Doctoral Thesis On Some Aspects of Bayesian Modeling in Reliability THESIS submitted for the award of the degree of DOCTOR OF PHILOSOPHY in STATISTICS By: Under the Supervision of: Md. Tanwir Akhtar Prof. Athar Ali Khan Department of Statistics and Operations Research Aligarh Muslim University Aligarh-202002 India 2015 List of Papers Papers included in the thesis Published Papers I. Akhtar, M. T. and Khan, A. A. (2014) Bayesian analysis of generalized log-Burr family with R. SpringerPlus, 3:185. II. Akhtar, M. T. and Khan, A. A. (2014) Log-logistic distribution as a reliability model: A Bayesian analysis. American Journal of Mathematics and Statistics, 4(3): 162-170. Communicated Paper I. Akhtar, M. T. and Khan, A. A. (2015) Hierarchical Bayesian approach with R and JAGS: Some Reliability Applica- tions. Journal of Data Science. Paper not included in the thesis Published Papers I. Khan, Y., Akhtar, M. T., and Khan, A. A. (2015) Bayesian Modeling of Forestry Data with R Using Optimization and Simula- tion Tools. Journal of Applied Analysis and Computation, 5(1): 38-51. Dedicated to My Family Especially My Father Dr. (Late) Md. Akhtar Hassan Preface There has been a dramatic growth in the development and applications of Bayesian statistics. One reason behind the dramatic growth in Bayesian modeling is the availability and development of computational algorithms to compute the range of integrals that are necessary in a Bayesian posterior analysis. Due to the speed and smoothness of modern computational tools and techniques, it is now possible to use the Bayesian paradigm to fit any statistical model whose likelihood and priors are specified or even very complex model that cannot be fit by alternative frequentist methods. Since the mid-1980s, the development of widely accessible powerful compu- tational tools and the implementation of Markov chain Monte Carlo (MCMC) methods have led to an explosion of interest in Bayesian statistics and modeling. This was followed by an extensive research for new Bayesian methodologies gener- ating the practical application of complicated models used over a wide range of sciences. During the late 1990s, BUGS emerged in the foreground. BUGS is a free software that could fit complicated models in a relatively easy manner, using stan- dard MCMC methods. Since 1998 and 2003, the WinBUGS and JAGS, respectively, has earned great popularity among researchers of diverse scientific fields. The present thesis entitled On Some Aspects of Bayesian Modeling in Reliability is a brief collection of modern methods and techniques for modeling of reliability data in Bayesian perspective. Bayesian modeling means that the process of fitting a probability model to a set of data and summarising the results by a probability distribution on the parameters of the models and on unobserved quantities, such as, prediction of new observations. The acceptance and appli- cations of Bayesian methods in virtually all branches of science and engineering vi 0. Preface have significantly increased over the last few decades. This increase is largely due to advances in simulation based computational tools for implementing Bayesian methods. Such tools are extensively used in this thesis. This thesis focuses on assessing the reliability of components or components of a system with particular attention to models containing covariates (or explanatory or regressor variables). Such models include failure time linear models, generalised linear models, and hierarchical models. Throughout the thesis, Laplace approx- imation, sampling importance resampling algorithm, and MCMC methods are used for implementing Bayesian analyses. MCMC makes the Bayesian approach to reliability computationally feasible and computationally straightforward. This is an important advantage in complex setting, e.g., hierarchical modeling, where classical approach fails or becomes too difficult for practical implementation. The objective of the present thesis is to apply Bayesian methods for model- ing of reliability data (discrete as well as continuous), with emphasis on model building and model implementation using the highly acclaimed, freely available LaplacesDemon and JAGS software packages, as run from within R. The bulk of this thesis forms a progression from the trivially simple to the moderately complex and cover linear, generalized linear, and hierarchical models. For fitting the model for reliability data, one needs a statistical computing environment. This environment should be such that one can write scripts to define Bayesian model, that is, specification of likelihood and prior distributions, use or write functions to summarize a posterior distribution, use function to simulate from the posterior distribution, and finally construct numerical as well as graphi- cal summaries to illustrate posterior inference. An environment that meet that requirements is the R system, which provides a wide range of functions for data manipulation, calculation, and graphics. Moreover, it includes a well-developed, simple programming language that users can extend by adding new functions. vii LaplacesDemon and JAGS are called from within R. JAGS is called by using its an interface from within R via R2jags package. Complete R, LaplacesDemon, and JAGS code concerning model building, priors specification, and model fitting for all analyses are provided. The interpretation of LaplacesDemon and JAGS output are also provided. All analyses using JAGS are directly compared with analyses using the same data using LaplacesDemon or standard R functions such as bayesglm or glmer. This thesis comprises of four chapters with a comprehensive list of bibliography provided at the end. Chapter 1 is an introductory chapter. This chapter covers basic concepts com- mon to all Bayesian analyses, including Bayes’ theorem, building blocks of Bayesian statistics, hierarchical Bayes, posterior predictive distribution, and model goodness of fit. The basic definitions of reliability, including reliability function, hazard function, mean time to failure, concept of censoring in Bayesian paradigm, are also included. This chapter also covers the tools and techniques for the Bayesian com- putation. Analytic approximation and simulation methods are covered here, but most of the emphasis is on Laplace approximation and MCMC method including Metropolis-Hastings algorithm and Gibbs sampler are currently the most powerful and popular techniques. An introduction to software packages R, LaplacesDemon, and JAGS used for Bayesian computations is also provided in this chapter. Important parametric reliability models and discrete as well as continuous reliability data analyses for component level data are presented and developed in Chapter 2. In this chapter, for the Bayesian computation of reliability data, R and JAGS code for binomial, Poisson, and lifetime models are developed. Both types of censored and uncensored data are considered for the Bayesian analysis. Analytic and parallel simulation techniques are implemented to approximate the marginal posterior densities of model parameters. viii 0. Preface Chapter 3 extends the Bayesian computation methods to the standard regression models used in reliability analysis using parametric models. In practical, logistic, Poisson, and lifetime regression models are considered for the analysis in Bayesian perspective. Real reliability data, which depends on concomitant variables are used. R and JAGS code are developed for the purpose of Bayesian computation. To app- roximate the posterior densities, both analytic and simulation methods are used. Chapter 4 introduces Bayesian computation methods for hierarchical models, which reveals the full power and conceptual simplicity of the Bayesian approach for the practical reliability problems. Also, Bayesian analysis of hierarchical models facilitate the joint analysis of data collected from similar components. For the Bayesian computation of hierarchical models, R and JAGS code are written and its output results are interpreted and provided. The research work presented in Chapter 2 and 3 of this thesis is based on my two published papers and the contents of Chapter 4 are based on a communicated paper, which are listed below. 1. Akhtar, M. T. and Khan, A. A. (2014). Bayesian Analysis of Generalized Log-Burr Family with R. SpringerPlus, 3:185. 2. Akhtar, M. T. and Khan, A. A. (2014). Log-logistic Distribution as a Reliability Model: A Bayesian analysis. American Journal of Mathematics and Statistics, 4(3): 162-170. 3. Akhtar, M. T. and Khan, A. A. (2015). Hierarchical Bayesian Approach with R and JAGS – Some Reliability Applications. Journal of Data Science. Acknowledgements All the praise and thanks are to ALLAH ‘the one universal Being’ who inspire entire humanity toward knowledge, truth and eternal commendation, who blessed me with strength and required passionate order to overcome all the obstacles in the way of this toilsome journey. In utter gratitude I bow my head before HIM. Words are not sufficient for expressing my intensity of sentiments. So, most humbly I express my deep sense of gratitude and indebtedness to my reverend supervisor Professor Athar Ali Khan who has always been a source of inspiration, whose constructive criticism, affectionable attitude, strong motivation and constant encouragement have added greatly to the readability and relevance of each chapter. I am grateful to him, who unsparingly helped me by sharing the in-depth knowledge of the subject, which has improved and made this thesis presentable. I will remain ever