A Bayesian justification of Cox's partial likelihood

In this paper, we establish both naive and formal Bayesian justifications of Cox's (1975) partial likelihood and its various modifications. We extend the original work of Kalbfieisch (1978), who showed that the partial likelihood is a limiting marginal posterior under noninformative priors for baseline hazards. We extend the result to scenarios with time-dependent covariates and time-varying regression parameters. We establish results for continuous time as well as grouped survival data. In addition, we present a Bayesian justification of a modified partial likelihood for handling ties. We also present tools for simplification of the Gibbs sampling algorithm for implementing partial likelihood based Bayesian inference in various practical applications. Copyright Biometrika Trust 2003, Oxford University Press.

[1]  M. Wulfsohn,et al.  Modeling the Relationship of Survival to Longitudinal Data Measured with Error. Applications to Survival and CD4 Counts in Patients with AIDS , 1995 .

[2]  N. Hjort.  Nonparametric Bayes Estimators Based on Beta Processes in Models for Life History Data , 1990 .

[3]  Jeremy M. G. Taylor,et al.  A Stochastic Model for Analysis of Longitudinal AIDS Data , 1994 .

[4]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[5]  B. Efron.  The Efficiency of Cox's Likelihood Function for Censored Data , 1977 .

[6]  M. Tanner,et al.  Maximization of the marginal likelihood of grouped survival data , 1994 .

[7]  D. Thomas,et al.  Simultaneously modelling censored survival data and repeatedly measured covariates: a Gibbs sampling approach. , 1996, Statistics in medicine.

[8]  M. Wulfsohn,et al.  A joint model for survival and longitudinal data measured with error. , 1997, Biometrics.

[9]  R. Prentice,et al.  Further results on covariate measurement errors in cohort studies with time to response data. , 1989, Statistics in medicine.

[10]  J. Kalbfleisch.  Non‐Parametric Bayesian Analysis of Survival Time Data , 1978 .

[11]  M. Davidian,et al.  Estimating the parameters in the Cox model when covariate variables are measured with error. , 1998, Biometrics.

[12]  Joseph G. Ibrahim,et al.  Bayesian Survival Analysis , 2004 .

[13]  D. Madigan,et al.  Bayesian Model Averaging in Proportional Hazard Models: Assessing the Risk of a Stroke , 1997 .

[14]  P. Gustafson.  A Bayesian analysis of bivariate survival data from a multicentre cancer clinical trial. , 1995, Statistics in medicine.

[15]  D. Oakes,et al.  Bivariate survival models induced by frailties , 1989 .

[16]  J. B Urridge,et al.  Empirical Bayes Analysis of Survival Time Data , 1981 .

[17]  J. Kalbfleisch,et al.  The Statistical Analysis of Failure Time Data , 1980 .

[18]  D. Ruppert,et al.  Measurement Error in Nonlinear Models , 1995 .

[19]  D J Sargent,et al.  A general framework for random effects survival analysis in the Cox proportional hazards setting. , 1998, Biometrics.

[20]  P Gustafson,et al.  Large hierarchical Bayesian analysis of multivariate survival data. , 1997, Biometrics.

[21]  D G Clayton,et al.  A Monte Carlo method for Bayesian inference in frailty models. , 1991, Biometrics.

[22]  Philip Hougaard,et al.  Analysis of Multivariate Survival Data , 2001 .

[23]  R. Prentice.  Covariate measurement errors and parameter estimation in a failure time regression model , 1982 .

[24]  Victor DeGruttola,et al.  Modeling The Relationship Between Progression Of CD4-Lymphocyte Count And Survival Time , 1992 .

[25]  W. Gilks,et al.  Adaptive Rejection Sampling for Gibbs Sampling , 1992 .