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