MODELING AND PLANNING ACCELERATED LIFE TESTING WITH PROPORTIONAL ODDS

Transcription

1 MODELING AND PLANNING ACCELERATED LIFE TESTING WITH PROPORTIONAL ODDS by HAO ZHANG A Disseraion submied o he Graduae School-New Brunswick Rugers, The Sae Universiy of New Jersey in parial fulfillmen of he requiremens for he degree of Docor of Philosophy Graduae Program in Indusrial and Sysems Engineering wrien under he direcion of Professor Elsayed A. Elsayed and approved by New Brunswick, New Jersey May, 007

2 ABSTRACT OF THE DISSERTATION Modeling and Planning Acceleraed Life Tesing wih Proporional Odds By HAO ZHANG Disseraion Direcor: Elsayed A. Elsayed Acceleraed life esing (ALT) is a mehod for esimaing he reliabiliy of producs a normal operaing condiions from he failure daa obained a he severe condiions. We propose an ALT model based on he proporional odds (PO) assumpion o analyze failure ime daa and invesigae he opimum ALT plans for muliple-sress-ype cases based on he PO assumpion. We presen he PO-based ALT model and propose he parameer esimaion procedures by approximaing he general baseline odds funcion wih a polynomial funcion. Numerical examples wih experimenal daa and Mone Carlo simulaion daa verify ha ii

3 he PO-based ALT model provides more accurae reliabiliy esimae for he failure ime daa exhibiing PO properies. The accuracy of he reliabiliy esimaes is direcly affeced by he reliabiliy inference model and how he ALT is conduced. The laer is addressed in he lieraure as he design of ALT es plans. Design of ALT es plans under one ype of sress may mask he effec of oher criical ypes of sresses ha could lead o he componen s failure. The exended life of oday s producs makes i difficul o obain enough failures in a reasonable amoun of esing ime using single sress ype. Therefore, i is more realisic o consider muliple sress ypes. This is he firs research ha invesigaes he design of opimum ALT es plans wih muliple sress ypes. We formulae nonlinear opimizaion problems o deermine he opimum ALT plans. The opimizaion problem was solved wih a numerical opimizaion mehod. Reliabiliy praciioners could choose differen ALT plans in erms of he sress loading ypes. In his disseraion we conduc he firs invesigaion of he equivalency of ALT plans, which enables reliabiliy praciioners o choose he appropriae ALT plan according o resource resricions. The resuls of his research show ha one can indeed develop efficien es plans ha can provide accurae reliabiliy esimae a design condiions in much shorer es duraion han he radiional es plans. iii

4 Finally, we conduc experimenal esing using miniaure ligh bulbs. The es unis are subjeced o differen sress ypes. The resuls validae he applicabiliy of he PO-based ALT models. iv

5 ACKNOWLEDGEMENTS I would like o express my deep graiude o my advisor Professor Elsayed A. Elsayed for his guidance, suppor, and inspiraion hrough his research. Professor Elsayed s enhusiasm and knowledge of he field have had a profound influence upon my graduae and general educaion. Wihou his coninuous encouragemen, his disseraion could no have been accomplished. My sincere hanks are exended o Professor David Coi, Professor Hoang Pham, and Professor John Kolassa for serving on my commiee and providing valuable suggesions. I also ake his opporuniy o hank Mr. Joe Lippenco for his generous help hroughou he experimen. I would like o exend my hanks o Professor Susan Albin, Professor Tayfur Aliok, Ms Cindy Ielmini and Ms Helen Smih-Pirrello for he suppors. v

6 TABLE OF CONTENTS ABSTRACT.ii ACKNOWLEDGEMENTS... v CHAPTER INTRODUCTION.... Research Background.... Problem Definiion Organizaion of he Disseraion Proposal... 9 CHAPTER LITERATURE REVIEW.... ALT Models..... Cox s Proporional Hazards Model..... Acceleraed Failure Time Models Proporional Odds Model and Inference Descripion of PO model Esimaing he Parameer in a Two-sample PO Model Esimaes of he PO model Using Ranks Profile Likelihood Esimaes of PO Model ALT Tes Plans AFT-based ALT Plans General Assumpions of AFT-based ALT Plans Crieria for he Developmen of AFT-based ALT Plans PH-based ALT Plans Conclusions vi

7 3 CHAPTER 3 ALT MODEL BASED ON PROPORTIONAL ODDS Properies of Odds Funcion Proposed Baseline Odds Funcion Approximaion Log-Likelihood Funcion of he PO-based ALT Model Simulaion Sudy Generaion of Failure Time Daa Simulaion Resuls Example wih Experimenal Daa CHAPTER 4 CONFIDENCE INTERVALS AND MODEL VALIDATION Confidence Inervals Fisher Informaion Marix Variance-Covariance Marix and Confidence Inervals of Model Parameers Confidence Inervals of Reliabiliy Esimaes Numerical Example of Reliabiliy Confidence Inervals Model Validaions Model Sufficiency Model Residuals Numerical Example of Cox-Snell Residuals CHAPTER 5 PO-BASED MULTIPLE-STRESS-TYPE ALT PLANS PO-based ALT Plans wih Consan Muliple Sress Types The Assumpions The Log Likelihood Funcion vii

8 5..3 The Fisher Informaion Marix and Covariance Marix Opimizaion Problem Formulaion Opimizaion Algorihm Numerical Example PO-based Muliple-Sress-Type ALT Plans wih Sep-Sress Loading Tes Procedure Assumpions Cumulaive Exposure Models Log Likelihood Funcion Fisher s Informaion Marix and Variance-Covariance Marix Opimizaion Crierion Problem Formulaion Numerical Example CHAPTER 6 EQUIVALENT ALT PLANS Inroducion Definiion of Equivalency Equivalency of Sep-sress ALT Plans and Consan-sress ALT Plans Numerical Example Simulaion Sudy CHAPTER 7 EXPERIMENTAL VALIDATION Experimenal Samples Experimens Seup Tes Condiions viii

9 7.4 Analysis of he Experimenal Resul CHAPTER 8 CONCLUSIONS AND FUTURE WORK Summary of Topics The PO-based ALT Model and Esimaion Procedures Confidence Inervals and Model Validaion The PO-based Muliple-sress-ype ALT Plans The Equivalency of ALT Plans Fuure Work REFERENCES CURRICULUM VITA PUBLICATIONS RELATED TO THE DISSERTATION... 8 ix

10 LIST OF FIGURES Figure 3. Odds funcions of four common failure ime disribuions... 5 Figure 3. Theoreical reliabiliy and prediced reliabiliy a sress level 0.5 by he PObased ALT model Figure 3.3 Theoreical reliabiliy and prediced reliabiliy a sress level 0.5 by he PHbased ALT model Figure 3.4 Reliabiliy comparison for he PO-based ALT model wih w = a sress level Figure 3.5 Reliabiliy comparison for he PH-based ALT model wih w = a sress level Figure 3.6 Reliabiliy comparison for he PO-based ALT model wih w = a sress level Figure 3.7 Reliabiliy comparison for he PH-based ALT model wih w = a sress level Figure 3.8 Reliabiliy comparison for he PO-based ALT model wih w = a sress level Figure 3.9 Reliabiliy comparison for he PH-based ALT model wih w = a sress level Figure 3.0 Reliabiliy comparison for he PO-based ALT model wih w = a sress level Figure 3. Reliabiliy comparison for he PH-based ALT model wih w = a sress level x

11 Figure 3. Plo of log odds funcions versus ime for Swarz s daa... 7 Figure 3.3 Plo of log cumulaive hazards funcions versus ime for Swarz s daa Figure 3.4 Kaplan-Meier reliabiliy vs. prediced reliabiliy a emperaure 5 C by he PO-based ALT model Figure 3.5 Kaplan-Meier reliabiliy vs. prediced reliabiliy a emperaure 5 C by he PH-based ALT model Figure 4. 95% reliabiliy confidence inervals a emperaure z = 5 C level Figure 4. Nelson-Aalen esimaors of Cox-Snell residuals Figure 5. Design of ALT plan wih wo sress ypes and hree levels for each sress ype Figure 5. A summary of he COBYLA algorihm Figure 5.3 Sep-Sress loading wih muliple-sress-ype... 7 Figure 5.4 Relaionship beween consan-sress and sep-sress disribuions... 9 Figure 6. Various loadings of a single ype of sress Figure 6. Two example seings of an ALT involving emperaure, curren, humidiy and elecric curren Figure 6.3 An example of equivalen consan ALT plan and sep-sress ALT plan Figure 6.4 Cumulaive exposure model... 5 Figure 6.5 Esimaed reliabiliy funcions from consan and sep simulaed daa Figure 7. Samples of he miniaure ligh bulbs Figure 7. Miniaure ligh bulbs esing se Figure 7.3 The layou of he acceleraed life esing equipmen Figure 7.4 The programmable power supply D xi

12 Figure 7.5 LabView applicaion inerface Figure 7.6 Log of cumulaive hazard rae plo Figure 7.7 Resul comparison xii

13 LIST OF TABLES Table 3. Mone Carlo simulaion daases... 6 Table 3. Sums of squared errors wih differen sress levels Table 3.3 Sums of squared errors wih differen w values Table 5. Proporions of unis allocaed o each sress level combinaion... 3 Table 6. Resuls of bisecional search Table 6. Model parameers for consan-sress ALT simulaion Table 6.3 Model parameers for sep-sress ALT simulaion... 5 Table 7. Summary of he experimenal daa... 6 Table 7. Esimaed parameers of he PO model xiii

14 CHAPTER INTRODUCTION. Research Background The developmens of new echnologies and global compeiion have emphasized he need for more accurae esimaion of reliabiliy of a produc, sysem or componen in a shorer ime. Tradiional life daa analysis involves analyzing failure daa in order o quanify he life characerisics of he produc, sysem or componen. In many siuaions i is very difficul, if no impossible, o obain such failure daa under normal operaing condiions because of he long life of oday's producs, he shor ime period beween design and release and he challenge of esing producs ha are used coninuously under normal condiions. Given his difficuly, reliabiliy praciioners have aemped o devise mehods o induce failures quickly by subjecing he producs o severer environmenal condiions wihou inroducing addiional failure modes oher han hose observed under normal operaing condiions. The failure daa obained under he severe condiions are used o esimae he life characerisics, and he reliabiliy performance of producs a normal operaing condiions. The erm acceleraed life esing (ALT) has been used o describe hose mehods. The accuracy of reliabiliy esimaion depends on he models ha relae he failure daa under severe condiions, or high sress, o ha a normal operaing condiions, or design sress. Elsayed (996) classifies hese models ino hree groups: saisics models,

15 physics-saisics models, and physics-experimenal models. Furhermore, he classifies he saisics models ino wo sub-caegories: parameric and nonparameric models. Wih parameric models, failure imes a each sress level are used o deermine he mos appropriae failure ime disribuion along wih is parameers. Parameric models assume ha he failure imes a differen sress levels are relaed o each oher by a common failure ime disribuion wih differen parameers. Usually, he shape parameers of he failure ime disribuion remain unchanged for all sress levels, bu he scale parameers may presen a muliplicaive relaionship wih he sress levels. Nonparameric models relax he requiremen of he common failure ime disribuion, i.e., no common failure ime disribuion is required. Cox s Proporional Hazards (PH) model (97, 975) is he mos popular nonparameric model. I has become he sandard nonparameric regression model for acceleraed life esing in he pas few years. This model usually produces good reliabiliy esimaion wih failure daa for which he proporional hazards assumpion holds and even when i does no hold exacly. In many applicaions, however, i is ofen unreasonable o assume he effecs of covariaes (sresses) on he hazard raes remain fixed over ime. Brass (97) observes ha he raio of he deah raes, or hazard raes, of wo populaions under differen sress levels (for example, one populaion for smokers and he oher for non-smokers) is no consan wih age, or ime, bu follows a more complicaed course, in paricular converging closer o uniy for older people. So he PH model is no suiable for his case. Brass (974) proposes a more realisic model: he Proporional Odds (PO) model. The Proporional Odds model has been successfully used in caegorical daa analysis (McCullagh 980,

16 3 Agresi and Lang 993) and survival analysis (Hannerz 00) in he medical fields. The PO model has a disinc differen assumpion on proporionaliy, and is complemenary o he PH model. I has no been used in reliabiliy analysis of acceleraed life esing so far. The accuracy of reliabiliy esimaion is a major concern in acceleraed life esing, since i is imporan for making appropriae subsequen decisions regarding prevenive mainenance, replacemen and warrany policies. In he acceleraed life esing experimens, a comprehensive reliabiliy esimaion procedure includes an appropriae ALT model and a carefully designed es plan in order o achieve high accuracy of he reliabiliy esimaes. An appropriae ALT model is imporan since i explains he influences of he sresses on he expeced life of a produc based on is physical properies and he relaed saisical properies. On he oher hand, a carefully designed es plan improves he accuracy and efficiency of he reliabiliy esimaion. The design of an acceleraed life esing plan consiss of he formulaion of objecive funcion, he deerminaion of consrains and he definiion of he decision variables such as sress levels, sample size, allocaion of es unis o each sress level, sress level changing ime and es erminaion ime, and ohers. Inappropriae decision variable values resul in inaccurae reliabiliy esimaes and/or unnecessary es resources. Thus i is imporan o design es plans o opimize he objecive funcion under specific ime and cos consrains.

17 4 Mos of he previous work on ALT plans involves only a single sress ype wih wo or hree sress levels. However, as producs become more reliable due o echnological advances, i becomes more difficul o obain significan amoun of failure daa wihin reasonable amoun of ime using single sress ype only. Muliple-sress-ype ALTs have been employed as a means o overcome such difficulies. For insance, Kobayashi e al. (978), Minford (98), Mogilevsky and Shirn (988), and Munikoi and Dhar (988) use wo sress ypes o es cerain ypes of capaciors, and Weis e al. (988) employ wo sress ypes o esimae he lifeime of silicon phoodeecors.. Problem Definiion In his disseraion, we invesigae wo relaed problems. The firs problem deals wih a nonparameric acceleraed life esing model, proporional odds model, and is applicabiliy for reliabiliy predicaion a normal operaing condiion. We also discuss is characerisics, robusness and is accuracy of reliabiliy esimaes. We begin by defining he odds funcion θ () reliabiliy funcion:, which is he raio of cumulaive disribuion funcion o he θ F () F () () = R() = F(), (.) or

18 5 Pr( T ) Pr( T ) θ () = =, (.) Pr( T > ) Pr( T ) where T is he failure ime of a es uni. Therefore he odds funcion θ () is lierally he odds of failure of a uni a ime. The odds funcion θ () is relaed o he hazard rae λ () a ime by θ () λ() = θ () +, (.3) or θ ( u) du 0 () e λ =. (.4) The proporional hazards model is a widely used nonparameric acceleraed life esing model, which assumes ha he covariaes, or sresses, ac muliplicaively on he hazards rae funcion, and is expressed as: k λ( ; z) = λ0( )exp( β z) = λ0( )exp β jzj, (.5) j=

19 6 where z is he vecor of he applied sresses, β is he vecor of unknown parameers and λ () 0 is an arbirary baseline hazard rae. However proporional hazards model may no be he mos appropriae model especially when he hazards raes a differen sress levels converge o he baseline hazard rae over ime. In his case, proporional odds model is more accurae for reliabiliy esimaion. Failure imes under wo differen sress levels are said o follow proporional odds model if θ(; z) = exp[ β ( z z)]. (.6) θ (; z ) The curren esimaion procedures in lieraures for he PO model are eiher oo complicaed for pracical acceleraed life esing use, or have no rigorous jusificaion for large sample properies. In lieraures, Dabrowska e al. (988) and Wu (995) propose esimaion mehods for only wo-sample daa. Pei (984) esimaes he parameers of he PO model using ranks, which ignores he acual observaions, and resuls in corresponding inaccurae esimaion. Murphy e al. (997) propose he esimaion mehod of he PO model based on profile likelihood. The likelihood is consruced from he condiional probabiliy densiy. Because he number of unknown parameers in he profile likelihood is exremely large, he esimaion calculaion is also exremely compuaionally inensive.

20 7 In his disseraion, a new approach for he reliabiliy esimaion of acceleraed life esing based on proporional odds model is proposed. The esimaes obained from his approach are verified using boh simulaion sudy and experimenal failure ime daa. We may use his new approach o predic he poin esimae of reliabiliy of he produc a any ime and any sress level. The inerval esimaes of model parameers and reliabiliy of he produc are also ineresing o us in his disseraion. We consruc he confidence inervals hrough he covariance marix obained by aking he inverse of Fisher informaion marix. In order o validae he assumpion of he proposed acceleraed life esing model and esimaion procedures, we use boh likelihood raio es o es he model sufficiency and Cox-Snell residual o verify he proporionaliy. The second major problem of his disseraion deals wih he accuracy of reliabiliy esimaion. In order o increase he accuracy of reliabiliy esimaes in acceleraed life esing problem, a carefully designed es plan is required. This es plan is designed o minimize a specified crierion, usually he variance of a reliabiliy-relaed esimae, such as reliabiliy funcion, mean ime o failure and a percenile of failure ime, under specific ime and cos consrains. Mos of he previous work on ALT plans involves only a single sress ype wih wo or hree sress levels. However, as producs become more reliable due o echnological advances, i becomes more difficul o obain significan amoun of failure daa wihin reasonable amoun of ime using single sress ype only. Muliple-sress-ype ALTs have been employed as a means o overcome such difficulies.

21 8 For insance, Kobayashi e al. (978), Minford (98), Mogilevsky and Shirn (988), and Munikoi and Dhar (988) use wo sress ypes o es cerain ypes of capaciors, and Weis e al. (988) employ wo sress ypes o esimae he lifeime of silicon phoodeecors. Unlike he case of he single-sress-ype ALT, lile work has been done on designing muliple-sress-ype ALT plans. Escobar and Meeker (995) develop saisically opimal and pracical plans wih wo sress ypes wih no ineracion beween hem. However, if prior informaion does no suppor he nonexisence of ineracion or if he so-called sliding level echnique canno be employed o avoid he poenial ineracion, hen he analysis based on he main effecs only could lead o serious bias in esimaion. Park and Yum (996) develop ALT plans in which wo sresses are employed wih possible ineracion beween hem wih exponenial disribuion assumpion. Elsayed and Zhang (005) consider opimum ALT plans wih proporional hazard model, which involve only wo sress ypes wih wo sress levels for each sress ype. In his disseraion, we also propose o design and develop opimum muliple-sress-ype acceleraed life esing plans based on he proporional odds model wih boh consansress loading and simple sep-sress loading. The objecive funcion is chosen o minimize he asympoic variance of reliabiliy funcion esimae a he design sress condiions. The plans deermine he opimum sress levels, he number of es unis allocaed o each sress level, he sress level changes and he corresponding changing imes, and/or he es erminaion ime. We adop he widely used cumulaive exposure model o derive he cumulaive disribuion funcion of he failure ime for a es uni

22 9 experiencing sep-sress loading. Since he model parameers are unknown before he es planning, we use he esimaes of hose parameers hrough a preliminary baseline experimens or hrough engineering experience o design he opimum es plans..3 Organizaion of he Disseraion Proposal The remainder of he disseraion proposal is organized as follows. Chaper provides a horough review of he curren lieraure of acceleraed life esing models and discusses he problems encounered in ALT modeling. In chaper 3, we propose he new approach for acceleraed life esing based on proporional odds model afer careful invesigaion of he properies of odds funcion. This new approach is verified by boh a simulaion sudy and an experimenal daa applicaion. In chaper 4, we consruc he confidence inervals for he model parameers and reliabiliy esimae a design sress condiions based on proporional odds model. We also provide mehods o invesigae he model sufficiency based on likelihood raio es and o validae he model assumpion based on Cox-Snell residuals. In chaper 5, we design opimum muliple-sress-ype acceleraed esing plans based on proporional odds model wih boh consan-sress loading and simple sepsress loading. Those carefully designed es plans provide he mos accurae reliabiliy esimaes a design sress condiions since he plans are achieved by minimizing he asympoic variance of reliabiliy esimaes a design sress condiion. A preliminary invesigaion of he equivalen ALT plans is presened in chaper 6. The resuls of his research enable reliabiliy praciioners o choose he appropriae ALT plan according o heir pracice and available resources. In chaper 7 we conduc experimenal esing using miniaure ligh bulbs as es unis and subjec hem o various es plans. We validae he

23 0 reliabiliy esimaes a differen sress condiions. In chaper 8, we give conclusive remarks of his research and presen fuure work relaed o he PO-based acceleraed life esing model and he proposed opimum muliple-sress-ype acceleraed life esing plans based on he proporional odds model.

24 CHAPTER LITERATURE REVIEW In his chaper, we presen a deailed overview of he reliabiliy models for acceleraed life esing and he moivaion of he new ALT mehod. We hen discuss he review of he ALT es plans, objecive consrains and hose limiaions. We begin by he review of he proporional hazards (PH) model and is parameer esimaion mehods in secion... In secion.., we presen a group of models, he acceleraed failure ime (AFT) models, and poin ou ha he Weibull disribuion is he only disribuion ha yields boh a proporional hazards and an acceleraed failure ime model. Afer ha, we inroduce he proporional odds (PO) model as a complemen o he PH model and several parameer esimaion mehods for he PO model in secion..3. Since he curren parameer esimaion mehods for he PO model are no suiable for acceleraed life esing, we conclude ha a new esimaion mehod of he PO model is needed for acceleraed life esing. In secion., we presen a horough review of lieraure abou ALT plans. The review indicaes ha here are no opimum plans for proporional odds cases when neiher he AFT nor he PH assumpions hold for he failure ime daa. Furhermore, curren exising ALT plans only consider single sress ype siuaions. However, as producs become more reliable hanks o echnological advances, i becomes more difficul o induce significan amoun of failure ime daa wihin he limied esing duraion using a single sress ype only. Muliple-sress-ype ALTs and ALT plans have o be applied o overcome such difficulies.

25 . ALT Models.. Cox s Proporional Hazards Model The mos widely used model describing he influence of covariaes on he failure ime disribuion is he proporional hazards (PH) or Cox model, inroduced by D. Cox (97). I was firs used in biomedical applicaions. The PH model has also been used as an acceleraed life esing model. Alhough i does no assume any form of he failure ime disribuion, i sill allows us o quanify he relaionship beween he failure ime and a se of explanaory variables, which are sresses in acceleraed life esing area. A brief descripion of he PH model follows. Le T denoe he failure ime, τ he righ censoring ime. The daa, based on a sample of size n, consiss of he riple ( i, Ii, z i), i =, K, n where i is he failure ime of he ih es uni under sudy, I i is he failure indicaor for he ih uni ( I i = if he failure has occurred, which means i τ, and I i = 0 if he failure ime is righ-censored, or i > τ ), and z = ( z, K, z ) is he vecor of covariaes or sresses for he ih uni, which affecs i i ki he reliabiliy disribuion of T, including emperaure, volage, humidiy, ec. Le λ( ; z ) be he hazard rae a ime for a uni wih a vecor of sresses, z. The basic PH model is as follows: λ(; z) = λ () c( β z), (.) 0

26 3 where λ 0( ) an arbirary baseline hazard rae; z = (,, K, ) a vecor of he covariaes or he applied sresses; z z z k β = (,, K, ) a vecor of he unknown regression parameers; β β β k c( β z) a known funcion k he number of he sresses. Because hazard rae funcion λ( ; z ) mus be posiive, a common feasible funcion for c( β z) is k c( β z) = exp( β z) = exp β jzj, j= yielding k λ( ; z) = λ0( )exp( β z) = λ0( )exp β jzj. (.) j= The main assumpion of he proporional hazards (PH) model is ha he raio of wo hazard raes of wo unis under wo sress levels z and z is consan over ime. In oher words:

27 4 λ( ; z ) λ ( )exp( β z ) k = = exp[ β j( zj z j)]. λ( ; z ) λ ( )exp( ) j= 0 0 β z This implies ha he hazard raes are proporional o he applied sress levels. Wihou he specificaion of he form of baseline hazard rae funcion λ 0( ), he inference for β could be based on a parial or condiional likelihood raher han a full likelihood approach. Suppose ha here are no ies beween he failure imes. Le < < L < D denoe he ordered failure imes wih corresponding sresses z, z, K, z D. If D< n, hen he remaining n D unis are censored. Define he risk se a ime i, R( i ), as he se of all unis ha are sill surviving a a ime jus prior o i. Then he parial likelihood, based on he hazard funcion as specified by Eq. (.), is expressed by L( β ) D β = i= exp( zi ). (.3) exp( β z ) j R( i ) j The log likelihood is obained as D D i i= i= l( β) = β z ln[ exp( β z )]. (.4) j R( i ) j

28 5 The parial maximum likelihood esimaes are found by solving a se of nonlinear equaions, which are obained by seing he firs derivaes of he log likelihood wih respec o he k unknown parameers β, β, K, βk o zero. This can be done numerically using a Newon-Raphson mehod or some oher ieraive mehods. Noe ha Eq. (.4) does no depend on he baseline hazard rae λ 0 (), so ha inferences may be made on he effecs of he covariaes wihou knowing λ 0( ). Due o he way failure imes are recorded, ies beween failure imes are ofen found in he daa. Alernae parial likelihoods have been provided by a variey of auhors (Breslow 974, Efron 977, Cox 97) when here are ies beween failure imes. Among hem, Efron suggess a parial likelihood as L( β ) = exp( β s ), [ exp( ) exp( z )] D i di j i= j= β z k R( i) k k D k d β i i where < < L < denoe he D disinc, ordered, failure imes, D d i is he number of failures a i, D i is he se of all unis ha failure a ime i, s i is he sum of he vecor z j over all unis ha failure a i, ha is s i = z, j j Di

29 6 R( i ) is he se of all unis a risk jus prior o i. Afer obaining he esimaes of he unknown parameers β, here are several ways o esimae he reliabiliy of unis a he design sress level z D. One way, wihou he explici specificaion of baseline hazard rae funcion, is based on Breslow s esimaor (Breslow 975) of he baseline cumulaive hazard rae. Breslow s esimaor of he baseline cumulaive hazard rae is given by ˆ () d, exp( ) i Λ 0 = ˆ i β z j R( i ) j which is a sep funcion wih jumps a he observed failure imes. Therefore he esimaor of he baseline reliabiliy funcion, R0( ) = exp[ Λ 0( )] is given by Rˆ ( ) = exp[ Λ ˆ ( )]. 0 0 This is an esimaor of he reliabiliy funcion of a uni wih a baseline se of sresses, z = 0. To esimae he reliabiliy funcion for a uni a he design sresses level z D, we use he esimaor R ˆ(; z ) = Rˆ () D 0 exp( ˆ β zd ).

30 7 Oher ways o esimae he reliabiliy funcion of unis a design sress require esimaing he baseline hazard rae funcion λ 0 () firs. Anderson (980) describes a piecewise smooh esimae of λ 0 () insead of sep funcion of Λ ˆ 0 (). He assumes λ 0 () o be a quadraic spline. Oher auhors (Elsayed 00) also sugges o use quadraic funcion o esimae λ 0( ). Based on he esimaor ˆ λ () 0, he esimaor of he reliabiliy funcion is given by ˆ ˆ R ( ; z ) exp[ exp( ) ˆ D = β z D λ 0( u) du]. 0 The PH model usually produces good reliabiliy esimaion wih failure daa for which he proporional hazards assumpion does no even hold exacly... Acceleraed Failure Time Models In he previous secion, we presen nonparameric mehods for acceleraed life esing which do no require any specific disribuional assumpions abou he shape of he reliabiliy funcion. In his secion, we presen a class of parameric models for ALT, named Acceleraed Failure Time (AFT) models, which have an acceleraed failure ime model represenaion and a linear model represenaion in log of failure ime. The acceleraed failure ime model represenaion assumes ha he covariaes ac muliplicaively on he failure ime, or linearly on he log of failure ime. Le T denoe

31 8 he failure ime and z a vecor of covariaes, he AFT model is defined by he relaionship R( ; z) = R [exp( δ z) ]. (.5) 0 The facor exp( δ z ) is called an acceleraion facor elling how a change in he covariae values changes he ime scale from he baseline ime scale. The model presened by Eq. (.5) implies ha hazard rae relaionship is given by λ( ; z) = λ [exp( δ z) ]exp( δ z). (.6) 0 The second represenaion of his group of models is he linear relaionship beween log failure ime and he sress values, and is given by lnt = μ + + ω z ψw, where ω is a vecor of regression coefficiens and W is he random error. The wo represenaions are closely relaed. If we le R0( ) o be he reliabiliy funcion of he random variable exp( μ + ψw ), hen he linear log failure ime model is equivalen o he AFT model wih δ = ω.

32 9 A variey of disribuions can be used for W or, equivalenly, for R 0 (). If R 0 () is assumed o be Weibull reliabiliy funcion, or W has a sandard exreme value disribuion, we will hen have a Weibull disribued AFT model. If R0( ) is assumed o be log-logisic reliabiliy funcion, or W has a logisic disribuion, we have a log-logisic disribued AFT model. If R0( ) is assumed o be log-normal reliabiliy funcion, or W has a normal disribuion, we have a log-normal disribued AFT model. Therefore he AFT model also implies ha he failure imes a differen sress levels follow a common disribuion wih he same shape parameer bu differen scale parameer, which has some relaionship wih he applied sresses. For he AFT models, esimaes usually mus be found numerically. When hese parameric AFT models provide a good fi o failure ime daa, hey end o give more precise esimaes of he reliabiliy. However, if a parameric model is chosen incorrecly, i may lead o consisen esimaes wih significan errors. We noe ha he Weibull disribuion is he only coninuous disribuion ha yields boh a proporional hazards and an acceleraed failure ime model. A PH model for T wih a Weibull baseline hazard λ () 0 = αλ α is λ z λ β z αλ β z. (.7) ( ; ) 0( )exp( = ) = ( α )exp( ) An AFT model for T wih a Weibull baseline hazard λ () 0 = αλ α is λ λ δ δ αλ αδ. (.8) ( ; ) 0[exp( ) ]exp( ) α z = z z = exp( z)

33 0 Comparing Eq. (.7) and Eq. (.8), we have β = αδ...3 Proporional Odds Model and Inference The PH model is a widely used nonparameric model for acceleraed life esing because of is relaive easiness of he esimaing procedure of parial likelihood and is large sample inference properies demonsraed using maringale heory. Moreover, reliabiliy praciioners have easy access o saisical sofware, including SAS, BMDP, and S-Plus, as described by Klein (997), for his model. Therefore, here is a empaion o use he PH model o analyze failure ime daa, even when he model does no fi he daa well. In many applicaions, however, i is ofen unreasonable o assume he effecs of covariaes on he hazard raes remain fixed over ime. Brass (97) observes ha he raio of he deah raes, or hazard raes, of wo populaions under differen sress levels (for example, one populaion for smokers and he oher for non-smokers) is no consan wih age, or ime, bu follows a more complicaed course, in paricular converging closer o uniy for older people. So he PH model is no suiable for his case. Brass (974) proposes a more realisic model as F(; z) β z F0 () = e. (.9) F( ; z) F ( ) 0

34 This model is referred o as he Proporional Odds (PO) model since he odds funcions, F () which are defined as θ (), under differen sress levels are proporional o each F ( ) oher. The proporional odds models have been widely used as a kind of ordinal logisic regression model for caegorical daa analysis, as described by McCullagh (980) and Agresi (00). Recenly he PO model has received more aenions for survival daa analysis. Dabrowska e al. (988) and Wu (995) propose esimaion mehods for only wo-sample life ime daa based on PO model. Pei (984) esimaes he parameer in he PO model using ranks, which ignores he acual observaions, and resuls in corresponding inaccurae esimaion. Murphy e al. (997) propose he esimaion mehod of he PO model based on profile likelihood. The likelihood is consruced from he condiional probabiliy densiy. Because he number of unknown parameers in he profile likelihood is exremely large, he esimaion calculaion is also exremely compuaionally inensive. The descripion of he PO model and he curren inference procedures of parameer esimaion are summarized as follows...3. Descripion of PO model Le T denoe he failure ime, τ he righ censoring ime. The daa, based on a sample of size n, consiss of he riple ( i, Ii, z i), i =, K, n where i is he failure ime of he ih uni under sudy, I i is he failure indicaor for he ih uni ( I i = if he failure has

35 occurred, which means i τ, and I i = 0 if he failure ime is righ-censored, or i and z (,, ) i = zi K zip is he vecor of covariaes, or sresses, for he ih uni. > τ ), F (; z) Le θ (; z) = F ( ; z) be he odds funcion, which is he raio of he probabiliy of a uni s failure o he probabiliy of is no failing, a ime for a uni wih a vecor of sresses, z. The basic PO model is given: θ( ; z) = θ ( )exp( β z), (.0) 0 where θ 0( ) an arbirary baseline odds funcion; z = (,, K, ) a vecor of he covariaes or he applied sresses; z z z k β = (,, K, ) a vecor of he unknown regression parameers; β β β k k he number of he sresses. For wo failure ime samples wih sress levels z and z, he difference beween he respecive log odds funcions log[ θ( ; z )] log[ θ( ; z )] = β ( z z )

36 3 is independen of he baseline odds funcion θ 0 (), and furhermore of he ime, hus one odds funcion is consanly proporional o he oher. The baseline odds funcion could be any monoone increasing funcion of ime wih he propery of θ 0 (0) = 0. When θ 0( ) = ϕ, PO model described by Eq. (.0) becomes he log-logisic acceleraed failure ime model (Benne 983), which is a special case of he general PO model. To invesigae he relaion beween he PO model and he PH model, we represen he PO model as described by Eq. (.0) wih hazard rae funcion. Afer mahemaical ransform, he PO model in Eq. (.0) can be represened by β z e λ 0() λ (; z) = ( β z e ) F ( ) 0, (.) where F0( ) is he baseline cumulaive disribuion funcion. I is easy o see, from Eq. (.), ha in wo failure ime sample seup, he model implies ha he raio of he hazard raes a differen sress levels converges o over ime. Thus he PO model is useful when he covariae (sress) effec on he hazard rae diminishes over ime...3. Esimaing he Parameer in a Two-sample PO Model

37 4 Dabrowska and Doksum (988) and Wu (995) give an esimaion procedure for he socalled generalized odds-rae models, of which he PO model is a special case, in wosample case. They show ha he esimaes are consisen and asympoically normal. They firs consider he generalized odds funcion for a random failure ime T as defined by c ( F ( )), c > 0 c Λ T ( c) = c ( F( )) log[ F ( )], c = 0 (.) where F ( ) is he cumulaive disribuion funcion for failure ime T. Noe ha Λ ( 0) is he cumulaive hazard rae, whereas Λ ( ) is he odds of he T failure before ime. For c oher han, Λ ( c ) also has an inerpreaion as an odds T T funcion for some siuaions. When considering experimens involving covariaes ha affec he disribuion of he failure ime, he generalized proporional odds-rae model is represened by Λ (; z c) = e β Λ (; z c), (.3) ( z z) T T where T is a random failure ime sample wih size n associaed wih covariae z, and T is a random failure ime sample wih size n associaed wih covariae z.

38 5 When c =, he generalized proporional odds-rae model as in Eq. (.3) is jus he proporional odds model as described in Eq. (.0). By rewriing (.3), hey obain c+ β ( z z) c+ ( FT ()) df () ( ()) () T e F T df T =. (.4) ( ) Based on Eq. (.4), he esimaor of z z e β ( z z), or ˆ( e β σ ), is given by β ( z z) ˆ( ) σ e = 0 0 ˆ ˆ ( c+ ) Ψ( F ( ))[ ( )] ˆ T F ( ) T df T, (.5) Ψ( Fˆ ( ))[ Fˆ ( )] dfˆ ( ) ( c+ ) T T T where Fˆ T () and Fˆ () T are lef-coninuous empirical disribuions based on he failure ime samples T and T, respecively, and Ψ() is some score funcion. Then he esimaor of β is given by ˆ β = ˆ σ e ( z z ) β ( z z ln[ ( ) )]. Under some mild regulariy condiions, Dabrowska and Doksum show ha / β ( z z ) β ( z z ( n ˆ ) n) ( σ ( e ) e ) + has an asympoically normal disribuion and an ( z z) efficien score funcion is given by he follow equaion for e β =,

39 6 ( u) ( c )( u) c + Ψ = +, ( z z) However, for e β, he efficien score funcion Ψ() remains unknown. Wu (995) consider he special case of he wo-sample proporional odds model and ( ) show ha an efficien esimaor of z z e β can be consruced based on he soluion of a pair of inegral equaions. Because of he special srucure of he proporional odds model, he soluion of he equaions can be obained in a closed form. Wu furher proves ( ) ha, by selecing any suiable and convenien score funcion, he esimaor of z z e β is asympoically normal. ( ) Alhough he asympoically efficien esimae of z z e β could be consruced explicily for his special wo-sample proporional odds model, his esimaor has limied usage for acceleraed life esing. Firsly, in he acceleraed life esing field, here are always more han wo samples, which means here are always more han wo sress levels. Furhermore his esimaion mehod can be used for he siuaion where here is only one sress ype. If here are wo sress ypes, such a emperaure and volage, or more, we ( ) β z z can no esimae he parameers vecor β from he esimaor of e. Also, an appropriae score funcion is difficul o deermine. Finally, his esimaion mehod doesn give an approach o he reliabiliy esimae a sress levels oher han he wo observed levels.

40 Esimaes of he PO model Using Ranks Pei (984) proposes an approximae esimaion mehod of proporional odds model for failure ime daa using he ranks of he failure imes, insead of he rue observaions, wihou specificaion of he forma of he baseline odds funcion. The approximaion is based upon a Taylor series expansion of he logarihm of he marginal likelihood abou zero, and can be used for censored failure ime daa. The approximae mehod is as follows. Le, K be observed uncensored failure imes and, K, be righ censored, D failure imes. Order he, j =, K, D, in he usual way o obain he ordered failure j D+ n < < imes () K ( D). The rank r j of j is defined in he usual way. The ranks, r j, of he censored failure imes j, j = D+, K, n, are defined as follows: j has rank r j if and only if j lies in he inerval ( ( r ), ( r + ) ), wih (0) = 0 and ( D + ) =. j j Inference for β in he proporional odds model described by Eq. (.0) is now based on he ranks esimae ˆ β [ ( ) ] a, (.6) R = Z B A Z Z where Z is he model marix, and a, A and B are defined as follows:

41 8 a is a n vecor wih a = ( c ) ξ, j j j r A is a n n symmeric marix wih ( A) = ( c )( c )( v ξ ξ ), ij i j ri, rj ri rj B is a n n diagonal marix wih ( B) = ( c )( ξ τ ), jj j rj rj where c j = 0 for j =, K, D and c j = for j = D+, K, n, j u i ξ j =, ( j =,, D), i= + u K i j u i τ j =, ( j =,, D), i= + u K i and

42 9 i j u k u k vij =, ( i j = D). k= + uk k= + uk The u j s are defined as follows: Le m be he number of he ( i = D+, K, n) having j heir ranks, r i, equal o j, j =, K, D, hen define uj = ( mj + ) + K( md + ), j =, K, D, so ha u j is he oal number of failures, censored or uncensored, greaer han or equal o ( j). i Pei (984) also gives he variance-covariance marix M Z B A Z = [ ( ) ] for he esimaor ˆR β based on ranks wihou proof. This esimaor looks appealing o reliabiliy praciioners since i gives a closed form of he esimaion. Bu i acually has lile usage for acceleraed life esing because of he common drawback of rank esimaor, inaccuracy, which is he corresponding resul of he fac ha hese rank esimaes are based on he ranks of he failure ime observaions insead of heir acual values. Meanwhile he esimaion procedure based on ranks does no provide any well-esed approach o he esimaion of he reliabiliy funcion of producs a he design sress level nor does i produce residuals for model checking using he rank esimaes, since no specificaion and inference abou he baseline odds funcion are discussed in his procedure Profile Likelihood Esimaes of PO Model

43 30 The esimaion mehod of he PO model based on profile likelihood is proposed by Murphy e al. (997). The likelihood is consruced from he condiional probabiliy densiy. The profile likelihood esimaion mehod of he PO model by Murphy e al. is summarized as follows. From he PO model described by Eq. (.0) and he relaionship beween odds funcion R( ; z) and reliabiliy funcion, θ (; z) =, we have R (; z) exp( β z) R (; z) =, (.7) θ ( ) + exp( β z) 0 Then we specify he condiional probabiliy densiy wih respec o he sum of Lebesgue and he couning measure by exp( β z) ϑ( ), (.8) [ θ ( ) + exp( β z)] [ θ ( ) + exp( β z)] 0 0 where θ0( ) is he lef limi of θ 0 () a. If θ 0 () is absoluely coninuous, he ϑ () is he derivaive of θ 0 () ; if θ 0 () is discree, hen ϑ() = Δ θ0() = θ0() θ0( ). The Eq. (.8) is he approximaion of he coninuous condiional probabiliy densiy f ( ; z ) defined as

44 3 θ (; z) f(; z) = R(; z) λ(; z) = R(; z) + θ ( ; z ) exp( β z) θ ( ) 0 (; z) exp( + β z ) θ0 ( ) = R = R θ () 0 (; z) exp( β z ) + θ0 ( ) exp( β z) θ ( ) =, [ θ ( ) exp( )] [exp( ) ( )] β z β z + θ0 We consider he use of he condiional probabiliy densiy as described in Eq. (.8) for righ-censored failure ime daa. If we le T denoe he failure ime, le τ o be he righ censoring ime. Then based on a sample of size n, he observaions consis of he riple ( i, Ii, z i), i =, K, n where i is he failure ime of he ih uni under sudy, I i is he failure indicaor for he ih uni ( I i = if he failure has occurred, which means i and I i = 0 if he failure ime is righ-censored, or i τ, > τ ), and z (,, ) i = z i K zki is he vecor of covariaes or sresses for he ih uni. Then he likelihood based on he condiional probabiliy densiy of Eq. (.8) for one observaion is exp( β z) ϑ( ) exp( β z) L (; θ0, β ) = [ θ0( ) + exp( β z)] [ θ0( ) + exp( β z)] [ θ0( ) + exp( β z)] I I. Murphy e al. prove ha if θ 0 () is known o be absoluely coninuous, hen, as in he case of densiy esimaion, here is no maximizer of he likelihood, bu if θ 0 () is known o be discree, hen a maximizer exiss. The likelihood funcion for he whole failure ime observaions can be wrien as

45 3 n exp( β z ) i Δθ0( ) i L(; θ0, β ) = i= [ θ0( i) + exp( β zi)] [ θ0( i ) + exp( β zi)] I i, (.9) where we have replaced ϑ by Δ θ0. The MLE of θ 0 will be a non-decreasing sep funcion wih seps a he observed failure imes. The profile log-likelihood for β is given by Pr lik ( β ) = log L( ; ˆ θ ( β), β ), n 0 where ˆ θ0( β ) maximizes he log-likelihood for a fixed β. The maximum profile likelihood esimaor, ˆ β, maximizes Pr lik ( β ). n The presence of he erms Δ θ0( ) in he likelihood of Eq. (.9) forces he esimaor of i θ 0 o have posiive jumps a he observed failure imes and have no jumps a any oher poins. Thus he number of unknown parameers is k, he number of covariaes, plus he number of observed failure imes. Murphy e al. also prove ha he maximum profile likelihood esimaor of β is consisen, asympoically normal, and efficien. Differeniaion of he profile likelihood yields consisen esimaors of he efficien informaion marix. Addiionally, a profile

46 33 likelihood raio saisic can be compared o perceniles of he chi-squared disribuion o produce asympoic hypohesis ess of he appropriae size. The calculaion of he maximum profile likelihood depends on he numerical algorihms, from which Murphy e al. choose a modificaion of he Newon-Raphson algorihm. The numerical algorihms can no always guaranee a convergen soluion, especially when he size of he unknown parameer is large. As menioned in he paper by Murphy e al, he number of he unknowns in he likelihood funcion of Eq. (.9) is k, he number of covariaes, plus he number of observed failure imes. As he size of failure ime sample increases, he maximizaion problem becomes exremely difficul. Anoher problem of his maximum profile likelihood esimaor of β is he fac ha he esimaor is a funcion of he failure imes only hrough heir ranks. The profile likelihood is he same wheher we use he acual failure ime observaions or replace hem by heir ranks. From he maximum profile likelihood esimaion, he esimaes of he baseline odds funcion can be only obained a he observed failure imes. As a resul, he reliabiliy esimaes also can be only obained a he observed failure imes, only explici reliabiliy funcion esimae is available. This problem also makes he maximum profile likelihood esimaion mehod no applicable o acceleraed life esing which requires he reliabiliy funcion esimaion of he uni a he normal design sress level.. ALT Tes Plans In order o increase he accuracy of reliabiliy esimaes in acceleraed life esing problem, a carefully designed ALT es plan is required. This es plan is designed o

47 34 minimize a specified crierion, usually he variance of a reliabiliy-relaed esimae, such as reliabiliy funcion, mean ime o failure and a percenile of failure ime, under specific ime and cos consrains. For easy implemenaion, commonly used consan sress ALT es plans consis of several equally spaced consan sress levels, each is allocaed a proporion of he oal number of es unis. Such sandard es plans are usually inefficien for esimaing reliabiliy of he produc a design sress. Earlier work by Meeker and Nelson (978) propose opimal saisical plans for consan sress ALTs which include only wo sress levels. Such plans lack robusness since he assumed life-sress relaionship is difficul, if no impossible, o validae. Meeker and Hahn (985) address his by proposing a 4:: allocaion raio for low, middle, and high sress levels and give he opimal low level by assuming he middle sress o be he average of he high and he low sress levels. In recen years, by considering oher es consrains and allowing non-consan shape parameer of he failure ime disribuion (Meeer and Meeker, 994), he use of compromised ALT plans for hree sress levels wihou opimizing he middle sress level and allocaion of es unis is advocaed. The exising ALT plans could be classified ino wo caegories: parameric ALT plans and nonparameric ALT plans. Parameric ALT plans are based on a pre-specified common reliabiliy disribuion and an assumed lifesress relaionship. These plans are ypically no robus o deviaions from he model assumpions and he model parameers. Recenly, Elsayed and Jiao (00) consider opimum ALT plans based on nonparameric proporional hazards model. These nonparameric ALT plans are based on he minimizaion of he asympoic variance of

48 35 he hazard rae esimae a design sress. The plans deermine he opimum hree sress levels and he allocaion of es unis o each sress level. Mos of he previous work on ALT plans involves only a single sress ype wih wo or hree sress levels. However, as producs become more reliable due o echnological advances, i becomes more difficul o obain significan amoun of failure daa wihin reasonable amoun of ime using single sress ype only. Muliple-sress-ype ALTs have been employed as a means o overcome such difficulies. For insance, Kobayashi e al. (978), Minford (98), Mogilevsky and Shirn (988), and Munikoi and Dhar (988) use wo sress ypes o es cerain ypes of capaciors, and Weis e al. (988) employ wo sress ypes o esimae he lifeime of silicon phoodeecors. Unlike he case of he single-sress-ype ALT, lile work has been done on designing muliple-sress-ype ALT plans. Escobar and Meeker (995) develop saisically opimal and pracical plans wih wo sress ypes wih no ineracion beween hem. However, if prior informaion does no suppor he nonexisence of ineracion or if he so-called sliding level echnique canno be employed o avoid he poenial ineracion, hen he analysis based on he main effecs only could lead o serious bias in esimaion. Park and Yum (996) develop ALT plans in which wo sresses are employed wih possible ineracion beween hem wih exponenial disribuion assumpion. Elsayed and Zhang (005) consider opimum ALT plans wih proporional hazard model, which involve wo sress ypes wih wo sress levels for each sress ype. Based on he employed acceleraed life esing model in designing an ALT, we classify he exising ALT plans ino wo caegories: Acceleraed failure ime model based (AFT-

49 36 based) ALT plans, Proporional hazards model based (PH-based) ALT plans. There is exensive research concerning he AFT-based opimum ALT which involves he mos widely used parameric acceleraed failure ime regression model wih he excepion of Elsayed and Jiao (00) who consider opimum ALT plans based on nonparameric proporional hazards model. There is lile research ha invesigaes he design of opimum ALT plans using muliple sress ypes. We now describe hese caegories in deails... AFT-based ALT Plans In general, AFT-based ALT plans use he parameric acceleraed failure ime (AFT) regression model. These ALT plans assume ha he failure imes follow a predeermined disribuion, e.g. Weibull disribuion, lognormal disribuion, exreme value disribuion, ec. Under his model, a higher sress level has he effec on reduced failure ime hrough a scale facor. This can be expressed in erms of reliabiliy funcion as R( ; z) = R [exp( δ z) ]. (.0) 0 The facor exp( δ z) is called an acceleraion facor. Since he performance of hese ALT plans is highly dependen on he assumed failure ime disribuion, he opimum plans hus obained are ypically no robus o he deviaions from he assumed disribuion.... General Assumpions of AFT-based ALT Plans

50 37 The AFT-based ALT plans use he following general assumpions: () The log-ime-o-failure for each uni follows a locaion-scale disribuion such ha y μ Pr( Y y) =Φ σ, where y = log( ) is he logarihm of failure ime, μ and σ are locaion and scale parameers respecively, and Φ( ) is he sandard form, cumulaive disribuion funcion, of he locaion-scale disribuion. () Failure imes for all es unis, a all sress levels, are saisically independen. (3) The locaion parameer μ is a linear funcion of sresses (z, z,...). Specifically we assume ha μ = μ( z ) = γ + γ z, for a one-sress ALT, or 0 μ = μ( z, z ) = γ + γ z + γ z, for a wo-sress ALT 0 (4) The scale parameer σ does no depend on he sress levels. (5) All unis are esed unil imeτ, a pre-specified censoring ime. The γ i s and σ are unknown parameers o be esimaed from he available ALT daa.

51 38... Crieria for he Developmen of AFT-based ALT Plans In developing preliminary es plans we assume consan sress levels during he ALT. The consan sress es plans in he lieraure can be furher classified as: () ALT plans o esimae perceniles of he life disribuion a specified design sress, and () ALT plans o esimae reliabiliy a specified ime and design sress based on differen opimizaion crierion.. ALT plans o esimae perceniles of he life disribuion a specified design sress Chernoff (96) considers maximum likelihood (ML) esimaion of he failure rae of an exponenial disribuion a he design sress level. The relaionship for he failure rae is assumed as a quadraic funcion of sress and an exponenial funcion of sress. He gives opimum plans boh for simulaneous esing wih Type I censored daa and for successive esing wih complee daa. Chernoff calls he plans Locally opimum because hey depend on he rue (unknown) parameer values. Mos opimum designs associaed wih nonlinear esimaion problems (including esimaion wih censored daa) resul in locally opimum designs. Mann e al. (974) consider linear esimaion wih order saisics o esimae a percenile of an exreme value (or Weibull) disribuion a design sress and obain opimal plans for failure daa wih censored observaions.

52 39 Nelson and Kielpinski (975, 976) obain opimum plans and bes radiional plans (radiional plans use equally spaced levels of sress wih equal allocaion of es unis o each sress level) for he median of a normal and lognormal disribuion. Their model assumes ha he normal disribuion locaion parameer μ (also he mean) is a linear funcion of sress and he scale parameer σ (also he sandard deviaion) does no depend on sress. They also assume simulaneous esing of all es unis and censoring a a prespecified ime. Nelson and Meeker (978) provide similar opimum es plans o esimae perceniles of Weibull and smalles exreme-value disribuions a a specified design sress when es unis are oversressed. They assume ha he smalles exreme-value locaion parameer μ (also he 0.63 percenile) is a linear funcion of sress and ha he scale parameer σ is consan. Using similar assumpions, Meeker (984) compares he saisically opimum es plans o more pracical es plans ha have hree levels of sress. Meeker and Hahn (985) provide exensive ables and pracical guidelines for planning an ALT. They presen a saisically opimum es plan and hen sugges an alernaive plan ha mees pracical consrains and has desirable saisical properies. The ables allow assessmen of he effec of reducing he esing sress levels (hereby reducing he degree of exrapolaion) on saisical precision. Jensen and Meeker (990) provide a compuer program ha allows he user o develop and compare opimum and compromise ALT plans. The program also allows he user o modify or specify plans and evaluae heir properies.

53 40 Previous work on planning ALTs assumes ha he scale parameer σ for a locaion-scale disribuion of log lifeime remains consan over all sress levels. This assumpion is inappropriae for many applicaions including acceleraed ess for meal faigue and cerain elecronic componens. Meeer and Meeker (994) exend he exising maximum likelihood heory for es planning o he nonconsan scale parameer model and presen es plans for a large range of pracical esing. The es seup assumes simulaneous esing of es unis wih ime censoring. Baron (99) proposes a variaion of he opimum ALT plans described by Nelson and ohers. He shows how o minimize he maximum es-sress subjec o meeing a cerain sandard-deviaion of he reliabiliy esimae a normal operaing condiions. Nelson, Meeker and ohers choose equal censoring imes a each sress level. This pracice does no compleely cover field applicaions. A long censoring ime ha yields sufficien failure a he lowes sress level is considered oo long a he highes sress level. Yang (994) proposes an opimum design of 4-level consan-sress acceleraed life es plans wih various censoring imes. The opimal plans choose he sress level, es unis allocaed o each sress, and censoring imes o minimize he asympoic variance of he MLE of he mean life a design sress and es duraion. The es plans derived are proven o be more robus han he 3-level bes-compromise es plans.. ALT plans o esimae reliabiliy a specified ime and design sress

54 4 Easerling (975) considers he deerminaion of a lower bound on reliabiliy a design sress condiions. He uilizes he probi model o show ha one can obain an appreciably improved lower bound on reliabiliy by allocaing es unis o sresses higher han he design sress, insead of esing all unis a sresses close o or a he design sress. Marz and Waerman (977) use Bayesian mehods for deermining he opimal es sress for a single es uni o esimae he survival probabiliy a a design sress. Maxim e al. (977) deermine opimum acceleraed es plans using he D-opimaliy crierion and assuming a bivariae exponenial or Weibull model. Meeker and Hahn (977) consider he opimum allocaion of es unis o oversress condiions when i is desired o esimae he survival probabiliy o a specified ime a a design condiion. The opimal crierion is o minimize he large sample variance under a logisic model assumpion. Based on he review of he lieraures, we summarize he following guidelines for planning ALTs: () Assuming a linear life-sress relaionship for he saisically opimum plan, ess a only wo levels of sress.

55 4 () Choosing he highes sress level o be as high as possible will increase precision and saisical efficiency. Bu he highes sress level should no be so high ha new failure modes would be inroduced and he assumed life-sress relaionship would be invalid. (3) Compromising es plans ha use hree or four levels of sress have somewha reduced saisical efficiency bu end o be more robus o misspecificaion of he model and is parameers. (4) More es unis should be allocaed o lower sress levels. There are wo reasons for his: small porion of unis will fail a he lower sress due o he limied es ime and inferences a lower sress levels are closer o he exrapolaion a design sress condiions. So more failures a lower sresses resul in a more accurae esimae of reliabiliy a normal condiions... PH-based ALT Plans Recenly Elsayed and Jiao (00) consider opimum ALT plans based on he nonparameric proporional hazards model. This ALT plan deermines hree opimum sress levels: high, medium and low levels, and opimum allocaions of unis o he hree sress levels such ha he asympoic variance of he hazard rae esimae a he design sress level is minimized. The consrains include he maximum available es duraion, oal number of es unis and he minimum number of failures a each sress level. The opimum ALT plans are based on he proporional hazards model. The procedure of he PH-based opimum ALT plans is summarized as follows.

56 43 The proporional hazards (PH) model is expressed as, λ( z ; ) = λ ( )exp( βz). (.) 0 The baseline hazard funcion λ 0( ) is assumed o be linear wih ime, λ () = γ + γ. (.) 0 0 Subsiuing Eq. (.) ino Eq. (.) yields λ( z ; ) = ( γ + γ )exp( βz). (.3) 0 Then he corresponding cumulaive hazard funcion Λ ( ; z) and reliabiliy funcion R(; z ) are obained respecively as, λ γ0 γ β, (.4) 0 Λ ( ; z) = ( u) du = ( + / )exp( z) R( ; z) = exp[ Λ ( ; z)] = exp[ ( γ + γ / )exp( βz)]. (.5) 0 For a failure ime sample ( i, Ii, z i), i =, K, n, where i is he failure ime of he ih uni under sudy, I i is he failure indicaor for he ih uni ( I i = if he failure has occurred,

57 44 which means i τ, and I i = 0 if he failure ime is righ-censored, or i > τ ), and z i is he sress level for he ih uni (only a single sress ype is considered in his siuaion). Based on he proporional hazard model and he baseline hazard funcion, he log likelihood for he failure ime sample is consruced as n l = [ I ln λ( ; z ) Λ( ; z )] i= n γ i = [ Iiln( γ0 + γi) + Iiβzi ( γ0i + )exp( βz)]. i= i i i i i (.6) The maximum likelihood esimaes ˆ γ ˆ ˆ 0, γ and β are he parameer values ha maximize he above sample log likelihood funcion and are obained by seing he hree firs derivaives of he log likelihood funcion wih respec o he parameers o zero, and solving he resulan equaions simulaneously. An acceleraed life es is planned o obain he mos accurae hazard rae esimae of he producs a he design sress level under given consrains (ime, cos, es unis, ec.). In order o achieve such objecive, he opimal crierion chosen is o minimize he asympoic variance of he hazard rae esimae a he design sress level over a prespecified period of ime T, ha is o minimize ˆ T T Var[ λ ( ; z )] [( ˆ ˆ D d = Var γ 0 + γ )exp( β zd)] d 0, (.7) 0 ˆ

58 45 where he variance Var[ ˆ λ ( ; z D )] is obained hrough dela mehod as T ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ z λ λ λ λ λ λ D Var[ λ( ; z )] [( ˆ ˆ D = Var γ0 + γ) e β ] =,,,, ˆ γ ˆ ˆ Σ ˆ ˆ ˆ 0 γ β γ0 γ β, (.8) and Σ is he variance-covariance marix of he parameer esimaes ( ˆ γ ˆ ˆ 0, γ, β ), which is calculaed as he inverse of he Fisher informaion marix. Le s consider he seup of an ALT plan, which is defined by. There is a single acceleraing sress ype z.. Tes can be run a hree or more sress levels. 3. Toal es duraion is limied o τ. 4. A oal of n es unis is available for esing. 5. The highes sress level z H is defined a he highes level a which he failure mode will no change. 6. Iniial values for he model parameer esimaes are given by a baseline experimen. Under he consrains of available es unis, es ime and specificaion of minimum number of failures a each sress level, he objecive of he ALT plans is o opimally allocae sress levels and es unis so ha he asympoic variance of he hazard rae esimae a he design sress is minimized over a pre-specified period of ime T. The opimum decision variables ( z, z, p, p, p ) are deermined by solving he following * * * * * L M 3 nonlinear opimizaion problem.

59 46 Min T 0 0 ˆ β zd f ( x ) = Var[( ˆ γ + ˆ γ ) e ] d Subjec o Σ= F 0< p <, i =,,3 i 3 i= p i = np Pr[ τ z ] MNF, i =,, 3 i i where MNF is he minimum number of failures and Σ is he inverse of he Fisher s informaion marix F. The above nonlinear opimizaion problem can be solved by numerical mehods..3 Conclusions A horough review of he lieraure abou ALT models indicaes ha widely-used nonparameric proporional hazards model can no be used for all failure ime daa, and proporional odds model is a good alernaive o proporional hazards model when he failure ime daa presen he propery of proporionaliy of odds funcion insead of hazard rae funcion. Unforunaely he curren parameer esimaion mehods of PO model are no applicable for acceleraed life esing. Therefore we resor o a new parameer esimaion mehod of PO model.

60 47 A horough review of he lieraure abou ALT plans indicaes ha here are no opimum plans for proporional odds cases when neiher he AFT nor he PH assumpions hold for he failure ime daa. Furhermore, curren exising ALT plans only consider single sress ype siuaions. However, as producs become more reliable hanks o echnological advances, i becomes more difficul o induce significan amoun of failure ime daa wihin he limied esing duraion using a single sress ype only. Muliple-sress-ype ALTs have been applied o overcome such difficulies. Bu lile work has been done on designing ALT plans wih muliple sress ypes. In his disseraion proposal, we propose o design and develop he PO-based ALT plans wih muliple sress ypes when neiher he assumpion of AFT model nor he PH model is valid.

61 48 3 CHAPTER 3 ALT MODEL BASED ON PROPORTIONAL ODDS In acceleraed life esing, i is more imporan o exrapolae he reliabiliy performance of producs a he design sress level from he failure ime daa a more severe sress condiions. Therefore reliabiliy praciioners are ineresed in esimaing he baseline funcion as well as he covariae parameers. Since he baseline odds funcion of he general PO models could be any monoonically increasing funcion, i is imporan o find a viable baseline odds funcion srucure o approximae mos, if no all, of he possible odds funcion. In order o find such a universal baseline odds funcion, we invesigae he properies of odds funcion and is relaion o he hazard rae funcion in his chaper. Based on hese properies, we propose a general form of he baseline odds funcion o approximae he odds funcion. We consruc he log-likelihood funcion for he PObased ALT model wih he proposed baseline odds funcion. The esimaes of he unknown parameers are obained hrough a numerical algorihm. We also demonsrae he PO-based ALT model by a simulaion sudy and an experimenal sudy. 3. Properies of Odds Funcion The odds funcion θ ( ) is defined as he odds on failure of a uni a ime, or he raio beween he probabiliy of failure and he probabiliy of survival, which is expressed as

62 49 F () R () θ () = =, (3.) F ( ) R ( ) R ( ) where F ( ) is cumulaive probabiliy funcion of he random failure ime T, and R() = F() is he corresponding reliabiliy funcion. From Eq. (3.), we could invesigae he properies of odds funcion hrough he reliabiliy funcion. Since mos failure ime processes are modeled on a coninuous scale, all he failure ime disribuions considered in his disseraion proposal are coninuous. From he properies of reliabiliy funcion and is relaion o odds funcion shown in Eq. (3.), we could easily derive he following properies of odds funcion θ ( ) : () θ (0) = 0, θ ( ) = ; () θ ( ) 0 for 0 ; (3) θ ( ) is a monoone non-decreasing funcion for 0, ha is θ ( ) 0. Proof: () Since R(0) = Pr( T > 0) =, hen θ (0) = = = 0; R(0) and Lim R ( ) = Lim Pr( T> ) = 0, hen θ( ) = Lim θ( ) = =. Lim Pr( T > ) () Since 0 R ( ) for 0, hen, so θ () = 0for 0. R () R () (3) Since R() = Pr( T > ) is a monoone nonincreasing funcion, we have R () 0. R ( ) Then θ () = [ ] = 0, so θ ( ) is a monoone nondecreasing funcion. R () [ R ()]

63 50 We also summarize he relaionship beween odds funcion and cumulaive hazard rae funcion Λ ( ), and hazard rae funcion λ ( ). Λ() (4) θ () = e, and Λ () = ln[ θ () + ] ; θ () (5) λ() = θ () +, and θ( ) = exp( λ( u) du). 0 Proof: Λ() (4) θ () = = = e, Λ () = ln[ θ () + ]. Λ() R () e f () R () {/[ θ () + ]} θ () (5) λ() = = = = R () R () {/[ θ() + ]} θ() +, and θ Λ() ( ) = e = exp( λ( u) du) Proposed Baseline Odds Funcion Approximaion Furhermore, ploing he odds funcions of some common failure ime disribuions as shown in Figure 3. provides furher clarificaion of he odds funcions.

64 5 0 Odds funcions of exponenial disribuions 0 Odds funcions of Weibull disribuions 8 6 μ = 300 μ = α =.5, β = α =.5, β = μ = Odds funcions of Gamma disribuions 00 α = 0.6, β = Odds funcions of Lognormal disribuions α = 0.7, β = α =, β = 0 60 μ = 5, σ = μ = 5, σ = 0 0 α = 4, β = 0 μ = 5, σ = Figure 3. Odds funcions of four common failure ime disribuions Based on he properies of he odds funcion and he plos in Figure 3., we could use a polynomial funcion o approximae he general baseline odds funcion in he PO models. The proposed general baseline odds funcion is θ () = γ + γ + γ +K Usually high orders are no necessary, second or hird order polynomial funcion is sufficien o cover mos of possible odds funcion. If he baseline odds funcion is assumed o be quadraic form, ha is

65 5 θ () = γ + γ. (3.) 0 we have he following heorem: Theorem: γ 0 and γ 0 in Eq. (3.). Proof: This heorem can be proven by conradicion. The values of γ and γ mus fall ino one of he following four cases:. γ < 0 and γ < 0 ;. γ < 0 and γ > 0 ; 3. γ 0 and γ < 0 ; 4. γ 0 and γ 0. If γ < 0 and γ < 0, hen θ () = γ + γ 0for 0. This is a conradicion o he 0 propery (). So case is no possible for baseline odds funcion. If case is rue, ha is γ < 0 and γ > 0, hen he baseline odds funcion θ () = γ + γ = ( γ + γ ) 0 has roos r = 0 and r = γ / γ. Since γ < 0 and γ > 0, we have r = γ / γ > 0. Then we obain he derivaive of he baseline odds funcion a r, θ 0( = r = 0) = γ < 0, which conradics o he propery (3) of odds funcion.

66 53 If case 3 is rue, hen he baseline odds funcion sill has wo roos r = 0 and r = γ / γ. Since γ 0 and γ < 0, we have r = γ / γ > 0. Then we obain he derivaive of he baseline odds funcion a r, θ 0( = r = γ/ γ) = γ+ γ( γ/ γ) = γ 0, which also conradics o he propery (3) of odds funcion. So he values of γ and γ mus saisfy γ 0 and γ 0. This heorem gives a bound for parameer esimaion in he laer secions of he disseraion proposal. 3.3 Log-Likelihood Funcion of he PO-based ALT Model We assume ha he baseline odds funcion is expressed in a quadraic form: θ () = γ + γ, γ 0, γ 0 (3.3) 0 where γ and γ are unknown parameers which we are ineresed o esimae and he inercep parameer is zero since he odds funcion crosses he origin according o propery () in secion 3.. The proporional odds model is hen represened by: θ( ; z) = exp( β z ) θ ( ) = exp( β z + β z + Lβ z )( γ + γ ), (3.4) 0 k k

67 54 where θ 0( ) an arbirary baseline odds funcion; z = (,, K, ) a vecor of he covariaes or he applied sresses; z z z k β = (,, K, ) a vecor of he unknown regression parameers; β β β k k he number of he sresses. Therefore he hazard rae funcion λ( ; z ) and cumulaive hazard rae funcion Λ( ; z ) are θ ( ; z) exp( β z)( γ + γ ) λ(; z) = =, θ( ; z) + exp( β z)( γ + γ ) + Λ ( ; z) = ln[ θ( ; z) + ] = ln[exp( β z)( γ + γ ) + ]. The parameers of he PO model wih he proposed baseline odds funcion for censored failure ime daa are obained as follows. Le i, i =, K, n represen he failure ime of he ih esing uni, z (,,, ) i = z i z i K zki he sress vecor of his uni, and I i he indicaor funcion, which is defined by I i if i τ, failure observed before ime τ, = I( i τ ) = 0 if i > τ, censored a ime τ.

68 55 The log likelihood funcion of he proposed PO-based ALT model is n l = I ln[ λ( ; z )] Λ( ; z ). (3.5) i i i i i i= i= n Subsiuing he hazard rae funcion and cumulaive hazard funcion ino Eq. (3.5) resuls in: e ( γ + γ ) l = I ln[ ] ln[ e ( + ) + ] n β zi n i β zi i γ i γ β z i i i= e ( γi + γi ) + i= n n β zi β zi β zi Ii e γ γi e γi γi e γi γi i= i= = {ln[ ( + )] ln[ ( + ) + ]} ln[ ( + ) + ] n n β zi Ii{ β zi ln( γ γi) ln[ e ( γi γi ) i= i= β z = i + ]} ln[ e ( γ + γ ) + ] i i (3.6) Taking he derivaives of he log likelihood funcion wih respec o he unknown parameers ( β, γ, γ ) respecively, we obain n n β zi n β zi i γ i γ i i γ i γ i = Iz i i Ii i i i i e β z = = i i i= e β z i i l z e ( + ) z e ( + ), β ( γ + γ ) + ( γ + γ ) + M n n β zi n β zi ki γ i γ i ki γ i γ i = Iz i ki Ii i i k i i e β z = = i i i= e β z i i l z e ( + ) z e ( + ), β ( γ + γ ) + ( γ + γ ) +

69 56 n n β zi n β zi i i Ii Ii β zi β zi i= i i= e i i i= e i i l e e =, γ ( γ + γ ) ( γ + γ ) + ( γ + γ ) + n n β zi n β zi i i i Ii Ii β zi β zi i= i i= e i i i= e i i l e e =. γ ( γ + γ ) ( γ + γ ) + ( γ + γ ) + The esimaes of he model parameers ( β, γ, γ ) are obained by seing he above derivaives o zero and solving he resulan equaions simulaneously. Unforunaely here are no closed form soluions for hese equaions. Therefore he soluions can be only obained by using numerical mehods such as Newon-Raphson mehod. To use Newon-Raphson mehod, we calculae he second derivaives of he loglikelihood funcion as follows. n β zi n β zi l zz i ie ( γi γi) zz i ie ( γi γi) = Ii β zi β zi i= [ e ( γi γi ) ] i= [ e ( γi γi ) ] + + β , n n β zi n β zi l e i e i = Ii + Ii + β z ( ) i β zi i= i i= [ e ( i i ) ] i= [ e ( i i ) ] γ γ + γ γ + γ + γ + γ +, n n β zi 4 n β zi 4 l 4i e i e i = Ii + Ii + β z ( ) i β zi i= i i= [ e ( i i ) ] i= [ e ( i i ) ] γ γ + γ γ + γ + γ + γ +,

70 57 n β zi n β zi zi i zi i = Ii β zi β zi i= [ e ( i i ) ] i= [ e ( i i ) ] l e e β γ γ + γ + γ + γ +, n β zi n β zi l zie i zie i = Ii β zi β zi β i= [ e ( i i ) ] i= [ e ( i i ) ] γ γ + γ + γ + γ + n β zi n β zi l zie i zie i = Ii β zi β zi γ i= [ e ( γi γi ) ] i= [ e ( γi γi ) ] β , n β zi n β zi l zie i zie i = Ii β zi β zi β i= [ e ( i i ) ] i= [ e ( i i ) ] γ γ + γ + γ + γ + n n β zi 3 n β zi 3 l i e i e i = Ii + Ii + β z ( ) i β zi i= i i= [ e ( i i ) ] i= [ e ( i i ) ] γ γ γ + γ γ + γ + γ + γ +. We hen consruc he Hessian marix as: l( β, γ, γ) l( β, γ, γ) l( β, γ, γ) γ γ β β β l( β, γ, γ) l( β, γ, γ) l( β, γ, γ) H ( β, γ, γ) = γ β γ γ γ l( β, γ, γ) l( β, γ, γ) l( β, γ, γ) γ β γ γ γ

71 58 Le x = β γ γ, x 0 is he iniial value of he parameers. The Newon-Raphson mehod sars wih his iniial value, and afer j seps of he algorihm, he updaed esimae of he parameers is given by x x H x u x, j+ = j ( j ) ( j ) l( β, γ, γ ) l( β, γ, γ ) l( β, γ, γ ) ux ( ) = x= x. j where j β γ γ The Newon-Raphson algorihm converges quie rapidly when he iniial value is no oo far from he rue value. When he iniial value is poor, he algorihm may move in he wrong direcion or may ake a sep in he correc direcion, bu overshoo he roo. The value of he log-likelihood funcion is compued a each sep o ensure ha he algorihm is moving in he correc direcion. If l( x k ) is smaller han ( x k ) l +, one opion is o cu he sep size in half and ry x x H x u x. k+ = k ( k ) ( k )/ 3.4 Simulaion Sudy In his secion, we compare he performance of he PO models wih ha of he PH models using simulaion daa. To validae he proposed PO-based ALT model, we apply boh he PO-based and PH-based ALT models o a group of simulaed failure ime daa ses. Ses of random failure ime daa from mixed log-logisic disribuion and Weibull disribuion

72 59 are generaed by Mone Carlo simulaion mehod. We now describe he deails of he simulaion sudy Generaion of Failure Time Daa By definiion, he reliabiliy funcion of he mixed disribuion, from which we generae he random failure ime daa, wih sress level z is w β z R( ) = + ( w)exp( αe ), (3.7) β z + ( γ e ) where α is a Weibull disribuion parameer, γ is a log-logisic disribuion parameer, and 0 w. The reason for choosing he log-logisic disribuion and he Weibull disribuion o consruc he mixed disribuion is due o he fac ha he log-logisic disribuion and he Weibull disribuion are special cases for he PO models and he PH models respecively as we discussed in Chaper. The firs par of Eq. (3.7) is a special case of he general PO models wih he baseline odds funcion θ0() = γ. While he second par, which is a Weibull disribuion reliabiliy funcion, is a special case of he general PH models. Equaion (3.7) becomes a pure log-logisic disribuion when w is, a pure Weibull disribuion when w is 0, or a mixed disribuion when w akes any value beween 0 and. If w is close o, he generaed failure ime daa from he mixed disribuion are

73 60 much like being drawn from a log-logisic disribuion. On he oher hand, if w is close o 0, he generaed failure ime daa from he mixed disribuion are much like being drawn from a Weibull disribuion. We expec ha he simulaion sudy reveals he resul ha if w is equal o, or close o, he PH-based ALT model is much beer han PO counerpar, and if w is equal o, or close o 0, he PO-based ALT model is much beer han PH model. When w is, solve from Eq. (3.7), we have β z e = [ R( ) ], γ or β z e = [( F( )) ]. (3.8) γ Based on Eq. (3.8), a random log-logisic failure ime i is generaed by Mone Carlo simulaion mehod as β z e i = [( rand( i)) ], (3.9) γ where rand( i ) is uniformly-disribued random variable beween [0, ].

74 6 Similarly, for w = 0, he Weibull disribued random failure ime i is generaed by β z e i = ln[ rand ( i)] (3.0) α For any 0 w, he mixed random failure ime i is generaed by he following equaion: i i rand3( i) w = i rand3( i) > w (3.) where rand( i), rand( i ), and rand3( i ) are uniformly-disribued random variables bounded in he inerval [0, ]. Using he following parameers: β = 30, γ = 800, and α = 0., he Mone Carlo random failure ime i ( n =, K,0 ) is generaed by Eq. (3.) wih w = and four sress levels: 0.3, 0.35, 0.4, and We generae hiry random failure imes for each of he sress levels. Similarly we could generae several daases wih differen w values as summarized in able 3..

75 6 Table 3. Mone Carlo simulaion daases w Daase I II III IV V VI VII 3.4. Simulaion Resuls Afer generaing he random failure ime daa, we apply he PO-based ALT model and PH-based ALT model o he daa ses respecively. The unknown parameers of he models are esimaed by he Maximum Likelihood Esimaion procedures as discussed in secion 3.3 for he PO-based ALT and Chaper for he PH-based ALT respecively. Then he reliabiliies a a specific sress level can be esimaed wih hose parameer esimaes for proporional odds model and proporional hazards model respecively. Since he failure ime daa are generaed from he mixed disribuion given in Eq. (3.7), we could obain he heoreical reliabiliy a any ime for any specified sress level using ha equaion. Then he reliabiliy esimaes obained from he PO-based ALT and he PH-based ALT models are compared wih he heoreical reliabiliy respecively. The comparisons are based on graphical echnique and he sum of squared reliabiliy errors (SSE) a he simulaed failure imes for he specified sress level. Firs, when w =, we apply he PO-based ALT model o he simulaed daa, we obain he following parameer esimaes:

76 63 ˆ β = 3.5 ˆ γ = 94 ˆ γ = Thereafer we use he esimaed parameers o predic he reliabiliy a he design sress level 0.5. Figure 3. shows he heoreical reliabiliy obained from he original parameers β = 30, γ = 800, and γ = 0 and he prediced reliabiliy obained from he esimaed parameers ˆ β = 3.5, ˆ γ = 94, and ˆ γ = a sress level 0.5. The sum of squared errors is Theoreical Prediced Reliabiliy Time Figure 3. Theoreical reliabiliy and prediced reliabiliy a sress level 0.5 by he PO-based ALT model Similarly we fi he same simulaed daa wih he PH-based ALT model; he heoreical and prediced reliabiliies are shown in Figure 3.3 wih sum of squared errors

77 Theoreical Prediced Reliabiliy Time Figure 3.3 Theoreical reliabiliy and prediced reliabiliy a sress level 0.5 by he PH-based ALT model The simulaion resuls show ha he PO-based ALT model provides more accurae reliabiliy esimaes han he PH-based ALT model when w =. The reliabiliy funcions a oher sress levels are ploed in Figure 3.4 hrough Figure 3. for he PO-based ALT and he PH-based ALT models respecively.

78 65 Reliabiliy Theoreical Prediced Time Figure 3.4 Reliabiliy comparison for he PO-based ALT model wih w = a sress level 0.45 Reliabiliy Theoreical Prediced Time Figure 3.5 Reliabiliy comparison for he PH-based ALT model wih w = a sress level 0.45

79 Reliabiliy Theoreical Prediced Time Figure 3.6 Reliabiliy comparison for he PO-based ALT model wih w = a sress level 0.4 Reliabiliy Time Theoreical Prediced Figure 3.7 Reliabiliy comparison for he PH-based ALT model wih w = a sress level 0.4

80 67 Reliabiliy Time Theoreical Prediced Figure 3.8 Reliabiliy comparison for he PO-based ALT model wih w = a sress level 0.35 Reliabiliy Time Theoreical Prediced Figure 3.9 Reliabiliy comparison for he PH-based ALT model wih w = a sress level 0.35

81 68 Reliabiliy Theoreical Prediced Time Figure 3.0 Reliabiliy comparison for he PO-based ALT model wih w = a sress level 0.3 Reliabiliy Theoreical Prediced Time Figure 3. Reliabiliy comparison for he PH-based ALT model wih w = a sress level 0.3

82 69 The sums of squared errors a differen sress levels are summarized in Table 3. for daase I using he PO-based ALT and he PH-based ALT models respecively. Table 3. Sums of squared errors wih differen sress levels Sress PO-based ALT PH-based ALT Similarly, we repea he above procedures wih differen w values. The sums of squared errors (SSE) for boh he PO-based ALT and he PH-based ALT model are summarized in Table 3.3.

83 70 Table 3.3 Sums of squared errors wih differen w values w PO-based ALT PH-based ALT Table 3.3 indicaes ha he PO-based ALT model provides more accurae reliabiliy esimaes when w approaches ; while he PH-based ALT model provides more accurae reliabiliy esimaes when w approaches 0. As he resuls of he simulaion sudy show, he choice of eiher he PO-based ALT model or he PH-based ALT model depends on he inheren propery of he failure ime daa. Neiher he PO-based ALT model nor he PH-based ALT model can be used o fi failure ime daa wihou checking he model assumpion. 3.5 Example wih Experimenal Daa

84 7 In his secion we use a se of experimenal daa as an example. The failure ime daa of MOS devices wih emperaure as he applied sress are provided by Swarz (986). The emperaure has five levels: 5 C, 00 C, 5 C, 50 C, and 5 C, where sress level 5 C is he design sress level of he devices. The failure imes i ( i =, K,99 ) of uni i is associaed wih is sress level z i. According o he Arrehenius model, we ransform he Celsius emperaure o /Kelvin as he sress z in he PO model. Therefore he ransformed five emperaure sress levels are: K, and K. K, 0. K, 0.5 K, The validiy of he PO-based ALT model relies heavily on he assumpion of proporionaliy of he odds funcions of unis a differen sress levels. Therefore, we plo he logarihm of he odds funcions a differen sress levels using produc-limi, or Kaplan-Meier, reliabiliy esimaes shown in Figure 3..

85 7 l ogodds ime emp Figure 3. Plo of log odds funcions versus ime for Swarz s daa From Figure 3., here is no significan deviaion from he proporional odds assumpion since he curves do no cross. We also plo he cumulaive hazards funcions a differen sress levels using Kaplan-Meier reliabiliy esimaes o validae he proporional hazard rae funcions as shown in Figure 3.3 Similar o he conclusions from Figure 3., here is no significan violaion of he proporional hazards assumpion. Therefore i is difficul o deermine wheher he inheren proporionaliy of Swarz s daa is closer o proporional odds assumpion or proporional hazards assumpion.

86 73 l ogcumh ime emp Figure 3.3 Plo of log cumulaive hazards funcions versus ime for Swarz s daa In order o deermine he appropriaeness of he PO-based ALT model or he PH-based ALT model for he reliabiliy predicaion a design sress level, we propose he following procedures:. We fi Swarz s daa using only failure ime daa a sress levels 5 C, 00 C, 5 C, and 50 C, wih he PO-based ALT model o esimae parameers of he model, and hen predic he reliabiliy a sress level 5 C.. Compare he Kaplan-Meier reliabiliy esimaes obained from acual failure ime daa a sress level 5 C wih hose reliabiliy esimaed by he PO-based ALT model and calculae he sum of squared errors (SSE). 3. Repea seps and using he PH-based ALT model. 4. Choose he model wih smaller SSE.

87 74 Applying he above procedures, we analyze he Swarz s daa as follows: Assuming ha he failure ime daase follows he proporional odds model, he log likelihood funcion is hen formulaed as 74 β 74 zi βzi i i i i i i i i= i=. l( β, γ, γ ) = I { βz + ln( γ + γ ) ln[ e ( γ + γ ) + ]} ln[ e ( γ + γ ) + ] In he above log likelihood funcion, only he failure ime daa a sress levels 5 C, 00 C, 5 C, and 50 C are used o esimae he hree unknown parameers of he model, which are used o predic he reliabiliy of he devices a design sress level 5 C. The prediced reliabiliy is hen compared o he Kaplan-Meier reliabiliy esimaes based on he acual failure ime daa obained a sress level 5 C. The MLE esimaes of ( βγ,, γ ) are: ˆ β = ˆ γ = ˆ γ = 5.8 Figure 3.4 shows he Kaplan-Meier reliabiliy and he prediced reliabiliy a emperaure level 5 C. The sum of squared errors (SSE), i.e. he sum of squared

88 75 difference beween he prediced reliabiliy and Kaplan-Meier reliabiliy esimae, is Reliabiliy Kaplan-Meier Prediced Time Figure 3.4 Kaplan-Meier reliabiliy vs. prediced reliabiliy a emperaure 5 C by he PO-based ALT model Similarly we fi he same failure ime daa wih he PH-based ALT model and compare he prediced reliabiliy esimaes wih he Kaplan-Meier reliabiliy esimaes obained from he acual failure daa a he sress level 5 C as shown in Figure 3.5.

89 Reliabiliy Kaplan-Meier Prediced Time Figure 3.5 Kaplan-Meier Reliabiliy vs. Prediced Reliabiliy a Temperaure 5 C by he PH-based ALT Model The sum of squared errors of he PH-based ALT model is.949, which is much larger han ha of he PO-based ALT model. We conclude ha he PO-based ALT model provides more accurae reliabiliy esimaes a normal operaing condiions han he PHbased ALT model for he given failure ime daa.

90 77 4 CHAPTER 4 CONFIDENCE INTERVALS AND MODEL VALIDATION Chaper 3 provides he poin esimae of he unknown parameers of he PO-based ALT model and he poin esimae of reliabiliy a design sress condiions. Some reliabiliy esimaion problems in many sysems require he inerval esimae of he reliabiliy a a specified sress level, especially he design sress level. In his chaper, we inroduce a mehodology for obaining he confidence inervals for he unknown parameers and he esimaed reliabiliy a he design sress level hrough Fisher informaion marix. We also describe he likelihood raio es and he modified Cox-Snell residuals o validae he model. Numerical examples are also given in his chaper o display he inerval esimaion procedure and he validaion procedures. 4. Confidence Inervals 4.. Fisher Informaion Marix We begin by he calculaion of Fisher informaion marix. Assume ha he baseline odds funcion is expressed in a quadraic form: θ () = γ + γ, γ 0, γ 0 (4.) 0

91 78 where γ and γ are consan parameers o be esimaed, and he inercep parameer is zero since he odds funcion crosses he origin according o he firs propery of odds funcion as described in Chaper 3. The proporional odds model is hen represened by: θ( ; z) = exp( β z ) θ ( ) = exp( β z + β z + Lβ z )( γ + γ ), (4.) 0 k k and he hazard rae funcion λ(; z ) and cumulaive hazard rae funcion Λ(; z ) are θ ( ; z) exp( β z)( γ + γ ) λ(; z) = =, θ( ; z) + exp( β z)( γ + γ ) + Λ ( ; z) = ln[ θ( ; z) + ] = ln[exp( β z)( γ + γ ) + ]. For a censored failure ime daa se, le i represen he failure ime of he ih uni, z = ( z, z, K, z ) he sress vecor of his uni, and I i he indicaor funcion, which is i i i ki if i τ (he censoring ime), or 0 if i PO-based ALT model is > τ. The log likelihood funcion of he proposed n l = I ln[ λ( ; z )] Λ( ; z ), (4.3) i i i i i i= i= n

92 79 Subsiuing hazard rae funcion and cumulaive hazard rae funcion ino he log likelihood funcion, we obain: n n β zi β zi i β zi γ γ i γ i γ i γ i γ i i= i= l = I { + ln( + ) ln[ e ( + ) + ]} ln[ e ( + ) + ]. (4.4) The unknown parameers ( β, γ, γ ) are obained by he maximum likelihood esimaion procedure. To deermine he confidence inervals of he esimaed parameers, we should firs consruc he Fisher informaion marix, which requires he firs and second derivaives of he log likelihood funcion l wih respec o each of hree parameers, respecively: n n β zi n β zi i γ i γ i i γ i γ i = Iz i i Ii i i i i e β z = = i i i= e β z i i l z e ( + ) z e ( + ), β ( γ + γ ) + ( γ + γ ) + n n β zi n β zi ki γ i γ i ki γ i γ i = Iz i ki Ii i i k i i e β z = = i i i= e β z i i l z e ( + ) z e ( + ), β ( γ + γ ) + ( γ + γ ) + n n β zi n β zi i i Ii Ii β zi β zi i= i i= e i i i= e i i l e e =, γ ( γ + γ ) ( γ + γ ) + ( γ + γ ) + n n β zi n β zi i i i Ii Ii β zi β zi i= i i= e i i i= e i i l e e =, γ ( γ + γ ) ( γ + γ ) + ( γ + γ ) +

93 80 n β zi n β zi l zz i ie ( γi γ i ) zz i ie ( γi γi ) = Ii β zi β zi i= [ e ( γi γi ) ] i= [ e ( γi γi ) ] + + β , n n β zi n β zi l e i e i = Ii + Ii + β z ( ) i β zi i= i i= [ e ( i i ) ] i= [ e ( i i ) ] γ γ + γ γ + γ + γ + γ +, n n β zi 4 n β zi 4 l 4i e i e i = Ii + Ii + β z ( ) i β zi i= i i= [ e ( i i ) ] i= [ e ( i i ) ] γ γ + γ γ + γ + γ + γ +, n n β zi 3 n β zi 3 l i e i e i = Ii + Ii + β z ( ) i β zi i= i i= [ e ( i i ) ] i= [ e ( i i ) ] γ γ γ + γ γ + γ + γ + γ +. The elemens of he Fisher informaion marix are he negaive expecaions of he second derivaives of he log likelihood funcion: l E[ ] 0 0 β l l F = 0 E[ ] E[ ], (4.5) γ γ γ l l 0 E[ ] E[ ] γ γ γ where Eg [ ()] = f() gd () (4.6) 0

94 8 g () is he negaive of he second derivaive of log likelihood funcion, or any elemen of he Fisher informaion marix, and f ( ) is he pdf of he failure ime disribuion, given by ( γ + γ e ) f() = R () = [ + ( γ + ) ] β z β z γ e. So he Fisher informaion marix F is a 3 3 symmeric marix. Since he baseline odds funcion is assumed o be independen of he sress vecor z in he PO model, he correlaions beween he sress coefficien β and baseline parameers γ, γ are zero. So he cells (, ), (, 3), (, ), and (3, ) of he Fisher informaion marix are zero as in Eq. (4.5). The calculaion of he Fisher informaion marix is performed numerically o inegrae Eq. (4.6) from ime 0 o ime T, which is large enough and more han he longes failure ime of he failure ime observaions. In he numerical calculaion procedure, he unknown parameers ( β, γ, γ ) of he proporional odds model are replaced by he esimaed values obained from he maximum likelihood esimaion procedure menioned in Chaper 3.

95 8 4.. Variance-Covariance Marix and Confidence Inervals of Model Parameers The asympoic variance-covariance marix Σ of he maximum likelihood esimaes of he parameers is he inverse of he Fisher informaion marix F : Σ 0 0 Σ = F = 0 σ σ 3. (4.7) 0 σ 3 σ 33 So he asympoic variances of he parameer esimaes ( ˆ β, ˆ γ ˆ, γ) for he PO-based ALT model are: Var( ˆ β ) = Σ ( j), j =, K, k, j Var( γ ) = σ, ˆ Var( γ ) = σ. ˆ 33 where Σ ( j) is he jh diagonal elemen of he marix Σ. Therefore he ( α)% confidence inervals of he parameers are: β : ˆ β ˆ j ± Zα / Var( β j), j =, K, k,

96 83 ˆ ± Z Var( ˆ ), γ : γ α / γ ˆ ± Z Var( ˆ ). γ : γ α / γ 4..3 Confidence Inervals of Reliabiliy Esimaes We use he dela mehod o calculae he asympoic variance of he esimaed reliabiliy a design sress level z D wih he PO-based ALT model. Under he PO model he reliabiliy funcion of any uni a he design sress z is D R (; z D) =, D + ( γ + β z γ ) e The asympoic variance of he esimaed reliabiliy is given by dela mehod as: R (; zd ) β M R (; z ) [ ˆ R R R R Var R( ; z )] = [ ] Σ D D ( ; zd) ( ; zd) ( ; zd) ( ; zd) K βk ( β, γ ˆ, γ) = ( β, ˆ γ, ˆ γ) β βk γ γ R (; zd ) γ R (; zd ) γ (4.8)

97 84 Therefore, he ( α)% confidence inervals of he reliabiliy a he design sress z is D R ˆ(; ) ˆ ˆ ˆ D Zα/ VarR [ (; D)], R (; D) + Zα/ VarR [ (; D)] z z z z. (4.9) 4..4 Numerical Example of Reliabiliy Confidence Inervals We use Swarz s daa (Swarz 986) as an example o calculae he 95% confidence inervals of unknown parameers and reliabiliy esimae a he design sress z = 5 C. The variance-covariance marix is D Σ= , Then he 95% confidence inervals of he parameers are: β : ˆ β ± Z ˆ α / Var( β) = ± = ( 65.94, 63.9) ˆ ± Z Var( ˆ ) = ± = (456.56,0.90) γ : γ α / γ

98 85 ˆ ± Z Var( ˆ ) = 5.8 ± = (4.08, 6.8) γ : γ α / γ The 95% reliabiliy confidence inervals a sress level z = 5 C, which are calculaed wih Eq. (4.9), are depiced in Figure 4.. As shown, he inervals inend o large when ime is larger. Therefore he reliabiliy esimaes are more accurae a he early sage. D 0.9 Prediced Lower Limi Upper Limi Reliabiliy Time Figure 4. 95% reliabiliy confidence inervals a emperaure z = 5 C level D 4. Model Validaions 4.. Model Sufficiency In Chaper 3, a polynomial baseline odds funcion is proposed o approximae he general odds funcion, θ () = γ + γ + γ +K. (4.0) 3 0 3

99 86 We also poin ou ha higher order may no be necessary and a quadraic baseline odds funcion is already sufficien in mos cases. In order o es he sufficiency of quadraic baseline odds funcion, we use he likelihood raio es. The hypohesis can be expressed as: H0: γ 3 = 0, versus H: γ 3 0. To es he null hypohesis H 0 agains he alernaive hypohesis H, we use he likelihood raio saisic = [ln (,, ) ln (,,, )], (4.) χlr L β γ γ L β γ γ γ3 where ( β, γ, γ ) is found by maximizing he log likelihood funcion wih consideraion of only γ and γ, and ( β, γ, γ, γ 3) is found by maximizing he log likelihood funcion wih γ 3 in he model. This saisic has an asympoic chi-squared disribuion wih degree of freedom. A large value of χ LR gives he evidence agains he null hypohesis H 0.

100 87 To es if we need higher order polynomial baseline odds funcion han cubic equaion, we could use he similar likelihood raio es. 4.. Model Residuals To es he validaion of he assumpion of proporional odds afer he fiing of he regression model, we need o calculae he residual of model fiing. In he usual linear regression seup, i is quie easy o define a residual for he fied regression model. In he proporional odds regression model, he definiion of he residual is no sraighforward. We could use Cox-Snell ype residual o assess he fi of a PO-based ALT model. The Cox-Snell residuals (Cox and Snell 968) could be used for assessing he goodness-of-fi of a Cox model. Wih some modificaions, we could also use he similar ype residual o assess he goodness-of-fi of a proporional odds model. We suppose a PO-based ALT model was used o fi he daase ( i, Ii, z i), i =, K, n. β We also suppose ha he proporional odds model (; ) z βz θ z = e θ () = e ( γ + γ ) has 0 been uilized. If he model is correc, hen, i is well known ha, if we make he probabiliy inegral ransformaion F ( i; z i) on he rue failure ime i, he resuling random variable has a uniform disribuion on he uni inerval. Therefore we can prove ha he random variable Y =Λ ( ; z ) has an exponenial disribuion wih hazard rae. i i Here, y =Λ ( z ; ) is he cumulaive hazard rae of failure ime a sress level z afer he parameer esimaes have obained.

101 88 Theorem: he random variable Y = Λ ( ; z ) has an exponenial disribuion wih hazard rae if proporional odds model is correc. i i Proof: If he model is correc, he random variable x = F ( ; z), which is obained from he probabiliy inegral ransformaion of he failure ime, follows a uniform disribuion on he inerval [0,]. Therefore he probabiliy densiy funcion of x is given by i i x [0,] f( x) = (4.) 0 oherwise Then he random variable Y = Λ ( ; z ), which is he cumulaive hazard rae funcion of i i failure ime, can be also expressed in erms of he random variable x = F ( ; z), i i Y = Λ ( ; z ) = ln[ F( ; z )]. (4.3) i i i i Wihou loss of generaliy, Eq. (4.3) can be simplified as y = ln( x). (4.4) So he random variable y is a funcion of he random variable x, which has a uniform disribuion on he inerval [0,], and is described by he probabiliy densiy funcion Eq.

102 89 (4.). We can obain he probabiliy densiy funcion of random variable y (DeGroo and Schevish 00): If a random variable X has a coninuous disribuion; he probabiliy densiy funcion of X is f ( x ) ; and anoher random variable is defined as Y = r( X). For each real number y, he cumulaive disribuion funcion of Y can be derived as follows: F( y) = Pr( Y y) = Pr[ r( X) y] = f( x) dx. (4.5) { xrx : ( ) y} Since he random variable Y is defined by Y = ln( X) as Eq. (4.4) here, hen Y mus belong o he inerval 0 Y <. Thus, for each value of Y such ha 0 Y <, he cumulaive disribuion funcion F( y ) of Y is derived from Eq. (4.5) as follows: F( y) = Pr( Y y) = Pr[ ln( X) y] = Pr[ln( X) y] y = Pr[( X) e ] y = Pr[ X e ] = y e 0 = e y f( x) dx (4.6) For 0 < Y <, he probabiliy densiy funcion f ( y ) of Y is df( y) y f ( y) = = e. (4.7) dy

103 90 Eq. (4.7) is he probabiliy densiy funcion of exponenial disribuion wih hazard rae. Proof is complee. Therefore we could uilize he above heorem o generae he Cox-Snell ype residuals for he PO-based ALT model o verify he assumpion of proporional odds. If he esimaes of he unknowns ( β, γ, γ ) from he posulaed model are ( ˆ β, ˆ γ ˆ, γ ), hen, he Cox- Snell ype residuals for he PO-based ALT model are defined as r =Λ ˆ ( ; z ) = ln[ ˆ( ; z ) + ] = ln[ e ( ˆ + ˆ ) + ], i =, K, n. (4.8) ˆ β z i i i i θ i i γ i γ i If he proporional odds model is correc and he esimaes ( ˆ β, ˆ γ ˆ, γ ) are close o he rue values of ( β, γ, γ ), hen he r i s should look like a censored sample from a uni exponenial disribuion. To check wheher he r i s behave as a sample from a uni exponenial disribuion, we compue he Nelson-Aalen esimaor (Klein and Moeschberger 997) of he cumulaive hazard rae of he r i s. If he uni exponenial disribuion fis he daa, hen, he Nelson- Aalen esimaor should be approximaely equal o he cumulaive hazard rae of he uni exponenial disribuion, Λ E () =. Thus, a plo of he esimaed cumulaive hazard rae of r i s, or Λ ˆ r( ri), versus r i should be a sraigh line hrough he origin.

104 9 The nonparameric Nelson-Aalen esimaor of he cumulaive hazard rae is obained as follows. To allow for possible ies in he daa, suppose ha he failures occur a D disinc imes < < L < D, and ha a ime i here are d i failures. Le Y i be he number of unis a risk a ime i, or he number of unis which are operaing properly a ime i. Then he Nelson-Aalen esimaor of he cumulaive hazard rae is defined as follows 0, if ˆ < Λ () = di / Yi, if i (4.9) The Nelson-Aalen esimaor has beer small-size performance han he esimaor based on he Kaplan-Meier esimaor. The Nelson-Aalen esimaor has wo primary uses in analyzing failure ime daa. The firs is selecing parameric models for he daa. For our case, a plo of Nelson-Aalen esimaors of r i s, or Λ ˆ r( ri), versus r i will be approximaely linear if he exponenial disribuion fis he daa r i s. The second use of he Nelson- Aalen esimaor is providing crude esimae of he hazard rae of he failure ime daa. This esimae is he slope of he plo obained above. For our case, he slope of he plo of Nelson-Aalen esimaors of r i s versus r i will be roughly if he daa are from an exponenial disribuion wih hazard rae Numerical Example of Cox-Snell Residuals

105 9 We use he simulaion daase I in Table 3. as an example o demonsrae he performance of he Cox-Snell residuals. We apply he PO-based ALT model o his daa se. The unknown parameers of he models are esimaed by he Maximum Likelihood Esimaion procedures as discussed in secion 3.3. We obain he following parameer esimaes: ˆ β = 3.5 ˆ γ = 94 ˆ γ = Therefore he Cox-Snell ype residuals for he PO-based ALT model are calculaed by r =Λ ˆ ( ; z ) = ln[ ˆ( ; z ) + ] = ln[ e ( ˆ + ˆ ) + ], i =, K, n. ˆ β z i i i i θ i i γ i γ i To check wheher he r i s behave as a sample from a uni exponenial disribuion, we rea hem as failure imes and compue he Nelson-Aalen esimaors of he cumulaive hazard rae of he r i s. The plo of he cumulaive hazard rae of he r i s versus he r i s is shown in Figure 4.. As shown in his figure, he sep funcion is he esimaed cumulaive hazard rae of he r i s and he sraigh line is a line wih he slope of. I is obvious ha he esimaed cumulaive hazard rae of r i s, or Λ ˆ r( ri), is approximaely a

106 93 sraigh line wih he slope of. Therefore, we conclude ha he proporional odds assumpion is valid for he given daa se. Figure 4. Nelson-Aalen esimaors of Cox-Snell residuals

107 94 5 CHAPTER 5 PO-BASED MULTIPLE-STRESS-TYPE ALT PLANS While acceleraed life es saves ime and expenses over esing a design sress condiions, he reliabiliy esimaes obained via exrapolaion in boh sress and ime are ineviably less accurae. One ineresing measuremen o obain more accurae esimaes is o devise a es plan ha ess unis a appropriaely seleced sress levels wih proper allocaion of es unis o each level. In oher words, an opimum acceleraed life esing plan will resul in more accurae esimaes of reliabiliy a design sress condiions. Design of ALT plans under one ype of sress may mask he effec of oher criical ypes of sresses ha could lead o he componen s failure. Therefore, i is more realisic o consider muliple sress ypes. In his chaper, we are invesigaing he design of opimum ALT plans based on he proporional odds model wih muliple sress ypes. 5. PO-based ALT Plans wih Consan Muliple Sress Types In his secion, we design consan muliple-sress-ype ALT plans based on he proporional odds assumpion. 5.. The Assumpions We assume he following condiions for he ALT plans wih muliple sress ypes.

108 95. There are k ypes of sress z = ( z, z, K, z k ) wih 3 levels for each sress ype used in he es.. The proporional odds (PO) model is employed o relae he reliabiliy a differen acceleraed sress levels and he design sress levels θ ( ; z ) = exp( β z + β z +Lβ z ) θ ( ). (5.) k k 0 where θ ( ; z) is he odds funcion of es unis a ime and sress vecor z ; θ 0( ) is he baseline odds funcion of he PO model; β, β, K, βk are unknown model parameers, which explain he effecs of sresses on he failure ime. 3. The baseline odds funcion is quadraic and given by θ () = γ + γ, (5.) 0 where γ and γ are unknown model parameers. 4. The number of sress level combinaion o deermined is 3 k as shown in Figure 5. for k =. The upper bounds of sresses are pre-specified as he highes sress levels beyond which he failure mode will change. While he lower bounds of wo sresses are specified as he design sress levels.

109 96 ( z z ) Upper, Upper, z z H p0 p p z M p 0 p p z L p 00 p 0 p 0 z D z D zl zm zh z Figure 5. Design of ALT plan wih wo sress ypes and hree levels for each sress ype 5. A oal of n unis are available for esing. The proporion of esing unis allocaed o he esing poin of he i h level of he firs sress ype z, i h level of he second sress ype z,, i k h level of he las sress ype z k is denoed by pii K i k, i = 0,, ; j =,, K, k, where he noaion of he levels 0,, are equivalen o he j noaion of he levels LM,, H, sanding for lower level, medium level, and high level respecively. Throughou he hesis, boh noaions are inerchangeable. 6. The es is erminaed a he pre-specified censoring ime τ. As defined by he above assumpions, he decision variables are he sress levels ( z L, z M, z H ), ( z L, z M, z H ),, ( z kl, z km, z kh ) and he proporions of es unis allocaed

110 97 o he 3 k sress combinaions pii K i, i 0,, ;,,, k j = j = K k. The oal number of decision variables is ( 3k + 3 k ). This facorial arrangemen of sress levels and es unis shown in Figure 5. is no saisically opimal. However, his arrangemen is moivaed by he acual pracice of reliabiliy engineers. Since any oher arrangemen wih he same number of esing poins will resul in more sress levels for each sress ype, i enables reliabiliy engineers o carry ou he enire es simulaneously by uilizing he available equipmen in an efficien manner and lead o subsanial savings in es ime and cos. In addiion, i allows for esing of ineracions afer he daa are colleced. 5.. The Log Likelihood Funcion Using he assumpion of proporional odds and he baseline odds funcion we obain he odds funcion a sress combinaion z = ( z, z, K, z ) k θ( ; z ) = exp( β z + β z + Lβ z )( γ + γ ). (5.3) k k Then he corresponding hazard rae funcion λ(; z ), cumulaive hazard funcion Λ(; z ), reliabiliy funcion R(; z ), and probabiliy densiy funcion f (; z ) are obained as follows: θ ( ; z) exp( β z)( γ + γ ) λ(; z) = =, (5.4) θ( ; z) + exp( β z)( γ + γ ) +

111 98 Λ ( ; z) = ln[ θ( ; z) + ] = ln[exp( β z)( γ + γ ) + ], (5.5) R (; z) = =, (5.6) θ( ; z) + exp( β z)( γ + γ ) + θ ( ; z) exp( β z)( γ + γ ) f(; z) = = [ θ( ; z) + ] [exp( )( γ + γ ) + ] β z. (5.7) Le i, i =, K, n represen he failure ime of he ih esing uni, z = ( z, z, K, z ) he sress vecor of his uni, and I i he indicaor funcion, which is defined by i i i ki I i if i τ, failure observed before ime τ, = I( i τ ) = 0 if i > τ, censored a ime τ. The log likelihood of he proposed ALT based on proporional odds model for his uni is l = I ln[ λ( ; z )] Λ( ; z ). (5.8) i i i i i i Subsiuing he hazard rae funcion in Eq. (5.4) and cumulaive hazard funcion in Eq. (5.5) ino above log likelihood resuls in: l = I [( β z ) + ln( γ + γ )] I ln[ e ( γ + γ ) + ] ln[ e ( γ + γ ) + ]. (5.9) ( β z i) ( β zi) i i i i i i i i i

112 99 Then he log likelihood funcion for he whole failure ime sample is l = l + l + L + l. n 5..3 The Fisher Informaion Marix and Covariance Marix For a single observaion riple (, I, z ), where z is a vecor of sress levels, he firs parial derivaives of log likelihood of his observaion wih respec o he model parameers are: ( β z) ( β z) γ γ γ γ = Iz Ii ( ) ( ) e β z e β z l ze ( + ) ze ( + ), (5.0) β ( γ + γ ) + ( γ + γ ) + M ( β z) ( β z) k γ γ k γ γ ( β z) ( β z) l z e ( + ) z e ( + ) = Izk I, (5.) β e ( γ + γ ) + e ( γ + γ ) + k ( β z) ( β z) l I Ie e =, (5.) γ ( γ + γ ) e ( γ + γ ) + e ( γ + γ ) + ( β z) ( β z) ( β zi) ( β zi) i i i i i ( ) β zi ( β zi) i e i i e i i l I I e e =, (5.3) γ ( γ + γ ) ( γ + γ ) + ( γ + γ ) +

113 00 Summing he above firs derivaives over all es unis and seing hem equal o zero will provide he equaions for solving he maximum likelihood esimaes of he model parameers ( β, β, K, βk, γ, γ). Assuming he correlaions only exis beween γ and γ, hen, for a single observaion riple (, I, z ), he second parial derivaives of he log likelihood wih respec o model parameers are ( β z) ( β z) l ze ( γ γ ) ze ( γ γ ) = I ( ) ( ) [ e β z ( ) ] [ e β z ( ) ] + + β γ + γ + γ + γ +, (5.4) M ( β z) ( β z) l zke ( γ γ ) zke ( γ γ ) = I ( ) ( ) k [ e β z ( ) ] [ e β z ( ) ] + + β γ + γ + γ + γ +, (5.5) ( β z) ( β z) l I Ie e = + + γ ( γ + γ ) [ e ( γ + γ ) + ] [ e ( γ + γ ) + ] ( β z) ( β z) i, (5.6) ( β z) 4 ( β z) 4 l 4I Ie e = + + γ ( γ + γ ) [ e ( γ + γ ) + ] [ e ( γ + γ ) + ] ( β z) ( β z), (5.7) ( β z) 3 ( β z) 3 l I Ie e = + +. (5.8) γ γ ( γ + γ ) [ e ( γ + γ ) + ] [ e ( γ + γ ) + ] ( β z) ( β z)

114 0 The elemens of he Fisher informaion marix for an observaion are he negaive expecaion of he above second parial derivaives: l τ l E = f(; ) d 0 β z, (5.9) β M l τ l E = f(; ) d 0 β z, (5.0) k βk l τ l E = f(; ) d 0 γ z, (5.) γ l τ l E = f(; ) d 0 γ z, (5.) γ l τ l E = f(; ) d γ 0 γ z, (5.3) γ γ where probabiliy densiy funcion is give by Eq. (5.7).

115 0 Wih layou of he ALT plan shown in Figure 5., he Fisher informaion marix for he sress combinaion z L is F LLL l( z ) L E 0 0 β L L M O 0 M M l( zl) 0 0 E 0 0 β = k. (5.4) l( zl) l( zl) E E γ γ γ l( zl) l( zl) E E γ γ γ Similarly he Fisher informaion marices for oher sress combinaions can be easily obained. Finally, he Fisher informaion marix for all es unis allocaed in Figure 5. is obained as F L np F. (5.5) = L L i = 0 ik i ik i ik which is a funcion of he model parameers ( β, β, K, βk, γ, γ), sress levels ( z L, z M, z H ), ( z L, z M, z H ),, ( z kl, z km, z kh ) and he proporions of es unis allocaed o he 3 k sress combinaions pii K i, i 0,, ;,,, k j = j = K k. The asympoic variance-

116 03 covariance marix Σ of he ML esimaes ( ˆ β, ˆ β ˆ ˆ ˆ, K, βk, γ, γ ) is he inverse of he Fisher informaion marix F Var( ˆ β) 0 L O 0 M M Σ= M 0 Var( ˆ β ) 0 0 = F k 0 L 0 Var( ˆ γ ˆ ˆ ) Cov( γ, γ) 0 0 Cov( ˆ γ ˆ ˆ L, γ) Var( γ), (5.6) which is a ( k + ) ( k + ) symmeric marix. Under he sandard regulariy condiions, he ML esimaes ( ˆ β, ˆ β ˆ ˆ ˆ, K, βk, γ, γ), which are calculaed based on he values of failure ime sample, are asympoically normally disribued wih mean ( β, β, K, βk, γ, γ ) and variance-covariance marix Σ. The asympoic covariance marix Σ depends on he inheren regression model, he layou of he ALT design and he parameers. Thus, for he POM-based ALT, if we have he iniial baseline esimaes for ( β, β, K, βk, γ, γ ), i is sraighforward o esimae he variance-covariance marix Σ numerically based on he iniial esimaes Opimizaion Problem Formulaion The appropriae opimizaion crierion for ALT plans depends on he purpose of he acceleraed life esing. As presened earlier, he possible crieria include he minimizaions of:

117 04 () he variance of esimae of a percenile of he failure ime disribuion a he design sress condiions, () he variance of reliabiliy esimae or hazard rae esimae a design sress condiions over a pre-specified period of ime, (3) he variance of an reliabiliy esimae or hazard rae esimae over a range of sress, (4) he variance of he esimae of a paricular parameer. In his secion, he opimizaion crierion is chosen o minimize he asympoic variance of he reliabiliy funcion esimae over a pre-specified period of ime T a he design sress condiions, ie, o minimize T [ ˆ Var R( ; z D)] d. 0 The variance of he reliabiliy funcion esimae a he design sress levels Var[ Rˆ ( z )] is obained by Dela mehod as D Var[ Rˆ ( ; z )] D R ( ; zd) R ( ; zd) R ( ; zd) R ( ; zd) = L Σ β βk γ γ R ( ; zd) R ( ; zd) R ( ; zd) R ( ; zd) L β β γ γ k ( β, β ˆ ˆ ˆ, K, βk, γ, γ) = ( β, β, K, βk, ˆ γ, ˆ γ) T (5.7) Wih he limiaions of available es unis n, es ime τ and specificaion of minimum failures a each sress combinaion, he objecive of he ALT plan is o opimally allocae he sress levels ( z L, z M, z H ), ( z L, z M, z H ),, ( z kl, z km, z kh ) and he proporions

118 05 of es unis allocaed o he 3 k sress combinaions pii K i, i 0,, ;,,, k j = j = K k, so ha he asympoic variance of he reliabiliy esimae a design sress levels is minimized over a pre-specified period of ime T. The opimum decision variables [( z L, z M ( z L, z M, z H ),, ( z kl, z km, kh solving he following nonlinear opimizaion problem, z H ), z ), pii K i, i 0,, ;,,, k j = j = K k] are deermined by Objecive funcion Min T f ( x) = [ ˆ Var R( ; z D)] d 0 Subjec o 0< p <, i = 0,,; j =,,, k ii Kik j K p ii K i k = z z z z z D L M H upper M zkd zkl zkm zkh zkupper np K [ R( τ z, z, K, z )] MNF, i, i, K i = 0,, ii ik i i ik k Σ= F where MNF is he required minimum number of failures Opimizaion Algorihm

119 06 The nonlinear opimizaion problem formulaed in he above secion is an opimizaion problem wih nonlinear objecive funcion, nonlinear consrains, and mulivariable decision variables. Since he objecive funcion is complicaed, we use a version of he mulivariable consrained search mehods, where no derivaives are required, o solve his opimizaion problem. The procedure is based on COBYLA (Consrained Opimizaion BY Linear Approximaions) opimizaion mehod proposed by Powell (99). This mehod is a sequenial search echnique which has proven effecive in solving problems wih nonlinear objecive funcion subjec o nonlinear consrains as well as linear and boundary consrains. Wihou loss of generaliy, we use he following consrained mulivariable nonlinear problem formulaion o illusrae he COBYLA algorihm: n Min f( x), x R, Subjec o ci ( x) 0, i =,, K, m (5.8) The COBYLA algorihm is based on he Nelder and Mead s mehod (Nelder and Mead 965) and he idea of generaing he nex vecor of variables from funcion values a he j verices { x : j = 0,, K, n} of a non-degenerae simplex in R n. In his case here are unique linear funcions, ˆf and { cˆ : i =,, K, m}, ha inerpolae he nonlinear objecive i

120 07 funcion f and nonlinear consrains { c : i =,, K, m} a he verices, and we approximae he calculaion (5.8) by he linear programming problem i Min ˆ n f( x), x R Subjec o cˆ i( x) 0, i =,, K, m (5.9) Changes o he variables are resriced by a rus region bound, which gives he user some conrol over he seps ha are aken auomaically and which respond saisfacorily o he fac ha here may be no finie soluion o he linear programming problem (5.9). The rus region radius ρ remains consan unil prediced improvemens o he objecive funcion and feasibiliy condiions fail o occur. Then he rus region radius is reduced unil i reaches a final value ha has o be se by he user. The COBYLA algorihm employs a meri funcion of he form Φ ( x) = f( x) + μ max{max{ c ( x ): i =,, K, m},0} (5.30) i in order o compare he goodness of wo differen vecors of variables. Here μ is a parameer ha is adjused auomaically. So we have Φ ( x) = f ( x ) whenever x is feasible, and holds. n x R is beer han n y R if and only if he inequaliy Φ ( x) <Φ( y ) The algorihm includes several sraegies, and is summarized wih he aid of Figure 5..

121 Figure 5. A summary of he COBYLA algorihm 08

122 09 Firsly we consider he Generae * x. The vecor of variables he linear programming problem (5.9). If he resulan * x is generaed by solving * x is no in he rus region, we redefine * x by minimizing he greaes of he consrain violaions cˆ x i = K m subjec o he rus region bound. * { i( ):,,, } The branch Δ ensures ha he simplex is accepable. The definiion of accepabiliy follows: For j =,, K, n, le σ j be he Euclidean disance from he verex x j o he j opposie face of he curren simplex, and le η be he lengh of he edge beween 0 x. We say ha he simplex is accepable if and only if he inequaliies j x and j σ αρ, j,,, n, j = K (5.3) η βρ hold, where α and β are consans ha saisfy he condiions 0 < α < < β. The vecor x Δ j is defined as follows. If any of he numbers { η : j =,, K, n} of Eq. (5.3) is greaer han βρ, we le l be he leas ineger from [, n ] ha saisfies he equaion l j η = max{ η : j =,, K, n}. (5.3) Oherwise we obain l from he formula

123 0 l σ j = min{ σ : j =,, K, n}. (5.33) The ieraion replaces he verex x l by face of he simplex ha is opposie he verex Δ x, so we require x Δ o be well away from he l x. Therefore we le v l be he vecor of uni lengh ha is perpendicular o his face, and we define he vecor Δ x by Δ 0 l x = x ±γρv, (5.34) where he sign is chosen o minimize he approximaion ˆ Δ Φ( x ) o he new value of he meri funcion, and γ is a consan from he inerval ( α,). Then he nex ieraion is j given he simplex ha has he verices { x : j = 0,, K, n, j l} and Δ x Numerical Example An acceleraed life es is o be carried ou a hree emperaure levels and hree volage levels for MOS devices in order o esimae is reliabiliy funcion a design emperaure level of 5 C and volage level of 5V. The layou of he es is shown as in Figure 5. where here are oally 9 sress combinaions. The es needs o be compleed in 300 hours. The oal number of unis available for esing is 00. To avoid he inducing of failure modes differen from ha expeced a he design sress levels, i has been deermined, hrough engineering judgmen, ha he emperaure level should no exceed

124 50 C and he volage level should no exceed 0V. The minimum number of failures for each sress combinaion is specified as 5 unis. Furhermore, we expec ha he acceleraed life esing provide he mos accurae reliabiliy esimae over a 0-year period of ime. The es plan is designed hrough he following seps:. According o he Arrehenius model, we ransform he Celsius emperaure scale o /Kelvin scale as he covariae z in he es plan. Then he design sress level of emperaure z D is /(5+76.3) or K of emperaure z upper is /( ) or K., and he upper bound of sress level. A baseline experimen is conduced o obain a se of iniial values of he parameers for he PO model wih quadraic odds funcion. These values are: β = 600 β = 0.0 γ = γ = 0 3. Under he consrains of available es unis and es ime, he objecive of he es plan is o opimally allocae he sress levels and es unis so ha he acceleraed life es provides he mos accurae reliabiliy esimae of he produc a design sress condiions, i.e. o ensure ha he asympoic variance of he reliabiliy funcion

125 esimae a design condiions is minimized over a pre-specified ime period of 0- year. The decision variables are he sress levels ( z L, z M, z H, z L, z M, and z H ) and he proporion of es unis ( p, i = 0,, ; j = 0,,, ) a each sress level ij combinaion as shown in Figure 5.. The values of he opimum decision variables are deermined by solving he following nonlinear opimizaion problem wih nonlinear consrains as well as linear and boundary consrains: Objecive funcion Min T f ( x) = [ ˆ Var R( z D )] d 0 Subjec o 0< p <, i = 0,,; j =,,3 ij j p ij =, i, j 5 C / z 73.6 / z 73.6 / z C, L M H 5 z z z 0, L M H np [ R( τ z, z )] MNF, i, j = 0,, ij i j Σ= F where MNF = 5, T = 0 years, n = 00, τ = 300 hours, x = ( z, z, z, z, z, z, p, p, p, p, p, p, p, p, p ). L M H L M H

126 3 4. We use he algorihm as described in secion 5..5 o solve his opimizaion problem. The algorihm is implemened using SAS/IML and SAS/OR. SAS/IML sofware is a powerful and flexible programming language (Ineracive Marix Language) in a dynamic, ineracive environmen. The fundamenal objec of he language is a daa marix. The programming is dynamic because necessary aciviies such as memory allocaion and dimensioning of marices are performed auomaically. 5. The opimum decision variables ha minimize he objecive funcion and mee he requiremen of he consrains are: Sress levels: TL = 76 C, T = 65 C, T = 44 C; M H VL = 5.35V, V = 7.39V, V = 9.5V. M H The corresponding proporions of unis allocaed o each sress level combinaion are are lis in he following able: Table 5. Proporions of unis allocaed o each sress level combinaion p T = 76 C T = 65 C T = 44 C ij L M H VL VM VH = 5.35V = 7.39V = 9.5V The objecive funcion value is

127 4 5. PO-based Muliple-Sress-Type ALT Plans wih Simple Sep-Sress Loading Acceleraed life esing (ALT) procedures are commonly used o evaluae he lifeime characerisics of highly reliable componens. If a consan sress ALT is used and some seleced sress levels are no high enough, here are many survived unis by he limied esing period, hus reducing he effeciveness of ALT. To ensure enough failed unis in a limied esing period, sep-sress acceleraed life esing (SSALT) has been developed. Mos of he previous work on designing ALT plans is focused on he applicaion of a single sress ype. Especially no previous work on sep-sress ALT plans has invesigaed he use of muliple sress ypes. Alhadeed and Yang (005) design opimal simple sepsress plan for cumulaive exposure model wih consideraion of single sress ype. Only opimal ime of changing sress level using log-normal disribuion is deermined in heir paper. Elsayed and Zhang (005) propose opimum simple sep-sress ALT plans based on nonparameric proporional hazards model. Alhadeed and Yang (00) also design opimal simple sep-sress plan for Khamis-Higgins model using Weibull disribuion. Teng and Yeo (00) presen a Transformed Leas-Squares (TLS) approach o drive opimum ALT plans. Khamis and Higgins (996) design he opimum sep-sress es using he exponenial disribuion. Bai e al. (989) discuss an opimal plan for a simple sep-sress ALT wih censoring for exponenial life daa. Miller and Nelson (983) obain he opimum simple sep-sress ALT plans where he es unis have exponenially disribued life. All of he previous work has only considered a single sress ype applicaion. However, he mission ime of oday s producs is exended so much ha i

128 5 becomes more difficul o obain significan amoun of failure daa wihin reasonable amoun of ime using single sress ype for simple sep-sress es. Muliple-sress-ype ALTs have been employed as a means of overcoming such difficulies. For insance, Kobayashi e al. (978), Minford (98), Mogilevsky and Shirn (988), and Munikoi and Dhar (988) use wo sresses o es capaciors, and Weis e al. (988) employ wo sresses o esimae he lifeime of silicon phoodeecors. In his secion, we propose muliple-sress-ype ALT plans based on he proporional odds (PO) model wih simple sep-sress loading. We do no make assumpions abou a common life disribuion of he es unis. The cumulaive exposure model is used o derive he life disribuion of es unis afer he sress level changing ime. The plans are opimized such ha he asympoic variance of reliabiliy predicion a design sress over a specified period of ime is minimized. We presen he proporional odds model and corresponding maximum likelihood esimaion for acceleraed life esing. We inroduce a simple sep-sress es based on PO model wih muliple sress ypes and explain how he cumulaive exposure model is applied o his case. A nonlinear opimizaion problem is formulaed o design he opimum ALT plans in his secion. Asympoic variance of he reliabiliy predicion a he design sress levels has been chosen as he objecive funcion of he nonlinear opimizaion problem. This opimizaion problem could be solved by direc search algorihm, such as he COBYLA algorihm proposed by Powell (99).

129 6 Sep-sress loading of acceleraed life esing is described as follows. A se of samples is subjeced o es saring wih specified low sresses. If we don have enough failures by a specified ime, he sress levels are increased and held consan for anoher period of ime. We repea his procedure unil enough failures achieved. The sep-sress ALT usually resuls in enough failures in a shorer period of ime han he consan-sress ALT does. Now we consider POM-based muliple-sress-ype ALT plans wih simple sep-sress loading. In he esing scheme, es unis are iniially placed on es a a low sress level vecor z L, where hese low sress levels saisfy z L z D, and run unil sress level changing ime τ. Then he sress levels are increased o he high level vecor z H and held consan unil a predeermined censoring ime τ. 5.. Tes Procedure. There are k ypes of sresses, z, z, K z k, applied o he es unis.. There are n es unis iniially placed under es a he low sress levels z = ( z, z, K, z ) unil changing ime τ. Then surviving unis a ime τ are L L L kl subjeced o higher sress levels z = ( z, z, K, z ) unil a pre-deermined H H H kh censoring ime τ. Figure 5.3 depics he es procedure for he case of k =. As shown in Figure 5.3. We sar he es wih all low sress levels and increase all he sress levels simulaneously a he sress changing ime τ. As an alernaive we

130 7 could increase he differen sress levels following some arbirary orders. How o deermine he opimum order is beyond he coverage of his disseraion and will be invesigaed as a opic of Equivalen ALT Plans in he fuure work as menioned in Chaper 6. In his disseraion we only consider increasing all he sress levels simulaneously since i is easy o carry ou. z H z L z D z H z L z D τ τ Figure 5.3 Sep-Sress loading wih muliple-sress-ype 3. There are n i failure imes observed while esing a he low sress levels and high sress levels respecively, i = L, H, and nl + nh n. 4. The objecive of he sep-sress ALT (SSALT) plan wih muliple sress ypes is o deermine he opimum values of he sress levels z L and z H, as well as he opimum changing ime τ so as o minimize he esimaion error of he reliabiliy esimaes a he design sress levels z. D

131 8 5.. Assumpions. The proporional odds (PO) model is assumed o fi he failure ime daa: θ( ; z) = exp( β z) θ ( ), (5.35) 0 where z = ( z, z, K, z ) k, which is a column vecor of sress levels; and β = (,, K, ), which is a column vecor of model parameers. β β β k. The baseline odds funcion θ 0( ) is quadraic and given by θ () = γ + γ, (5.36) 0 where γ and γ are unknown parameers, here is no inercep erm because he odds funcion always crosses he origin. 3. The failure imes of he es unis are saisically independen Cumulaive Exposure Models To analyze he failure ime daa from a sep sress es, we need o relae he life disribuion under sep-sresses o he disribuion under consan sresses. We adop he mos widely used cumulaive exposure model o derive he cumulaive densiy funcion

132 9 of he failure ime for a es uni experiencing sep-sress loading since he odds funcion changes immediaely afer τ, he sress level changing ime under sep-sress loading. The cumulaive exposure models assume ha he remaining life of a es uni depends only on he exposure i has seen, and he uni does no remember how he exposure was accumulaed. Figure 5.4 shows he relaionship beween consan-sress and sep-sress disribuions. F i () F() z H z L s τ τ Figure 5.4 Relaionship beween consan-sress and sep-sress disribuions Le Fi ( ) denoe CDF of ime o failure for unis run a consan sress z i, i = L, H. Following proporional odds model, hey are given by: L F ( ) = exp[ ln{ θ ( ) e β z + }], (5.37) L 0 H F ( ) = exp[ ln{ θ ( ) e β z + }]. (5.38) H 0

133 0 The es runs under sress z L o ime τ a sep. The populaion CDF of unis failed by ime in sep is: F ( ) F( ) = L τ < (5.39) Tes unis a sep have an equivalen saring ime s, which would have produced he same populaion cumulaive failures. Thus, s is he soluion o he following equaion F () s = F ( τ ), (5.40) H L or equivalenly β z H β zl exp[ ln{( γ s+ γ s ) e + }] = exp[ ln{( γ τ + γ τ ) e + }], (5.4) The above equaion is obained by subsiuing Eq. (5.37) and Eq. (5.38) ino Eq. (5.40). Therefore he populaion CDF of unis failing by ime τ is F () F( s) = H τ +, τ. (5.4) In summary, he equivalen populaion cumulaive disribuion funcion of unis failing under muliple sep-sresses is

134 FL (), < τ F () = FH ( τ+ s), τ (5.43) 5..4 Log Likelihood Funcion A es uni experiences one of wo possible ypes of failure paerns: (a) i eiher fails under sress level z L before he sress is changed a ime τ, or (b) i does no fail by ime τ and coninues o run eiher o failure or o censoring ime τ a sress level z H. The following provides he log likelihood of a single observaion (ime o failure). Firsly we define he indicaor funcion I = I( τ) in erms of he sress changing ime τ by: I if τ, failure observed before ime τ, = I ( τ ) = 0 if > τ, failure observed afer ime τ. and he indicaor funcion I = I( τ ) in erms of he censoring ime τ by: if τ, failure observed before ime τ, I = I( τ ) = 0 if > τ, censored a ime τ. where τ τ. Following proporional odds model and Eq. (5.43), we have

135 β zl ( γ+ γe ) β zl + γ e + f(; z L) =, for < τ [( γ ) ] f(; zh ) = [( γ ) ] β zh ( γ+ γ ) e β zh + γ e +, for τ β zl Λ (; zl) = ln[( γ + γ ) e + ], for < τ Λ = + + β zh (; zh ) ln[( γ γ ) e ], for τ where = τ + s. Therefore he log likelihood of one single observaion can be expressed as l = ln L( ; z, z ) = I { I [ln f( ; z )] + ( I )[ln f( ; z )]} ( I ) Λ( ; z ) L H L H H β zl = II{ln( γ + γ ) + β z ln[( γ + γ ) e + ]} L β zh ( I) I{ln( γ γ ) β zh ln[( γ γ ) e ]} + + β zh ( I) ln[( γ γ ) e ] (5.44) The firs parial derivaives wih respec o he model parameers are:

136 3 β zh lnl II IIe ( I) I ( I) Ie γ γ + γ H ( + ) e + γ + γ ( + β z ) e + β zl = + β z L γ γ γ γ β zh ( I ) e β zh ( γ + ) e + γ (5.45) ln ( ) ( ) γ γ + γ ( + ) + γ + γ ( + ) + β zl β zh L II II e I I I I e = + β z L β z H γ γ e γ γ e ( γ + ) + β zh ( I) e β zh γ e (5.46) ln L I I ( + ) e ( I ) I ( + ) e = II + I I β ( ) ( ) β zl zl γ γ zh γ γ zl ( ) zh β zl β zh γ+ γ e + γ + γ e + I z γ γ e β ( ) H ( + ) β zh ( γ + γ ) e + zh β zh (5.47) The las equaion could be decomposed ino ln L IIz ( + ) e ( I) Iz ( + ) e β ( γ γ ) ( γ γ ) β zl L γ γ H γ γ = IIz L + ( I ) IzH β zl β zh + e + + e + I z e β ( ) H ( γ + γ ) β zh ( γ + γ ) e + zh β zh M

137 4 ln L IIz ( + ) e ( I) Iz ( + ) e = IIz + I I β ( γ γ ) ( γ γ ) k β zl kl γ γ kh γ γ Lp ( ) zkh β zl β zh + e + + e + I z e β ( ) kh ( γ + γ ) β zh ( γ + γ ) e + zh β zh The above equaions, when summed over all es unis and se equal o zero, provide he maximum likelihood esimaes for he model parameers Fisher s Informaion Marix and Variance-Covariance Marix If we only consider he correlaions among γ and γ, he second parial derivaives wih respec o he model parameers are as shown as: β zl ln L II II e ( I) I = + β z γ ( γ+ γ) L [( γ ( + γ ) e + ] γ+ γ ) ( ) ( ) + +, [( γ γ ) ] [( γ γ ) ] β zh β zh I I e I e β zh β zh + e + + e + (5.48) 4 β zl ln L 4II II e 4( I) I = + β z γ ( γ+ γ) L [( γ ( + γ ) e + ] γ+ γ ) ( ) ( ) + +, [( γ γ ) ] [( γ γ ) ] 4 β zh 4 β zh I I e I e β zh β zh + e + + e + (5.49)

138 5 3 β zl ln L II II e ( I) I = + β z ( ) L [( γ ( + γ ) e + ] ) γ γ γ + γ γ + γ ( ) ( ) + +, [( γ γ ) ] [( γ γ ) ] 3 β zh 3 β zh I I e I e β zh β zh + e + + e + (5.50) + + β zl β zh ln L IIz L( γ γ) e ( I) Iz H( γ γ ) e = β zl β zh β [( γ+ γ ) e + ] [( γ + γ ) e + ] ( I ) z ( + ) e [( γ ) ] β zh H γ γ β zh + γ e +, (5.5) M + + β zl β zh ln L IIz kl( γ γ) e ( I) Iz kh ( γ γ ) e = β zl β zh βk [( γ+ γ ) e + ] [( γ + γ ) e + ] ( I ) z ( + ) e [( γ ) ] β zh kh γ γ β zh + γ e +. (5.5) The above derivaives are given in erms of he random quaniies I, I and sress levels z L, H z as well as he model parameers. The elemens of he Fisher informaion marix for an observaion are he negaive expecaions of he second parial derivaives: l τ l E = f(;, ) 0 L H d β z z, (5.53) β M

139 6 l τ l E = f(;, ) 0 L H d β z z, (5.54) k βk l τ l E = f(;, ) 0 L H d γ z z, (5.55) γ l τ l E = f(;, ) 0 L H d γ z z, (5.56) γ l τ l E = f(; L, H) d γ 0 γ z z, (5.57) γ γ where he probabiliy densiy funcion f( ; z, z ) is given as he derivaive of he equivalen cumulaive densiy funcion in Eq. (5.43). L H The above equaions show he componens of he Fisher s informaion marix for a single observaion. Since all n es unis placed under he sep-sress es experience he same es condiions, he Fisher s informaion marix for he sample is jus he summaion of all unis:

140 7 F l E 0 0 β L L M O 0 M M l 0 0 E 0 0 n β = k. (5.58) l l E E γ γ γ l l E E γ γ γ The Variance-Covariance marix for maximum likelihood esimaes ( ˆ β, K, ˆ β, ˆ ˆ k γ, γ,) is defined as he inverse marix of he Fisher s informaion marix: Var( ˆ β) 0 L O 0 M M Σ= M 0 Var( ˆ β ) 0 0 = F k 0 L 0 Var( ˆ γ ˆ ˆ ) Cov( γ, γ) 0 0 Cov( ˆ γ ˆ ˆ L, γ) Var( γ). (5.59) 5..6 Opimizaion Crierion In order o obain he mos accurae reliabiliy esimae under he consrains of esing condiions, such as ime, cos, number of available unis, ec, we choose o minimize he asympoic variance of he reliabiliy esimaes a he design sress levels over a prespecified period of ime, i.e., minimize

141 8 T T ˆ Var[ R( ; z )] d = Var[ ˆ ] d. (5.60) β D ( ˆ γ + ˆ γ ) e + z D 0 0 As shown earlier, he asympoic variance of he reliabiliy esimae a he design sress levels is derived as: Var[ Rˆ ( ; z )] D R ( ; zd) R ( ; zd) R ( ; zd) R ( ; zd) = L Σ β βk γ γ R ( ; zd) R ( ; zd) R ( ; zd) R ( ; zd) L β β γ γ k ( β, β ˆ ˆ ˆ, K, βk, γ, γ) = ( β, β, K, βk, ˆ γ, ˆ γ) T 5..7 Problem Formulaion The problem is o design an opimum muliple-sress-ype ALT based on proporional odds model using simple sep-sress loading wih consideraion of censoring and he consrains of es unis, censoring ime and specificaion of minimum number of failures a he low sress levels, such ha he asympoic variance of he reliabiliy esimae a he design sress levels is minimized over a pre-specified period of ime T. This opimum ALT plan gives he mos accurae reliabiliy predicion a he design sress levels over he pre-specified period of ime T. The decision variables include he low sress levels z L and sress changing ime τ. The high sress levels z H are given as he highes possible sress levels beyond which he failure mode will change. The opimum soluion of z L and τ is deermined by solving he following nonlinear opimizaion problem.

142 9 Objecive funcion Min T T f ( x) = [ ˆ Var R( ; zd)] d = [ 0 Var 0 ˆ ] d β zd ( ˆ γ + ˆ γ ) e + Subjec o npr[ τ ; z ] MNF, L Σ= F. where zl zl M x = =, τ zkl τ and MNF is he minimum required number of failures a low sress levels. The opimum soluion depends on he values of model parameers ( β, γ, γ ). A design using he pre-esimaes of he model parameer is called a locally opimum design (Chernoff, 96) and is commonly adoped by Bai and Kim (989), Bai and Chun (99), and Nelson (990). We also assume ha he baseline esimaes of ( β, γ, γ ) are available hrough eiher preliminary es or engineering experience obained prior o he design of he opimum ALT plan Numerical Example

143 30 A muliple-sress-ype acceleraed life es wih simple sep-sress loading is o be carried ou for MOS capaciors in order o esimae heir life disribuion a design sress levels: emperaure of 50 C and volage of 5V. The es needs o be compleed in 300 hours. The oal number of available esing unis is 00. To avoid failure modes oher han hose expeced a he design sress levels, i has been deermined, hrough engineering judgmen, ha he applied esing emperaure level should no exceed 50 C and he volage level should no exceed 0V. The required minimum number of failures a he low sress levels is specified as 50. Furhermore, he objecive of he acceleraed life es is o provide he mos accurae reliabiliy predicaion a he design sress levels over a 0- year period of ime. The opimum ALT plan is deermined as follows:. According o he Arrehenius model, we ransform he Celsius emperaure scale o /Kelvin as he covariae in he ALT model, i.e., he design sress level of emperaure is z D = /( ) = / 33.6 z H = / 53.6 K. K, and he highes sress level of emperaure is. The PO model is used o fi he failure ime daa. The model is given by: θ ( z ;, z) = exp( β z+ β z) θ ( ), 0 where z is he sress level of emperaure, z is he sress level of volage, β, β are unknown parameers, and θ 0 () is he baseline odds funcion, which is given by θ () = γ + γ. 0

144 3 3. A baseline experimen is carried ou o obain he iniial values for he model parameers. These values are lised below: ˆ β 600 ˆ β 0.0 ˆ γ ˆ γ The problem is o opimally design a muliple sep-sresses ALT o fi failure ime daa wih ype I censoring, under he consrains of available es unis, censoring ime and minimum required number of failure unis a low sress levels, such ha he asympoic variance of he reliabiliy esimae of he produc a design sress levels is minimized over a pre-specified period of ime T, which is 0 years in his case. The opimum decision variables, including he low sress levels z * L, * z L and sress changing ime τ * are deermined by solving he following nonlinear opimizaion problem: Objecive funcion Min T f ( x) = Var[ 0 ˆ ] d β zd ( ˆ γ + ˆ γ ) e + Subjec o npr[ τ ; z ] MNF, L

145 3 / 53.6 z / 33.6, L 5 z 0, L τ < τ, Σ= F. where z z x = =, L L z L τ τ zd / 33.6 z D = z = D 5, z H z / 53.6 z H 0 H = = T = 87600, τ = 300, MNF = We use he algorihm as described in secion 5..5 o solve his opimizaion problem. The algorihm is implemened using SAS/IML and SAS/OR. SAS/IML sofware is a powerful and flexible programming language (Ineracive Marix Language) in a dynamic, ineracive environmen. The fundamenal objec of he language is a daa marix. The programming is dynamic because necessary aciviies such as memory allocaion and dimensioning of marices are performed auomaically. 6. The opimum soluions ha minimize he objecive funcion and saisfy he consrains are

146 33 z = 3 C, * L * z = 8.7V, and τ = 7 hours. * L

147 34 6 CHAPTER 6 EQUIVALENT ALT PLANS 6. Inroducion The significan increase in he inroducion of new producs coupled wih he significan reducion in ime from produc design o manufacuring, as well as he increasing cusomer s expecaion for high reliabiliy, have promped indusry o shoren is produc es duraion. In many cases, acceleraed life esing (ALT) migh be he only feasible approach o mee his requiremen. The accuracy of he saisical inference procedure obained using ALT daa has a profound effec on he reliabiliy esimaes and he subsequen decisions regarding sysem configuraion, warranies and prevenive mainenance schedules. Specifically, he reliabiliy esimae depends on wo facors, he ALT model and he experimenal design of es plans. Wihou an opimal es plan, i is likely ha a sequence of expensive and ime-consuming ess resuls in inaccurae reliabiliy esimaes and misleading he final produc design requiremens. Tha migh also cause delays in produc release, or he erminaion of he enire produc. Tradiionally, ALT is conduced under consan sresses during he enire es. For example, a ypical consan emperaure es plan consiss of defining hree emperaure levels (high, medium, and low) and es unis are divided among hese hree levels where unis a each level are exposed o he same emperaure unil failure or unil he es is erminaed. The es resuls are hen used o exrapolae he produc life a normal

148 35 condiions. In pracice, consan-sress ess are easy o carry ou bu need more es unis and long ime a low sress level o yield enough number of failures. However, in many cases he available number of es unis and es duraion are exremely limied. This has promped indusry o consider sep-sress es where he es unis are firs subjeced o a lower sress level for some ime; if no failures or only a small number of failures occur, he sress is increased o a higher level and held consan for anoher amoun of ime; he seps are repeaed unil all unis fail or he predeermined es ime expires. Usually, sep-sress ess yield failures in a much shorer ime han consansress ess, bu he saisical inference from he daa is more difficul o make. Moreover, since he es duraion is shor and a large proporion of failures occur a high sress levels far from he design sress level, much exrapolaion has o be made, which may lead o poor esimaion accuracy. On he oher hand, here could be oher choices in sress loadings (e.g., cyclic-sress and ramp-sress) in conducing ALT experimens. Each sress loading has some advanages and drawbacks. This has raised many pracical quesions such as: Can acceleraing es plans involving differen sress loadings be designed such ha hey are equivalen? Wha are he measures of equivalency? Can such es plans and heir equivalency be developed for muliple sresses especially in he seing of sep-sress ess and oher profiled sress ess? When and in which order should we change he sress levels in muli-sress muli-sep ess? Figure 6. shows various sress loading ypes as well as heir adjusable parameers. These sress loadings have been widely uilized in ALT experimens. For insance, saicfaigue ess and cyclic-faigue ess (Mahewson and Yuce, 994) have been frequenly

149 36 performed on opical fibers o sudy heir reliabiliy; dielecric-breakdown of hermal oxides (Elsayed, Liao and Wang, 006) have been sudied under elevaed consan elecrical fields and emperaures; he lifeime of ceramic componens subjec o slow crack growh due o sress corrosion have been invesigaed under cyclic sress by NASA (Choi and Salem, 997). These ypes of sress loading are seleced based on he simplificaion of saisical analyses, and familiariy of exising analyical ools and indusrial rouines wihou following a sysemaic refinemen procedure. Due o budge and ime consrains, here is an increasing necessiy o deermine he bes sress loading ype and he associaed parameers in order o shoren he es duraion and reduce he oal cos while achieving reliabiliy esimae wih equivalen accuracy o ha of consan sress esing. Research on ALT Plans has been focused on he design of opimum es plans for given sress loading ype. However, fundamenal research on he equivalency of hese ess has no ye been invesigaed in he reliabiliy engineering. Wihou he undersanding of such equivalency, i is difficul for a es engineer o deermine he bes experimenal seings before conducing acual ALT.

150 37 Figure 6. Various loadings of a single ype of sress Furhermore, as is ofen he case, producs are usually exposed o muliple sress ypes in acual use such as emperaure, humidiy, elecric curren and various ypes of vibraion. To sudy he reliabiliy of such producs, i is imporan o subjec es unis o muliple sress ypes simulaneously in ALT experimens. For consan-sress ess, i migh no be difficul o exend he saisical mehods in he design of opimum es plans for single sress o muliple sresses scenarios. However, he problem becomes complicaed when ime-varying sresses such as sep-sresses are considered. For example, in a muliplesress-ype muli-sep es, issues such as when and in which order he levels of he sresses should be changed are challenging and unsolved. Figure 6. illusraes wo experimenal seings ou of housands of choices as one can imagine in conducing a muliple-sress-ype muli-sep ALT. In general, an arbirary selecion from combinaions of muliple sress profiles may no resul in he mos accurae reliabiliy esimaes,

151 38 especially when he effecs of he sresses on he reliabiliy of he produc are highly correlaed. Therefore, opimizaion of es plans by uning he high dimensional decision variables under ime and cos consrains needs o be carefully invesigaed from he perspecive of saisics, operaions research and engineering physics. Figure 6. Two example seings of an ALT involving emperaure, curren, humidiy and elecric curren 6. Definiion of Equivalency We will consider differen opimizaion crieria depending on he ype of sress loading and he objecive of he es. The opimizaion crieria o be considered are he minimizaion of he asympoic variance of he maximum likelihood esimae (MLE) of: () he specified failure ime quanile a normal operaing condiions; () he mean ime o failure a normal operaing condiions; (3) he reliabiliy funcion or (cumulaive disribuion funcion) a a given age of he produc a normal operaing condiions; (4) he hazard funcion over a specified period of ime a normal operaing condiions;

152 39 (5) he model parameer(s) (for muliple parameers, consider D-opimaliy or A- opimaliy). D-opimaliy is he crierion ha deermines how well he coefficiens of he design Approximaion are esimaed. I requires changes in he locaions of he sampling poins o maximize his crierion which in urn maximizes he confidence in he coefficiens of he approximaion model. A-opimaliy is based on he sum of he variances of he esimaed parameers for he model To sudy he equivalency among ALT plans involving differen sress loadings, several definiions are explored. Some feasible definiions are: Definiion. Two ALT plans are equivalen if hey generae he same values of he opimizaion crierion during he esing wih Type I censoring. Definiion. Two ALT plans are equivalen if difference beween he esimaed imes o failure by he wo plans a normal operaing condiions are wihin δ %, where δ is an accepable level of deviaion. In he following secions, we invesigae he equivalen ALT plans based on he firs definiion. 6.3 Equivalency of Sep-sress ALT Plans and Consan-sress ALT Plans The firs scenario of equivalen ALT plans considered is he case of he equivalency of sep-sress ALT plans and consan-sress ALT plans. Since consan-sress ess are he mos commonly conduced acceleraed life ess in indusry and heir saisical inference has been exensively invesigaed, we focus on deermining he equivalen sep-sress

153 40 ALT plan. The consan-sress ALT plan serves as he baseline es resul for comparison wih he sep-sress plan. More imporanly, he consan-sress ALT plan requires longer es duraion when compared wih oher es plans. Therefore, he efficiency of equivalen plans will also be measured by he percen reducion in es ime. In his secion, we are ineresed in deermining he minimized es duraion of he sep-sress ALT plan, which is equivalen o he baseline consan-sress ALT plan.. Figure 6.3 An example of equivalen consan ALT plan and sep-sress ALT plan As a preliminary invesigaion of he equivalen ALT plans, we consider he following simplified single sress ype case. As shown in Figure 6.3a, he baseline opimum consan-sress ALT plan is deermined wih he pre-deermined censoring ime τ. The baseline consan-sress ALT plan is a good compromise opimum plan (Nelson 004). The adjecive good compromise is due o he fac ha only he low sress level z L is deermined hrough he opimizaion process in order o minimize he asympoic variance of he reliabiliy esimaes a he design sress level z D. I assumes ha he high sress level z H is chosen o be he highes sress level beyond which anoher failure mode will be inroduced. The inermediae sress level z = ( z + z ) / is midway beween M L H

154 4 will be inroduced. The inermediae sress level z = ( z + z ) / is midway beween M L H he low sress level z L and he high sress level z H. The opimum z L is obained such ha he asympoic variance of ML esimaes of he reliabiliy funcion a he design sress level z is minimized. The allocaion of fracion of es unis o he sress levels D z L, z M and z are 4 / 7 H p L =, p M = / 7, and p H = / 7 respecively. This unequal allocaion is a compromise ha exrapolaes reasonably well. For a sample of n es unis, he allocaion is n = 4 n/ 7, n = n/ 7, and n = n/ 7. Therefore he opimizaion L M problem can be formulaed by following he same procedure described in Secion 5..4 wih k = : H Objecive funcion Min T f ( x) = [ ˆ Var R ( ; z )] d C C D 0 Subjec o Σ = C F C p = 4/7, p = / 7, p = / 7 L M H z D zl, zh zupper =, z = ( z + z ) / M L H np [ R( τ ; z )] MNF L np [ R( τ ; z )] MNF M np [ R( τ ; z )] MNF H L M H where

155 4 lc( z ) L E 0 0 β lc( z ) L lc( z ) L FC = n L 0 E E γ γ γ lc( zl) lc( zl) 0 E E γ γ γ lc( zm) E 0 0 β lc( zm) lc( zm) + n M 0 E E γ γ γ lc( z ) M lc( z ) M 0 E E γ γ γ lc( zh) E 0 0 β lc( zh) lc( zh) + n H 0 E E γ γ γ lc( zh) lc( zh) 0 E E γ γ γ x = z L The only decision variable is he low sress level z L in he above opimizaion problem. Le * z L denoe he opimum soluion, and f z denoe he minimum asympoic * C( L) variance of he reliabiliy esimaes given by he consan-sress ALT plan. Compared wih he consan-sress ALT plan, he sep-sress ALT plan can subsanially shoren he es duraion. The censoring ime τ of he sep-sress ALT plan shown in Figure 6.3b is less han he censoring ime τ of he consan-sress ALT plan shown in Figure 6.3a. The objecive of he equivalency of ALT plans is o deermine he minimum

156 43 τ which resuls in he equivalen asympoic variances of he reliabiliy esimaes a he design sress level obained from he wo es plans. To obain he opimum asympoic variance of he reliabiliy esimaes a he design sress level for any given τ of he sep-sress ALT plan, we formulae he opimizaion problem following he procedures described in secion 5..7: Objecive funcion Min T f ( x ) = [ ˆ Var R ( ; z )] d S S D 0 Subjec o npr[ τ ; z ] MNF npr[ τ ; z, z ] MNF z = z upper Σ = S F S where x z = τ l S E 0 0 β l S l S FS = n 0 E E γ γ γ l S l S 0 E E γ γ γ

157 44 MNF and MNF are he minimum required number of failures a low sress level and high sress level respecively Solving he above opimizaion problem, we obain he opimum soluion: * * z, τ, and he achieved minimized asympoic variance f (, ; ) S z τ τ for given τ. Following he * * definiion of he equivalen ALT plans in secion 6., he minimum censoring ime τ * of he sep-sress ALT plan is defined as: τ = inf{ τ f ( z, τ ; τ ) f ( z )}. (6.) * * * * S C L We use he following bisecional search procedure o deermine he minimum censoring * ime τ : Sep. Solve he nonlinear opimizaion problem defined in he consan-sress ALT plan, and find f z, choose a small value number ε arbirarily. * C( L) Sep. Le i =, τ L = 0, τ H i = τ, and τ = ( τ + τ ) = τ /. L H Sep 3. Deermine f ( z, τ ; τ ) by solving he above nonlinear opimizaion problem S * * i defined in he sep-sress ALT plan. Sep 4. Le i = i+. If f ( z, τ ; τ ) f ( z ), hen le * * i * S C L τ L i = and τ = ( τ + τ )/; i τ L H oherwise le τ H i = and τ = ( τ + τ )/. i τ L H

158 45 Sep 5. Deermine f ( z, τ ; τ ) by solving he above nonlinear opimizaion problem S * * i defined in he sep-sress ALT plan. If τ τ ε and i i f ( z, τ ; τ ) f ( z ), * * i * S C L * i hen sop and le τ = τ ; oherwise coninue Sep Numerical Example A consan-sress acceleraed life es is conduced a hree emperaure levels for MOS devices in order o esimae is reliabiliy funcion a design emperaure level 5 C. The es needs o be compleed in 00 hours. The oal number of available es unis is 00. The highes emperaure level ha es unis can experience is 50 C. I is expeced ha he ALT will provide he mos accurae reliabiliy esimae over a period of 0000 hours. A consan-sress ALT plan can be designed hrough he following seps:. According o Arrehenius model, he scaled sress z = 000 /( emp+ 76.3) is used. Then he design sress level z D = 3.35 and he upper bound is z =.9. upper. A baseline experimen is conduced o obain a se of iniial values of he parameers for he PO model wih quadraic odds funcion. These values are: β =.8, γ = 0, and γ = Following he formulaion in secion 6.3 for he consan-sress ALT plan, he nonlinear opimizaion problem is expressed as follows: Objecive funcion

159 46 Min 0000 f ( ) [ ˆ C x = Var RC( ;3.35)] d 0 Subjec o Σ = C F C p = 4/7, p = / 7, p = / 7 L M H 3.35 = zd zl, z = z =.9, zm = ( zl + zh) / np [ R( τ ; z )] 90 L np [ R( τ ; z )] 50 M L M np [ R( τ ; z )] 40 L H H upper where lc( z ) L E 0 0 β lc( z ) L lc( z ) L FC = n L 0 E E γ γ γ lc( zl) lc( zl) 0 E E γ γ γ lc( zm) E 0 0 β lc( zm) lc( zm) + n M 0 E E γ γ γ lc( z ) M lc( z ) M 0 E E γ γ γ lc( zh) E 0 0 β lc( zh) lc( zh) + n H 0 E E γ γ γ lc( zh) lc( zh) 0 E E γ γ γ

160 47 n = 00, τ = 00 x = z L 4. The soluion of he opimizaion problem is: z = 3. and f (3.) = * L C In order o shoren he es period and sill obain he equivalen asympoic variance, we consider he sep-sress ALT plan. The nonlinear opimizaion problem of he sep-sress ALT plan is formulaed as follows: Objecive funcion Min 0000 f ( ) [ ˆ S x = Var RS( ;3.35)] d 0 Subjec o npr[ τ ; z ] 50 npr[ τ ; z, z ] 40 z =.9 Σ = S F S where x z = τ

161 48 l S E 0 0 β l S l S FS = n 0 E E γ γ γ l S l S 0 E E γ γ γ n = 00 * The minimum censoring ime τ of he sep-sress ALT plan is deermined by: τ = inf{ τ f ( z, τ ; τ ) f ( z )}. * * * * S C L Following he bisecional search algorihm (ε is arbirarily chosen as ) in Secion 6.3, he resuls are lised in Table 6.. Table 6. Resuls of bisecional search i i τ f ( z, τ ; τ ) S * * i * From Table 6., he minimum censoring ime τ * = 53. The corresponding opimum low sress level is z = 3.5; he opimum sress changing ime is τ = 46.

162 Simulaion Sudy In his secion, we conduc a simulaion sudy o verify he resulan consan ALT plan and sep-sress ALT plan in he previous secion are equivalen indeed in erms of he esimaion accuracy. The ses of simulaion daa following boh consan ALT plan and sep-sress ALT plan are generaed by Mone Carlo simulaion mehod based on he following reliabiliy funcion. R () = + ( γ )exp( β z). (6.) I can be easily verified ha he assumpion of he PO model is valid for he failure ime samples generaed by he above reliabiliy funcion. The corresponding PO model can be expressed as θ ( ; z) = θ ( )exp( βz) = ( γ )exp( βz). (6.3) 0 The oal sample size is 00 and divided ino hree sress groups wih consan ALT plan: n L = 00, p = 800 / 7 4, n = 57, and n = 9. L M H The model parameers used o generae he Mone Carlo simulaion failure imes are summarized in Table 6. for he consan ALT plan.

163 50 Table 6. Model parameers for he consan-sress ALT simulaion Common parameers Censoring ime: τ = 00 ; Design sress level: z = 3.5; Baseline parameers: β =.8, γ = 0 D Sress Low Medium High Sress level z L = 3. z M =.5 z H =.9 Sample size n L = 4 n M = 57 n H = 9 Daase I II III Rewriing Eq. (6.3) by solving for, we obain = F ( )exp( β z) γ [ F ( )] (6.4) Therefore 4 simulaed failure imes are generaed for low sress level by he following equaion i rand( i)exp( β zl) =, for i =, K,4 (6.5) γ [ rand( i)] where rand( i ) is uniformly disribued random variables on he inerval (0, ). And 57 simulaed failure imes are generaed for medium sress level based on he equaion i rand( i)exp( β zm ) =, for i = 5, K, 7 (6.6) γ [ rand( i)]

164 5 As well here are 9 simulaed failure imes generaed for high sress level derived from he equaion i rand( i)exp( β zh ) =, for i = 7, K, 00 (6.7) γ [ rand( i)] Any simulaed failure ime greaer han 00 will be censored. Similarly, he model parameers for he sep-sress ALT simulaion are lised in Table 6.3. Table 6.3 Model parameers for he sep-sress ALT simulaion Common parameers Censoring ime: τ = 53; Design sress level: z = 3.5; Sress changing ime τ = 46 ; Baseline parameers: β =.8, γ = 0 D Sress Low High Sress level z = 3.5 z =.9 Sample size n = 00 Daase VI V Using he model parameers in Table 6.3, he Mone Carlo simulaion daa for he sepsress ALT are generaed based on he cumulaive exposure model, which is explained in Chaper 5 and shown in Figure 6.4. To be more specific, he procedure of generaion of he Mone Carlo simulaion daa for he sep-sress ALT is as follows. A oal of 00 failure imes are generaed based on he equaion

165 5 i rand i = = γ [ rand( i)] ( )exp( β z), for i,, 00 K (6.8) F i () F() z z s τ τ Figure 6.4 Cumulaive exposure model. Among he 00 failure imes simulaed in sep, all failure imes greaer han τ are discarded. 3. Suppose ha only n failure imes are kep in sep. The remaining 00 n failure ime are generaed based on he equaion rand( i)exp( β z ) = + τ, for = +, K, 00 (6.9) i s i n γ [ rand( i)] 4. Among he 00 n failure imes simulaed in sep 3, all failure imes smaller han τ are discarded and all failure imes greaer han τ are censored. The resuls of he reliabiliy esimaions from he simulaed failure ime daa based on he consan-sress ALT plan and he sep-sress ALT plan are shown in Figure 6.5.

166 consan sep Reliabiliy Esimaion Figure 6.5 Esimaed reliabiliy funcions from consan and sep simulaed daa As shown in Figure 6.5, he esimaed reliabiliy funcions are so close ha he equivalency of he consan-sress ALT plan and sep-sress ALT plan are verified.

167 54 7 CHAPTER 7 EXPERIMENTAL VALIDATION The objecive of his chaper is o validae he PO model and he opimal ALT plans based on his model by conducing an acceleraed life esing in he Qualiy and Reliabiliy Engineering Laboraory of he Indusrial and Sysems Engineering Deparmen. 7. Experimenal Samples The experimen is designed o sudy he effecs of emperaure and volage on he lifeime disribuion of he miniaure ligh bulbs, o predic heir reliabiliy a normal operaing condiions using he PO model, and o design opimum ALT plans based on he PO model. Each experimenal se has a board ha conains up o 3 miniaure ligh bulbs depending on he applied sress levels. The se is placed in a emperaure and humidiy chamber where humidiy is held consan. The es unis are Chicago Miniaure 606-CM49 ype miniaure ligh bulbs. The design work condiions of his ligh bulb are: volage is vols, and curren is 0.06 amps. Mechanically, ligh bulbs consis of a meal base, which iself consiss of a screw hread conac (aached elecrically o one side of he filamen), insulaing maerial and an elecrical "foo" conac (he lile brass bulge on he boom which is elecrically conneced o he oher end of he filamen). The meal conacs a he base of he bulb are

168 55 conneced o wo siff wires ha go o he cener of he bulb and, in urn, hold he filamen. The bulb iself is he glass housing ha no only shields he filamen from oxygen in he amosphere bu also holds in an iner gas, usually argon. The filamen is he par of he bulb ha does he work o creae ligh. I is made up of a long, exremely hin (abou.0 inch) lengh of ungsen, a very versaile meal. The ypical filamen in a household bulb is over six fee long and is ighly wound o form a double-coil. When elecriciy is passed hrough he bulb, he elecrons (curren) vibrae hrough he filamen, creaing very high emperaures: up o 4,000 degrees F. This emperaure is needed o cause he aoms in he filamen o gain energy and hen o emi phoons in sufficien quaniies o bahe he area in a useful amoun of visible ligh. Mos of he phoons emied by a bulb are infrared. Tungsen is one of he only widely available meals ha can wihsand such emperaures, bu in he presence of oxygen i will cach fire and burn iself up. This is why he bulb is evacuaed and filled wih an iner gas. Ligh bulbs evenually fail due o one of four modes:. Breakage of he glass bulb accouns for a small porion of failures, especially in moor vehicles.. Sealing Failure occurs when he bulb's amospheric seal is broken and oxygen eners he bulb. The filamen burns up insanly. Such failures occur when bulbs are screwed ino sockes oo ighly.

169 56 3. Long Term Failure occurs when he filamen evenually becomes so faigued ha is elecrical resisance increases o he poin ha curren will no flow. The inside of he bulb ges very dark and he elecrical conac on he base sars o burn. 4. Thermal shock is he mos common failure mode of bulbs. As soon as he swich is urned on he bulb flashes brighly and hen fails. The reason mos bulbs fail in his manner is hermal shock. When he swich is urned on, full curren suddenly his he filamen a he speed of ligh. This sudden, massive vibraion causes he filamen o wildly bounce. This mechanical movemen causes meal faigue (jus like bending a paper clip unil i breaks) ha resuls in breakage of he filamen. The resuls of hree experimens are used in esimaing he parameer of he PO so ha he reliabiliy of miniaure ligh bulbs is esimaed using he PO a he same sress condiions of he fourh experimen. The reliabiliy is hen compared wih he experimenally obained reliabiliy from he resul of fourh experimen. The deails of experimen are presened in he nex secion. Typical se up ha shows all bulbs working is shown in Figure 7. while Figure 7. shows some failed bulbs.

170 57 Figure 7. Samples of he miniaure ligh bulbs Figure 7. Miniaure ligh bulbs esing se

171 58 7. Experimens Seup In order o coninuously monior he failure imes of esing componens and o conrol he applied facors, an auomaic acceleraed life esing environmen is designed. Figure 7.3 depics he layou of he experimenal equipmen. The NI PCI 69 is a mulifuncion analog, digial and iming daa acquisiion board wih 3 analog inpu, 48 digial I/O and. The SCB-68 is a shielded I/O connecor block wih 68 screw erminals for easy signal connecion o a Naional Insrumens 68- or 00-pin DAQ device. The SCB-68 feaures a general breadboard area for cusom circuiry and sockes for inerchanging elecrical componens. The wo expansion boards for he compuer are used o rerieve he informaion of he curren saus of esing componens and he esing environmen. Figure 7.3 The layou of he acceleraed life esing equipmen NI 69 DAQ board is conneced o SCB 68 I/O connecor block, which is hen wired o he resisor board and he es unis.

172 59 Figure 7.4 shows he programmable power supply used o provide differen volage sress levels. Figure 7.4 The programmable power supply NI LabView sofware is used o develop he applicaion used o coninuously monior he saus of he es unis. The failure ime daa are auomaically saved as spreadshee file by LabView applicaion. Figure 7.5 shows he graphic user inerface of he programmed LabView applicaion.

173 60 Figure 7.5 LabView applicaion inerface 7.3 Tes Condiions Acceleraed ess are performed a DC volage sress of 3.5 V ~ 5V and emperaure beween 75 o C and 00 o C. Temperaures less han 75 o C and above 00 o C are no appropriae since i is difficul o observe he miniaure ligh bulb failure in a reasonable es ime and exrapolaion can no be jusified respecively. As a compromise, he range of 75 o C ~ 00 o C is known as he appropriaed emperaure range o observe failures of miniaure ligh bulbs, The experimen is conduced a four differen sress levels ( 75 o C, 3.5V ), ( 75 o C, 5V ), (50 o C, 3.5V ), (50 o C, 5V ), and (50 o C, V ). Miniaure ligh bulbs are esed a each