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Accelerated TestingStatistical Models, Test Plans,and Data AnalysesWAYNE NELSONConsultant, Schenectady, NYA JOHN WILEY & SONS, INC., PUBLICATION

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Accelerated Testing

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Accelerated TestingStatistical Models, Test Plans,and Data AnalysesWAYNE NELSONConsultant, Schenectady, NYA JOHN WILEY & SONS, INC., PUBLICATION

Copyright 0 1990,2004 by John Wiley & Sons, Inc. All rights reserved.Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.No part of this publication may be reproduced, stored in a retrieval system or transmittedin any form or by any means, electronic, mechanical, photocopying, recording, scanningor otherwise, except as permitted under Sections 107 or 108 of the 1976 United StatesCopyright Act, without either the prior written permission of the Publisher, orauthorization through payment of the appropriate per-copy fee to the CopyrightClearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax(978) 750-4470. Requests to the Publisher for permission should be addressed to thePermissions Department, John Wiley At Sons, Inc., 1 I1 River Street, Hoboken, NJ 07030,(201) 748-6011, fax (201) 748-6008.Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their bestefforts in preparing this book, they make no representation or warranties with respect to the accuracyor completeness of the contents of this book and specifically disclaim any implied warranties ofmerchantability or fitness for a particular purpose. No warranty may be created or extended by salesrepresentatives or written sales materials. The advice and strategies contained herein may not besuitable for your situation. You should consult with a professional where appropriate. Neither thepublisher nor author shall be liable for any loss of profit or any other commercial damages, includingbut not limited to special, incidental, consequential,or other damages.For general information on our other products and services please contact our Customer CareDepartment within the US. at 877-762-2974, outside the U S . at 317-572-3993or fax 3 17-572-4002.Wiley also publishes its books in a variety of electronic formats. Some content that appears in print,however, may not be available in electronic format.Library ojcongress Cataloging-in-Publication is available.ISBN 0-47 1-69736-2Printed in the United States of America.I0987654321

This book is gratefully dedicated to the many clients whose fruitfulcollaboration and challenging applications stimulated my interest anddevelopments in accelerated testing, and to the many colleagues who kindlyprovided me with examples, references, suggestions, and encouragement.

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ContentsPreface, xi1. Introduction and Background, 11.2.3.4.5.6.Survey of Methodology and Applications, 3Types of Data, 12Types of Acceleration and Stress Loading, 15Engineering Considerations, 22Common Accelerated Tests, 37Statistical Considerations, 43Problems, 492. Models for Ufe Tests with Constant Stress, 511. Introduction, 51Basic Concepts and the Exponential Distribution, 53Normal Distribution, 58Lognormal Distribution, 60Weibull Distribution, 63Extreme Value Distribution, 65Other Distributions, 688. Life-Stress Relationships, 719. Arrhenius Life-TemperatureRelationship, 7510. Inverse Power Relationship, 8511. Endurance (Fatigue) Limit Relationships and Distributions, 9212. Other Single Stress Relationships, 9513. Multivariable Relationships, 9814. Spread in Log Life Depends on Stress, 105Problems, 1072.3.4.5.6.7.3. Graphical Data Analysis, 1131. Introduction, 1132. Complete Data and Arrhenius-Lognormal Model, 1143. Complete Data and Power-Weibull Model, 1284. Singly Censored Data, 134vii

viiiCONTENTS5. Multiply Censored Data, 1396. Interval (Read-Out) Data, 145Problems, 1544. Complete Data and Least Squares Analyses, 1671. Introduction, 1672. Least-Squares Methods for Lognormal Life, 1703. Checks on the Linear-LognormalModel and Data, 1824. Least-Squares Methods for Weibull and Exponential Life, 1895. Checks on the Linear-WeibullModel and Data, 2036. Multivariable Relationships, 210Problems, 2295. Censored Data and Maximum Ukelihood Methods, 2331.2.3.4.5.Introduction to Maximum Likelihood, 234Fit the Simple Model to Right Censored Data, 242Assess the Simple Model and Right Censored Data, 255Other Models and w e s of Data, 265Maximum Likelihood Calculations, 284Problems, 3026. Test Plans, 317Plans for the Simple Model and Complete Data, 317Plans for the Simple Model and Singly Censored Data, 328Evaluation of a Test Plan by Simulation, 349Survey of Test Plans, 3615. ML Theory for Test Plans, 364Problems, 3711.2.3.4.7. Competing Failure Modes and Size Effect, 377Series-System Model, 378Series Systems of Identical Parts, 383Size Effect, 385Nonuniform Stress, 3875. Graphical Analysis, 3926. ML Analysis for Competing Failure Modes, 4077. ML Theory for Competing Modes, 413Problems, 4171.2.3.4.8. Least-Squares Comparisons for Complete Data, 4251.2.3.4.5.Hypothesis Tests and Confidence Intervals, 426Graphical Comparisons,429Compare Log Standard Deviations, 434Compare (Log) Means, 437Compare Simple Relationships, 441

CONTENTS6. Compare Multivariable Relationships, 445Problems, 4489. Maximum Likelihood Comparisons for Censored and Other Data,4511.2.3.4.5.Introduction, 451One-Sample Comparisons, 452Two-Sample Comparisons, 458K-Sample Comparisons, 465Theory for LR and Related Tests, 470Problems, 48810. Models and Data Analyses for Step and Varying Stress, 4931. Survey of Theory for Tests with Varying Stress, 4942. Step-Stress Model and Data Analyses, 4953. Varying-StressModel and Data Analyses, 506Problems, 51311. Accelerated Degradation, 5211. Survey of Applications, 5212. Degradation Models, 5233. Arrhenius Analysis, 534Problems, 544Appendix A. Statistical Tables, 549Al. Standard Normal Cumulative Distribution Function O(u), 550A2. Standard Normal Percentilesz p , 552A3.A4.M.A6a.A6b.A7.Standard Normal Two-Sided Factors Kp,552t-Distribution Percentiles r(P;v), 553Chi-square Percentiles ( P ; v )554,F-Distribution95% Points F(0.93vl,v2), 556F-Distribution 99% Points F(O.W,V ,V ),558Probability Plotting Positions Fi 10O(i-OS)/n, 560References, 561Index, 579ix

Preface to the Paperback EditionFirst published in 1990, this book remains the most comprehensive presentationof statistical models and methods for accelerated test data. It is gratifying that ithas been widely used and praised by practitioners and researchers in statistics andengineering. This paperback edition is available at a bargain price thanks to thefine work of Mr. Steve Quigley, Ms. Susanne Steitz, and the Wiley staff.For subsequent advances in accelerated testing, the reader may wish to consult:Meeker, W.Q. and Escobar, L.A. (1998), Statistical Methods for ReliabilityData, Wiley, New York, www.wiley.com. In particular, their degradation models and corresponding statistical methods and Chapter I 1 of this book overlaplittle and together comprise a basic introduction to accelerated degradation.Nelson, Wayne (2004), “A Bibliography of Accelerated Test Plans,” over 100references, available from the author, WNconsult@aol.com.Since 1990, commercial software for analysis of accelerated test data has continued to advance. To reflect these advances, Table 1.1 of Chapter 5 and corresponding text have been updated. Note that confidence limits using a normal approximation to the sampling distribution of a maximum likelihood estimator andits asymptotic standard error are not current best practice. Instead, one should usesoftware that calculates confidence limits using the likelihood ratio, as describedin Section 5.8 of Chapter 5 , as these intervals are now known to be a better approximation in virtually all applications.B. NELSONWAYNEConsulting and TrainingWNconsult@aol.comScheneciady, New YorkJune 2004X

PrefaceProduct reliability contributes much to quality and competitiveness.Many manufacturers yearly spend millions of dollars on product reliability.Much management and engineering effort goes into evaluating reliability, assessing new designs and design and manufacturing changes, identifyingcauses of failure, and comparing designs, vendors, materials, manufacturingmethods, and the like. Major decisions are based on life test data, often froma few units. Moreover, many products last so long that life testing at designconditions is impractical. Many products can be life tested at high stress conditions to yield failures quickly. Analyses of data from such an acceleratedtest yield needed information on product life at design conditions (lowstress). Such testing saves much time and money. This book presents practical, modern statistical methods for accelerated testing. Up-to-date, it provides accelerated test models, data analyses, and test plans. In recent years,much useful methodology has been developed, and this book makes it available to practitioners. This book will contribute to more efficient acceleratedtesting and to valid and more accurate information.This book is written for practicing engineers, statisticians, and others whouse accelerated testing in design, development, testing, manufacturing, quality control, and procurement. It will aid workers in other fields concernedwith regression models for survival, for example, in medicine, biology, andactuarial science. Also, this book is a useful supplement for statistics and engineering courses, as it presents many stimulating real examples, emphasizespractical data analysis (employing graphical methods and computer programs), and shows how to use versatile maximum likelihood methods forcensored data.This book is organized to serve practitioners. The simplest and most useful material appears first. The book starts with basic models and graphicaldata analyses, and it progresses through advanced maximum likelihoodmethods. Available computer programs are used. Each topic is selfcontained for easy reference, although this results in some repetition. Thusthis book serves as a reference or textbook. Derivations are generally omitted unless they provide insight. Such derivations appear in advanced sectionsfor those seeking deeper understanding or developing new statistical models,xi