Bayesian Inference and Computation in Reliability and Survival Analysis

Preface
	Outline of This Book Volume
List of Reviewers
Acknowledgments
Contents
Contributors
About the Editors
Part I Reliability Data Analysis
	A Bayesian Approach for Step-Stress-Accelerated Life Tests for One-Shot Devices Under Exponential Distributions
		1 Introduction
		2 Model Description
		3 Maximum Likelihood Estimation
		4 Bayesian Approach
			4.1 Normal Prior
			4.2 Jeffreys Prior
		5 Simulation Study
		6 Data Analysis
		7 Concluding Remarks
		References
	Bayesian Estimation of Stress–Strength Parameter for Moran–Downton Bivariate Exponential Distribution Under Progressive Type II Censoring
		1 Introduction
		2 Model and Notations
		3 Bayesian Framework
			3.1 A Markov-Chain Monte Carlo (MCMC) Process
			3.2 Plug-In Bayesian Estimate of δ
			3.3 Mean-Value Monte Carlo Method
			3.4 Importance Sampling Estimation
		4 Monte Carlo Simulation Study
		5 Numerical Example
		6 Concluding Remarks
		References
	Bayesian Computation in a Birnbaum–Saunders Reliability Model with Applications to Fatigue Data
		1 Introduction
		2 The Birnbaum–Saunders Distribution
		3 Bayesian Computation and Reliability Model
		4 Application to Fatigue Data
		5 Conclusions, Discussion, and Future Research
		References
	A Competing Risk Model Based on a Two-Parameter Exponential Family Distribution Under Progressive Type II Censoring
		1 Introduction
		2 Competitive Risk Models
		3 Maximum Likelihood Estimation
			3.1 Special Family 1: Weibull Distribution
			3.2 Special Family 2: Burr XII Distribution
		4 Bayesian Estimation
			4.1 A Markov-Chain Monte Carlo Process
		5 Simulation Studies
		6 An Illustrative Example
		7 Conclusion
		Appendix
			Proof of Proposition 1
			Proof of Proposition 2
			Proof of Proposition 3
			Proof of Proposition 4
		References
Part II Stochastic Processes in Reliability Analysis
	Bayesian Computations for Reliability Analysis in Dynamic Environments
		1 Introduction and Overview
		2 Modulated Nonhomogeneous Poisson Processes for Rail Track Failures
			2.1 Bayesian Analysis of the Modulated NHPP
				2.1.1 Data Augmentation for Sampling from p(Λ0(t)|β,D)
				2.1.2 General Data Augmentation Algorithm
		3 Markov Modulated Markov Processes
			3.1 The Bivariate Markov Model
			3.2 Bayesian Analysis of MMMPs
		4 Numerical Illustrations
			4.1 A Markov Modulated Poisson Process Model for Software Failures
			4.2 A Markov Modulated Compound Poisson Process Model for Power Outages
		5 Concluding Remarks
		References
	Bayesian Analysis of Stochastic Processes in Reliability
		1 Introduction
		2 Stochastic Processes
			2.1 Distributions Associated with Stochastic Processes
		3 Intensity Functions
		4 Bayesian Inference
		5 Homogeneous Poisson Process
		6 The Power Law Process
		7 Software Reliability Models
			7.1 Jelinski–Moranda Model
			7.2 Littlewood–Verrall Model
			7.3 Goël–Okumoto Model
			7.4 Musa-Okumoto Model
		8 Self-Exciting Point Processes
		9 Conclusion
		References
	Bayesian Analysis of a New Bivariate Wiener Degradation Process
		1 Introduction
		2 Bivariate Degradation Model
			2.1 Model
			2.2 Reliability Function
		3 Statistical Inference
			3.1 Prior Specification
			3.2 Gibbs Sampling
		4 Data Analysis
		5 Conclusion
		Appendix
		References
	Bayesian Estimation for Bivariate Gamma Processes with Copula
		1 Introduction
		2 Gamma Process with Copula
			2.1 The Likelihood Function
		3 Markov Chain and Monte Carlo Procedure
			3.1 Blocking
			3.2 Updating Bi|D, Bj, θ (i≠j)
			3.3 Updating Copula Parameter
			3.4 Model Comparison
		4 Numerical Analysis
			4.1 Simulation Study
			4.2 Numerical Example
		5 Concluding Remarks
		Appendix
		References
Part III Biomedical Data Analysis
	Review of Statistical Treatment for Oncology Dose-Escalation Trial with Prolonged Evaluation Window or Fast Enrollment
		1 Introduction
		2 Dose-Escalation Algorithm
			2.1 The 3+3 method
			2.2 Model-Based Method
				2.2.1 Continual Reassessment Method
				2.2.2 Bayesian Logistic Regression Model
			2.3 Toxicity Interval-Based Method
				2.3.1 Modified Toxicity Probability Interval (mTPI)
				2.3.2 Bayesian Optimal Interval Design (BOIN)
		3 Time-to-Event Consideration
			3.1 The 3+3 Method
			3.2 CRM/BLRM
				3.2.1 Weighted Likelihood Function Method (TITE-CRM)
				3.2.2 TITE-CRM with Suspension Rule
				3.2.3 TITE-CRM with Predictive Risk
				3.2.4 TITE-CRM with Cycle Information
				3.2.5 TITE-CRM with Adaptive Time-to-DLT Distribution
				3.2.6 BLRM Adaptation
			3.3 Model-Assisted Method
				3.3.1 R-TPI
				3.3.2 TITE-BOIN
				3.3.3 BOIN12
				3.3.4 Imputation of Unobserved DLT Data
			3.4 Use Kaplan-Meier Method to Derive Fractional DLT for Pending Subjects
		4 Summary
		References
	A Bayesian Approach for the Analysis of Tumorigenicity Data from Sacrificial Experiments Under Weibull Lifetimes
		1 Introduction
		2 Model Specification
		3 Bayesian Approach
			3.1 Laplace Prior
			3.2 Normal Prior
			3.3 Beta Prior
			3.4 Prior Belief on pi
		4 Simulation Study
		5 Sensitivity Analysis on Prior Accuracy
		6 Application to Tumorigenicity Data from Sacrificial Experiments
		7 Concluding Remarks
		References
	Bayesian Sensitivity Analysis in Survival and Longitudinal Trials with Missing Data
		1 Introduction
		2 Sensitivity Analysis for Censoring in Survival Trials
			2.1 Delta-Adjusted Imputation and Jump-to-Reference
			2.2 Estimation of Survival Functions
			2.3 Inference Using the Bootstrap Method
		3 Sensitivity Analysis in Longitudinal Trials
			3.1 Models under the MAR
			3.2 Control-Based Imputation Methods
			3.3 Bayesian Sensitivity Analysis
		4 Examples
			4.1 A Time-to-Event Trial Example
			4.2 A Longitudinal Study Example
		5 Summary and Discussions
		References
	Bayesian Analysis for Clustered Data under a Semi-Competing Risks Framework
		1 Introduction
		2 Models and Methodologies
		3 Data and Bayesian Analysis
			3.1 Breast Cancer Data
			3.2 Bayesian Inference
			3.3 Results
		4 Concluding Remarks
		Appendix
		References
	Survival Analysis for the Inverse Gaussian Distribution: Natural Conjugate and Jeffrey's Priors
		1 Introduction
			1.1 Parameterizations
			1.2 Development of Bayesian Models
				1.2.1 Bayesian Survival Analysis
				1.2.2 Natural Conjugate Prior
				1.2.3 Jeffrey's Prior
		2 Gibbs Sampling Algorithm
		3 Monte-Carlo Simulation
			3.1 Selection of Hyperparameters
				3.1.1 Simulation Results
			3.2 Comparison at Different Censoring Levels
		4 Illustrative Example
		5 Concluding Remarks
		References
	Bayesian Inferences for Panel Count Data and Interval-Censored Data with Nonparametric Modeling of the Baseline Functions
		1 Introduction
		2 Models and the Observed Likelihoods
			2.1 Gamma Frailty Poisson Process for Panel Count Data
			2.2 The PH and PO Models for General Interval-Censored Data
		3 Modeling the Baseline Functions Nonparametrically
		4 Data Augmentation
			4.1 For Panel Count Data
			4.2 For Interval-Censored Data
		5 Bayesian Computation
			5.1 Prior Specification
			5.2 Gibbs Sampler for Panel Count Data
			5.3 Gibbs Sampler for Interval-Censored Data
		6 Simulation Study
			6.1 Panel Count Data
			6.2 Interval-Censored Data under the PH and PO Models
		7 Real-life Data Application
			7.1 The Patent Study
			7.2 The Bladder Tumor Study
			7.3 Breast Cosmesis Data
		8 Discussion
		References
	Bayesian Approach for Interval-Censored Survival Data with Time-Varying Coefficients
		1 Introduction
		2 Bayesian Approach for Clustered Interval-Censored Data
			2.1 Model and the likelihood
			2.2 Prior
			2.3 Posterior Computation
				2.3.1 Birth Move
				2.3.2 Death Move
		3 Bayesian Approach for Spatially Correlated Interval-Censored Data
			3.1 Model Specification
			3.2 Prior Distributions
			3.3 Posterior Inference
				3.3.1 Sample D
				3.3.2 Sample
				3.3.3 Sample Ω
		4 Illustrative Examples
			4.1 Dental Health Data
			4.2 Smoking Cessation Data
		5 Discussion and Remarks
		References
	Bayesian Approach for Joint Modeling Longitudinal Data and Survival Data Simultaneously in Public Health Studies
		1 Introduction
		2 Data and Preliminary Data Analysis
		3 Statistical Models
			3.1 Separate Modeling of Longitudinal Continuous Data
			3.2 Separate Modeling of Time-to-Event Data
			3.3 Joint (Simultaneous) Modeling of Longitudinal Continuous Data and Time-to-Event Data
		4 Results
			4.1 Results from Separate Linear Mixed-Effects Model on CD4 Longitudinal Data
			4.2 Results of Separate Cox Proportional Hazards Regression
			4.3 Results of Joint Modeling of Longitudinal CD4 and Time-to-Death
		5 Conclusions and Recommendations
		References
Index