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Proportional Hazard Model Applications in Reliability

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Proportional Hazard Model Applications in Reliability Proportional Hazard Model Applications in Reliability A Dissertation Presented by Alexandre C. Mendes to The Graduate School of Engineering In partial fulfillment of the requirements for the degree of Doctor of Philosophy in Industrial Engineering Boston, Massachusetts April 2014 ABSTRACT This dissertation proposes two main methods as a modification of the semi-parametric Proportional Hazard Model (PHM) with innovative application for reliability testing. The first method developed uses a median of lifetime survival history for subjects with multiple occurrences to be modeled using the non-recurrent PHM method. The second method developed proposes censoring of influential observations, applying recurrent PHM theory. Both methods are validated using small electromechanical appliances with covariates identifying typical user performance as part of accelerated reliability testing. The Median PHM method is developed to allow the treatment of time-dependent covariates and competing failure types, which is fairly often necessary to properly model reliability studies. The advantages of this method are discussed and verified to provide the statistics or reliability engineer practitioners with the ability to evaluate multiple stresses, multiple covariates in accelerated reliability testing without the need to fit a physical model for the failure type and define a parametric distribution to model the acceleration parameter of interest. The analysis is presented in terms of hazard ratios that have immediate application for categorical covariates test levels and is easily extended for continuous covariates. Immediate plots for reliability and hazard curves are produced to enhance the analysis of the reliability model. Mendes develops the likelihood function with time-dependent covariates and presents a method to lay out the data and correct the input for the time-dependent covariate value derived from multiple measures for the II original subject with multiple failure occurrences. Different models per failure type are evaluated. The second method, Censored PHM method, is proposed to provide help when too many repeated events for a given subject may bias the analysis using the PHM for recurrent events. The method censors the multiple occurrences at a given threshold that might be related to a reliability product requirement or increased dependency between failure measures. The likelihood function for this method is simpler as the retrieval for covariate levels, since the censored subjects are maintained in the modeling analysis as part of the risk set. Fixed Effects and Stratified Cox models are applied to treat the recurrent events. Data set arrangements for recurrent events are available in the literature; however the author reviews additional techniques to separate failure modes and properly assign categorical and continuous time-dependent covariate values when both methods are used. A complete guide to the modeling steps for the Median and Censored PHM methods proposed for reliability modeling is presented in the conclusion of this study. III ACKNOWLEDGEMENTS After more than 4 years of diligent academic work while performing my professional duties as an engineer, the motivation to keep me going to write this dissertation was due to my faith in God. My adviser Professor Nasser Fard and my dissertation committee members Professor Sagar Kamarthi and Professor Tucker Marion guided me throughout the process. Professor Fard has helped me to make capital decisions on the line of research and expand my knowledge and ability to develop and apply statistical concepts in reliability. During the course of this work, Professor Fard was able to provide me with paper publications opportunities and presentations with International Society of Science and Applied Technologies (ISSAT) and RAMS (Reliability and Maintainability Symposium), and at a Joint Statistical meeting as part of the American Statistical Association (ASA). Moreover, we have submitted a sequence of 3 journal papers with International Journal of Quality, Reliability and Safety Engineering (IJRQSE), of which 2 papers have already been published. These papers enabled the sequential assembly of the dissertation literature research and knowledge for the proposed methods. I was able to count on very reliable and knowledgeable professionals during the course of this work and am hopeful my research has sparked their interest in statistical modeling. I want to thank my wife Sarah who has accepted my innumerous trade-offs when I had to choose to do work on the dissertation and related papers to develop this research. IV I am very proud of my education with Northeastern University and thankful for its ability to accommodate part-time PhD students in the Department of Mechanical and Industrial Engineering via solutions such as video-streaming. For that reason I was able to pursue and defend this dissertation. V TABLE OF CONTENTS ABSTRACT II ACKNOWLEDGEMENTS IV TABLE OF CONTENTS VI SUMMARY IX LIST OF FIGURES XIII LIST OF TABLES XV CHAPTER 1 INTRODUCTION 18 1.1 OVERVIEW 18 1.2 MOTIVATION 21 1.3 OPEN RESEARCH ISSUES 22 1.4 PROBLEM DEFINITION AND OBJECTIVE 23 1.5 PROPOSED SOLUTION 24 1.6 DISSERTATION CONTRIBUTION 25 CHAPTER 2 LITERATURE REVIEW 27 2.1 INTRODUCTION TO SURVIVAL ANALYSIS 27 2.2 RELIABILITY ANALYSIS 29 2.3 DATA SET ARRANGEMENT 39 2.4 ACCELERATED FAILURE TIME (AFT) MODELS 42 2.5 KAPLAN-MEIER AND LIFE-TABLE METHODS 45 2.6 PROPORTIONAL HAZARD MODEL (PHM) 48 2.6.1 ESTIMATION OF THE COVARIATES VECTOR 51 2.6.2 ESTIMATION OF THE BASELINE HAZARD RATE 57 2.6.3 TIME-DEPENDENT COVARIATES 59 2.6.4 RECURRENT EVENTS 65 2.6.5 COMPETING RISKS 73 CHAPTER 3 PROPOSED METHODS 78 3.1 BACKGROUND ON PROPOSED APPROXIMATION FOR RECURRENT EVENTS 78 3.2 PROPOSED METHOD TO EVALUATE RECURRENT EVENTS USING PHM 87 VI 3.3 MAXIMUM LIKELIHOOD DISCUSSION 91 3.3.1 TREATING TIME-DEPENDENT COVARIATES 100 3.4 DATA ARRANGEMENTS FOR CENSORING 104 CHAPTER 4 EXPERIMENTAL WORK 108 4.1 EXPERIMENTAL DESIGN 108 4.1.1 FAILURE TYPE 1 111 4.1.2 FAILURE TYPE 2 112 4.1.3 FAILURE TYPE 3 113 4.2 COVARIATES 114 4.2.1 MODEL 117 4.2.2 POWER MODE 119 4.2.3 IDLE DESIGN 120 4.2.4 TRIALS DESIGN 121 4.2.5 TOTAL COUNT 122 4.2.6 PATTERN 124 4.3 EXPERIMENTAL MATRIX 125 4.4 MODELING STEPS 127 4.4.1 RECURRENT EVENTS 131 4.4.2 APPROXIMATION TO NON-RECURRENT EVENTS – MEDIAN PHM 133 4.4.3 APPROXIMATION TO CHRONIC EVENTS – CENSORED PHM 134 CHAPTER 5 METHODS ANALYSIS 136 5.1 THE DATA SETS 136 5.1.1 RAW DATA SET DESCRIPTIVE ANALYSIS 139 5.1.2 TREATED RAW DATA SET DESCRIPTIVE ANALYSIS 143 5.1.3 MEDIAN DATA SET DESCRIPTIVE ANALYSIS 149 5.1.4 CENSORED DATA SET DESCRIPTIVE ANALYSIS 156 5.2 MODELING DISCUSSION 162 5.2.1 MEDIAN DATA SET MODELING ANALYSIS 163 5.2.1.1 Proportionality Assumption 163 5.2.1.2 Model Screening Failure Type 1 166 5.2.1.3 Model Assessment Failure Type 1 175 5.2.1.4 Model Screening Failure Type 2 197 5.2.1.5 Model Assessment Failure Type 2 205 5.2.1.6 Model Screening Failure Type 3 210 5.2.1.7 Model Assessment Failure Type 3 216 5.2.1.8 Model Screening All Failure Types Combined 221 5.2.2 CENSORED DATA SET MODELING ANALYSIS 226 5.2.2.1 Fixed Effects Model Screening for All Failure Types 227 5.2.2.2 Fixed Effects Model Assessment for All Failure Types 230 5.2.2.3 Stratified Cox Model Screening for All Failure Types 237 5.3 METHODS COMPARISONS 243 5.3.1 MEDIAN AND CENSORED METHODS VERIFICATION 244 VII 5.3.2 MODEL EXTENSION 247 5.3.3 IMPROVED MODELS 252 CHAPTER 6 CONCLUSIONS 260 6.1 RESEARCH ACHIEVEMENTS 260 6.2 METHODOLOGY CONTRIBUTION 268 6.3 SUGGESTIONS FOR FUTURE WORK 279 REFERENCES 285 VIII SUMMARY Reliability prediction for small appliances has received particular attention during the past few years as consumer expectations for functionality and reliability are driving future purchases and influencing new consumers via web content sharing. Companies are ever looking to improve their product performance to generate greater revenue opportunities and improve customer loyalty. Consumers conversely are more knowledgeable of product attributes and demanding higher quality and reliability. With the explosion of social media communication, open forums can be used to praise or damage a brand name. Especially for small appliances with high aggregate value; reliability has been elevated as the one metric to define a company’s performance. Reliability prediction during the design phase is achieved via typical physical models that are limited to evaluate a few stress levels at a time and require correct specification of a statistical function to describe the time varying parameter. Unfortunately, there is excessive focus on determining the acceleration factor; Mendes and Fard underscore the importance of determining the hazard for specific covariates used to determine reliability of the subjects. Typical reliability parametric methods require correct specification for the hazard function and do not allow regression for time-dependent covariates. For applications in reliability there is a high chance of a need to address time-dependent covariates that are related directly to the environment to which the machine is operated or with the characteristics of the household or location, prone to be changing. IX This research proposes the use of the Proportional Hazard Model (PHM), first developed by (Cox, Regression Models and Life-Tables 1972), typically applied in the medical and biostatistics fields, to assess time-to-event response variable for the electromechanical industry. This is a rather innovative approach since it uses the median of current events to calculate the likelihood function of the PHM and therefore its coefficients estimated regression values. In this research, the method is expanded to handle time-dependent covariates with recurrent events and competing failure modes.
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