Great strides in the analysis of survival data using Bayesian methods have been made in the past ten years due to advances in Bayesian computation and the feasibility of such methods. In this chapter, we review Bayesian advances in survival analysis and discuss the various semiparametric modeling techniques that are now commonly used. We review parametric and semiparametric approaches to Bayesian survival analysis, with a focus on proportional hazards models. Reference to other types of models are also given.


Keywords:

beta process;
Cox model;
Dirichlet process;
gamma process;
Gibbs sampling;
piecewise exponential model;
Weibull model

/pdf/bayesian-survival-analysis-4yeebbv71t.pdf

Bayesian Survival Analysis

We investigate two-stage parametric and two-stage semi-parametric estimation procedures for the association parameter in copula models for bivariate survival data where censoring in either or both components is allowed. We derive asymptotic properties of the estimators and compare their performance by simulations. Both parametric and semi-parametric estimators of the association parameter are efficient at independence, and the parameter estimates in the margins have high efficiency and are robust to misspecification of dependency structures. In addition, we propose a consistent variance estimator for the semi-parametric estimator of the association parameter. We apply the proposed methods to an AIDS data set for illustration.

https://www.jstor.org/stable/pdfplus/2533269.pdf

Inferences on the association parameter in copula models for bivariate survival data.

Several authors have constructed nonparametric Bayes estimators for a cumulative distribution function based on (possibly right-censored) data. The prior distributions have, for example, been Dirichlet processes or, more generally, processes neutral to the right. The present article studies the related problem of finding Bayes estimators for cumulative hazard rates and related quantities, w.r.t. prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments. A particular class of prior processes, termed beta processes, is introduced and is shown to constitute a conjugate class. To arrive at these, a nonparametric time-discrete framework for survival data, which has some independent interest, is studied first. An important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated, for example, time-inhomogeneous Markov chains. We find posterior distributions and derive Bayes estimators in such models and also present a semiparametric Bayesian analysis of the Cox regression model. The Bayes estimators are easy to interpret and easy to compute. In the limiting case of a vague prior the Bayes solution for a cumulative hazard is the Nelson-Aalen estimator and the Bayes solution for a survival probability is the Kaplan-Meier estimator.

/pdf/nonparametric-bayes-estimators-based-on-beta-processes-in-3m25hu01te.pdf

Nonparametric Bayes Estimators Based on Beta Processes in Models for Life History Data

(1976). Survival Distributions: Reliability Applications in the Biomedical Sciences. Technometrics: Vol. 18, No. 4, pp. 501-501.

Survival Distributions: Reliability Applications in the Biomedical Sciences

This article presents a nonparametric Bayesian estimator of a survival curve based on incomplete or arbitrarily right-censored data. This estimator, a Bayes estimator under a squared-error loss function assuming a Dirichlet process prior, is shown to be a Bayesian extension of the usual product limit (Kaplan-Meier) nonparametric estimator.

Nonparametric Bayesian Estimation of Survival Curves from Incomplete Observations

This paper discusses shrinkage estimation in nonparametric Bayesian survival analysis using censored data. The shrinkage estimators proposed are based on estimating the parameter measure of a prior Dirichlet process in a nonparametric Bayesian survival curve estimator which is the posterior mean of this process. The shrinkage is toward a prior family of exponential survival curves. The estimators are then compared by simulation with the wholly nonparametric estimator of Kaplan-Meier and the maximum likelihood estimator for the exponential family. These comparisons are done in cases where the exponential assumption is both

V. Susarla

Papers

Nonparametric Bayesian Estimation of Survival Curves from Incomplete Observations

Shrinkage estimation in nonparametric Bayesian survival analysis: a simulation study