Contents

Preface

Acknowledgments

1. Reliability Concepts and Reliability Data

1.1. Introduction

1.2. Examples of Reliability Data

1.3. General Models for Reliability Data

1.4. Repairable Systems and Nonrepairable Units

1.5. Strategy for Data Collection, Modeling, and Analysis

2. Models, Censoring, and Likelihood for Failure-Time Data

2.1. Models for Continuous Failure-Time Processes

2.2. Models for Discrete Data from a Continuous Process

2.3. Censoring

2.4. Likelihood

3. Nonparametric Estimation

3.1. Introduction

3.2. Estimation from Singly Censored Interval Data

3.3. Basic Ideas of Statistical Inference

3.4. Confidence Intervals from Complete or Singly Censored Data

3.5. Estimation from Multiply Censored Data

3.6. Pointwise Confidence Intervals from Multiply Censored Data

3.7. Estimation from Multiply Censored Data with Exact Failures

3.8. Simultaneous Confidence Bands

3.9. Uncertain Censoring Times

3.10. Arbitrary Censoring

4. Location-Scale-Based Parametric Distributions

4.1. Introduction.

4.2. Quantities of Interest in Reliability Applications

4.3. Location-Scale and Log-Location-Scale Distributions

4.4. Exponential Distribution

4.5. Normal Distribution

4.6. Lognormal Distribution

4.7. Smallest Extreme Value Distribution

4.8. Weibull Distribution

4.9. Largest Extreme Value Distribution

4.10. Logistic Distribution

4.11. Loglogistic Distribution

4.12. Parameters and Parameterization

4.13. Generating Pseudorandom Observations from a Specified Distribution

5. Other Parametric Distributions

5.1. Introduction

5.2. Gamma Distribution.

5.3. Generalized Gamma Distribution.

5.4. Extended Generalized Gamma Distribution

5.5. Generalized F Distribution

5.6. Inverse Gaussian Distribution

5.7. Birnbaum–Saunders Distribution

5.8. Gumpertz–Makeham Distribution

5.9. Comparison of Spread and Skewness Parameters

5.10. Distributions with a Threshold Parameter

5.11. Generalized Threshold-Scale Distribution

5.12. Other Methods of Deriving Failure-Time Distributions

6. Probability Plotting

6.1. Introduction

6.2. Linearizing Location-Scale-Based Distributions

6.3. Graphical Goodness of Fit.

6.4. Probability Plotting Positions

6.5. Probability Plots with Specified Shape Parameters.

6.6. Notes on the Application of Probability Plotting

7. Parametric Likelihood Fitting Concepts: Exponential Distribution

7.1. Introduction

7.2. Parametric Likelihood

7.3. Confidence Intervals for θ

7.4. Confidence Intervals for Functions of θ

7.5. Comparison of Confidence Interval Procedures

7.6. Likelihood for Exact Failure Times

7.7. Data Analysis with No Failures

8. Maximum Likelihood for Log-Location-Scale Distributions

8.1. Introduction

8.2. Likelihood

8.3. Likelihood Confidence Regions and Intervals

8.4. Normal-Approximation Confidence Intervals

8.5. Estimation with Given σ

9. Bootstrap Confidence Intervals

9.1. Introduction

9.2. Bootstrap Sampling

9.3. Exponential Distribution Confidence Intervals

9.4. Weibull, Lognormal, and Loglogistic Distribution Confidence Intervals

9.5. Nonparametric Bootstrap Confidence Intervals

9.6. Percentile Bootstrap Method

10. Planning Life Tests

10.1. Introduction

10.2. Approximate Variance of ML Estimators

10.3. Sample Size for Unrestricted Functions

10.4. Sample Size for Positive Functions

10.5. Sample Sizes for Log-Location-Scale Distributions with Censoring

10.6. Test Plans to Demonstrate Conformance with a Reliability Standard

10.7. Some Extensions

11. Parametric Maximum Likelihood: Other Models

11.1. Introduction

11.2. Fitting the Gamma Distribution

11.3. Fitting the Extended Generalized Gamma Distribution

11.4. Fitting the BISA and IGAU Distributions

11.5. Fitting a Limited Failure Population Model

11.6. Truncated Data and Truncated Distributions

11.7. Fitting Distributions that Have a Threshold Parameter

12. Prediction of Future Random Quantities

12.1. Introduction

12.2. Probability Prediction Intervals (θ Given)

12.3. Statistical Prediction Interval (θ Estimated)

12.4. The (Approximate) Pivotal Method for Prediction Intervals

12.5. Prediction in Simple Cases

12.6. Calibrating Naive Statistical Prediction Bounds

12.7. Prediction of Future Failures from a Single Group of Units in the Field

12.8. Prediction of Future Failures from Multiple Groups of Units with Staggered Entry into the Field

13. Degradation Data, Models, and Data Analysis

13.1. Introduction

13.2. Models for Degradation Data

13.3. Estimation of Degradation Model Parameters

13.4. Models Relating Degradation and Failure

13.5. Evaluation of F(t)

13.6. Estimation of F(r)y

13.7. Bootstrap Confidence Intervals

13.8. Comparison with Traditional Failure-Time Analyses

13.9. Approximate Degradation Analysis

14. Introduction to the Use of Bayesian Methods for Reliability Data

14.1. Introduction

14.2. Using Bayes’s Rule to Update Prior Information

14.3. Prior Information and Distributions

14.4. Numerical Methods for Combining Prior Information with a Likelihood

14.5. Using the Posterior Distribution for Estimation

14.6. Bayesian Prediction

14.7. Practical Issues in the Application of Bayesian Methods

15. System Reliability Concepts and Methods

15.1. Introduction

15.2. System Structures and System Failure Probability

15.3. Estimating System Reliability from Component Data

15.4. Estimating Reliability with Two or More Causes of Failure

15.5. Other Topics in System Reliability

16. Analysis of Repairable System and Other Recurrence Data

16.1. Introduction

16.2. Nonparametric Estimation of the MCF

16.3. Nonparametric Comparison of Two Samples of Recurrence Data

16.4. Parametric Models for Recurrence Data

16.5. Tools for Checking Point-Process Assumptions

16.6. Maximum Likelihood Fitting of Poisson Process

16.7. Generating Pseudorandom Realizations from an NHPP Process

16.8. Software Reliability

17. Failure-Time Regression Analysis

17.1. Introduction

17.2. Failure-Time Regression Models

17.3. Simple Linear Regression Models

17.4. Standard Errors and Confidence Intervals for Regression Models

17.5. Regression Model with Quadratic μ and Nonconstant σ

17.6. Checking Model Assumptions

17.7. Models with Two or More Explanatory Variables

17.8. Product Comparison: An Indicator-Variable Regression Model

17.9. The Proportional Hazards Failure-Time Model

17.10. General Time Transformation Functions

18. Accelerated Test Models

18.1. Introduction

18.2. Use-Rate Acceleration

18.3. Temperature Acceleration

18.4. Voltage and Voltage-Stress Acceleration

18.5. Acceleration Models with More than One Accelerating Variable

18.6. Guidelines for the Use of Acceleration Models

19. Accelerated Life Tests

19.1. Introduction

19.2. Analysis of Single-Variable ALT Data

19.3. Further Examples

19.4. Some Practical Suggestions for Drawing Conclusions from ALT Data

19.5. Other Kinds of Accelerated Tests

19.6. Potential Pitfalls of Accelerated Life Testing

20. Planning Accelerated Life Tests

20.1. Introduction

20.2. Evaluation of Test Plans

20.3. Planning Single-Variable ALT Experiments

20.4. Planning Two-Variable ALT Experiments

20.5. Planning ALT Experiments with More than Two Experimental Variables

21. Accelerated Degradation Tests

21.1. Introduction

21.2. Models for Accelerated Degradation Test Data

21.3. Estimating Accelerated Degradation Test Model Parameters

21.4. Estimation of Failure Probabilities, Distribution Quantiles, and Other Functions of Model Parameters

21.5. Confidence Intervals Based on Bootstrap Samples

21.6. Comparison with Traditional Accelerated Life Test Methods

21.7. Approximate Accelerated Degradation Analysis.

22. Case Studies and Further Applications

22.1. Dangers of Censoring in a Mixed Population

22.2. Using Prior Information in Accelerated Testing

22.3. An LFP/Competing Risk Model

22.4. Fatigue-Limit Regression Model

22.5. Planning Accelerated Degradation Tests

Epilogue

Appendix A. Notation and Acronyms

Appendix B. Some Results from Statistical Theory

B.1. cdfs and pdfs of Functions of Random Variables

B.2. Statistical Error Propagation—The Delta Method

B.3. Likelihood and Fisher Information Matrices

B.4. Regularity Conditions

B.5. Convergence in Distribution

B.6. Outline of General ML Theory

Appendix C. Tables

References

Author Index

Subject Index

This text is printed on acid-free paper. images

Copyright © 1998 by John Wiley & Sons, Inc. All rights reserved.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United Stales Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers. MA 01923, (978) 750-8400. fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue. New York NY 10158-0012, (212) 850-60! I. fax (212) 850-6008. E-Mail: PERMREQ@WILFY.COM.

Library of Congress Cataloging-in-Publication Data:

Meeker. William Q.

Statistical methods for reliability data / William Q. Meeker. Luis A. Escobar.

p. cm. — (Wiley series in probability and statistics.

Applied probability and statistics)

“A Wiley-Interscience publication.”

Includes bibliographical references and index.

ISBN 0-471-14328-6 (cloth : alk. paper)

1. Reliability (Engineering)—Statistical methods. 1. Escobar.

Luis A. II. Title. III. Series.

TS173.M44 1998

620′.00452—dc21 97-39270

To Karen, Katherine, and my parents
W.Q.M.

To Lida, Juan, Catalina, Daniela, and my mother Inés
L.A.E.

Preface

Over the past 10 years there has been a heightened interest in improving quality, productivity, and reliability of manufactured products. Global competition and higher customer expectations for safe, reliable products are driving this interest. To meet this need, many companies have trained their design engineers and manufacturing engineers in the appropriate use of designed experiments and statistical process monitoring/control. Now reliability is being viewed as the product feature that has the potential to provide an important competitive edge. A current industry concern is in developing better processes to move rapidly from product conceptualization to a cost-effective highly reliable product. A reputation for unreliability can doom a product, if not the manufacturing company.

Data collection, data analysis, and data interpretation methods are important tools for those who are responsible for product reliability and product design decisions. This book describes and illustrates the use of proven traditional techniques for reliability data analysis and test planning, enhanced and brought up to date with modern computer-based graphical, analytical, and simulation-based methods. The material in this book is based on our interactions with engineers and statisticians in industry as well as on courses in applied reliability data analysis that we have taught to MS-level statistics and engineering students at both Iowa State University and Louisiana State University.

Audience and Assumed Knowledge

We have designed this book to be useful to statisticians and engineers working in industry as well as to students in university engineering and statistics programs. The book will be useful for on-the-job training courses in reliability data analysis. There is challenge in addressing such a wide-ranging audience. Communications among engineers and statisticians, however, is not only necessary but essential in the industrial research and development environment. We hope that this book will aid such communication. To produce a book that will appeal to both engineers and statisticians, we have placed primary focus on applications, data, concepts, methods, and interpretation. We use simple computational examples to illustrate ideas and concepts but, as in practical applications, rely on computers to do most of the computations. We have also included a collection of exercise problems at the end of each chapter. These exercises will give readers a chance to test their knowledge of basic material, to explore conceptual ideas of reliability testing, data analysis, and interpretation, and to see possible extensions of the material in the chapters.

It will be helpful for readers to have had a previous course in intermediate statistical methods covering basic ideas of statistical modeling and inference, graphical methods, estimation, confidence intervals, and regression analysis. Only the simplest concepts of calculus are used in the main body of the text (e.g., probability for a continuous random variable is computed as area under a density curve; a first derivative is a slope or a rate of change; a second derivative is a measure of curvature). Appendix B and some advanced exercises use calculus, linear algebra, basic optimization ideas, and basic statistical theory. Concepts, however, are presented in a relaxed and intuitive manner that we hope will also appeal to interested nonstatisticians. Throughout the book we have attempted to avoid the heavy language of mathematical statistics.

A detailed understanding of underlying statistical theory is not necessary to apply the methods in this book. Such details are, however, often important to understanding how to extend methods to new situations or developing new methods. Appendix B. at the end of the book, outlines the general theory and provides references to more detailed information. Also, many derivations and interesting extensions are covered in advanced guided exercises at the end of each chapter.

Particularly challenging exercises (i.e., exercises requiring knowledge of calculus or statistical theory) are marked with a triangle (images). Exercises requiring computer programming (beyond the use of standard statistical packages) are marked with a diamond (images).

Special Features of the Book

Special features of this book include the following:

1. We emphasize general methods that can be applied to the wide range of problems found in industrial reliability data analysis—specifically, nonparametric estimation of a failure-time distribution function, probability plotting, and maximum likelihood estimation of important reliability characteristics (failure probabilities, distribution quantiles, and hazard functions), and associated statistical intervals. In the basic chapters (3, 6, 7, 8, 17, and 19), we apply these methods to the most frequently encountered models in reliability data analysis. In special chapters (which can be skipped without loss of continuity or understanding), we apply the general methods to important but less frequently occurring situations (e.g., problems involving truncation and prediction).

2. Throughout the book we use computer graphics for displaying data, for displaying the results of analyses, and for explaining technical concepts.

3. We use simulation methods to complement large-sample asymptotic theory (practical sample sizes are often small to moderate in size). We explain and illustrate modem, more accurate (but computationally demanding) methods of inference: likelihood and bootstrap methods for constructing statistical intervals.

4. For both nonparametric and parametric analyses, we illustrate the use of general likelihood-based methods of handling arbitrarily censored data (including left, right, and interval censoring with overlapping intervals) and truncated data that frequently arise in statistical reliability studies.

5. We provide methods for planning reliability studies (length of test, number of specimens, and levels of experimental factors).

6. We cover methods for analyzing degradation data. Such data are becoming increasingly important where there are requirements for extremely high reliability.

7. Almost all of our examples and exercises use real data, including many data sets that have not previously appeared in any book. In order to protect proprietary information, some data have been changed by a scale factor and. in some cases, generic product names have been used (e.g.. Device-A. Component-B. Alloy-A).

8. Numerical examples in this book were done using the S-PLUS system for graphics and data analysis (a product of MathSoft, Inc.. Seattle. WA). A suite of special S-PLUS functions was developed in parallel with this book. Although we have not included explicit information about software use in the chapters, the suite of special S-PLUS functions and a listing of the S-PLUS commands used to do the examples in the book are available from the authors via anonymous ftp at the Wiley ftp site. Instructions about how to access the software are given below.

How to Download the Software Examples

The Wiley public ftp site includes special S-PLUS function examples created for the applications discussed in this book. The files can be accessed through either a standard ftp program or the ftp client of a Web browser using the http protocol. You can access the files from a Web browser through the following address:

http://www.wiley.com/products/subject/mathematics

On the Mathematics and Statistics home page you will see a link to the ftp Software Archive, which includes a link to information about the book and access to the software.

To gain ftp access, type the following at your Web browser’s URL address input box:

ftp://ftp.wiley.com

You can set an ID of anonymous; no password is required.

The files are located in the public/sci_tech_med/reliabilitv directory. Be sure to also download and read the README.TXT file, which includes directions on how to install and use the program.

If you need further information about downloading the files, you can reach Wiley’s tech support line at 212-850-6753.

Other Software to Use with the Book

Today there are many commercial statistical software packages. Unfortunately, only a few of these packages have adequate capabilities for doing reliability data analysis (e.g., the ability to do nonparametric and parametric estimation with censored data). Nelson (1990a, pages 237-240) outlines the capabilities of a number of commercial and noncommercial packages that were available at that time. As software vendors become more aware of their customers’ needs, capabilities in commercial packages are improving. Here we describe briefly the capabilities of a few packages that we and our colleagues have found to be useful.

MINITAB (1997), SAS PROC RELIABILITY (1997), SAS JMP (1995), S-PLUS (1996), and a specialized program called WinSMITH (Abemethy 1996) can do nonparametric and parametric product censored data analysis to estimate a single distribution (Chapters 3, 6, 7, and 8). SAS JMP can also analyze data with more than one failure mode (Chapter 15). MINITAB, SAS PROC RELIABILITY, SAS JMP, and S-PLUS can do parametric regression and accelerated life test analyses (Chapters 17 and 19), as well as semiparametric Cox proportional hazards regression analysis. SAS PROC RELIABILITY can, in addition, do the nonparametric repairable systems analyses (Chapter 16).

Overview and Paths Through the Book

There are many possible paths that readers and instructors might take through this book. Chapters 1–16 cover single distribution models without any explanatory variables. Chapters 17–21 describe failure-time regression models. Chapter 22 presents case studies that illustrate, in the context of real problems, the integration of ideas presented throughout the book. This chapter also usefully illustrates how some of the general methods presented in the earlier chapters can be extended and adapted to deal with new problems.

Chapters 1–3 and 6–8 provide basic material that will be of interest to almost all readers and should be read in sequence. Chapter 4 discusses parametric failure-time models based on location-scale distributions and Chapter 5 covers more advanced distributional models. It is possible to use only a light reading of Chapter 4 and to skip Chapter 5 altogether before proceeding on to the important methods in Chapters 6–8. Chapter 9 explains and illustrates the use of bootstrap (simulation-based) methods for obtaining confidence intervals. Chapter 10 focuses on test planning: evaluating the effects of choosing sample size and length of observation. Chapters 11–16 cover a variety of special more advanced topics for single distribution models. Some of the material in Chapter 5 is prerequisite for the material in Chapter 11, but it is possible simply to work in Chapter 11, referring back to Chapter 5 only as needed. Otherwise, each of Chapters 10 through 14 has only material up to Chapter 8 as prerequisite. Chapter 15 introduces some important system reliability concepts and shows how the material in the first part of the book can be used to make statistical statements about the reliability of a system or a population of systems. Chapter 16 explains and illustrates the fundamental ideas behind analyzing system-repair and other recurrence data (as opposed to data on components and other replaceable units).

There are several groups of chapters on special topics that can be read in sequence.

Accelerated testing. Chapter 17 introduces models and methods for regression analysis (assessing the effects of explanatory variables) for failure-time data. Chapter 18 introduces physically based reliability models used in accelerated testing. Chapter 19 shows how to analyze data from accelerated life tests. Chapter 20 describes test planning for regression and accelerated test applications with censored data.

Degradation analysis. Chapter 13 provides methods for analyzing degradation reliability data. Use of degradation data in accelerated tests is covered in Chapter 21. Chapter 22 contains a case study that describes a method for planning accelerated degradation tests.

Bayesian methods. Chapter 14 introduces concepts and applications of Bayesian methods for failure-time data. A case study in Chapter 22 extends these ideas to regression with an application to accelerated life test data analysis.

Appendix A provides a summary and index of notation used in the book. Appendix B outlines the general maximum likelihood and other statistical theory on which the methods in the book are based. Appendix C gives tables for some of the larger data sets used in our examples.

Use as a Textbook

A two-semester course would be required to cover thoroughly all of the material in the book. For a one-semester course, aimed at engineers and/or statisticians, an instructor could cover Chapters 1–4, 6–8, and 17–19, along with selected material from the appendices (according to the background of the students), and a few other chapters according to interests and tastes.

This book could be used as the basis for workshops or short courses aimed at engineers or statisticians working in industry. For an audience with a working knowledge of basic statistical tools, Chapters 1–3, key sections in Chapter 4. and Chapters 6–8 could be covered in one day. If the purpose of the short course is to introduce the basic ideas and illustrate with examples, then some material from Chapters 17–20 could also be covered. For a less experienced audience or for a more relaxed presentation, allowing time for exercises and discussion, two days would be needed to cover this material. Extending the course to three or four days would allow covering selected material in Chapters 9–22.

WILLIAM Q. MEEKER
LUIS A. ESCOBAR

Ames, Iowa
Raton Rouge, Louisiana
April 1998

Acknowledgments

A number of individuals provided helpful comments on all or part of draft versions of this book. In particular, we would like to acknowledge Chuck Annis, Hwei-Chun Chou, William Christensen, Necip Doganaksoy, Tom Dubinin. Michael Eraas. Shuen-Lin Jeng, Gerry Hahn, Joseph Lu, Michael LuValle, Enid Martinets, Silvia Morales. Peter Morse, Dan Nordman, Steve Redman, Ernest Scheuer. Ananda Sen. David Steinberg, Mark Vandeven, Kim Wentzlaff, and a number of anonymous reviewers. Over the years, we have benefited from numerous technical discussions with Vijay Nair. Vijay provided a number of suggestions that substantially improved several parts of this book. We would also like to acknowledge our students (too numerous to mention by name) for their penetrating questions, high level of interest, and useful suggestions for improving our courses.

We would like to make special acknowledgment to Wayne Nelson, who gave us detailed feedback on earlier versions of most of the chapters of this book. Additionally, much of our knowledge in this area has it roots in our interactions with Wayne and Wayne’s outstanding books and other publications in the area of reliability and reliability data analysis.

Parts of this book were written while the first author was visiting the Department of Statistics and Actuarial Science at the University of Waterloo and the Department of Experimental Statistics at Louisiana State University. Support and use of facilities during these visits are gratefully acknowledged.

We greatly benefited from facilities, traveling support, and encouragement from Jeff Hooper and Michèle Boulanger at AT&T Bell Laboratories, Gerald Hahn at General Electric Corporate Research and Development Center, Enrique Villa and Victor Pérez Abreu at CIMAT in Guanajuato, México, Guido E. Del Pino at Universidad Católica in Santiago, Chile, Héctor Allende at Universidad Técnica Federico Santa María in Valparaiso, Chile, Yves L. Grize in Basel, Switzerland, and Stephan Zayac. Ford Motor Company.

Dean L. Isaacson, Head, Department of Statistics, Iowa State University, Lynn R. LaMotte, former Head, and E. Barry Moser, Interim Head, Department of Experimental Statistics, Louisiana State University, provided helpful encouragement and support to both authors. We would also like to thank our secretaries Denise Riker and Elaine Miller for excellent support and assistance while writing this book.

Finally, we would like to thank our wives and children for their love, patience, and understanding during the recent years in which we worked most weekends and many evenings to complete this project.

W. Q. M.
L. A. E.

CHAPTER 1

Reliability Concepts and Reliability Data

Objectives

This chapter explains:

Basic ideas behind product reliability.

Reasons for collecting reliability data.

Distinguishing features of reliability data.

General models for reliability data.

Examples of reliability data and the motivation for the collection of the data.

A general strategy that can be used for data analysis, modeling, and inference from reliability data.

Overview

This chapter introduces some of the basic concepts of product reliability. Section 1.1 explains the relationship between quality and reliability and outlines how statistical studies are used to obtain information that can be used to assess and improve product reliability. Section 1.2 presents examples to illustrate studies that resulted in different kinds of reliability data. These examples are used in data analysis and exercises in subsequent chapters. Section 1.3 explains, in general terms, important qualitative aspects of statistical models that are used to describe populations and processes in reliability applications. Section 1.4 emphasizes the important distinction between studies focusing on data from repairable systems and nonrepairable units. Section 1.5 describes a general strategy for exploring, analyzing, and drawing conclusions from reliability data. This strategy is illustrated in examples throughout the book and in the case studies in Chapter 22.

1.1. INTRODUCTION

1.1.1 Quality and Reliability

Rapid advances in technology, development of highly sophisticated products, intense global competition, and increasing customer expectations have put new pressures on manufacturers to produce high-quality products. Customers expect purchased products to be reliable and safe. Systems, vehicles, machines, devices, and so on should, with high probability, be able to perform their intended function under usual operating conditions, for some specified period of time.

Technically, reliability is often defined as the probability that a system, vehicle, machine, device, and so on will perform its intended function under operating conditions, for a specified period of time. Improving reliability is an important part of (he larger overall picture of improving product quality. There are many definitions of quality, but general agreement that an unreliable product is not a high-quality product. Condra (1993) emphasizes that “reliability is quality over time.”

Modern programs for improving reliability of existing products and for assuring continued high reliability for the next generation of products require quantitative methods for predicting and assessing various aspects of product reliability. In most cases this will involve the collection of reliability data from studies such as laboratory tests (or designed experiments) of materials, devices, and components, tests on early prototype units, careful monitoring of early-production units in the field, analysis of warranty data, and systematic longer-term tracking of products in the field.

1.1.2 Reasons for Collecting Reliability Data

There are many possible reasons for collecting reliability data. Examples include the following:

Assessing characteristics of materials over a warranty period or over the product’s design life.

Predicting product reliability.

Predicting product warranty costs.

Providing needed inputs for system-failure risk assessment.

Assessing the effect of a proposed design change.

Assessing whether customer requirements and government regulations have been met.

Tracking the product in the field to provide information on causes of failure and methods of improving product reliability.

Supporting programs to improve reliability through the use of laboratory experiments, including accelerated life tests.

Comparing components from two or more different manufacturers, materials, production periods, operating environments, and so on.

Checking the veracity of an advertising claim.

1.1.3 Distinguishing Features of Reliability Data

Reliability data can have a number of special features requiring the use of special statistical methods. For example:

Reliability data are typically censored (exact failure times are not known). The most common reason for censoring is the frequent need to analyze life test data before all units have failed. More generally, censoring arises when actual response values (e.g., failure times) cannot be observed for some or all units under study. Thus censored observations provide a bound or bounds on the actual failure times.

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