regression based on Bayesian P-splines , 2006, Comput. Stat. Data Anal..

[24]  Philippe Lambert,et al.  Archimedean copula estimation using Bayesian splines smoothing techniques , 2007, Comput. Stat. Data Anal..