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Bayesian Survival Analysis

verfasst von: Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

Verlag: Springer New York

Buchreihe : Springer Series in Statistics

Enthalten in: Professional Book Archive

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Survival analysis arises in many fields of study including medicine, biology, engineering, public health, epidemiology, and economics. This book provides a comprehensive treatment of Bayesian survival analysis.
Several topics are addressed, including parametric models, semiparametric models based on prior processes, proportional and non-proportional hazards models, frailty models, cure rate models, model selection and comparison, joint models for longitudinal and survival data, models with time varying covariates, missing covariate data, design and monitoring of clinical trials, accelerated failure time models, models for mulitivariate survival data, and special types of hierarchial survival models. Also various censoring schemes are examined including right and interval censored data. Several additional topics are discussed, including noninformative and informative prior specificiations, computing posterior qualities of interest, Bayesian hypothesis testing, variable selection, model selection with nonnested models, model checking techniques using Bayesian diagnostic methods, and Markov chain Monte Carlo (MCMC) algorithms for sampling from the posteiror and predictive distributions.
The book presents a balance between theory and applications, and for each class of models discussed, detailed examples and analyses from case studies are presented whenever possible. The applications are all essentially from the health sciences, including cancer, AIDS, and the environment. The book is intended as a graduate textbook or a reference book for a one semester course at the advanced masters or Ph.D. level. This book would be most suitable for second or third year graduate students in statistics or biostatistics. It would also serve as a useful reference book for applied or theoretical researchers as well as practitioners.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

The analysis of time-to-event data, generally called survival analysis, arises in many fields of study, including medicine, biology, engineering, public health, epidemiology, and economics. Although the methods we present in this book can be used in all of these disciplines, our applications will focus exclusively on medicine and public health. There have been several textbooks written that address survival analysis from a frequentist perspective. These include Lawless, Cox and Oakes (1984), Fleming and Harrington (1991), Lee (1992), Andersen, Borgan, Gill, and Keiding (1993), and Klein and Moeschberger (1997). Although these books are quite thorough and examine several topics, they do not address Bayesian analysis of survival data in depth. Klein and Moeschberger (1997), however, do present one section on Bayesian nonparametric methods. Bayesian analysis of survival data has received much recent attention due to advances in computational and modeling techniques. Bayesian methods are now becoming quite common for survival data and have made their way into the medical and public health arena.

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

2. Parametric Models

Abstract

Parametric models play an important role in Bayesian survival analysis, since many Bayesian analyses in practice are carried out using parametric models. Parametric modeling offers straightforward modeling and analysis techniques. In this chapter, we discuss parametric models for univariate right censored survival data. We derive the posterior and predictive distributions and demonstrate how to carry out Bayesian analyses for several commonly used parametric models. The statistical literature in Bayesian parametric survival analysis and life-testing is too enormous to list here, but some references dealing with applications to medicine or public health include Grieve (1987), Achcar, Bolfarine, and Pericchi (1987), Achcar, Bookmeyer, and Hunter (1985), Chen, Hill, Greenhouse, and Fayos (1985), Dellaportas and Smith (1993), and Kim and Ibrahim (2001).

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

3. Semiparametric Models

Abstract

Nonparametric and semiparametric Bayesian methods in survival analysis have recently become quite popular due to recent advances in computing technology and the development of efficient computational algorithms for implementing these methods. Nonparametric Bayesian methods have now become quite common and well accepted in practice, since they offer a more general modeling strategy that contains fewer assumptions. The literature on nonparametric Bayesian methods has been recently surging, and all of the references are far too enormous to list here. In this chapter, we discuss several types of nonparametric prior processes for the baseline hazard or cumulative hazard, and focus our discussion primarily on the Cox proportional hazards model. Specifically, we examine piecewise constant hazard models, the gamma process, beta process, correlated prior processes, and the Dirichlet process. In each case, we give a development of the prior process, construct the likelihood function, derive the posterior distributions, and discuss MCMC sampling techniques for inference. Several applications involving case studies are given to demonstrate the various methods.

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

4. Frailty Models

Abstract

In studies of survival, the hazard function for each individual may depend on a set of risk factors or explanatory variables but usually not all such variables are known or measurable. This unknown and unobservable risk factor of the hazard function is often termed as the individual’s heterogeneity or frailty—a term coined by Vaupel, Manton, and Stallard (1979). Frailty models are becoming increasing popular in multivariate survival analysis since they allow us to model the association between the individual survival times within subgroups or clusters of subjects. With recent advances in computing technology, Bayesian approaches to frailty models are now computationally feasible, and several approaches have been discussed in the literature. The various approaches differ in the modeling of the baseline hazard or in the distribution of the frailty. Fully parametric approaches to frailty models are examined in Sahu, Dey, Aslanidou, and Sinha (1997), where they consider a frailty model with a Weibull baseline hazard. Semiparametric approaches have also been examined. Clayton (1991) and Sinha (1993, 1997) consider a gamma process prior on the cumulative baseline hazard in the frailty model. Sahu, Dey, Aslanidou, and Sinha (1997), Sinha and Dey (1997), Aslanidou, Dey, and Sinha (1998), and Sinha (1998) discuss frailty models with piecewise exponential baseline hazards. Qiou, Ravishanker, and Dey (1999) examine a positive stable frailty distribution, and Gustafson (1997) and Sargent (1998) examine frailty models using Cox’s partial likelihood. In this chapter, we present an overview of these various approaches to frailty models, and discuss Bayesian inference as well as computational implementation of these methods.

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

5. Cure Rate Models

Abstract

Survival models incorporating a cure fraction, often referred to as cure rate models, are becoming increasingly popular in analyzing data from cancer clinical trials. The cure rate model has been used for modeling time-to-event data for various types of cancers, including breast cancer, non-Hodgkins lymphoma, leukemia, prostate cancer, melanoma, and head and neck cancer, where for these diseases, a significant proportion of patients are “cured.” Perhaps the most popular type of cure rate model is the mixture model discussed by Berkson and Gage (1952). In this model, we assume a certain fraction π of the population is “cured,” and the remaining 1 – π are not cured.

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

6. Model Comparison

Abstract

Model comparison is a crucial part of any statistical analysis. Due to recent computational advances, sophisticated techniques for Bayesian model comparison in survival analysis are becoming increasingly popular. There has been a recent surge in the statistical literature on Bayesian methods for model comparison, including articles by George and McCulloch (1993), Madigan and Raftery (1994), Ibrahim and Laud (1994), Laud and Ibrahim (1995), Kass and Raftery (1995), Chib (1995), Chib and Greenberg (1998), Raftery, Madigan, and Volinsky (1995), George, McCulloch, and Tsay (1996), Raftery, Madigan, and Hoeting (1997), Gelfand and Ghosh (1998), Clyde (1999), and Chen, Ibrahim, and Yiannoutsos (1999). Articles focusing on Bayesian approaches to model comparison in the context of survival analysis include Madigan and Raftery (1994), Raftery, Madigan, and Volinsky (1995), Sinha, Chen, and Ghosh (1999), Ibrahim, Chen, and Sinha (2001b), Ibrahim and Chen (1998), Ibrahim, Chen, and MacEachern (1999), Sahu, Dey, Aslanidou, and Sinha (1997), Aslanidou, Dey, and Sinha (1998), Chen, Harrington, and Ibrahim (1999), and Ibrahim, Chen, and Sinha (2001a).

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

7. Joint Models for Longitudinal and Survival Data

Abstract

Joint models for survival and longitudinal data have recently become quite popular in cancer and AIDS clinical trials, where a longitudinal biologic marker such as CD4 count or immune response can be an important predictor of survival. Often in clinical trials where the primary endpoint is time to an event, patients are also monitored longitudinally with respect to one or more biologic endpoints throughout the follow-up period. This may be done by taking immunologic or virologic measures in the case of infectious diseases or perhaps with a questionnaire assessing the quality of life after receiving a particular treatment. Often these longitudinal measures are incomplete or may be prone to measurement error. These measurements are also important because they may be predictive of survival. Therefore methods which can model both the longitudinal and the survival components jointly are becoming increasingly essential in most cancer and AIDS clinical trials.

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

8. Missing Covariate Data

Abstract

Missing covariate data often arise in various settings, especially in clinical trials, epidemiological studies, and environmental studies. In the frequentist setting, it is well known (see Little and Rubin, 1987) that analyses based on complete cases can and will often result in inaccurate estimates of coefficients and their standard deviations. In the Bayesian context, complete case analyses will often lead to posterior distributions with properties that are quite different than those based on the observed data posterior, i.e., the posterior distribution using all of the cases. Thus, it becomes increasingly important in these situations to develop methods which incorporate the missing data into the analysis. The missing data problem has received much attention under the frequentist paradigm for a wide variety of models because of its common occurrence in many studies. Little (1992) gives an excellent review of developments for a variety of missing data problems. A recent book by Schafer (1997) examines frequentist and Bayesian approaches for missing data problems involving normal and categorical data, but it does not discuss survival models.

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

9. Design and Monitoring of Randomized Clinical Trials

Abstract

Bayesian approaches to the design and monitoring of randomized trials has received a great deal of recent attention. There are several articles discussing Bayesian design and monitoring of randomized trials, including Berry (1985, 1987, 1993), Freedman and Spiegelhalter (1983, 1989, 1992), Spiegelhalter and Freedman (1986, 1988), Spiegelhalter, Freedman, and Blackburn (1986), Breslow (1990), Jennison and Turnbull (1990), Thall and Simon (1994), Freedman, Spiegelhalter, and Parmar (1994), George et al. (1994), Grossman, Parmar, and Spiegelhalter (1994), Lewis and Berry (1994), Spiegelhalter, Freedman, and Parmar (1993, 1994), Rosner and Berry (1995), Adcock (1997), Joseph and Belisle (1997), Joseph, Wolfson, and DuBerger (1995), Lindley (1997), Pham-Gia (1997), Simon (1999), Simon and Freedman (1997), Weiss (1997), Bernardo and Ibrahim (2000), and Lee and Zelen (2000). In this chapter, we will review some of the basic approaches to the Bayesian design and monitoring of randomized trials as well as compare the Bayesian and frequentist approaches.

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

10. Other Topics

Abstract

In this last chapter, we discuss various other topics in Bayesian survival analysis. Specifically, we discuss semiparametric proportional hazards models built from monotone function, accelerated failure time models, Dirichlet process and Polya tree priors in survival analysis, Bayesian survival analysis using multivariate adaptive regression splines (MARS), change point models, poly-Weibull models, hierarchical survival models using neural networks, Bayesian model diagnostics, and future research topics.

Joseph G. Ibrahim, Ming-Hui Chen, Debajyoti Sinha

Backmatter

Titel: Bayesian Survival Analysis
verfasst von: Joseph G. Ibrahim
Ming-Hui Chen
Debajyoti Sinha
Verlag: Springer New York
Electronic ISBN: 978-1-4757-3447-8
Print ISBN: 978-1-4419-2933-4
DOI: https://doi.org/10.1007/978-1-4757-3447-8

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Parametric Models

3. Semiparametric Models

4. Frailty Models

5. Cure Rate Models

6. Model Comparison

7. Joint Models for Longitudinal and Survival Data

8. Missing Covariate Data

9. Design and Monitoring of Randomized Clinical Trials

10. Other Topics

Backmatter