AD-AI64 407 EFFECTS OF ASSUMING INDEPENDENT COMPONENT FTUETMS-Y7IF THEY ARE ACTUA (U) OHIO STATE UNIV RESEARCHFOUNDATION COLUMBUS N L MOESCHBERGER ET AL DEC 95

UNCLASSIFIED AFOSR-TR-85-i215 AFOSR-82-0387 F/G 12/1 NI

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MICROCOPY RESOLUTION TEST CHARTNATION4AL BUREAU OF STANDARDS- 1963-A

FOSR.TR- 85 - 1 2 15

RF Project 763265/714837Annual Report

CD EFFECTS OF ASSUMING INDEPENDENT COMPONENT FAILURE TIMES,IF THEY ARE ACTUALLY DEPENDENT, IN A SERIES SYSTEM

Melvin MoeschbergerDepartment of Preventive Medicine

and L)II.

John Klein " LECTEDepartment of Statistics FWB1 2 086 .

For the Period

October 1, 1984 - October 31, 1985

AFOSR/NMBuilding 410

Bolling Air Force BaseWashington, DC 20332

Grant No. AFOSR 82-0307

December 1985

,~~~lThe Ohio State UniversityIIV IIResearch Foundation

1314 Kinnear RoadColumbus, Ohio 43212

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FSR.-TR, 8 - 1 9 1 r qAD R4. TiTtLIE (dad Subtle#) S. 'TYPE OF

r REPORT 4 PEI[ROD COVI[RIEO

Effects of Assuming Independent Component Failure Annual ReportTimes, If They Are Actually Dependent, in a Series 10/1(84 to 10/31/85System a. PERFORMING 040. REPORT NUMNUlR

7. AUTHOR(a) 4. CONTRACT OR GRANT NUMteIRe)

Melvin L. Moeschberger and John P. Klein Grant No.AFOSR 82-0307

S. PERFORMING ORGANIZATION NAME AN ADORSS 10. PROGRAM ELEMENT. PROJECT. TASKThe Ohio State University Research Foundation AREA & WORK UNIT NUMSERS

1314 Kinnear Road Ok OA1 (5Columbus, Ohio 43212

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IS. SUPPI..MANTARY NOTES

Is. KEY WORDS (Centlem. ovn~e *ide, If 0aeeeew aed idemtst? by. week ...&..e)modeling series systems, system reliability, competing risks, bivariateexponential distributions, test for independence, consistency of the productlimit estimator, dependent risks, estimation of component life

20. ABSTRACT (Centfawe M roeveWe ide It 00e00040F mad I e. t bV 6e44 n-m.)The overall objective of this proposal is to investigate the robustness to

departures from independence of methods currently in use in reliability studies* when competing failure modes or competing causes of failure associated with a

single mode are present in a series system. The first specific aim is toexamine the error one makes in modeling a series system by a model which assumesstatistically independent component lifetimes when in fact the component life-times follow some multivariate distribution. The second specific aim is to

(continued)

DDO 1473 EDTIN, of I Nov o s OSOLETE Unclassified

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Block 20- Abstract (Continued)

assess the effects of the independence assumption on the error in estimatingcomponent parameters from life tests on series systems. In both cases,estimates of such errors will be determined via mathematical analysisand computer simulations for several prominent multivariate distributions.A graphical display of the errors for representative distributions will bemade available to researchers who wish to assess the possible erroneousassumption of independent competing risks. A third aim is to tighten thebounds on estimates of component reliability when- the risks belong to ageneral dependence class of distributions (for example, positive quadrantdependence, positive regression dependence, etc.). Major decisions in-volving reliability studies, based on competing risk methodology, havebeen made in the past and will continue to be made in the future. Thisstudy will provide the user of such techniques with a clearer understand-ing of the robustness of the analyses to departures from independent risks,

an assumption commonly made by the methods currently in use.

I

SECURITY CLASSI CATIOIS fl P? . C h-fwr... n.w. *, " .;

Annual Report for the3 Air Force Office of Scientific Research

by

The Ohio State University Research Foundation1314 Kinnear Road

Columbus, Ohio 43210

for

Melvin L. MoeschbergerAssociate Professor of Preventive Medicine

and

John P. KleinAssistant Professor of Statistics

Title: Effects of Assuming Independent Component Failure Times,if They are Actually Dependent in a Series System

Period Covered: October 1, 1984 to October 31, 1985

Principal Investigators: Melvin L. Moeschberger and John P. Klein

Phone: (614) 421-3878 or (614) 422-4017

Business Contact: Phillip V. Spina

Phone: (614) 422- 3,730

Date of Submission: October 26, 1985

Melvin L. Moeschberger, h.D. Mai D. Keller, M.D., Ph.D.'I. "DChairperson of Preventive Medicine

JorP KlaffPh.D J ds# ustagi, Ph.D.0C airperson of Statistics

AIR 1FORCZ O&TiCE C ' T!7TT prrnn iNOTICE CF " .

This ten-

L ta, luch:iical ingor.ation Diviz1ou

6a

!oeschberger, Melvin L.

A. Trable of Contents

Teopic Pages

Abstract 3Specific objectives 4Introduction to Problem-and significance of Study 5-7Progress Report onThird Year's Work 8Methods9Literature Cited 10-12

S Appendices ABC,D,E 13

V

Acce-,on ForNTIS CRA&f

DTIC TA13U "VUO .. ced

By.................

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(3)

Moeschberger, 1. vin L.

W. TECHNICAL SECTION

(I. Abstract

) The overall objective of this poposal is to investi-gate the robustness to departures from independence of methodscurrently in use in reliability studies when competing failuremodes or competing causes of failure associated with'a singlemode are present in a series system. The first specific aimis to examine the" rror one makes in modeling a series aystemby a model which assumes statistically independent componentlifetimes when in fact the component lifetimes follow somemultivariate distribution. -The second specific aim is to assessthe effects of the independence assumption on the error inestimating component parameters from life tests on seriessystems. In both cases, estimates of such errors will be deter-mined via mathematical analysis and computer simulations for

V several prominent multivariate distributions. A graphical dis-play of the-errors for representative distributions will bemade available to researchers who wish to assess the possibleerroneous assumption of independent competing risks. A thirdaim is to tighten the bounds on estimates of component relia-bility when the risks belong to a general dependence class ofdistributions (for example, positive quadrant dependence, posi-tive regression dependence, etc.). Major decisions involvingreliability ptudies, based on compting risk methodology, havebeen made in the past and will contLnue to be made in the future.This study will provide the user of such techniques with aclearer understanding of the robus less of the analyses to de-partures from independent risks, an assumption commonly madeby the methods currently in use.

V.

4

Moeschberger, Melvin L.

I1. Specific Objectives

The overall objective is to investigate the robust-ness to departures from independence to methods currently inuse in reliability studies when competing failure modes or com-peting causes of failure associated with a single mode arepresent in a series system. We shall also refer to such com-petitive events as competing risks. The approach will be throughthe investigation of certain aspects of specific parametric multi-variate distributions or by classes of distributions which areappropriate in reliability analyses when there are competingrisks present.

The specific objectives are:

1) to assess the error incurred in modeling systemlife in a series system assumed to have indepen-dent component lifetimes when in fact the com-ponent lifetimes are dependent.

2) to assess the error in estimating component param-eters (i.e., component reliability, mean com-ponent life, etc.) in a series system employingeither parametric or nonparametric models whichassume independent component failure times whenin fact the lifetimes are dependent and followsome plausible multivariate distribution.*

3) to derive bounds on component reliability whenthe failure modes are dependent and fall in aparticular dependence class (e.g., positive quad-rant dependence, positive regression dependence,etc.).

4) to develop tests of independence, based on datacollected from series systems, by making some

:.U restrictive assumption about the structure of thesystems. **

* A plausible parametric multivariate distribution will beone that satisfies one of the following conditions:

i) the distribution of the minimum of the componentfailure times closely approximates widely accept-ed families of system life distributions.

or ii) the marginal distributions closely approximatethe distributions of component failure times inthe absence of other failure modes.

**This objective has been added to the original objectives be-cause it answers a natural question raised by our preliminaryinvestigation.

.Moeschberger, Melvin L.

ZUX. Zntzoduatdon to Pobeum and Sig nene of Study

avin Weinbe- *(1978) in an edtralcomment in the'blshd ~prfoeding of a worksh opo virnT.. Dilogi-calNaad d Cometing Risks noted -thit Jsthe question of

Mting rslks will not quiety. go away: correctios for .cm-petimg risks zhould be applied routinely to data." Theprbeof m ,tinZ sks commly arises in.a wide range of experi-montal 4situatiJons. Although we shall con our attetonin the following discussion to those situatio involvingseries systems in which competing failure modes or competingcauses of fa l associat with a single mode are present,it is ce y true that we might just- as easily speak ofctriAls, anal eimnts, or other-me ial and bio-

"logial studies whir. competing events int our study oft'i 4 a event of... nt .(cf. agokcos (1979).

Coside lecoi or mechanical ystems, such as-atelite t misson eqpment,. computers, aircraft, missiles

and , weaponr csisitstrng of .several componets in series.Usually e "component will have a random life length and thelife of the entLie system will end with the failure of theshdest lived MRmponent, We will examine two sttuations mopeclosely in which "ccpe .,. 'isks play a vital role.

" First, suppose we are'attempting to evaluate system lifefkom knowledge of the individual component lifetimes. Suchan evaluatin will utiLze either an a.nalysis involving math-&matial statis computer simulation. At a recent

c g n ane.on .. odeling and Siilation, McLem (1981) pesnteda scb" to simulate thd life of a-issl4eI whIch consisted ofmny major- cmponent in seIes.- The failure distribution asso-ciated Vith each' component was assumed to be Icawn (usuallye4p all K WIb1u.) -:To aSive at the MySte aU ds-tIo! Iti, the components were. assumed to act indep.endent .y ofeach "Ctbt. Reali y, ts my or may not be the case.

the omgponet lifetimeS were dependent for any reason, thecompted sysl-m failure distribution (as well as Its subsequent

eIparinet such 'as sgytg Mean life and system rolfiab1ty'fora fied tm) wo!ld ony crudely a =o *im the true

ditrbuio.The first speicific aim of this PropSal is7 to.acetain the error incurred in mdling system life in a

aerieNs systeim assumed to have idependent. component lifetmes(i.e., risks) when, in fact, tbhe risks, axe e.

Second, suppose we wish to evaluate some aspect of thedis-tIbu tion of a particular failure mode based on a typicallife tst of a series systs. The response of interest is the

4A.i until failure of a particular. mode of interst. Frequentlythis response will niot be observyable due to the occur-ence ofsome other event which precludes failure associated with themade of Interest. We shall to= such competing events whichintsupt our study of the main failure modes of interest ascrisks.

SII-

6

moeschberger, Melvin L.

Compt g risks &rise in such relability studies when

pre-detirmin*d period of observation has expiced(Tpe Z censoring).

21" the study is terminate due to a pre-detencuined numberof failures of the particular failure mode of interestbeing obeved (Tye IZ censoring).

3) some systems fail because comonnt other than theoe of interest xalfunctioi.

4) the of interest fais from some cause otherthan the one of interest.

In al four situations, one may think of the* main event ofinterest as being censored, i.e., not fully observa"le. In thefirst two situations, the time to occurene of'the event ofinterest shou'l be Iependent of the" ceno-ing mehanism. Insuch instarmces, the methodology for estimating relevant reliabili-ty probabilities has received onsdeable attention (cf. Davidand Ifoeschberger (1978)', Xalbfeish and Prentice (1980), E'.andt-Jonson nl Johns ('1980), Mann, Schafe, Singp=a.a (1974)and BaloW" and -Poschan (1975) for references and discussion'-Tz thethird situation, the t failure of theinterest may or a not be endepenent of the failure times of

-*othe cii-ionets , the system. For exaqple, there may becomm envomental factors such as extr temperature ichmay affect 'the l4fetime -of several copnents. hs the questionof dependent'competing risks is raised. A *si observ.ation

y be made with respect -to tb fourth situation, viz., failureU "times associated'with different failure modes of a single com-panent may be dependent. or a very special type of dependence,.themodels discussed by Marshall-Olcin (1967), Langberg, Pros.banand Quinzy (1978), and Langberg, Proschan, and Quinzy (1981)alow one to* cmuIert; eedn models into independetoes

Ifr. no asptLcns whatever axe made about the type of"depezaence between the distri-buton of potential failumre times,there appears to be little -hope of estimating relevant componentpa ze. In some situations, one may be appreciably mnsled(cf. Tsiatis (.975), -Peterson, (1976)). However, s ZasnerlIng(1980) so clearly. points ou. in his review of Birnbaum's (1979)

'r z~wiograph

*ther seems to be a need for some robustnessstudies. How far might one be off. quantita-tively, if his analysis is based on incorrectassumptions?"

The second specific aim will address this importantissue. First if a specific parametric model which assumes

.7

Moeschberger, Melvin L.

i-ependent risks has been used in the analysis, it wouldbe of interest to know how the error in estimation- saffected by this assumption of independence. That is, ifindependent specific parametic distrbutions A=e assumedfor the failure t4mes associated with different failuremodes when w really should use a bivariate (or multiva-iate)Aistribution, then what is the .magnitude of the. e=rCr ines~tiating .ccau ,e|"nt prm? Secondly, onm xWWish toallow for a less stringent type of model assumption, and askthe sawe question with regard to the estimation e=ror. That"is, if a nmetrIc analysis is performed, assuming in-dependent risks, when some types. of depe nencies may bepresent, then what is the Magnitude of the estimation error?

The third specific aim will attempt to obtain bounds onthe Ponp ot reliability when the failure times belong toa bzad dependepce class (e.g., association, positive qu t.dep ence, positive regression dependence,. etc.), Moredetails will be presented in the methods section.""

in sminry, ommpating risk analyses have been perfomedin the past and wil continue to be perfomed in the future.This study will provide the user of such techniques with aClearer understanding of the robustness to departures from

I idependent risks, an assumption which most of the methodscurrently in use assume.

3 * 1

,fK

Woeschberger, Melvin L. 8.

IV. Progress Report on Third Year's Work

A summary of the first and second year's work is reported in the annual reports

f dated October 26th 1983 and October 26th 1984 respectively. We believe that during the

past three years we have made substantial progress in dealing with the objectives as

outlined on page 4. In addition to the papers and articles referred to in the first two

annual reports, we would like to mention some of the more recent work.

First, the recently published paper which investigates the problem of improving the

product-limit estimator of Kaplan and Meier (1958) when there is extreme independent

right censoring is presented in Appendix A. This paper looks at several techniques for

completing the product limit estimator by estimating the tail probability of the survival

curve beyond the largest observed death time. Two methods are found to work well for a

variety of underlying distributions. The first method replaces those censored

observations larger than the biggest death time by the expected order statistics,

conditional on the largest death, computed from a Weibull distribution. The Weibull is

chosen since it is known to be a reasonable model for survival in many situations.

Parameters of the model are estimated in several ways, but the method of maximum

likelihood seems to provide the best results. The second method replaces the constant

value of the product limit estimator beyond the last death time by the tail of a Weibull

survival function. Again parameters are estimated by a variety of methods with the

maximum likelihood estimators performing the best.

Second, a paper which obtains bounds on the component reliability, based on data

+' from a series system, for the Oakes (1982) model has been revised. Since this model has

the same dependence structure as the random effects model with w having a gamma

* distribution, these bounds are good for a general class of distributions. The bounds,

4i which are determined by specification of a range of coefficients of concordance, are

found by solving a differential equation in the observable system reliability and crude life

on one hand and the unobservable component survival function on the other hand. This

revision is reproduced in Appendix B.

- * --Iv.-/- . .

Moeschberger, M Ivin L. 9.

Third, we have submitted an overview paper for publication which summarizes

some of the work performed during the past three years. The results of this paper were

presented in an invited talk to the Eastern North American Region of the Biometrics

Society at Raleigh, North Carolina in the Spring of 1985. (See Appendix C for a copy of

this paper.)

Finally, a paper has been developed which discusses some general properties of a

random environmental stress model. Estimation of parameters under the Gamma stress

model is considered, and a new estimator based on the scaled total time on test

transform is presented (See Appendix D). Part of the results in this paper were presented

at the International Statistical Institute meeting in Amsterdam in August, 1985. A copy

of that contributed paper is found in Appendix E.

V. Methods

We refer to pages 8-52 of the original proposal for a discussion of the general

methodology.

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Moeschberger, Melvin L

LITERATURE CITED

3Ab~er.N., Langberg, N.A.& Leon,'R.V. and Proschant F. (1978).Two Concepts of Positive dependence, with applications inimltivariate analysis. Statistics Report No. 14486, FloridaState University* Tallahassee.

Barlow, R.E. and PCOSchan, F. (1]975). Statistical Thelory of~~ Reliability and Life Testing - Probablity 1odl. 131t,

Rinehat and Winston, Inc., New Yorkc-.

41 Basu, A.P. and Klein, J.P. (1.982). Some recent results in competingrisks. Survival AnalYsis. Crowley and Johnson, Editors, 216-229.

~ Berman.S.N. (1963).. Notes on extreme values, competing risks, andsemi.-Markov processes. Ann. math. Statist. 34, 1U04-1106.

134=aum r~ Z. (1979). On *the Mathematics of Competing Risks.~ U.S. Dept. of HEW. Publication No. 79-1351.

Chiangs C.L. (1968). Introduction to Stochastic Processes inBiostatistics. Wiley* NewV WERE

Cozx, D.R. (1962). Renewal TheoEy. Methuen, London.

~ David, S.A. (1974). Parametric approaches to the theory of competingrisks.' Pp. 275-290 in: Proschan and Serfling (1974).

David# H.A. and Moeschberger, t4.L. (1978). Theory of Comm~tingRisks. Griffin, London.L

~~ Zasterling, R. G. (1980). Book review of Z .W. Biznbaum monogaph.Techcustics 22: 131-1.32.

El A dt-Johnsn R.C. and Johnson, N.L. (1980). Survival Modelsand Data Aaly1sis. Wiley, Nov York.

Fisher, L. and Xanarek, P. (1974). Presenting censored sikrvivaldata when censoring'and'survival times may not be independent.Reliabilit and Biometry, Statistical Analysis of Lifelength.

Fishma n, G.S. (1973). Concepts and Methods in Discrete EventDigital. Simulation. -Wily, New YorX.

I Fridayr D.S. and Patel, G.P. (1977). A bivariate exponential modelwith applications to reliability and computer generation of randomvariates. The=r and Applications of Reliability, Vol. I. C.P.Taokos and I.N. ShZIPi Editors,#2-58

Gail, 14. (1975). A& review and critique of some models used incometig risk analysis. siometrics 31, 209-222.

Gibbons, J.D. (1971). No nparametric Statistical Inference.McGraw-Hill Book Company, New York.

Gumbel, E.J. (1960). Bivariate exponential distributions. J_ Aer.

Statist. Assoc. 55- 698-707.

Voel, D.G. (1972). A representation of mortality data by competingrisks. Biometrics 28, 475-488.

~~ Johnsou, X.!.. and Kotz, S. (1972). Distributionls in Statistics:Continuous Mul-tivarlate Distributions. Wley, New York.

Kalbfleisch, J.D. and Prentice, R.L. (1980). The StatisticalAnalysis of Failure Time Data. Wiley, New York.

Kaplan, E.L. and Meier, P. (1958). Nonparamatric estimationfrom Incomplete observations. J. Amer. Statist. Assoc. 53.4S7-481.

Klein, J.P. and Basu A.P. (1981). Replacing Dependent Systemsby Idependent Systems in a Competing Risk Framework withApplications o Submitted for publication.

Kotz, S. (1974). -Multivariate Distributions at a Cross Roadin Statistical Distributions in Scientific Work VI. Reidell

i Pubi ng Co., Boston' 247-270.

Lagakos, S.W. (1979). General right censoring and its impact onthe analysis of survival data. Biometrics 35: 139-156.

Langberg, N.° Proschan, F. and Quinzi, A.J. (1978). ConvertingDe et Models into independent Ones, Preserving Essential

~ Features A=n-Prob., 6, 174-181.

Langbezg, N., Proschan, F. and Quinzi, A.J. (1981). Estmtn•-Dependent Lifelengths with Applications to the Theory ofCopetn Risks ; Ann. -Statist. 9:157-167.

Lee L. and Thompson, W.A., Jr. (1974). Results on failure ti andpattern for'the series system. Pp 291-302 in: Proschan and SerflUng(1974).

Mann, N.R. and Grubbs, F.E. (1974). Approximately optimum con-fidence bounds for system reliability based on component test data.Technometrics 16:335-347.

Mann, l.R., Schafer, R.Z., and Singp=walla, N.D. (1974) Methodsfor the Statistical Analysis of Reliability and Life Data, Wiley, N.Y.

Marshall, A.W. and Olkin, I.. (1967). A Multivariate ExponentialDistributi=n J. Amer. Statist. Assoc. 66, 30-40.

McLean, T.J. (1981). Customer's Risk Evaluation. Paper presented toModeling and Simulation Conference. Pittsburgh, Pa.

MNedehall, W. and Lehman, E.g., Jr. (1960). An approximation tothe negative moments of the positive blki, , al useful in lifetsting. Tachnometrics 2&227-242.

12.

Moeschberger, Meivin L.

Miller, D.R. (1977). A note on independence of multivariate life-S times in competing risks. Ann Statist.S- 576-579.

Moeschberger, M.L. and David, N.A. (1971). Life tests under coM-petLg causes of failure and the theory of -ompeting risks.Biowistrics 27, 909-933.

Moesoberger, M.L. (1974). Life tests under dependent Competing- causes of failure. Technometrics 16, 39-47.

Oakes, D. (1982). A model for association in bivariate survivaldata. J. Roy. Statis. Soc. 44:414-422.

Peterson, A.V., Jr. (1976). Bounds for a joint distribution functionwith fixed sub-distribution functions: Application to competingrisks. Proc. Nat. Acad. Sci. 73. 11-13.

Peterson, A.V., Jr. (1978). Dependent competing risks: bounds fornet survival functions with fixed crude survival functions.Environmental International 1: 351-355.

Prentice, R.L., Kalbfleisch, J.D., Peterson, A.V., Jr., Flournoy,S N.v Farewell, V.T., and Breslow, N.E. (1978). The analysis offailure times in the presence of competing risks. Biometrics 34:

541-554.Proschan, F. and Serfling, R.J. (Eds.) (1974). Reliabilit. andBiometXy: Statistical Analysis of Lifelength. Soorety for'ncis'r-al and Applied HMat ematics, -P-ladelphia, Pennsylvania.

Rose D.M. (1973). Investigation of dependent competing risks.Ph.D. dissertation, Universi:ty of Washington.

Thompson, W.A., Jr. (1979). Technical nete: competing risk pre-sentatin of reactor safety studies Nuclear Safeqty 20: 414-417.I!,. ?siatis , A. (1975). A nide-ifiability aspect of the problem

of reing risks. Proc. Nat. Ac d. Sci. 72, 20-22.

Tsitis, A. (1977). Nonidenifability problems with the reliabilit.approach tC Competing risks. Technical Report No. 490, UniversityOf Wisconsin-zadison.

Winberg, Alvin (1978). Editorial. Environmntal International1:285-287.

13.

APPENDIX A

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1.EP19OR 144.11116 2. GOVT ACCESSION NO 3. RECIPI(NT'S CATALOG MUMOIER

U.TTE(ndS816 S. type or NIKPoMv & PIEWOO covIEMI

A Omparison of several methods of estimting th AnnAualsurivl untin he there is extrem ight 10/1/84 to 10/31/85

censoring AL PECRIORNING ONG. REPORT MUNDmER_______________________________________ 714837 JWI 763265a. coTraACT on GRANT mumBeoveJ

M. L. 1m 9-serr and 3dmn P. Xlein AFSOR -82-0307

9- PER1FORMING ORGANIZATION MA149 AND LOOREKSS IS. PROUMEEET RJC.TSAREA &KUNT NWER.SK

1314 Kinanear RoadAEAO:. 98(bluibus, Ctdio 43212 ACES

Buailding 410M 446 OFA9Doli 9AF. ahi~tnD. C. 20132 7____________

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BuildixM 410Bolling AMB I". ~jSLCAO/WNAIGWashingtcm DC 2033216. CISTRIOUTION STATEMCNT (of We Repwq

Distribuitiam =Wlited; qvwrved for public release.

17. INSTRINUTION STATEMENT (of Me abllro anlow in MteM N IIWO diem RtpuQ

II. SUPPLEMENTARY NOTES

19. Key WORDS (@maItmuS wo ftme.. ad. It 808nMY Md fdowdfy 6Y te NookI)~

justed K(apiat-Maier Survival ~tnaiuBias of Survival Functicui,Life-Testing, survival Analysis, Right cansoring.

tdere is mftv=n cersoring n the right, the Kplan-ier product limit~ estimtoris Jownto be a biased estimator of the survival functicn. several

ficatica of the Kaplanr-aie estiznstor are exandined and ciarsd withrespect to bias anid mean squared error.

DD I JAN72 1473 ESOITION or I MNV41 Sis OSSOLE16TE UnclassifiedSE9CURI1TY CLASSIFICATION OF THIS PAGE (WWO. M. 00#0.

Iboimia 41, 253-l5March 1985

A Comparison of Several Methods of Estitating the SurvivalFunction when There Is Extreme Right Censoring

NL L Mow4er O Joib P. MeDepartments of Preventive Medicine' and Statistics, The Ohio State University,

320 West 10th Avenue, Columbus, Ohio 43210, U.S.A.

SUMMARYWhen there is extreme censoring on the right the Kaplan-Mew product-limit estimator is known tobe a biased estimator of the survival fuancion. Several modifications of the Kaplan-Meier estimatorae examained and compared with respect to bia and mean squared error.

In human and animal survival studies, as well as in life-testing experiments in the physicalsciences, one method of estimating the underlying survival distribution (or the reliabilityof a piece of equipment) which has received widespread attention is the Kaplan-Meierproductlimit estimator (Kaplan and Meier, 1958).

For the situation in which the longest time an individual is in a study (or on test) is notfailure time, but rather a censored observation, it is well known that theme are many

complex problems associated with any statistical analysis (Lapkos, 1979). In particular,the Kaplan-Meter product-limit estimator is biased on the low side (Gross and Clark,1975). In the cue of many censored observations larger than the largest observed failuretime, this bias tends to be worme. Fstimated mean survival time and selected percentiles, aswell as other quantities dependent on knowledge of the tail of the survival function, willalso exhibit such biam.

A practical situation which motivates this study is a large-scale animal experimentconducted at the National Center for Toxicological Research (NCTR), in which mice werefed a particular dose of a carcinogen. The goal of the experiment was to assess the effectsof the carcinogen on survival and on age-pecific tumor incidence. Toward this end, micewere randomly divided into three groups and followed until death or until a prespecifiedgroup censoring time (280, 420, or 560 days) was reached, at which time all those still alivein a given group were sacrificed. Often there were many surviving mice in all three groupsat the sacrifice times.

In general, we consider an experiment in which n individuals are under study andcensoring is permitted. Let 41, ... , 4., denote the m ordered failure times of those mindividuals whose failure times are actually observed (t) ,c ... is t)). The remainingn - m individuals have been censored at various points in time. It will be useful to introduce

. the notation Sj to denote the number of survivors just prior to time f(j); that is, S is thenumber of individuals still under observation at time tj), including the one that died at4j). Then the Kaplan-Meier product-limit estimator (assuming no ties among the tj)) of

Key word" Adjusted Kaplan-Meer survival estimion; Bia of survival fnction; Life-testin& Rightcenoing; Survival analysis

253

A0QI

254 Biometrics, March 1985

the underlying survival function, P(t) Pr(T > t, is

for t < 41)

t) - (- 1)/S for 4) Is t < tJ+,, (I){0t for I S I(. I)

forj- 1... m, wheret4.+-t, if the longest time an individual is on study is a censoringtime or t4.) - co if the lonrst time an individual is on study is a death.

This paper first proposes, in §2, some methods of "completing" the Kaplan-Meierestimator of the survival function by (i) replacing those censored observations that arelarger than the last observed fuilure time by their expected order statistics; (ii) using aWeibull distribution to estimate the tail probability P(t), for t > t4; and (iii) employing amethod suggested by Brown, Hollander, and Korwar (BHK) (1974). The second purposeis to demonstrate the magnitude of the bias and mean squared error (MSE) of the Kaplan-Meier estimator and to compare all methods of "completing" P(t) in the context of theaforementioned mouse study, utilizing simulated lifetimes from exponential Weibull,lognormal, and bathtub-shaped hazard function distributions. These results are presentedin §3.

2. Completion of Kalha-Miar Prodmet-LUmk Esdints

2.1 Expeced Order Statistics

One method of attempting to "complete" Pt), t > , would be to "estimate" the failuretimes for those censored observations that are larger than the longest observed lifetime. Letn, be the number of censored observations larger than 4t.. A theorem regarding theconditional distributions of order statistics states that for a random sample of size n froma continuous parent, the conditional distribution of T.), given T, - ,, u > n - nc,is just the distribution of the (u - n + nk)th order statistic in a sample of size n, drawn fromthe parent distribution truncated on the left at t - , (see David, 1981, p. 20).For computational purposes, take t, as an estimate of the (n - n,)th order statistic. Then

find the expected value of the n, order statistics from the parent distribution truncated onthe left at t,. Since the Weibull distribution with survival function P(t) - exp(-tk/0) hasbeen widely accepted as providing a satisfactory fit for lifetime data, it seems reasonable toemploy the results of Weibull distribution theory to complete P(t), t > t,. (It should benoted that any distribution which is reasonable for the specific situation may be used.) Theexpected values of Weibull order statistics up to sample size 40 for location parameterequal to I and shape parameter equal to .5 (0.5)4(1)8 may be found in Harter (1969). Forlarger sample sizes, he states a recurrence relation which may be used.

To compute expected values of the n, order statistics in question, values for k and a mustbe chosen. One approach is to use the maximum likelihood estimators, k and , computed

gby using all observations to estimat k and 0. A second approach, due to White (1969),uss least squares estimates of k and 9 obtained by fitting the model

lf(4j)) - (I/k) In 0 + (i/k) ln(H())J (2)

to the tj/s, where H(%)) is the estimated cumulative hazard rate at 4, obtained from theKaplan-Meier estimator. In our Monte Carlo study, we found the maximum likelihoodestimators performed better than the least squares estimators in all cases. Consequently,the meihod of least squares will be dropped from future discussion in this paper.

The survival function for a Weibull random variable, truncated on the left at t,, is*P(t) _ CX((Ik - tk)/el, t > 4'. (3)

Esimar of Survivl with Right Censoring 255

So, by the theorm on oder statistics stated at the beginning of this section, the conditionalSsdibutm of T, given T(.. - 4.. (u - n -n, + 1,... , n) will be approximatedby the (u - n + n,)th order statistic in a sample of n, drawn from (3). For simplicity, letjm u - n + n, so that j- 1.... Now the expected value of theJth order statistic from(3) is

- f yk+ f~IkpylAPyJeIkk/)dy (4)

where P(y) - ex(-/O), y- (tk - )I/k o 0 and Tj,, is thejth order statistic in a sampleof size n. Equation (4) can also be written as

E(Tj-a,) - (0','(r .t)/kPzIIPzI 'z'd (5)

where P(z) _ ex(zX-Z Z( y/) Ik O. Now E(T.) may be crudely estimated by'Ii[E(Zt.. + t11,11 (6)

where F(Z.p) is the expected value of the jth order statistic from a sample of size n,determined from Harr's (1969) tables or recurrence relation, and J and k are maximumlikelihood estimators of 9 and k, respectively.

Them n, estimated expected order statistics may then be treated as "observed" lifetimesin adjusting (or "completing") the estimated survival function computed in (1). The areaunder the estimated survival function up to t, remains unchanged. The area under theextended estiated suviva function boned on the , estimated expete order statits is

then added to the initial area to obtain a more precise estimate ofJV(t) [estimated orderstatistic (EOS) extension].

2.2 Wei/bd Maximum Likdeihood Techniques

A straghforward approach to completing P(t) is to setP(t) - exp(-tk/9) for t > t. (7)

Estimates of k and 9 based on all observations can be obtained by either the maximumlikelihood (WTAIL) or the least squares method. However, our study found the completionusing maximum likelihood estimators was always better'in terms of bias and mean squared

One suggestion for ostensibly improving this estimator would be to "tie" the estimatedtail to the product-limit estimator at t,. Two methods were attempted to accomplish thisgoel. First, the likelihood was maximized with respect to k and 0 subject to the constraintthat exp(-t,/O) - Pt). This method will be referred to as the restricted MILE tail probabilityestimate (RWTAIL extension). Second, a scale-shift was performed on the tail probabilityin (7) to tie it to the product-limit estimator. This method led to higher biases and meansquared errors of the survival function and will be dropped from further discussion in thispaper.

2.3 BHK-Tp Methods

The Brown-Hollander-Korwar completion of the product-limit estimator sets

P(t) - ex(-t/O9) for t> t, (8)

256 Biomeric, March 1985

where 0 satisfies P ) - CX(-t ). In the BHK spirit we ti to complete P(t) by aWeibull fiuction which used estimates of k and 0, k and 0*, that satisfied the followingtwo relations:

P(4~.0) - eXp-t(A,/9*)and

The latter method also led to consistently poor performance and the results will not bepresente&

Tab!51 100 (and MSEB1003)for eAvnat me= sw i twmv for WW nous mu of (o'm fm

Wdbili Rentictedhbm %midmod w'rAIL WebuUIc d BILK aff 0msc ecow RWTAIL

Duibsias x at 560 daw K-M eauemica gitomi m extson

Wuibu 400 18.7 -2.000P -1.462 -. 1016 .131 .206(4.034)r (2.271) (1.172) (1.160)b (1.543)

k -. 5 So0 223 -2.802 -2.078 -. 176' .208 .299(7.886) (4.498) (1.922)P (2.344) (3.292)

600 25.5 -3.625- -2.704 -.187 .344 .479(13.179) (7.522) (3.025)P (4.275) (6.031)

400 24.6 -. 991" -. 047 -. 046 016' .0379(1.013) (.215)b (.257) (.275) (.343)k-I 50 32.6 -1.63? -. 049 -. 047' .073 .116(2.696)" (.416)' (.535) (.O) (.705)600 39.3 -2.339w .022b .034 .140 .214(S.592) (.596)' (.987) (1.023) (1.353)400 7.5 -.036 .036w -. 005 .0036 .004(.012r' (.053) (.013) (.014) (.014)

k,4 500 34.6 -. 314 1.S7 -. 020 .014' .019(.109) (2XI30) (.036)P (.041) (.044)600 59.9 -. 903 5.982 .144 .028b .039

(.822) (41.430) (4.168) (.147)P (.157)

Loma 400 20.6 -. 86r -. 1786 -. 544 -. 586 -.412(.7r)- (.179)' (.363) (.403) (.267)

ck-I 500 29.0 -!.427w -. 150' -. 865 -. 918 -. 696(2.060) (.323' (.853) (.938) (.644)

600 36.9 -2.079v -. 022 -1.234 -1.281 -1.038(4.345)" (.571)P (1.679) (1.800) (1.301)

400 8.6 -. 070 .129" -. 047 -.053 -. 027'(.014)P (.056) (.014) (.014)' (.014)'

k=4 500 29.1 -. 330 1.033" -. 170 .181 -. 135'(.118) (1.459) (.051) (.055) (.043)

600 4.5 -. 853 4.430v -. 391 -. 392 -. 356'(.734) (23.159) (.199) (.199) (.17m)

Bbb 400 18.6 -1.069 -. 185 -. 170 1.12r .063'(1.173) (.234)b (260) (1.745) (.361)

p-I S00 26.1 -1.722" -1.59 -. 202 1.523 .0"'6(2.996) (.427) (.560) (3.230) (.608)

600 32.6 -2.45? -. 362 -. 310 1.761 .047'(6.043) (.727)' (.982) (4.490) (1.254)

400 8.1 -3.716" -I.543 -1..47 -. 936 .3436(3.213)" (2.463) (2.476) (1.0 1) (.544)P

p =.4 500 13.3 -2.370" -1.826 -1.814 .825 .58P(5.649)r (3.472) (3.446) (1.03I)' (1.303)

600 18.7 -3.07? -2.191 -2.175 -. 75 .S41'(9.466)" (5.013) (4.983) (I.285)' (2.792)

wrn amum mtmbod.Woom admaiui mediad.

* 4

Estimators ofSurvival with Right Censoring 257

3. A Comperhain of the Varloi Method.U A simulation study of data such as that collected at NCTR was performed. Three groupsof 48 lfetimes were simulated with ail testing stopping at 280, 420, and 560 days,respectivey, for the three group&. Distributions with mean survival times of 400, 500, and600 days were used. The generated lifetimes greater than or equal to the sacrifice time foreach particular group were considered as censored. The remaining set of obseved lifetimes,along with the number censored at the three sacrifice times, constituted a single sample.For each of the distributions studied 1000 such samples were enerated. Weibull distribu-tions with shape parameters .5, decreasing filure rate, 1, constant failure rate, and 4,

I~a/I0 (ad ME/10~)forTab 2

Awl100(a MM 10 fr dnwng9Mt pwmefor venou mahods of complktwn

Eitiimed Weibull WeibuilDhsuda oa staaMi 58AE ~r

(25.189.3)8 (16.424) (7.524)0 (10.812)k - .5 500 -7.655w -4620 1.897 AV.642

(58.604)r (22.711) (24.276) (14.319)P (21.442)600 -10.306 -6390 2.213 .734 1.064

(10621) (42~49 (36.895) (25.419)b (37.911)400 -3.610w 104 .248 .084 .067

(13.035r (1.892 (2.423) (1.980) (2.945)kmi 500 -5.913" .096t I28 .121 .306

(34.963)r (2.995r (4.681) (4.361) (5.903)600 -8.216w 146 .610 .418 .550

(67.459)r (4.1"1b (9147) (8.331) (10.792)(.038)P (236) (.060) (.047) (.063)

k=4 500 -1.195 5.324r -. 031 -.026.04(.2) (33.091)" (.146) (.141)b (.177)

600 -2-554 17.913w A12 .090.08(6.524) (355.02)r (.794) (.676) (.641!b

LOsInanm 400 -2.62r" .0 44 6 -1.263 -1.758 -.967(6.906)" (1.526?P (1.979) (3.407) (1.673)

k- 1 So0 -4.680w lip -2.354 -2.718 -1.908(21.902) (2.708)P (6.153) (7.909) (4.751)

600 -6.73w .759b -3.507 -3.766 -2.980(45.373r' (4.764?P (13.123) (14.M1) (101257)

400 -.085 .161 -. 038 -.162" -.02 0b(.060? (.409r" (.08 ) (.065) (.093)

k=4 500 -1151 3.722" -. 584 -. 657 -.4840(1.566) (17.654r" (.403) (.495) (.318r

600 -2.621 13.695w -1.214 -1.236 -1.1S5P(6.872) (210.30)" (1.616) (1.662) (1.498r

Bathtub 400 -3.6291 -. 177 .0536 -. 104 .105(13.167r" (1.717)P (2.052) (2.058) (3.190)

PM.1 So0 -6.068" -.457 -. 071 _108604q(36.826)" (1.955?P (4.702) (3.6 19) (5.245)

600 -7.997" -.31S .043 -24 -.014b(63.954r" (4.330t (7.786) (7.608) (9.923)

40 -.347 .143t .276 1.154C .981-(.273?0 .34 (1.078) (3.877 (4.747)-

pm.4 So0 -1.425 .521 .764 1.699 1.71r"(2.035) (1.540)b (2.067 (8.574) (10.714)"

600 -3.5W4 -. 137 .13? 2M34 2.450(12.28) (1.804?P (2.352) (17.530) (22.456)

Sm e~atiiton motbodL'Wan atmndion tuethod.

* ~ -~ , J.J .... ,..,~ . 4.,.

3 258 Biometrics, March 1985

increasing failure rate, were used. Lognormal distrbutions, failure rate changes fromincreasing to deceasi with first two moments comparable to the above Weibuil distri-butions with k - 1 and k - 4, were also used. Finally, a bathtub hazard model of Glaser(1980), failure rate changes from decreasing to increasing, was used. This distribution is amixture of an exponential of parameter X with probability I - p and a gamma withparameter X and index 3 with probability p. Mixing parameters ofp- .1 and p - .4 wereused.

The bias and MSE for the estimation of the tail probabilities, ie., the completed portionof the product-limit estimator, were calculated for each hypothesized distribution and foreach competing method of completion. Since these results were extremely similar to thosefound in estimating mean survival time, f -o P(t) di, we show only the bias and MSEof each competing estimator of is in Table 1. This also allows us to demonstrate themagnitude of the bias and MSE of the product-limit estimator of g. The bias and MSE forestimating the 90th percentile are also presented for the various estimation methods inTable 2. As one would expect, the Kaplan-Meier (K-M) estimator performs considerablymore poorly than the other estimation schemes. The BHK extension does very well if theunderlying distribution is exponential or lognormal with first two moments compatiblewith the exponential. BIK does reasonably well for the bathtub-shaped hazard model, butit performs very poorly for the Weibull with increasing failure rate and for the lognormalwith first two moments compatible with the WeibuiL

The remaining three extensions (EOS, WTAIL, and RWTAIL) appear to be somewhatcomparable. Each of them is best under certain circumstances although many times thebiases and MSEs are so close to one another that they are essentially equivalent. Only theEOS extension has the desirable property of never being worst. It usually is competitivewith the method that is best. Ordering the extensions from the standpoint of simplicity,from simplest to most complex, we have BHK, WTAIL, RWTAIL, and EOS.

In summary, the Kaplan-Meier estimator should probably be extended in the presenceof extreme right censoring. The choice of extension depends on one's knowledg of thedistribution of lifetimes une consideration and the extent of computer facilities, available.If the data follow an exponential-type distribution or if no computer facilities are present,the BHK method is the extension of choice due to its simplicity. If the data exhibit anoncontant failure rate and computer facilities are available, then the RWTAIL or EOSextensions seem to be advisable.

AcNoWLBDGWAMT

This work was suppurted in part by the Air Force Office of Scientific Research underContract AFOSR-82-0307.

* R*SUMt

On mit que l'estimateur de Kapian-Meier an un esziiateur baj de la fonction de survie quand le

Mer ont amns t ensompaea du point de uue de kn bii cationd moyem quade apqueln

~Brown, B. W., Jr., Hollander, M., and Korwar, R. M. (1974). Nonparametri towt of hidependencefor censored dat., with applications to bet trnsplant studies. In Reiability and BiometryStatistical Analysis of ,ifeleng h, F. Prouchan and R. J. Soling (eds), 291-302. il&dephiLSociety for Industrial and Applied Mathematic.

David, H. A. (1981). Order Statistics. New York Wiley.Gla=', R. E. (1980). Bathtub and related failure rate c t ra. Journal of the Amercan

Statistical Association 75, 667-672.

" Estimators of Survival with Right Censoring 259

Grown. A. J. and Clark, V. A. (1975). Survival Distributions: Reliability Applications in the BiomedicalSciences. New York: Wiley.

Harter, H. L (1969). Order Statistics and Their Use in Teoting and Estimation. Vol. 2. AerospaceResuarch Laboratories, Office of Aerospac Research, United States Air Force.

Kapa, E. L and Meier, P. (1958). Nonparametric estimation from incomplete observations. Journalof the Amerimn Statisticl Aociation 51 457-481.

Lapkos S. W. (1979). General right censoring and its impact on the analyms of survival data.Biometrcs 35, 139-156.

White, J. S. (1969). The moments of logWeibull owder statistics. Technometrics 11, 373-386.

Received June 1980, revised May 1984.

A .%

A.-.

~Appendix B

pC

w % .

SECURITY CLASSIFICATION O THIS PAGE (3W%.., 0.,. £.,.,.dj, i READ ,.STRUCTIO.S.:Y REPORT DOCUMENTATION PAGE RAD INSTRUTINSFO _

BEFORE COMPLETING FORMI. REPORT NUMBER Z. GOVT ACCESSION NO. I. RECIPIENT*S CATALOG NUMBSER

4. TITLE[ (id Suwbtitle) S. TYPE OF REPORT 6 PERIOD COVEREO

BOUNDS ON NE1 SURVIVAL PROBABILITIES 10/1/84 to l0,31/85FOR DEPENDENT COMPETING RISKS 10/1/84_to_________

4. PERFORMING OiG. REPORT NUMBER

714837 Ma 7632647. AUTHOR(@) 6. CONTRACT OR GMANT UMUERe.)

John Klein and M. L. Moeschberger AFOSR-82-0307

9. PERFORMING ORGANIZATION NAME ANO AOORESS 10. PROGRAM ELEMENT. PROJECT. TASKPEIFOHMINGAREA 4 WOiRK UNIT NUMdBERS

The Ohio State University

1314 Kinnear RoadColumbus, Ohio 43212

It. CONTROLLING OFFICE MAME A ADOORESS 12. REPORT DATE

AFOSR-NM November 26, 1985

Building 410 13. MUMER Of PAGES

Bollin7 AFB, Washinzton. D.C. 20332 1914. MONITORING AGENCY NAME 4 AOOREWSSt Ei

ralet fem CoEnvwllE Ofice) IS. SECURITY CLASS. (of Wo rpo )

AFOSR-PKZ

Building 410 UnclassifiedBolling' AFB Ise. OIECLASSIFICATION10011"GRADING

Washington, D.C. 20332 SCHEDULE

1 I,. DISTRIBUTION STATEMENT (r thi. froprt

Distribution unlimited; approved for public release

17. OmSTRUTION STATEMENT (of the ab. atrt .tmw In t 20, It ,Ioent kwa R.,pEI

L. SUPPLEMENTARY NOTES

.4

19. KEY WORDS (CwWenuo on r.vwo side It necessarAnd Idenify by Meok n nmbeu)

Key words: Competing Risks, Product Limit Estimator, Net Survival Function,Coefficient of Concordance

20. ABSTRACT (Centfnu. a, erse ltie It necesary and Identify by block ft"ONA)

Improved bounds on the marginal survival function based on data from acompeting risk experiment are obtained. These bounds are obtained byspecifying a range of possible concordances for the risks. These boundsare tighter than those of Peterson (1976). A comparison to otherexisting bounds is also made.

,5,,

T DD I , 1473 oTO. o, I Nov 05 iS OBSOLErE UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE (when De fIrae, d)

BOUND. ON N.T S UlVIVAI. loIoASIIJTIK.;

FOR IDPI.NI)ENT C(MI'TIN(; RISKS

by

John P. Klein and M. L. HoeschbergerOeparcmenc ot Scaciscics Deparcment of Preventive MedicineThe Ohio State University The Ohio State University

SUMM1ARYImproved bounds on the marginal survival funcction based on data from a

competing risk experiment are obtained. These bounds are obtained by specifying

a range of possible concordances for the risks. These bounds are tighter than

those of Peterson (1976). A comparison to other exiscing bounds is also made.

Key Words: Competing Risks, Produce Limit Escimacor, Net Survival Function,

Coefficient of Concordance.

C

t. INTRODUCTION

A common problem in survival analysis is to estimate the marginal

survival function of the time, X, until some event such as remission,

component failure, or death due to a specific cause occurs. Often obser-

Xvation of this main event of interest is impossible due to the occurrence

of a competing risk at some time Y X, such as censoring, failure of a

different component in a series system, or death from some cause not

29. related to the study. Standard statistical methods, which assume these

competing risks are independent, estimate the marginal survival function

by the Product Limit Estimator of Kaplan and Meier (1958). This estimator

has been shown to be consistent for the marginal survival function by

Langberg, Proschan and Quinzi (1981) when the risks follow a constant

sum model defined by Williams and Lagakos (1977). When the risks are

not in the class of constant sum models, the Product Limit Estimator

4 is inconsistent and, in such cases, the investigator may be appreciably

misled by assuming independence.

*% %

-kc % . : . .

2

In the competing risks framework we observe T minimum(XY) and

I = x(X < Y) where X(-) denotes the indicator function. Tsiatis (1975)

and others have shown that the pair (TI) provides insufficient information

to determine the joint distribution of X and Y. That is, there exists both

an independent and a dependent model for (X,Y) which produces the same

joint distribution for (T,I). However, these "equivalent" independent

Aand dependent joint distributions may have quite different marginal

distributions. Also, due to this identifiability problem, there may be

several dependent models with different marginal structures which will

yield the same observable information, (T,I): In light of the consequences

*of the untestable independence assumption in using the Product Limit

estimator to estimate the marginal survival function of X, it is important

to have bounds on this function based on the observable random variables

I(T,I) and some assumptions on the joint behavior of X and Y.Peterson (1976) has obtained general bounds on the marginaL survival

function of X, S(x), based on the estimable joint distribution of (TI).

Let Ql(x) - P(T > x, I - 1), and Q2 (x) - P(T > x, I - 0 ) be the crude

survival functions of T. His bound, obtained from the limits on the joint

distribution of (X, Y) obtained by Frichet (1951), is

-J Ql(x) + Q2 (x) < S(x) <.Ql(x) + Ql(O). (1.1)

Since these bounds allow b~r any dependence structure, they can be very wide

and provide little useful information to an investigator.

Fisher and Kanarek (1974) have obtained tighter bounds on S(x) in

terms of a dependence measure a. Their model assumes that simultaneous

"I'U

" .,* 4*., .f -** .,.-. ,., . -. - , : -., . ,, , 4 . ., . . . ..

Cal

to the occurrence of Y an event occurs which either stretches or contracts

the remaining life of X by an amount associated with a. That is,

P(X > xlY - y < x) a P(X > y*+ a(x-y)IY > y + a(x-y)). A large a, for

example, implies that a small survival after censoring is the same as a-times

as much survival if censoring was not present. They show that if a is assumed

known, then the marginal survival function can be estimated from the

observable information. Also these estimates, SM(x),are decreasing in a.

For their bounds, the investigator specifies a range of possible values

aL < a so that S (x) < S(x) < S (x).

Recently, Slud and Rubenstein (1983), have proposed general bounds.

They show that knowledge of the function

S(x) - lira- P(x < X < x +6 ( > x, Y < x)(x) - P(x < X < x +SX > x, Y > x)

if along with the observable information (T,I) is sufficient to uniquely

determine the marginal distribution of X. These estimates Sp(t) are

decreasing functions of p for fixed x. Their bounds are obtained by

specifying a range of possible values p1(x) 1 0(x) P2 (x) so that if

p(x) is the true function S0 2 (x) < S(x) < So1 (x).

In this paper we obtain different bounds'on the marginal survival

function by assuming a particular dependence structure on X and Y. These

bounds are functions of the observables (T,I) and a familiar dependence

measure, the concordance probability between X and Y. Tn Section 2 we

describe this model in detail. In Section 3 we derive the bounds and show

"p consistency when the dependence parameter is known. In section 4 these

bounds are compared to those obtained by Peterson, Fisher and Kanarek,

and Slud and Rubenstein.

4

I [I. THE MODEL

The dependence structure we shall employ to model the joint survival

was first introduced by Clayton (1978) to model association in bivariate

lifetablet and, later, by Oakes (1982) to model bivariate survival data.

Let S(x) - P(X > x), R(y) - P(Y > y),with S(O) - R(O) - l,be the

continuous univariate survival functions of the death and censoring times,

respectively. For 9 >1 define F(x.y) = P(X > x, Y > y) by

F(x,y) - + 0-1 (2.1)

This joint distribution has marginals S and R. As 8-1, then (2.1) reduces

to the joint distribution with independent marginals. For 04-,F(x,y)

min(S(x), R(y)) the bivariate distribution with maximal positive association

for these marginals. The probability of concordance is 6/(0 + i) so that

Kendall's (1962) coefficient of concordance is T - (8 - 1)/(e + 1) which

spans the range 0 to 1.

This model has a nice physical interpretation in terms of the

functions A(x IY - y) and X(xIY > y), the hazard functions of X given Y y

and X given Y > y, respectively. From (2.1) one can show that

A(xIY - y) - BA(xlY > y)

or

P(X > xlY - y) -[P(X > xjY > y))? (2.2)~!

For 8 >1 the hazard rate of survival if censoring occurs at time y is

8 times the hazard rate of survival if censoring does not occur at

time y. This implies that the hazard rate after censoring occurs is

hy

., *P4.

. 5

3 accelerated by a factor of 0 over the hazard rate if censoring had not

occurred. Also when 0 - 1, (2.2), reduces to the condition required by

Williams and Lagakos (1977) for a model to be constant sum and hence for

the usual product limit estimator of S(t) to be consistent (See Basu and

Klein (1982) for details).

Oakes (1982) also shows that (2.1) can be obtained from the following

random effects model. Let S*(x) - exp (- - and let R*(y) beSW(x

similarly defined. Let W have a gamma distribution with density

1 1-

g(w)caw _-1 •- w and conditional on W - w let X,Y be independent with

survival functions {S1x)}w and {R*(y))w . Then, unconditionally, X,Y have

the joint survival function F(x,y) given by (2.1).

For fixed marginals S and R the joint probability density function,

f(x,y), can be shown to be totally positive of order 2 for all 9 > 1.

This implies that (X,Y) are positive quadrant dependent. In particular,

one can show that for S,R fixed the family of distributions

F - (F(x,y): 0 > 11 is increasing positive quadrant dependent in e as

4, defined by Ahmed, et al. (1979).

IIS

6

1I1. BOUNDS ON MARGINAL SURVIVAL

Suppose that X and Y have the joint distribution (2.1) and let

T min (X,Y), then the survival function of T is

I -1 18-

F(T) = [[-jI + -- -1 (3.1)

and the crude density function associated with X,

dq1 (t) - -± P(T < t, X < Y), is given by

qI(t) - 8(t) [F(t)P, (3.2)

where s(t) - -dS(t)/dt.

Now consider the differential equation

s(t)/58 (t) - q (t)/OF(t)f (3.3)

and suppose 6 is known. Then the solution of (3.3) for S(t) is

(t) 1 + (0-1) (u) d if 8 >1SOt)= +(Ol)0 [F(u)p u

(3.4)

t q 1 (u)exp( - I F(u- ) du) if e 1.

0

The functions F(-) and qI(*) are directly estimable from the data one

sees in a competing risks experiment. Let T1 9 .... T denote the observed

test times of n individuals put on test and let Ii. i - 1, ..., n be 1 or 0

according to whether the Ti was an observation on X or Yi. respectively.

L".

1 7n n

Define F(t) - x(Ti > t)/n and Ql(t) X (Ti < t, 1).

Then if 9 is known, a natural estimator of Se(t) is

t dQl" (u) 1Se(t) - (1+ ((-1) -() f (-1) if e >

t 0 d (u) f6(3.5)

= exp( - A ) if 60 F~u

For e - 1, this estimator is of the form of the hazard rate estimator

proposed by Nelson (1972). The estimators (3.5) can be expressed in

the following form for computation purposes.

T T(i)- i -I(i) " 1~(3.6)i 1- 1

(1 + (0-l)n9- (n-i+l) -.T T(i)<tZ(1)-l

if 9>1

where T( , ... , T(n ) are the ordered death times.

For 6 known and if the true underlying joint distribution of (XY)

is of the form (2.1) then S (t) is a consistent estimator of S(t) as shown

by the following theorem.4

Theorem 1. Let (X,Y) have the form (2.1) with marginals S(t), R(t)

respectively. Let 6 > I be known. Then on the set where S(t) > 0 we have

S( (t) S(t) a.s.

*- : *

Proof:

For 0 1, the result follows by a theorem of Langberg, Proschan and

Quinzi (1981). Suppose that e > 1. Note that Ql(t) Ql (t) a.s. and

F (u) - F(u) a.s. by the strong law of large numbers. Since Se(t) is a

SdQiu)

continuous function of 1 in the support of P(u),it suffices to show~0 [F(u) ]e

t dQ(u) t dQj(u)1 - - a.s.

0 [F(u)] 0 [F(uA9

Now, after an integration by parts,

r t d^l (U) t) t

dQl-u Ql(t)te ^ e Q(u)d(_ )0 [F(u) (F(t)) 0 (u)

Q1 (t) t t

-- ; [Ql(u) - Q1 (u)]d( 8 1 ) + I Q(u)d( )"F@(u) F (u)

-lt Q1 (t) 1

R (u)] 0 [l(u) Q(u)]d( u)

t

+ .dQ1(u)0 F (u)

By the dominated convergence theorem

t tim dQ1 (u) 0 (~)I im f = I dQl(u) a.s.,

So(U) [F(u)p

4;

9

li= Q1(c) - Q1(t)

and i sup {1Q1 (U) - q1 (u)I} - 0, a.s.

n-ma

Hence, applying the above results to (3.7), the result now follows: //

To obtain bounds on the net survival function based on data from a

competing risks experiment, we proceed as follows. First, note that from

(3.5) it is true that S (t) is a decreasing function of 6 for fixed t.+ t (u)du

Also, as 0 -~ 1 we have S0()t exp - udlu)

0

which provides an upper bound. Notice that this upper bound corresponds

to an assumption of independence. As 9 * m one can show that S (t) + F(t)

which corresponds to Peterson's (1976) lower bound.

In practice the above bounds, with e - 1, -, while shorter than

Peterson's bounds, may still be quite wide.

Tighter bounds may be obtained by an investigator specifying

a range of possible values for e. if the sample size is sufficiently

large and _ then e2(t) < S(t).! S (t). Specifying el, e21 e2 2

is equivalent to specifying a range of values TI < T < T2 for the

coefficient of concordance T since 8 - (l+-)/(l--T). Hence the primary

value of Se(t) is in putting bounds on S(t) rather than on estimation of

S(t).

10

f - ~ a

10

9 IV. EXAMPLE AND COMPARISONS

To illustrate the bounds obtained in the previous section, consider

the mortality data reported in Hoel (1972). The data was collected on a

group of RFM strain male mice who were subjected to a dose of 300 rads of

radiation at age 5-6 weeks. There were three competing risks, thymic

lymphoma, reticulutum cell sarcoma, and other causes of death. For

illustrative purposes we consider reticulum cell sarcoma as the risk of

interest.

Table 1 reports the value of S6 (t) for concordance T - (0 - )/(0 + 1).

The value of Se(t) at T - 0 corresponds to Nelson's (1972) hazard rate

estimator assuming independence. Peterson's upper and lower bound

(T 1 1) are also reported as are Fisher and Kanerek's bounds and the Slud

and Rubenstein bounds for several values which reflect a positive

association between risks.

From Table 1 we first- note that Peterson's bounds are very wide.

Substantial improvement is obtained if one assumes a non-negative

dependence structure between risks (See Table 2). Further tightening

of these bounds is achieved by assuming that T is in the range 0 to .5

where the width of the boundaries is at most about 50% of that of Peterson's

bounds.

Substantial improvement in the general bounds is also obtained by

the bounds of Fisher and Kanerek or Slud and Rubenstein. The bounds of

Fisher and Kanerek assume a specific censoring pattern and require a

specification of a stretching constant a. Without some additional informa-

tion, such specification may be impossible. Slud and Rubenstein's bounds

are for the general dependence structure. Their bounds require the

C.p

11

specification of the o(t) function. This function is a quantity which

is not easily conceptualized by investigators from either a statistical

or biological perspective. This makes it questionable whether reasoable

upper and lower bounds on p(t) can be extracted from one's prior beliefs.

The major advantage of the bounds printed in this paper is that they

require only the specification of an upper and lower concordance, a

measure quite familiar to most investigators and easily explainable to

nonstatisticians.

NIT

I$.

4 m

>M J

C%4 0 0C" -: 4

* 12

-- 4 N% a0 (4 P. (-4 0~~ a-.~ P1 4 .-4 0 0

ao C

aD . C-4 C-4 Go- PlOr, 0% 7 Nt C4

0% C" 04 v- O

V1 0% .- I% 0% (4 0

a 0 0% C.. 4 0 0V,' e, a* N

% 4 0 Wn % -4 N4 &-I

c4 P10 NO (470

C-4 IA t 4AV-1 It P

0" 0- %a 0 .-4 %(-4 %Q

A 0 -4 %a Cn.4 V-4 %0 (*4 %0

* P0% u- I 0P Nc 1 N

Ch 'TrbC4 N

it. z1 - . '0 0 1 Pt4' 4 'O '%N '

Uc # - -- % - -

Gd -40 4'1 0 e4'

.0 00- P '

C4 W, 0% r-' 0' 1 -4 c 0

co % N M n 04 P.P.

U3~P. 0- C4 C4e .%!W1 ( 4 C4I cc 0

0a "0P N I 0 a-. V. .*~a r ' % N e.'04 a-

-4 ~~1" 1u CP fP1(4.4 00

V3 e1- 0 P 4 p. N0 mI LA -i- 0

w % - fl- - - - -4

en0 N A In0Z a -4 0 3so 0. . .- .. 0

42 C4

a* %0 Nv N4 P10 V1

ka c% 0 ( 0 -7 V .

el C1 -. 10 P1. 0w1 N

0~~ ~k 41I - % P1. % N G6 7 % %04 00T e .0

W AWCO 0 0) *. . . . .. . . . . 00) -I - - -4C-

41 P1 N 00 P I

3 0. N 0%0 cc t1 G

m A%- -

14

Table 2

RELATIVE SIZE OF THE BOUNDS ON NET SURVIVAL

FOR AN ASSUMED DEPENDENCE STRUCTURE

AS COMPARED TO PETERSON'S BOUNDS

Time 0 < T < 1 0 < T < .5 0 < T < .7

V 350 .9707 .0879 .2674

525 .9352 .2449 .5931

600 .7338 .5171 .6787

620 .6722 .5120 .6298

650 .5009 .4420 .4870

675 .3831 .3576 .3797

700 .2883 .2767 .2833

750 .0600 .0600 .0600

N

, .. .... _.I__ .. ... .

* 15

References

I. Ahmed, A. H., Langberg, N.A., Leon, R.V., and Proschan, F. (1979).Partial Ordering of Positive Quadrant Dependence, withApplications. Florida State University Technical Report.

2. Basu, A. P., and Klein, J. P. (1982). Some Recent Results inCompeting Risks Theory. Survival Analysis, Crowley andJohnson, eds. Institute of Mathematical Statistics, Hayward,CA, 216-229.

3. Clayton, D. G. (1978). A Model for Association in BivariateLifetables and its Application in Epidemiological Studies ofFamilial Tendency in Chronic Disease Incidence, Biometrika 65,141-151.

4. Fisher, L., and Kanarek, P. (1974). Presenting Censored SurvivalData When Censoring and Survival Times May Not Be Independent.Reliability and Biometry Statistical Analysis of Lifelength,Proschan and Serfling, eds. SIAM, Philadelphia, 303-326.

5. Frechet, M. Sur les Tableaux de Correlation Dont les Harges SontDonnees. Annales de l'Universitg de Lyon. Section -A, Series3. 14, 53-77.

6. Hoel, D. G. (1972). A Representation of Mortality Data by CompetingRisks. Biometrics 28, 475-488.

7. Kaplan, E. L., and Meier, P. (1958). Nonparametric Estimation fromIncomplete Samples. JASA 53, 457-481.

8. Kendall, H. G. (1962). Rank Correlation Methods, 3rd ed., Griffin,London.

9. Langberg, N., Proschan, F., and Quinzi, A. J. (1981). EstimatingDependent Life Lengths, With Applications to the Theory ofCompeting Risks. Ann Stat 9, 157-167.

10. Nelson, W. (1972). Theory and Application of Hazard Plotting forCensored Failure Data. Technometrics 14, 945-966.

11. Oakes, D. (1982). A Model for Association in Bivariate SurvivalData. JRSSB 44, 412-422.

12. Peterson, A. V. (1976). Bounds for a Joint Distribution Functionwith Fixed Subdistribution Functions: Applications to CompetingRisks, Proc. Nat. Acad Sci 73, 11-13.

13. Slud, E. V. and Rubenstein, L. V. (1983). Depending Competing Risksand Summary Survival Curves. Biometrika, 70, pp. 643-650.

!AU-

* 16

14. Tsiatis, A. (1975). A Nonidentifiability Aspect of the Problem ofCompeting Risks. Proc. Nat. Acad. Sci. 72, 20-22.

15. Williams, J. S., and Lagakos, S. W. (1977). Models for CensoredSurvival Analysis: Constant Sum and Variable Sum Models.Biometrika 64, 215-224.

I

A

. , APPENDIX C

4.:

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1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER/: ]

4. TITLE (and Suhfule) S. TYPE OF REPORT & PERIOO COVERED

Independent or Dependent Competing Risks? 10/1/84 to 10/31/85-- Does it Make a Difference

6. PERFORMING O1G. REPORT NUMBER____714837/7632657. AUTHOR(&) S. CONTRACT OR GRANT NUMBER(s)

" .Melvin L. Moeschberger and John P. Klein Grant No.AFOSR 82-0307

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK

The Ohio State University Research Foundation 92

1314 Kinnear Road 92

Columbus, Ohio 43212I I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATEAFOSR-NM November 26, 1985Building 410 13. NUMBER OF PAGES

Boiling AFB, Washington, D.C. 20332 4414. MONITORING AGENCY NAME & ADDRESS(II different from Controlin4 Office) IS. SECURITY CLASS. (of this report)

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* Bolling AFB iSa. OECLASSIFICATION/DOWNGRADING

Washington, D.C. 20332 SCHEDULE

IS. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different froim Report)

ISI. SUPPLEMENTARY NOTES

19. K Y WORDS (Continue on reverse side it necessary and identify by block number)

Competing risks; Component life; Modeling series systems; Robustness studies;System reliability; Gumbel bivariate exponential; Downton bivariate exponential;Oakes bivariate exponential.

20. ABSTRACT (Continue on reverse side it necessary end identify by block manber)

This article investigates the consequences of departures from independence whenthe component lifetimes in a series system are exponentially distributed. Suchdepartures are studied when the joint distribution is assumed to follow eitherone of the three Gumbel bivariate exponential models, the Downton bivariateexponential, or the Oakes bivariate exponential model. Two distinct situationsare considered. First, in theoretical modeling of series systems, when the dis-

*tribution of the component lifetimes is assumed, one wishes to compute system

reliabilit and mean system life. (continued)

DD NFORM 1473 EDITION OF I NOV 65 IS OBSOLETE,~ - AN 73 UnclassifiedSECUITY CLASSIFICATION OF THIS PAGE (When Dole Entetd

6N

nhEPENDE~r OR DEPDE]r COMPE RISKS?DOES IT MAKE A DIM FNC

John P. Klein M.L. bergerDepartent of Statistics Department of Preventive MedicineThe Ohio State University The Ohio State UniversityColumbus, Ohio 43210 Columbus, OtLio 43210

ABST"ACr

This article i.nvestigtes the consequences of departures fromindepexience when the coponent lifetimes in a series system areexponentially distributed. Such departures are studied when thejoint distribution is assumed to follow either one of the threeGumbel bivariate exponential models, the Dwton bivariateexponential, or the Oakes bivariate exponential. model. I~odistirnt situations are considered. First, in theoretical modeling

0 of series systems, when the distribution of the capnent lifetimesis assumed, one wishes to compute system reliability and mean

psystem life. Second, errors in parametric and onparaetricestimation of component reliability and capoimt mean life arestudied based on life-test data collected on series systems whenthe assumption of independence is made erroneously. Systems withrto conponents are studied.

KEY WORDS: Competing risks; Component life; Modeling seriessystems; Robustness studies; system reliability;

*Gumbel bivariate exponential; Downton bivariate

exponential; Oakes bivariate exponential.

I~~ 1.____

i ". In troductio

Consider a systm consisting of several compoets Linked in

series. For sxd a system the failure of any am of the componentscauses the system to fail. Th a biological or medical contxt

we can consider the coonents to be different lethal diseasesanWor different reasons for rveval fimm the study. In a clini-cal trials ftueworic the primary response of interest, death or

remission, and censoring can be considered as components of the

system. MThs general formulation has been detailed in the theoryNI-" of competing risks (cf. David and Moeschberger (1978)).

A common assumption in such a foriulation is that the

component lifetimes are statistically independent. Several

audv= have shown that based n data fzria series syste only,

this assumption, by itself, is not testable because there is no

way to distingupish betien, ineaat or. depwjdent component

Lifetimes (see Basu (1981), Basu and: K.eint (1982), Miller (1977),Peterson (1976), etc.). Hiever, several authors (see Lagakos

(1979) p. 152 and Easterling (1980) p. 131) have pointed out the

need to determine, quantitatively, how far off one might be if an

analysis is based on an incorrect asiuuption of iependence.

I To study the effects of erronously assuming independence we

shall assume that each of the components have exponentially dis-

tributed lifetimes uhen tested separately and that the property of

marginal exponentiality will be preserved even though somedependence may be induced when the components are linked in series.

The assumption of y ed component lifetimes

has been made by Man and Grubb (1974), %tan finding confidence

bounds on system reliability, Boarazi and Kendall (1970), sben

estimating co nerit lifetimes frcm system data, and Miyamrau(1982), Wan oambinb component and sytem data. (See Barlow and

P C 178) or Mann, Schaffer, and Singzala (1974) for a

_0'

Ia" comuplete review.) We shall modal the depadence structure bythe three models of 0 wb@3 Q96CI, a model proposed by Do.intn

(197I), and a mode3 deseribed by Oakes (1982). These models arebriefly described In Section 2.

The effects of a departure from t e assumption of idepedent

ocmenvt lifetimes will be addessed for two distinct situations.

The first situation arises in modeling the performance of a

theoretical series system constrced from two cmpoents. fiere,

based on testing each component separately or on engineering

design principles, it is reasonable to assume that the componentsare exponentially distributed w.th krzo parameter values. Based

on this information, we wish to predcit parameters suh as the

mean life or reliability of a series system consrted from

these oompwift. In Section 3 we describe how these quantities

are affected by departures fro m a~u

The seond situatio involves mking inferences about

component lifetm distributio fr data collected n series

systens. Ccmonly, data collected on such systems are analyzed

by assuming a constant-sum model, of Qtich indeperience is a

special case (compare William and Lagakos (1977) and L Maos and

Williams (1978)). In Section 4.1 we study the properties of the

maximum likelihood estimators of the component mean life calcu-

lated under an erroneous assumption of independent exponential

component lifetimes as mentioned above. Because of the wide

spread use of the nonparetric estimtor proposed by Kaplan-tHmier (1958) for the cponent reliability we study in Section 4.2,

its proerties, the m: z l r% liabiIt e exsanential

and irdependn is incrrcty s .

2. The ?tMdels

Considiez a two component series system with am~pamet life

lengths x,, x2. suppose that each has an eniwital urvival

6 ,2

Ifunction

(2.1) Ti(t) aPOC.j ;- t) 2 exp-11 t) Ai :0 0, t > 0.

Tis; assztsion is made an the basis of extensive testing of

each omponent separately or on kzowledge of the underlying

mechanism of failure.

z To examine the effects of a departure from rcepe we-

consider five bivariate exponential models, each with marginals

equivalent to (2.1). The first three,m dels are due to Gumbel

(1960); the last tw m:dels are due to Downton (1970) and Oakes

(1982).

2.1 Gumbel's Model A

For this model the joint survival ftizcticn is

(2.2) P(X > X.1 X2 > 2 ) = e(-- "x - Y- 2 - .2x 2 ),

x , I 2 !.0, X , X2 >, < _ 1 2 11i' 2 -

The correlation betwee X1 , X2 is

P 1 _ 2 exp(l12 A12) Ei2(- 1121A12) - ,

where Ei(z) = exp(-u) du is the -integrated logrithmI UFor this model o varies from - .40365 to 0 as X12 decreases from

X2 to 0. It is never positive. The regression X2 on X is non-

leanear with

E(X Jxz x2) = (A1 + 12x2 - '12/A2)/(xl + X 12x) 2

2.2 Gumbel's Model B

For this model the joint suVival function is

(2.31 _P(XlX 1 1 X2 > xY2 ) exp(- .1xY ,X2X 2 )(l + 4p(1-exp(- 1xl))

01- exp(- AX2 x2)), Al l 2 .:, x1; x 2 , 0, - l/' C p _ 1/l.

The coct1ietjn, P, my be positive or negative. Meu regressicn.

-3-

9.~ ~~~ rp'S ~ ' i

Iof X2 on X, is again ronineaxw'witfx

E0C11 X2 ax 2) a( 1 +2P -40 exp(-7 X x2))/Al.

The effects of a departure frcm independence on modeling system

reliabillity and estimating component reliabilitie has been studied

in detail in Mbeschberger and K~lein (1984).

2.3 Gumbel's Model C

For this model the joint survival function is

P(l>x1') > Y~ = exP Xlxl)m + (AX2x2 )mlm}

The correlation is 1x2> .m 1xX2>a

0 = ( 4 + -2 ) ( c o; s s i ) 3 d o - 1

u.tLich varies from 0 to 1. For this mo~del m z1 corresponds to

independence and as m -

(2.S ) P(X 1 > x1 , X2 > x2 ) - minimum [exp(- Xlxl) I exp(- X2x2)J

the Flrichet (1958) upper bound for these marginals.

2.4 Downton's Model

DoI4nton (1970) suggests modeling bivariate exponential systems'

by a successive damage model. -This mo~del assumes that in a tw.o

caqponent system the times between successive shoks on each

componnt ha e deendent expnetial distributions- and -that the

nuber of shocks requir-ed to cauzse each amen to fail fo1Jlows

a bivariate geometric distribution. The joint probabilitydensity function of- the coMoent life times is

(2.S5) foc XP. Y- I' A(3~XxP(- X2x2 )) I,(2sP/l-2"1.

J.. P *** P

AW %p-*

Qwr is the modified Bese1 wtion of the first kind ofore eo n Op :l. The cltrela-

tion b enX, X2 is P WKIc spans the interval L,13. As - 1

the joint survival funtion of X, X2 approaches the upper

Frichet distribution (2.5). For this modelE(XlIX2 = x2) = l- P A IP /Xr

2.5 Oakes' Model

Oakes (1982) has pzvposed a model for bivariate survival

data. This model was first proposed by Clayton (1978) to model

association in bivariate lifetables. Special cases of Oakes'

general model have been suggested by Lindley and Sinrurwal-la (1985)

and Hutchinson (1981).

For ths model the joint survival pobability is

(2.7) P(Xl> x1l, X2 > x2 ) =exp(.X(e-)x 1 ) 2)-]

)l' ;k2 1 , ' >1 , x 2 >.

For 8 = 1, X1, X2 are independent and P(X > X. X2 > x2 ) - (2.S)

as e --. For this model Kendall's (1962) coeffcient of concordance

is T (e-l)/(e+l) ;hich spans the range 0 to 1. The correlation,

P, also spans the range 0 to 1 and is found numerically.

This model has the following physical interpretation. Let

r( xIX2 = x2 ) and r(xIX2 > x2 ) be the conditional failure rates

ofx, givenX 2 x2 and X2 > 2 " Men r(xiX2 = x2 ) =6 r(x.IX2 >x 2 ).

The model can also be derived from a random effects model.

This fozmilation assmes that 4en the conponents are tested

separately under ideal oonditions the coponent srvival functions

ae Si(t) z expt- exp(xit(9-l)) + 1], i = 1, 2, and that when the

t cmonents are put in a series systm in the opem-ting

enviroment-t ere is a indon factor W QJdch simltanemusly changes

each comonent life distribution to. SW. WhataSo

distribution with density function M0€) aw e then,

V.,

I-5.4 .

'.4

uWcOnditicnally, the joint survival fumction (2.7) hlds.

2.6 Frif6et Bouds

Fret C19S8) obtained bounds of the joint suvival trtions

Qiich can be obtained for any set of marginal distributions. For

exponential marginals these are

': tIA fV2 (e1, a) <-(X 2 X2 )

MINIMUM (e ,e .

. For this set of marginals the lower Frhet distribution hascorrelation -. 694 and the upper Frichet distribution has

correlation 1.0. These aze the minimal and maximal corrlations

o__ for exponential mrgials.

3. Errors in Mobdeling Systi Life

L Suppose that based on extensive testing or based on theore-

tical considerations each of the to components in a series

system is known to have an exponential distribution, (2.1) with

marginal means i/AI, I/X2, respectively. It is of interest to

predict the system reliability R(t) = P(X 1 > t, X2 > t) and the

system mean life p F(t) dt. If the investigator assu.mes that

the two components are independent then the system reliability is

(3.1) 1(t) = exp(- (Xl + \2)t) and system mean life is 11: l/(Xl + k2)"

If the c zponents are rat independent, but in fact follow one

of the models in Section 2, then a measure of the effects of

incorrectlUy awningn IM ten Mie t) M1't) - r (t))!'T1 t) and

is P -' tl 1u',' brpa tn& S*y ste relIability and system mean

life, respectively, Qwire Y~t) and Uz are omp~uted under the

approriae dependent mo~de l ue ofAt) can be =qpted

' : directly ftcm (2.2), (2.3), (2.4), (2.6) (by mkmirAl integration)

or fromu (2.7). E for - are giv-en: in Appendix 1 Al

rK

A l I A

csswns f(or A(t) and 4 deper on the values of X' and 12 only

throvf th rai X1 / X2 aKndfrK < h vle ar equiva-lot to t oe for r 1 For the upperPkechet di istiution,

K.

&Ct) 0 P7 W 1 uere K 1and tp zthe pointQw-e re(t) :P. Also"/KforK>1. Forthelower rechet

distribution

K K 1-: A~p) p- ['p - 'p p- h f p *p -1_)>0

t,.n o t - -I Table 3. o

=ion of the equality XK + X : . Table I gives th values ofA(tp) x lOO% and 6 x lOOt for p= .9, .7, .5, .3, . for the upper

and lower Frichet distributions.From Table 1 we see that the largest perrent error occurs

when the parmters are equal. Also for fixed K them is rela-ot tively smaI error in estimating system reliability by modeling

a dependent system by an independent system ,ben F(t) is large.

For smller values of system re iAbility one can be appreciablymisled. Errors in estimating system mean life appear to be

substantial unless one component has considerably longer marginallife than the second one. In that instance, one can see instiun-

* tively that the correlation would have a minimal impact.Figud.res lA -C and 2A-,2C are. plq of A(tp) ,p : .2,,.,S,..75,

,. and X2 :1, 1.S for the five zzdelsdescribed in Secti6 2.Figures 3A,. 38 ame plots of. 6 for. all five models as a, functiion ofthe corzlat.on. Frcm these plots,.nte that for positivecor"la-

" tions the most modelim g occurs for the Qumbel C zmdel. Forrelatively am correlation, -- ,2 . p .25 thee may still .be. a

-7-

At

IN

moderate modeLi, erro, an the ordwe at least + 10% for pre-

dictiM system rel iabil ity, at lCt) z .25, or F'Ct) z .5and for

estiamting the mean system life.

TABLE 1UPPER AND LOWER BOUNDS ON THE PERCET ERROR IN MOOEINS SYSTEM LIFE

FITis.? -W.7 M(20S FT43 FT.. EAN LIFE

LOE tPPER LJER UPPER LOVECR UPPER LO NER UPPER LONER UPPR LOWER UPPER

NN 00 NN WAf saN NU NS mmN US NI no DW 10101 MND

-~ ~ ~ t --- - - - -

1 -. 2. 5.41 -Le-I 19.5-2 -17.1 4t.42 -1 .00 3- -1F00.0 2. -3. 10.02 -0.26 3.37 -L 3 -15.27 25. -100.00 ".3 -100.0 11.44 -U.07 50.003 -0.22 2.67 -2.06 1.33 -12.9 13.92 -51.52 3512 -100.00 77.3 -34.3 33.354 -0.19 2.13 -2.44 7.39 -11.02 14.87 -44.11 27.23 -100.00 3.49 -32.3 25.005 -0.16 1.77 -2.2 4.12 -..57 12.25 -33.33 22.22 -100.00 44.73 -31.04 20.006 -0.14 1.52 -1.17 5.23 -1.45 10.41 -33.91 1877 -100.00 3o." -29.53 16.677 -0.13 1.33 -1.67 4.56 -7.55 9.05 -30.33 16.24 -100.00 33.35 -23.18 14.293 -0412 1.18 -1.51 4.04 -4.82 8.01 -27.42 14.31 -100.00 29.15 -26.9 12.50Y -4.11 1.06 -1.37 3.63 -6.22 7.11 -25.02 12.79 -100.00 25.3? -25.93 11.11

10 -0.10 0.94 -1.24 3.30 -5.71 6.50 -22.99 11.57 -100.00 23.28 -24.90 10.00

11 -4.09 0.80 -1.17 3.02 -5.28 5.95 -21.27 10.55 -100.00 21.13 -24.12 9.0912 -0.06 0.1 -1.08 2.78 -4.I1 3.48 -19.78 9.70 -100.00 19.38 -23.34 1.33

13 -0.08 0.76 -1.01 2.58 -4.59 5.00. -18.49 3.9 -100.00 17.88 -22.42 7.6914 -0.07 0.70 -0.9 2.41 -4.30 4.73 -17.13 3.34 -100.00 16,39 -21.96 7.14

is -0.07 0.64 -0.90 2.25 -4.03 4.43 -16.35 7.82 -100.00 13.43 -21.35 6.67

16 -0.06 0.42 -0.83 2.12 -S.83 4.16 -15.45 7.34 -100.00 14.50 -20.79 4.25

17 -0.06 0.59 -0.80 2.00 -3.43 3.93 -14.45 6.92 -100.00 13.65 -20.26 3.88

18 -0.06 0.56 -0.76 1.89 -3.45 3.72 -13.93 6.54 -100.00 12.88 -19.76 3.56

19 -0.04 0.33 -0.73 1.80 -3.29 3.53 -13.27 6.20 -100.00 12.20 -19.30 3.26

20 -0.05 0.50 -0.49 1.71 -3.14 3.36 -12.47 5.90 -922 1139 -18.87 5.00

21 -0.05 0.41 -0.66 1.63 -3.00 3.20 -12.13 5.43 -8.34 11.03 -13.44 4.76

22 -0.05 4.4k. -0.64 i.5 -2.18 3.0. -11.63 537. -4.73 03 3 -1.0 4.55

23 -0.05 0.44 -0.41 1.30 -2.74 2.93 -11.14 5.14 -41.41 10.07 -17.71 4.35

24 -0.04 0.426 -0.59 1.44 -2.44 2.81 -10.74 4.3 -71.34 9.45 -17.37 4.17

23 -0.04 0.41 . -0.57 .1.38 -254 2.70 -10.34 4.74 -75.49 9.26 -17.04 4.00

V! 7

4 . MNrs in Estain Ccopment Parameters

•.1 Paratric a eion

rn this section we examine effects of incorrectly assuming

irdepend ece on the magnitude of the estimation error in esti-

mating the first component mean life based on data from series

systems. Suppose that n series systems are put on test. For

each system we observe the system failure time and uhch camponent

caused the failure. Let ni dentoe the number of systems Qiere.th,

the system failure was caused by failure of the i !orponent,i = 1, 2, and let T be the total time on test for all n systers.

If we assume that the coaponent lifetimes are independent and

exponentially distributed then e e and David (1971) show

that the maximzu likelihood estimator of ul, the first componentmean life is

(4.1) 11 Tht. for n, > O.

This estimator is asymptotically unbiased and for n finite

E(uI) = E(T)-E(I/InJr. > 0) due to the independence of T and nI.

Suppose now that the two component lifetimes are not inde-

pendent but follow one of the models discribed in Section 2. If

we incorrectly assune independence then a measure of the excess

bias due to incorrectly assuming independence is

B = [E(ullDepenlent model) E(J independence)]/u I . For each of

the dependent models under consideration T and nI are independent.

For large n, B converges to (O/p- ul)/ 1 Where is is the mean-, steu .if e aid p'is the bily the first ccmponent fails

first, computed under the dependent model. For finite n,

E(U1 ) zn u E (Wnjn1 ,> () oiuuted undler the appropriate model-

whre Ep (/nln ) ( )PK(>p)K/K / (J_( 1 -P)n)" Expressions

for u and p are given in Appendix 1 and Appendix 2, respectively.

Z-9-

e r dePend o! Al, A2 only t ugh the i K 2.

For all models, p a 1/2 m K = 1.For the upper p-rchet distribution p z 0 if K < 1; 1/2 if K =;

and 1 if K > 1. Hernce for K < I no failures f Mm the f irst wcO-

ponent are ever observed so that the modeling error B becomes

infinite for all n. ForK> 1, p = land ui = P SO that

B = Cl - ECil[Tndependence)/U1) qldch tends to 0 as n . In

this case the models with correlation ranging from ( to 1 have B

increasing forp< p 0 and decreasing for p > p0 " For the wer

Frkchet distribution, p is the value of X ,tich solves the

equation X + XK - 1 = 0. For K< 1 we havep < 1/2 and for K > 1

we have p> 1/2. Table 2 gives the value of Bfor n = 2S S,0-

for the two Fr~chet distributions. It also gives the arxixmn

modeling error for the Guibel C model 4uch is an indication of

maximal excess mo~deling ervr.

From Table 2 we note that the dependee structure erts a large

effect on estimating the smaller of the two cop ient mum and

that either effect is uo-st exaggerated .for small sampieszs

For K > 1 there is very little sample size effect on the modeling

error. For K strictly bigger than cne the maxinum bias under the

Gwibel C model decreases with K and the correlation at uhich this

maxinun is attained also decreases to 0. Figures A-4C are plots

of B as n - - for K 3/2, 1, 2/3, resppcitvely. Figures 5A-SC

are plots of B for n 10 for K = 3/2, 1, 2/3, respectively.

TAKE 2MATIvE KNOlEN 00 IN ESTINATINS T11 HE

LMt -MMWl C L. CLOER Wsum C

K Nu lll AM MO HAS lIUNI 110.1 I0 I IAS NUll Moot I BIAS

1/10 -147.44 o44o.e 1.000 *looee 4.09 *el.o 1.041 -ogo o5.44 oo44o 1.Oj0

II 9 -12." 4a4o4 1.000 oo444 -7.3 t444o4 1.000 -44o4 -57.34 #o4m 1.000 444

I/ 1 -123.37 o 1oo44 .000 444o44 -74.68 ooa4 t.000 4o4444 -54-.73 4 o 1.000 44#4o

1/ 7 -107.2 .. 4o 1.000 4e4 -49.03 04o4* 1.000 o...o -55.37 woo 1.000 e444

1 4 -93.44 #44,* 1.000 t4444 -4.34 404#4 1.000 4044#e -54.43 woo# 1.000 0#4441/ 5 411.91 440 1.000 04444 -61.54 044444'1.000 @444O -14.13 #"4o". 1.000 "4oeH

11 4 -71.45 0404 1.000 44404 -57.50 L44444 1.000 .e.o -51.24 eeee 1.000 00440e

I/ 3 -61.78 .ooo.o 1.000 04#4" -53.00 444444 1.000 0 -440.73 oo4444 1.000 0044*

It 2 -52.14 *o. 1o 1.000 #o464# -47.31 t.o444 1.000 *4oooo -43.01 o4ooo 1.000 044,44

1 -40.90 105.99 1.000 105.99 -39.45 102.13 1.000 102.13 -38.43 100.00 1.000 100.00

2 -32.4 -2.79 0.510 10.82 -32.84 -1.04 0.310 11.50 -32.12 0.00 0.528 11.91

3 -28.23 -t.82 0.400 5.33 -21.39 -0. 9 O .34 -21.39 0.00 0.424 4.15

4 -2.43 -1.3310.346 3.39 -25.74 -4.32 0.342 3.74 -23.3 0.00 .M 4.01

S -23.2 -.0 0.312 L23 -23.80 -0.41 0.330 L71 -U." 0.06 0.341 2.91& -22.0 -4.0 4.31 1.3 -2L2 -0.34 0.307 L09 -2L.34 LOO6.o19 2.261 -20.77 -4.71 0.I 1.46 -20." 0.29 0.20 1.6#A -21.1t LOG M.Us 1.831 -11.49 -4.07 0.237 1.20 -19.92 -4.2A 0.2n7 1.40 -20.04 L~OO 0.21" 1.53

S9 -t0.70 -4.3910.246 1.01 -19.00 -0.23 0.266 1.09 -19.13 O.O 0.27" 1.31

10 t -17.11 -.33 0.237 OX8 -11.20 -4.20 4.237 t.03 -tIL33 LO.O 0.V1 114

I

ASMIIC IG THE FIOPJT UNIT ESTIIIATOR

.~L 6W C LMOli ~ lll 9XNM ww ElK 0 M IM R O JiW S IG O Meit AND IGl SO RHlOw INS Ilsi# °00.0 -45.8; .O 0 4 6-0 OO.00 00 .00 "lOZ.0 R33 K.00 23K.3a

11/ 1 "l0.00 42.81.000 42.81 -100.00 10.0 t.000 100.00 -100.00 2U.33 1.000 233.331I 7 -100.00 42.06 1.000 42.86 -100.00 100.00 1.00 100.00 -100.00 233.33 1.000 233.33

1/ 6 -100.00 42.96 1.000 42.6 -100.00 100.00 1.00 100.00 -100.00 233.33 1.000 233.13t/ 5 -100.00 42.86 1.000 42.86 -100.00 100.00 1.000 100.00 -100.00 233.33 1.000 233.3311 4 0tO0.00 42.0& 1.000 42.86 -t00.00 t0O.00 1.000 100.00 -100.04 233.33 1.000 233-3311 3 -100.00 42.84 1.000 42.86 -100.00 100.00 1.000 100.00 02.70 233.33 1.000 233.33

11 2 -100.00 42.86 1.000 42.86 -100.00 100.00 1.000 100.00 -24.21 233.33 1.000 233.331 -100.0 11.52 1.000 19.52 -t"0.0 41.42 1.000 41.42 -9.63 6L27 1.000 62.572 -100.00 0.00 0.21 3.17 -24.67 0.00 O.5 7.65 -4.40 0.00 0.323 13.67S-100.00 0.0 0.424 2.0? -14.1" 0.00 0.424 4.10 -2.3 0.00 0.424 7.22

4 -63.03 0.00 .33 1.3 -10.7 0.00 0.373 2.0 -2.11 0.01 .73 4.743 -1.67 0.000.341 1.0L -6.42 0.00 0.341 1.1 -44. 0.0 0.341 3L476 -i.3 0.00 0.31 .79 -4.1 0.00 0.31? t.54 -34.91 0.000.3t? 2.707 -1.11 0.00 0.303 0.4 -5.36 0.00 00.3 1.26 -23.71 6.0O .303 Li?8 -1.03 0.00.290 0.34 -5.0? 0.00 0.290 1.0 -24.4 0.00 0.20 1.939 -0.92 0.00 .27? 0.46 -4.50 0.000.271 0.0 -21.31 0.00 0.279 1.57

10 -0.12 0.00 0.27t 0.44 -4.03 0.00 0.27t 0.78 -16.69 0.00 0.271 1.37

IFigures SA-SC for p = .25, 6A-4C for p . and 7A-7C for

p .7S ,are plots of &1(p) for the S models and k = 3/2, 1, 2/3.

As in the previous figures one can see that for even a smaU

departure from independence the relative effect of dependence

can be quite large.

4.2 NoprmercEstimation

A second approach to the, pzobin of estimating c~onentparameters is via the q estimator of Kaplan andMaeier (1958). Investigators Qo routinely use n'npaz'mwtz'itechniques may take this approach in hopes of obtaining estimatorsthat are robust with respect to the assumption of exponentiality.Howver this estimator is not necessarily robust to the assumptionof independence.

The product limit estimator, assuming independent risks isconstructed as follows. Suppose that n systems are put on testand let ril, ... , r. be the renks of the ordered ni failures

from cause i, X(1) x X.()' aong allnorder lifetimes.Xi(1), ... ./ni , -th alThe estimator of the t r,,liability for the ith component is

4.2.1 Si(x) 1 if <XC xi(l)

i(i,x) xI x(j(1 )I! n-r.. '

jzl n- r. +1

whtere j(ix) is the largest value of j for which xj(j) < x. This

estimator is asympotically unbiased when the component lifetimesare independent.

When the risks are dependent Klein and e (1983)show that S(t) is not estimating the marginal component relia-bility, but rather-it is estimating consistently another survivalfunction

4.2.2 r.X) = ex( x d Qjt where.

rkt) z (jmjniu %X,X2 ) 2- t) and Q.C~t)z

P(minOC,X 2) < t, innX 1,X2) a X1), 1,z 2. Exrsios for

*1.

i. *

(kt) for the fJ..v models of iatergt am. given in Appendix 3.A measure of the affect of dependene in~ using the Prioduct

limit estbmator wil dependent risks is alCp) 2(Ir9 Ct I - p)/pe, wiexv tp is th, time whiere the true omvonent reliability is p.

di Cp) is again only a function of k 2 l)2.For the upper

LIZ Fr~.het distribution9 a 1Cp) = p 4l - 1. for k <l

P-1 / 2 2.I for k a since

a for k >1

for k -c. I RIM)=2. for all t since the first clomponent never

fails, while for k > I. all failures are due to the first olmponent.

For' those moels with corrilation nin the riange CC - lJfi (P)

is increaszi for correlatons less than P* and deczeasiW fbr

correlationis gmeter than P~ when k > 1. For the lower Frele

distribution Ff1 (t) p ep f du for p t.(1Y

For ~ -l otherwise.

Table 3 shows the value of 1 (p) x i~o% for p z .7,. .S, .3 for the

tw.o Freci distributions. o k > 1, the mnaxim=u value under

the Guiibel C mocdel is also given. As in the parametric estimation

Iproblem the largest e-ors are incurred ,ben k < 1. In all cases

the effect of a departurie fromn independence is the largest when

p is small (i.e. for large t). The effect decreases as k

u ~~vasels reflecting the fact that when kc X ) > X' 2 the majorityof the system failures are due to the failure of the first

coponent.

S.Colsos

The r~sults presented in this paper show that for afl. fivebivariate exponetial models one may be appreciably misaled byfalsely assuning inependence of component lifetimes in a serie-ssystem. The amount of error inaurred in modeling system reliabilitnot only depends upon the corrlelation between component l.ifetim'es

but also on the level of system reliability. The errlor inmodeling mean system life similarly depends upon the correlationand the l.ength of mean system life. Both quantities depend on therelative magmitudes; of the paramters.

For thie dual pcobLen of estimting component reliabilitybased an daa frcz a series system, it appears that departuresfrcz independence ame of pwel MMonsq~eCe. Both parem'icand rnputric estimators of relevant comonent parameters

are incnsistent. Bias increases dramatically as the correlationgets further frm zezm. However, the five models do not e~dibitappreciable differences in bias and mean squared error as

correlation changes. This suggests that these mo~dels may belongto a large class of bivariate exponential distributions 4-tichpossesses the properties exhibited here.

ACIWLDGE r

Rlesearc Sponsored by the Air Force Office of ScientificResearch under cotract AEMS-8 2-0307.

Baz'lcw, RE. and Proschan, F. (19S75.). Stat-isticAl. oflat and Life Testin. Holt Rir 7, and Winstn.

Basu, A.P. C1981). rdentifiahlity problems in the theory ofu t and ouplemntary risks - a surey.. Statistical

Distriutiom in Scient fic Work (Taillie, Patil",miandIHsaari,&ds.), WRiht, Ebll-: Reidel Publishing Co., 335-348.

Basu, A.P., and Klein, J.P. (1982). Some Recent Results inCompeting Risks Theory. Survival Anal is, Crowley axd Johnson,eds. Institute of athaat atis t , Hayerd, CA, 216-229.

Binbau, Z.W. (1979). On the Mathamatics of Competing Risks.U.S. Dept. of HEW. Publication No. 79-1351.

Boardman, T.J. and Kendell, B.J. (1970). Estimation in compoundexponential failure models. Techrometrics 12, 891-900.

Clayton, D.G. (1978). A Model for Associationin Bivariate Life-tables and its Application in Epidemiological Studies ofFamilial Tendency in Chronic Disease Incidence, Bicnmtria65, 141-151.

David, H.A. and Meschberger, M.L. (1978). The Theory of

Downton, F. (1970) . Bivaz.ate exponential distributions inreliability theory. J. Roy. Statist. Soc. B. 32: 408-417.

EasterLing, R.G. (1980). Book review of Z.W. irnha u monograph.Techncnetrics 22, 131-132.

E1r6chet, M. Sur les Tableaux de Correlation Dont les Margis SontDonnees. Annales de l'Univer'sit de L . Section A, Series3, 14, S3-77.

Gw"I~e, E.J. Ci(96a1). - ivariate expnential ds'btosJ. Amer. Statist. Assoc. SS, 698-707.

KIrtchlnsn, :r. P. (981). comuourKI Gin Sivariate Distribution.Metri-a, 28, 263-271.

aplan, E.eL. and Mier, P. ,,S58). Nonpaanstri. stimatian fromuiculete observations. J. -Anw'. Statist. Assoc. 53, 41S7-481.

Kl~ein, J.P. and rtwesdiber e, f.L.- (1284). Asymptotic biasof the product limit ewtimator nider dependent ccupe&tzag risks.IAER Trwnft1!2ns, 9, 1-7.

Lindley,* D. and Singpuzalla, N. (1984). Mu~ltivariate distribuztionsfor the reliabilIity of a systemu of components Sharing a C00110environmient. Presented at the Intenational Conference anReliability an Quality Control.

Mann, N.R. and Grubbs, F.E. (1974). Approximately optin con-fidence bo~unds for system reliability based on component testdata. Techncmetrics 16, 33S-347.

Mann, N.R., Schafer, R.E., and Sirgpurwella, N.D. (1974). M~ethodsfor the Statistical Analysis of Reliability and Life Data.Wiley, New Yorkc.

Hiyauuizr, T. (1982). Estinating comonent failure rates fromcombined1 component and systaem data: exponentially distri.-buted component lifetimes. Techncetrics 24: 313-318.

Miller, D.R. (1977). A no~te on independence of m~ultivariate life-times in competing risks. Ann. Statist. S: 576-S79.

Moeschberger, 1.L. and David, H.A. (1971). Life tests undercompeting ca),ses of failure and the theory of competing risks.Biametrika 27, 909-933.

l'beschberger, M.L. and Kl~ein, J.P. (1984). Consequences ofdepartures from independence in exponential series systems.Teohrxomevrics 26: 277-284.

Oakes, D. (1982). A model for association in bivariate survivaldata. J. Roy. Statist. Soc. B. 44: 414-422.

Pterson, A.V., Jr. (1976). Bounds for a joint distribu~tionfuznction with fixed sub-distrition ftzntions: Application tocompting risks. Proc. Nat. Aca. Sci. 73: 1.1-13.

willias J.S.. and LaguJm, S.W. (1977). Models for censoredsurvival analysis: Constnat-sun and variable-sun mo~dels.Biametrikca 64: 215-22'4.

r le F 0 .

jorm~s fr aqtedSyteu Life,

Gal A- w ,X)

qr1,'2 r12

wheare ;F() is the survival function of a standard norwal randO

variable.

Ganbel B- 1 + 12(A.2

Ga~bel. C -- C*)-/M (AL.3)

M5 E)1 A 2 ) -p 1x 2

+* [(X +X,) 4/)0QX]X 1124 2 2P 2 x

2 ~XC(X I+,k2) - 40 X1x2

Oakes -foundi by numeri.cal integrat ion

Iq ,.

II

A~pnd - Formulas for p: P0C' 2X2,

Guutbel A 1 /0C <X)=12 + (-A 1 xp((Al+'X2 )2T(A~ j > 0

where IC.) is the survival function of a stanard ro l.

random variable. WA. 1)

Gumbel B - P(X1 < X2) A 4o A2(C --,2 )

(A2.2)

Gmbel C - P(X < X2 ) z Am

(A2.3)

Downton - P(X1 < X2) 2A 1A.(-P)

A'4

2) 2

- ', x1 A2 )(,-A 2 + V%/(A+A 2 ,2 - 1 x2

(A2.4)

Oakes - P(X1 .< X2) found numerically.

j --

-P R

Appendix 3

Gunbel A - ff~x = (( x-1 A3. 1)

Gwnbel B - ff(xW) exp- JCi + 4o~l-exp(-)L t)(1-2exp.JA1tM) 4A3.2)0 l+ 40C1.'exp(-%2 t))(..exp(--X t))

Gumbei C - H(x) =exp~) X m (- (A3.3)

M'4mton. - Found numerically due to no close foru solution for

Oakces - =) exp) " x~expQ1 Ce-.)t) dt

* .. ~~ ~ % . % 1 ,

Rela tive ErmrE in Modeling System Reliability att -2

U,)

CD0O ~

C) a---- GUMBEL 5+ ----CUNSEL C

X ---- OOWNTON

LJO

CJ

0 5

CORS.

FViure 1B

Relative Error. in Modeling System Reliability at tp -25

A 1 :1 2 : 1.5

U,)

Q

-;

cfl(~j KEY

----CUMBEL+ ----CUNSEL CX ---- OOWNTON

Lji ~---- OAKES

LI,

CD

-LJ.

Ln

0?'o7 '0

0.0 -. 2 .G0250

SoA

FIGLRE 2A

* Relative Error in Mdeling System Reliability at t .

A = 1.0, A2 :1.5

Io

In

.%.

COT KEYcc .. - ... -- - uMS L A

C) -- a --- ... UM L 8+ .... UMSEL C

t -j x ---- OOWNTON• ,.LLJ ----. OAKE.SI -

-

CD

'-b •5 - 0 2 5 a0 •.00 :.,0 2S 0 •50S 0'7S L.00i ; 'CIORR.

5, O::I~A

|ma)

]. - -p ' " • .. - . ' . - % ° * - - . - - . . " .. " - "* -- N . . . ' 0% % % % . % , -'

.. . ... . " .iLUO idii i Iii i b~ii di l'l 'h - . . .t 1 "il

m..mu ,

LIURE 2B

Relative Error in modeling System Reliability at t .5

A 1 1.0, A2 1.5N

C-/

SKEY

x ----OOdNTONLU j ---- OAKES

! co

II

"~~~~LJ X -.. )M O

T, I . S -02 .... 0R~r 25O o10

.0" tJ-Io,2J U0

MJ r

9

CORR

FZGJRE Il' I.7

Relative Error in Modeling System Reliability at tp.

U A1:1.0, 12 = 1.0

CD

0

+ --- UME0,

0

--

KEYW---OAE

4. ... GUM8EL

-J .UJ

A; 0

.0 .so -. 2S o.00z0-.oj .o0b . O R

FIGIRE 2C

Relative Error in Modeling System Reliability at t p .75

A1 - 1.0, A2 1.5

CD

's

_ ,

o0

" (:][--" - C .. UM13EL A

+ .... GU eEL C" ' x ----0o o ..t o .0C-

0

°~ ~ ~ ~ 2. U '0. S b." .

-J.S" J s b, . o o . . . . . : s . . ... . . .. . . . . . . . . . . . .- - G"lS: ;2-A " "- O R .

FIQJRE .uA

Relative Err ini Modeling Mean System Life

3 1 1.0, A 2 1.

UL-u

0 + ---- CUMBEL CLLJ X---- OUNTONlip. --- OCAKES

LU 0

cc1

0)

0)

G'- .50 -. 2 0.00 *--0.2S . So*5 .7 '1'.

FIG=R 38

j RelAtiVOemwrU in ModefLg tMan SYsteM Life

L.0 )l2 x .

~tz.

LLj X -- CONO

-LJD

cru

-0s S o' .2 .S 0.7 A

COR

riGJRr 4A

Asyiptotic Modeling Error in Estivating i& for K: .

I.-

01

0 KE

jjO

0-..GN ..+----GUN8(L C

CD x OUNTO,4C -- OAKES

L-Jo

UI

a:O.O 0;7

IRC01 ,I

FIGLRE 48

Asyptotic Modeling Error in Esticating u, for K 1

C3

U"

8-o

0CD

Cu

Al + ..ZOW"OTL'S---Was

---GOWNYO"'

.4I0 -0.20 .. 0 . .- 0;20 .. 4 . 60- CORR.:1.

Asymptotic Modleling Errvr in Estimating v, for K: .67

CI C.0

Cu

0;

KIE

" U*"L

0+

x --UONO0)k

-b o S 0002 )",07S 10

COR..

~ilk

FrGU1E SA

S i tuml S a w l 1e 'S i z e ( N = 1 0 ) M o el i n g E r r r i n E s t i m n t i zn g m 1 f o r K 1 .

CD

o KEY

+ --GUI40EL CC) X ---- OOwmNTO

--- COE

0L

COA

Smal Srmzple Size (N--10) Mrxljeling Th'it. in r-*srimrilg Upl K c1

(D

Ccc

( KEY0 0 ---- IJNSEL A&J .... CUPISEL 9

+----GUMSCX--- -DONto'

a

0O.50 0. 2S 00 0.2S .0650 .007 1.00CORR.

~11 SmpleSizeFIGURE 5C

Smal Samle Sze Mdelig or (N=10) in Estimating ui, K =.67

'),

Id,

_j -- ---- CUMBEL 4LJ. 0 ---- ut6l 8

+ --- CUMBELC0 x -.-OOWNbON

0- -

r-

-0 ti 0 . 00

1*CORR I

FIQJRE 6A

Asymptotic Error in the Nonparametric Estintor" of theFirst Component Survival Futn .ion at

p .25, K= 1.5.

'-I4

,* .- ,

-I 'A

' . ,

2S 0.0 -. 2S .... to • .- S 1 00

&Za

1,.

,I/

S !.

'-50 -b.s 0 .0O0 O'.2 5 .0.50 O. 75 1.0

rrGUR 6s

Asymptotic Error in the Nonparaetric Fstiinator-of theFirst rixuponent Survival runction at

t:p .25, K =1.

CDpc0

*$ .,;-

0

(Vu KEYC3 - 0---- GUBL6. A

CD &----UMIIEL B+ ----GUMBEL C

-JO-LJI

cc

CD

h.n

2s- 1 .0

so07

J-- 0.257 0.75, 1.00CORRO05

FTQ.JRE 6C

kwnmtuti: Errvr irs the Nonparametric r-tiiiiator of theN Fifnt Cnomponent Survival r~irri(on a~tt * 25, K .67.

C)

LO

U,)

U')-T Thr!_SCDO

KEY

0o a--CME

& --CIE+ -lGSE

L 0

0.

-~ '0.50 -0.25 0.00 0.25 05 .510-. ~ CORR.

ASVrnPtotic ILryor in ctit t'nparamti-ic :...tujtknir* .f. theg ~Fimst ~CUponenr Surviv&l !'.a-criun ar

t p .25, K z.5

yo-

0.

+

ul/

-C2 IN2.

ri(*WI: 69

A-.ympt.)ic Crr'or ini the Nonpcukm~tric itat~" hfirst rc.~nix-nent Survival Function Iat

.S. K, 1.JPp

C)

+ ---- GUMBEL C)( ---- O~WNTO4

!-

C)

cc

.0;.

riWRE 6c

Asvmpot ic Error in the INbnparametric Estimator of theI First Component Survival Function at

t .5, K .67.

cop

0

U

oD .- 0--- wUI8ELC) a---CUM!E *J

+ --- GU 5EL C

0

0

-0.50 -0.2S 0.00 0.25 05 .510I CORAR.

VV

ri(WE 7A

Asvmptoti Ermor in the Nonparatric: stimator of the!First (omjxrnent Survival funcrfion -it

t p .75, K z I.S.

-4

c.

0I

±ou

a-

"I

h'

,I.,

-.0 -0.25 '0.-00 0.25 0.50 0.7S . coCORR.

..... - S. S -

rXM~: 7B

Awimprot Lc Lrrz in the lNbnparamitric F.,t im'~tor 'i'the

Fin'~t Oosp.nent Survival rUww:rifl .-it

t .7 5, X:1.

U"

0

crl0

o KETcn. o a---GUlUELA

L) + ---- GuMBEL Cx ---- DGWNrom

---- OAKCES

cc~

-j.a:J

QA

CORR

FIGURE XC

Asvuptotic Error in the. Nonparmtric Estimator Of the'First Comnponent Survival 'Function at

t .7S,'K .67.

CDp

=0

CD

0n

pL X ---- ONO

>- )g----UOAKES

cr

c;

(0S

'-050 -0. 25 0.00 Q2IS 05 .510CORA.

161M~, ~ ~ .~- 4~J*~ ~

24.

i

SECURITY CLASSIFICATION OF THIS PAGE (Vl en Data o rEnted), RREPORT DOCUMENTATION PAGE READ CMSTRUCTIONS

BEFORE COMPL.EING FORM

. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENTS CATALOG NUMBER

4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

A Random Environmental Stress Model for 10/1/84 to 10/31/85Competing Risks

6. PERFORMING OG. REPORT NUMBER_________________________________________ 714837/763265

7. AUTHOR(#) S. CONTRACT OR GRANT NUMBER(.)

John P. Klein and Sukhoon Lee Grant No.AFOSR 82-0307

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKThe Ohio State University Research Foundation 9 2 AREA & WORK UNIT NUMBERS

1314 Kinnear Rd., Columbus, Ohio 43212

It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATEAFOSR-NM November 26, 1985Building 410 13. NUMBER OF PAGESBolling AFB, Washington, D.C. 20332 41

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AFOSR-PKZ UnclassifiedBuilding 410

Bolling AFB 1s. DECLASSIFICATION/DOWNGRADING

Washington, D.C. 20332 SCHEDULE

IS. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, It different from Report)

IS. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side If neceseary end identify by block number)

20. ABSTRACT (Continue an reverse side If neeeeary and Identify by block number)

A random environmental effects model is proposed for competing risks experi-ments. The model assumes a random stress, Z, which changes the scale parameterof each of the assumed Weibull times to occurrence of the risks. Some generalproperties of the model are discussed, and specific properties for a Uniform orGamma stress model are presented. Estimation of parameters under the Gammastress model is considered, and a new estimator based on the scaled total time ontest transform is presented.

DO ,FJORA. , 1473 £oiTIOm OF I NOV 65 IS OBSOLETEUnclassIf Ted

i! , SECURITY CLASSIFICATION OF THIS PAGE (Whenm Dee IEnted)

A RANDOM ENVIRONMENTAL STRESS MODEL FOR COMPETING RISKS

John P. Klein and Sukhoon Lee

Depatmet of Statistics~The Ohio State University

~Columbus, Ohio 43210

ABSTRACT

A random environmental effects model is proposed for competing risks experiments. Themodel assumes a random stress, Z, which changes the scale parameter of each of the assumed

Weibull times to occurrence of the risks. Some general properties of the model are discussed, and

specific properties for a Uniform or Gamma stress model are presented. Estimation of parametersunder the Gamma stress model is considered, and a new estimator based on the scaled total time ontest transform is presented.

IN'MODUCTION

The problem of competing risks. arises naturally in a number of engineering or biologicalexperiments. In such experiments, for some items put on test, the primary event of interests (suchas death, component failure, etc.) is not observable due to the occurrence of some competing riskof removal from the study (such as censoring, failure from a different component, etc.).Competing risks arise in an engineering context in analyzing data from

(a) series systems,

(b) field tests of equipment with a fixed test time and a random or staggered entry into

the study, or

(c) systems with multiple failure modes.Competing risks arise in biological applications in analyzing data from

Ja . . . . - . . . .. E -. - , € .. .

(a) clinical trial with a fixed trial duration and staggered entry(b) clinical trials with some patients withdrawing from the trial prior to response(c) studies of the time to death from a variety of causesA common assumption made in analyzing competing risks experiments is that the potential

(unobservable) times to occurrenc of the competing risks are independen This assumption is nottestable due to the identifiability problem. That is, for any dependent competing risks model, thereexists an independent competing risks model which yields the same observables. (See Basu andKlein (1982) for details.) However, Moeschberger and Klein (1984) show that an investigator canbe appreciably misled in modeling competing risks by erroneously assuming indepencence.

In this paper we present a model for dependence between the various risks by assuming thatdependence is due to some common environmental factor which effects the potential times tooccurrences of each risk. In section 2 we present the model and study its properties for bivariateom s-series and parallel systems. In section 3, we consider estimation of the model parameters for

competing risks systems.

2. THE MODEL

For simplicity we shall consider the problem of bivariate systems and discuss our model interms of engineering applications. We assume that under ideal, controlled conditions, as one mayencounter in the laboratory in the testing or design stage of development, the time to failure of the

t two components, to be linked in a system, are X0 and YO. We suppose that under these

conditions, X0 ,Y0 have survival functions F0 , GO on [0, .). We assume that both X0 andY0Sfollow a Weibull form with parameters (ill, X1) and (12, X2 ), respectively, That is, F0 (x) = exp(-

) 1x'll). The Weibull distribution, which may have increasing ('Ti > 1), decreasing (11 < 1) or

constant failure rate (11 = 1) has been shown experimentally to provide a teasonable fit to manydifferent types of survival data. (See Bain (1978)). We now link the two components into a

MRsystem in such a way that under ideal lab conditions the two components are independent

Now suppose that the above system (X0 , YO) is put into operation under usage conditions.

We suppose that under such conditions the effect of the environment is to degrade or improve each

component by the same random amount That is, the effect of the environment is to select arandom factor, Z, from some distribution, H, which changes the maginal survival functions of the

two components to F0 and G0 Z. A value of Z less than one means that component reliabilities are

simultaneously improved, while a value of Z greater than one implies a joint degradation. The

3

resulting joint reliability of the two components' lifetimes, (XY) in the operating environment is

F(xy) = E[exp(-Z(X xT'l 1+ k2Y'1 2 ). (2.1)

This model has been proposed by Lindley and Singpurwalla (1984) in the reliability contextwhen F0 , Go are exponential and H( ) follows a gamma distribution. This basic dependence

structure was also proposed by Clayton (1978) to model associations in bivariate survival data, andlater by Oakes (1982) to model bivariate survival data. Hutchinson (1982) proposed a similar

model when H( ) has a gamma distribution and F0 (t) = G0 (t) = exp(-t).

The model described above for a general distribution of the environmental stress has aparticular dependence structure which we summarize in the following lemmas.

Lemma 1. Let (X,Y) follow the model (2.1) where Z is a positive random variable with finite

r s(- + .)th inverse moment. Then

TI 112- -S"1l2 -(r1l +s"2)

E(XrYs) =X k2 M + rill1) ( + sr 2 ) E(Z (2.2)

The proof follows by noting that, given Z = z, (XY) are independent Weibulls with parameters

(I 1, ;L1 z) and (112 ,X;2 z), respectively and E(XrVZz) = 1 z r(1+ r/t11) with a similar

expression for Ys. When the appropriate moments existwe have

(A) E(X) =E(X0) E(Z )

(B) V(X)=E(X2) Var(Z'l )+E(Z 1 )2ar(X ),

-11111 -1/11(C) Coy (XY) = E(X0) E(Yo) Cov(Z , Z - which is greater than 0.

f :11 = 112 =1 then the correlation between (X,Y) is

r(l+ Itri)2 Var(Z 1 /11)

Var(Z l /il) r(1+2l)+ (r(1+21) - r(+fI)2) E(Z"b11)2

In this case the correlation is boundedcabove by r(+1/Itl)2 / r(1+2/T1). Figure I shows the

maximal correlation as a functcion of -n forn rc(0, 10). Note that this maximalcorrelation is an

FIGURE 1UPPER BOUND ON MAXIMRL CORRELATION FOR RANDOMI ENVIORNMENT MODEL.

0

c,

Lo

4to

*4Ln

c'j

00'. O0 1. 00 2,. o0 3'.0O0 Ll' 00 5'.0O0 6'.O00

":" COHkON WEIBULL 54RP.

• ::

4

increasing function of Tl. One can also show that F(x, y) is positive quadrant dependent for any

111,'12"

Exact expressions for competing risks quantities of interest can be computed when aparticular model is assumed for the distribution of Z. We shall consider the gamma and uniform

models. Consider first the gamma model with hz(z) = ba za" I exp(-bz)/7(a), z > 0. For this

r' model, the joint survival function is

baF(x,y) = (2.3)

[b+ .lX + X2y a

which is a bivariate Burr Distribution (see Takahasi (1965)), the marginal distributions are

univariate Burr distributions with

4/TE(X) = (Xl/b) h(1+I11,)r(a- 1i1ll) r(a), if a> 1/1,

-2/l r(l+2tnI)r(a-2/1l) r(i+1/ll)r(a-ll) .2}, if a> 2/111r (a) r(a)

-" with similar expressions for E(Y), Var(Y). The covariance of (XY) is

-171 -/1 r(a-ltrll-1r2) -r(a-l/T12) r(a-l/Tl)

Cov(X,Y) = (X1/b) (X2/b) r(l+XtT1) r(1+1/r2){i~i r(a) r(a)

for a > l/111 + 1/112. For the gamma model, the reliability function for a bivariate series system is

given by

L Rs(t) = (l+(%1/b)t 1+ (X2 /b)t 2 )-a, (2.4)

and for a parallel system by

= ~ T (l(X/~1 112 112 112)a(2)Rp(t) =(+(X/b)t+ (1+ (X2/b)t )" - (1+( X.I/b)t2+ (X2 /b)t )(25)

Figures 2A-E are plots of the series system reliability for .1= 1, X2 2 and several combinations

' 5

of 1 ,1112. Each figure shows the reliablity for a = 1/2, 1, 2, 4, and the independent Weibull

model. In all cases, b = 1. For these figures we note that for fixed X1, X2 11 1, 112, t, the series

system reliability is a decreasing function of the shape parameter a. Figures 3A-E are plots of the. parallel system reliability (2.5) for the above parameters. Again, the reliability is a decrtasing

function of a. Also in both the series and parallel system reliability, the shape of the reliability

function is quite different from that encountered under independence.The gamma model is a reasonable model for the environmental stress due to its flexibility and

the tractability of the model in obtaining close form solutions for the relevant quantities and inestimating parameters. However, in some cases, such as when the operating environment is

1.; always more severe than the laboratory environment, the support of H may be restricted to somefixed interval. A possible model for such art environmental stress is the uniform distribution over

[ab]. For this model, the joint survival function is

I1 T12 'Il 12(exp (-b(X 1 x + X2y )) - exp(-a( lx +

F(x,y) = (2.6)

./, :,,, (b-a)()liX +)L2 y "/I(11/1 -)T1

"5: E(X) = )X1 (I+1/'11)' 1 (b -a /{(11l-1)(b-a)] if ill 1

S-lnn(b/a)/[X 1 (b-a)] if 11=1,

V,- 2/11 (11-2)/111 (1 l-2)T.bVat(X) 2 bl F )(l+2/1ll) (b] 2 - a1 1

2 .:2/()L12ab) - An(b/ a)2/[(b-a) 7.1]2 if Ill= 1

fn(b/a) H. (b-a) (bl/2 +al/2.X1

'Y'- 7- 7 -

RD-fl164 417 EFFECTS OF ASSUMNING INDEPENdDENT COMPONENT FAILURE TINES 2/-~ IF THEY ARE ACTJA. (U) OHIO STATE UNIV RESEARCH

FOUNDATION COLUMBUS N L MOESCH8ERGER ET AL. DEC 85

mhhhhmhhhmmulI lflfloflflflflflf

iiiii'Jim0 :1 2 5

11.25 1111. 111i6

MICROCOPY RESOLUTION TEST CHARTNATIOAL BUREAU OF STANDARDS-1963-A

FIGURE 2 RSERIES SYSTEM RELIABILITY UNDER GAMMA(A.1) MODEL

FOR THE ENVIRONMENTAL STRESS.

* KEYo ---- SHAPE 1/2

&---- SHAPE I

co

Lf)Cuj

0

9xoo 1.0 23.0 4.00500TIME.

FIGURE 2 BSERIES SYSTEM RELIABILITY UNDER GAMMA(A.1) MODEL

FOR THE ENVIRONMENTAL STRESS.X1=1-0. )12=2. 0. 'ni=1 0. n22.0.

0

Ry KEY--- SHPE1/2

o+ ---- SAPE2

Dco

-

0

CC,

cnJ0r

'Do 1.00 2.00 3.00 4.00 5.00TIME.

.;;S&gg z

FIGURE 2 CISERIES SYSTEM RELIABILITY UNDER GAMMA(R,l) MODEL

FOR THE ENVIRONMENTAL STRESS.

0

LnKEY

CD 0---- SHAPE 1/2A ---- SHAPE I+ ---- SHAPE 2X ---- SHAPE 4W ---- INDEP

z

>co

Ct)

LA

0

0. 00 .020 'o .050

TIE

FIGURE 2 0SERIES SYSTEM RELIABILITY UNDER GAMMA(R,1) MODELFOR THE ENVIRONMENTAL STRESS.),i=1. X2=2.O, ni=1/2. 12=1/2.

KEYO ---- SHAPE 1/2

.-- SHAPE I+ ---- SHRPE 2X.-.- SHAPE

S----INDEP

ZCD

-

>Mo

,U>0

'b. oo 1o.00 2.00 3.00 '.00 5.00TIME.

.~N,

FIGURE 2 ESERIES SYSTEM RELIABILITY UNDER GAMMA(A,1) MODELFOR THE ENVIRONMENTAL STRESS.

,i= l.0, >.2=2.O0, ni=1/2. 12=2.0.

* KEYC6 ---- SHAPE 1/2-- .. SHAPE I

+ ----SHAPE 2

---- NDEP

ZCD

Cc

co

co

Cuj

0

Ib ,0 1.00 2.00 3.00 4.00 5.00TIME.

FIGURE 3 RPARALLEL SYSTEM RELIABILITY UNDER GAMMA(A,1) MODELFOR THE ENVIRONMENTAL STRESS.

* KETCD---- SAPE 1/2

Cfo

MDbL fCC

TIE

FIGURE 3 BPARALLEL SYSTEM RELIABILITY UNDER GAMMA(A,1) MODELFOR THE ENVIRONMENTRL STRESS.)j=1.0, >2=2.0, '=i.0. 12=2.0.

KEY

0"0 ---- SHAPE 1/2---- SHAPE I+ ....- SHAPE 2

X ---- SHAPE 9---- INDEP

Cf,

z

i_ TIMLO

- C

co

NN

cb. o 1.00 2-.00 3.00 4.00 5.00

a TIME.

FIGURE 3 CPARALLEL SYSTEM RELIABILITY UNDER GAMMAIA,1) MODELFOR THE ENVIRONMENTAL STRESS.Xi=10. )2=2.0, i=2.0. n2=2.0.

0

0

*KEYS---- SHPE 1/2A ---- SHAPE I

+ ....- SHAPE2X .- .- SH APE

;" --... INDEP

Z(F

m

z

0'j

TIE

cr,

C;_

0

0b.o0 1.00 2.00 3.00 4.00 5.00TIME.

FIGURE 3 0PARALLEL SYSTEM RELIABILITY UNDER GAMMAIA,l) MODELFOR THE ENVIRONMENTAL STRESS.

== .0, xa= ,0, 11/2. '42=1/2.°0

I-

* p KEY-- .. SHAPE 1/2---- SHAPE I

+ ----SHAPE 2X ---- SHAPE*I ---- I NDEP

CDz

uU

-j

CcD

>0

UV)

lo Io0 11.00 2".00 3".00 4.00 5".00TIME.

Aa 9W, ~

and

1112-11 2 111112-l'2

r(1+tili)r(+111 2) 111112 (b -aCov(XY) xi 1/111 __ __ ____ ____ (111112 -111-112) (__

x1 l- x2 121 1-

111 111 112112111112 (b -a A( -a

)i 1 1, 11 1 , 1 /1

(b-a) 2

111-1 111-1

r (1+/1 1) r ((21 1 -1)/Tl) fln(bla) 1112 (bi -a1 ) /1 J 1f/JJ1

x(1ltrU X 11011-1) (b-a T11 (-)2 ~ ( a)fnba

if II* 1/111=

1li l Iln bla)2 l

r (ab/l) Il V(b)n ( a 2nba

For ~ ~ ~ ~ ~ ~ ~ ~ ~~i tIii mel the. reblt ucinfrasxe ytmi

R5(t) - exp(-b(X 1 11 2 11t exp(-a( XI t 11 2 112)] (2.7)

CtA~t2 +~}

4

~7

U and for a parallel system is

Rp(t) -[exp(-b(Xl t 1) -exp(-a X1 t) + [exp(-b X2t 12 ) -exp(-a k2t 12)(b-a) X Ill (b-a) 2t72 Rs(t)

VFigures 4A-E show the reliability for a series system ard figures 5A-E for a parallel system

under the uniform model for various combinations of )L1, 2, ! 1' 2 a,b. Notice that when A =

.25, B = .75, which corresponds to an operating environment which is less severe than the test

environment, the system reliability is greater than that expected under independence, while when

(a,b) = (1.25, 1.75) or (1., 2), which corresponds to an environment more severe than the test

environment, the system reliability is smaller. Also when the (ab) contains 1, which corresponds

to an environment which incurs the possibility of no differential effect from that found in the

laboratory, there is little difference in the dependent and independent system reliability.

3. Estimation of Parameters Under Gamma Model

Consider the model (2.3) with 1 112 =1. For this model, the reliablity for a series system

is

Rs(t) (I + tXI )-a. (3.1)b

Notice that this model depends only on two parameters 0= (X1+ ;2)/b and a so that if we had data

only from systems on test in the operating environment, the only identifiable parameters are a, 0,

NO

FIGURE 4 RSERIES SYSTEM RELIABILITY UNDER UNIF(R,B) MODELFOR THE ENVIRONMENTAL STRESS.

I- I l0, X2=2.I0, nj I.O. 1 . 0

* KEY- ---- R=O.25.B=O.75

,- ...R=0.75. B 1.25x ---- R=1. 00, B=2.O00{

.----INDEP 1."

U v-C-O

C.' .

M- A-

9b. 00 1.00 2.00 Doo 4.00 s.00:,l TIME.

FIGURE '4 BSERIES SYSTEM RELIABILITY UNDER UNIF(A.B) MODEL

FOR THE ENVIRONMENTAL STRESS.)j=.O >2=2.0, i=1O '2=2.0.

0

I' KEY

(b :::RuO. 25.8=0. 75AL A 0 75.B1.25----Ru:1. 25, .1. 75

X--A I 00,B-2.00....- INDEP

C,,

z(D

LL_ !r

-

V)

('U

0.0102.0 30 .050

TIME

FIGURE 4 CSERIES SYSTEM RELIABILITY UNDER UNIF(AB) MODELFOR THE ENVIRONMENTAL STRESS.

I=., .=2.0, ni=2.0. '22.0.

0

U)

KEY+ --.A-O.25.BuO.75

X ... . 0 .~ B-2.0

S----INDEP

I-

z

-JL- Wcc

>0

'00 .0 20 30 .050

TIE

FIGURE 4j DSERIES SYSTEM RELIABILITY UNDER UNIF(A.B) MODEL

FOR THE ENVIRONMENTAL STRESS.

0

PI - KEYC8; CD---- R-0.25.8-..75

A---- A=0.75. Ba1. 25

E---- INDEP

F-

zLL_ Uf)

-

cC

U, o .020 30 l 050

TIME

AzVI

FIGURE 5 RPARALLEL SYSTEM RELIABILITY UNDER UNIF(A,B) MODELFOR THE ENVIRONMENTAL STRESS.S)'=l.O- , X,2= 2. 0, 111 0 2=1.0.

0

co

*. KEYD ---- A-O.25.B-O.75A-- A-O.75 B-1.25+ ---- A- 1. 25, 8 1 •75X -- ,A! .00, 8-2. 00

.. -0NDEP

0,

i -

0

co

c

L17('a

* C3

060

b ,00 1.00 2.00 3.00 4.00 5.00TIME.

FIGUR9E 5 BPARALLEL SYSTEM RELIABILITY UNDER UNIF(IIB) MODEL

FOR THE ENVIRONMENTAL STRESS.I0 22 . 111.0. 1= .0

uo KEY

o 0 ----AuO.25 8=0.75

+ ----INDEP8-.7

zD

c

>-co

Cr)-(NJ

c'J

rb.0 1 .00 2 .00 3 1.00 '4 .00 5.00TIME.

CD~o - - -, - -

FIGURE 5 CPARALLEL SYSTEM RELIABILITY UNDER UNIF(A,B) MODEL

FOR THE ENVIRONMENTAL STRESS.

KEY~----Ruo.25. B.O.75::--AR0.75. Do1. 25

+---Ru 1. 25. Bo..75

C,

Ln-J

1I

FIGURE 5 0PARALLEL SYSTEM RELIABILITY UNDER UNIF(A,B) MODEL

FOR THE ENVIRONMENTAL STRESS.-i=1.0, >2=2.0, n"=1/2. n2=1/2.

0)

U)

* KEY

-- .--- - OAu .25 B-O.75

X --- A.. 100.3-2.00

S----INDEP

zco

MoL)

0-0

~cr)

Win)

04

'-b.O0 1.00 2.00 3'.00 4.00 5.00TIME.

FIGURE 5 EPARALLEL SYSTEM RELIABILITY UNDER UNIF (A.B) MODELFOR THE ENVIRONMENTAL STRESS.

)i1. 2=2.0. Ili 112. '12 2. 0.

04

0

FKU,

* KEYo O---- AO.25.B.O.75

£k ---- A-0 75, Ba..25+g ----A.!. 25. 8.1. 75X -- :-A*1.OO.B-2.OO

1 0

-

CD

4--,-

cu,

C.00 1 .00 2 .00 3 .00 4 .00 5 .00TIME.

8

11, not XI, X, , a, b. However, in many instances we have extensive data on the performance of

the components in the lab under ideal operating conditions so that one may consider X1, X2 , 11 to be

S known based on estimates from this data. We shall focus on the problem of estimating 0 and a,

based on data on the system failure times collected in the operating environment. Let tl, ..., tn be

the failure times for n such systems put on test, and, let wi = till, i = 1, ..., n.

Prior to attempting to estimate (a, 0), we would like to check if the model (3.1) is feasible.

~A graphical check of this model can be done through the scaled total time on test (SITOT) plot of

Barlow and Campo (1975). The STIOT for W is

I t Rs(t)dt0

Gw(t) = -'I(1- = l-(1-t)(a'l)/a for a> 1. (3.2)

Rs(t)&0

Note that (3.2) depends only on a. Figure 6 shows the form of the STTOT for several values of a.

Notice that for all a, the STrOT is below the 450 line (which corresponds to exponential system

life) since the hazard rate of the series system is decreasing. Let

i

Tn(W(i)) = Z, WOj) + (n-i)W(i), (3.3)j=l

where W(1)5 W(2):5... <W(n) are the ordered systems failure times be the total time on test at

W(i). The empirical IfTOT plot then plots (i/n, Tn(W(i))/T(W(n))) which can be compared to

figure 6 for a graphical check of the model. Also, crude estimates of a can be obtained by*(1

If

FIGURE 6SCALED TOTAL TIME ON TEST TRANSFORMFOR GAMMA MODEL.

0

KET* --- A-1 2SX ---- A- 13•0" ---- Am2. 0A ---- Am 3.00

+ --A-.0O

ItJ

0-4C

I--

cc

u-jo

V)

0 0 0. 20 0'.0 O 0. 60 0'.80 1.O00.. • T_I . , ,,, ', , ,',. ., '-", %.." r ". ".'. ' ,'. .,-. ' .,. -,...€ .,c ..- ..-n-.- .' . ..

comparing the empirical and theoretical STOT plots. When there is no random environmental

effect and the components are independent, then the empirical SITOT plot should look like the 450

line. Also as a tends to infinity this plot approaches the 450 line.

We now consider several estimates of a and 0. The log likelihood for the model (3.1), based

on a sample of size n, isn

L(a,e) = n n a + n ln 0 -(a+l)1 An (1+0 W i) (3.4)i=1

so tha

na/aa L(aO) = nla - I In (1 + 0 Wi) (3.5)

i=l

nand aIaO L(a,O) = n/O - (a+l) i w/i(l+ Owi) (3.6)

i=l

I For (3.5) we note that the maximum likelihood estimator of a

amle -= (3.7)nIn (1 + O Wi)

and the maximum likelihood estimator of 0 is the solution to

n n W.+1) ( )=0. (3.8)

S In ( 1 + O Wi) I + OWi

One can show that ( is positive if ni w,2 > 2( 1 w) 2 . (3.9)

In such case erl € is obtained by solving* |

-" '" " * " ..' " .w * ' "; " ' " '" i :" ' : i "

po

10

(3.8) numerically.

A second estimator of (a, 0) is the method of moments (mine). Since E(W) = [0(a-1)]- 1

and E(W2 ) - 2[ 02 (a-1)(a-2)] "1 where a > 2, we have

I wi2 I wi2-2(y.w,) 2

a +e 1 + (3.10) and O(( ) (3.11)I wi2 - 2(7. wi2) I wi(l wi)2

provided that (3.9) holds. If (3.9) does not hold, then this estimator does not exist

A third estimator was suggested by Berger (1983) in a different context He suggested

estimating 0 a modified methods of moments estimator ber = (a w) - 1, (3.12)

where w = wi/n,

which is used as the true value of 0 in the likelihood (3.4) so that the estimator of a is the solution

to

i nn (1 w i )+wa(a+ l ) lwi(ww- An+1 -_-_2_ = 0 (3.13)

aw wa lswi/(aw)

A final estimator is based on the STTOT plot Let Ci = Aln(l-i/n) andD i =

6 n(l-Tn(W(i))/Tn(W(n))), i = I, ... , n-1. If (3.2) holds, then we should have

~D i - (l-I/a)Ci, i =I..,n-1, (3.14) r

so the value of a which minimizes

n-ii (Di - (1-1/a) q) 2 is a reasonable estimator of a

The resulting estimator is als = C 2 (3.15)XCI - CDi

which is in the parameter space if C 2 > ,C Di. A better estimator should be obtained by

weighting the Di's differently since for i < j, Var (Di) < Var (Dj). The variance of Di depends on

the unknown parameter a so we weight by the variance of Di computed under an assumed

exponential distribution. The variance of Di in that case is

i 1Vi = i=1, ..., n-1 (3.16)

j= n-lj)

~ so that the weighted least squares estimator of a is

al Ci2 C/V if Y Ci2/Vi > z CiDi/Vi. (3.17)

Vi Vii

Once we have obtained a by either of tht. two least squares estimators, we substitute this value into

(3.6) and solve this equation numerically for 0s or Owls.

The condition Z Ci2/Vi > £: Ci Di/Vi includes a few more possible samples than the

condition (3.9) for the other three estimators. However, those samples which satisfy I Ci2 /Vi >

tot Ci Di / V i for which (3.9) fails to obtain yield very large estimates of 0. Since a reasonable

model for T when 0 and a are not estimable is the independent Weibull series system which has

Ssystem reliability very close to (3.1) when a is very large, this is not a problem. Figures 7a and 7b

are scaled total time on test plots from two simuilated samples of size 30 from (3.1) with a = 3, 0=

1. Looking at figure 7a, we see that the estimated scaled total on test doesn't look too different

from the 45* so that an exponential model might not be unreasonable. For this data set only the

weighted least squares estimator exists and it yields aWLS = 45.33 and 0= .0567. For the data in

l figure 7b all estimates exist, and we have

: 7 .

FIGURE 7 RSCALED TOTAL TIME ON TEST PLOTFOR SIMULATED DATA.

Lu

Li')

c'j

'0.0 0.20 0'.40 0'.60 0.80 1.00*~ I/N.

II

FIGURE 7 BSCALED TOTAL TIME ON TEST PLOTFOR SIMULATED DATA.

LUo

CIc-

0

Lo

lb.oo 0.20 0'.40 0'.60 0'.80 11.00I /N.

12

Ond e = .93 amle = 2.98

S 0 rme = .491 atm e = 4.86

Ober - .7 2 0 aber - 7.02

013 = .739 . a 5= 3.58

Owls = .970 awls - 2.89

To study the properties of these estimators, a small scale Monte Carlo study was performed.

Random samples of size n = 15, 30, 50, 75, or 100 were generated with X1 + X2 = 3, b = 3, so 0

=I and a = 2, 3, 5. 1000 samples were generated for each combination of n and a. The bias,

standard deviation of the estimates and n, the number of samples where the estimator exists is

reportedin table 1 for a, table 2 for 0, and in table 3 for an estimator of the system reliability

Sobtained from (3.1) at to = 9.085. The true system reliability at to is .8255 when a = 2, .75 when a

.. =3, and .619 when a = 5. Also reported in each table is the bias and standard deviation of the least

square and weighted least square estimators when they are restricted to those samples where the

other estimators exist.

From these tables we note that Berger's modified estimator performs very poorly. Also the

weighted least squares estimator allows for estimation of parameters in many more samples when n

Sis small. In general the maximum likelihood estimator outperforms the other estimators, however,

when the weighted least squares estimator is restricted to those samples where the maximum

likelihood estimator exists, this estimator performs much better when n is small The somewhat

better performance of the MLE in terms of bias is deceptive since some of the estimates of a are less

than one, which implies that the mean system reiliability is infinite. Also the weighted least squares

estimator of system reliability seems to outperform the other estimators of the system reliability in

spite of its relatively poor performance as an estimator of 0. Our recommendation is to use the

weighted least squares estimator since it more often provides estimators of the relevant parameters

and is somewhat easier to compute.

(ABLE 1SIAS AND STANDARD DEVIArION(SD) OF ESTIMATORS OF A

MAXIMUM WEIGHTED METHOD OFLIKELIHOOD LEAST SQUARES LEAST SQUARES MOMENTS BERGER'S METHOD

1;~ N 10 BI AS so M SIAS SD Mi BIAS SD m B IAS SD M BIAS SD5L5 79 4.2 9 8S2 4.8 41. /62 6.8 49. 770 9.1 37. 770 14.4 65.

766 1.3 5. 715 3.5 16.

2 TO 916 2.8 20. 953 4.7 37. 877 6.4 52. 916 6.1 32. 916 10.0 52.

912 1.1 3. 857 5.4 51.950 979 5.8 114 99 1.7 10. 956 4.0 16. 979 8.5 131. 979 15.4 241.976 1.0 3. 952 3.5 12.

2 75 996 1).9 4. 998 1.0 3. 974 2.4 12. 996 2.2 4. 996 4.5 9.

996 1.0 S. 972 2.4 12.2 100 999 0).3 3. 1000 1.7 35. 989 1.5 9. 999 1.7 5. 999 3.7 9.

I 999 0.6 2. 989 1.5 9.3 15 642 7.3 39. 753 36.4 843. 653 13.1 149. 643 16.8 77. 643 26.0 14.

636 1.3 9. 573 10.7 159.3 30 809 5.7 30. 870 13.8 141. 768 11.9 100. 910 9.9 104. 809 17.7 68.

804 1.8 6. 731 11.1 102..3 50 916 3.6 18. 935 6.9 65. 864 6.4 33. 916 6.6 25. 916 12.5 42.

912 2.9 29. 851 -7 32.3 75 96! 2.5 14. 977 2.8 17. 925 11.6 144. 963 4.7 24. 963 9.6 38.

958 1.3 5. 923 11.6 144.S 100 979 1.7 7. 989 2.1 12. 956 3.7 19. 978 3.0 9. 978 7.2 15.

979 1.3 5. 952 3.6 19.5 15 520 38.7 573. 665 8.2 53. 558 30.3 493. 522 69.8 925. 522 112.9 1505.

516 -0.7 5. 458 1.9 15.3 30 674 20.4 148. 752 9.3 69. 669 13.7 109. 674 31.7 202. 674 56.3 347.

660 3.2 29. 601 9.4 86.() 801 7.6 39. 850 9.0 97. 756 13.4 88. 801 11.4 52. 801 23.4 91.

787 2.0 10. 722 8.6 56.3 75 893 12.8 139. 415 9.0 94. 827 6.6 22. 893 15.3 122. -9'3 32.6 260.

878 2.9 16. 714 5.9 20.I 3 lo,:' 91 9.5 T1 1-3 19.6 307. 3' 11.0 91. I 92 13.0 1) 0. 992. 27 1 3.

9 879 13.7 23. 2a1 9.5 79.

ILK

TABLE 2BIAS AND STANDARD DEVIATION(SD) OF ESTIMATORS OF e

MAXIMUM WEIGHTED METHOD OF

LIKELIHOOD LEAST SQUARES LEAS) SQUARES MOMENTS BERGER'S METHOD

A N M B8IAS SD M ASD M -- [ASSD M.BIAS S BD M BIAS SO

l1 769 o.356 1.702 852 -.102 0.742 782 -.192 0.729 770 -.683 0.205 770 -. 803 0.122766 -.027 0.025 715 -. 156 0.701

2 30 916 0.112 0.919 9533 -. 135 0.580 877 -.254 0.596 916 -.623 0.192 916 -.798 0.095

912 -. 100 0.567 857 -.239 0.584S 2 50 979 0.016 9.648 989 -.126 0.492 956 -.263 0.514 979 -.575 0.184 979 -.792 0.079

976 -.115 0.486 952 -.260 0.5132 75 996 -.025 0.522 998 -.125 0.432 974 -.247 0.475 996 -.341 0.174 996 -.790 0.065

996 -. 124 0.431 972 -.246 0.4742 100 999 -.019 0.437 1000 -.101 0.381 989 -.216 0.423 999 -.508 0.153 999 -.785 0.052

1000 -.101 0.381 989 -.216 0.4233 13 642 0.691 1.900 753 0.210 1.049 653 0.119 1.031 683 -.513 0.348 643 -.705 0.203

636 0.390 1.040 573 0.245 1.0393 30 809 0.175 1.049 870 0.000 0.757 769 -.096 0.745 910 -.469 0.338 809 -.725 0.160

904 0.074 0.740 731 -.057 0.7433 5') 916 0.075 0.766 935 -.012 0.618 864 -.L12 0.663 916 -.404 0.333 916 -.718 0.135

912 0.011 0.609 851 -.100 0.6603 75 963 0.030 0.624 977 -.027 0.555 925 -.144 0.603 963 -.375 0.322 963 -.717 0.120

958 -.010 0.546 923 -.142 0.6033 100 978 -.028 0.515 989 -.075 0.472 956 -.165 0.511 978 -.343 0.297 978 -.716 0.104

979 -.055 0.465 952 -.161 0.5093 15 522 1.352 3.I09 665 0.7t5 L.&09 558 0.578 1.536 522 -.238 0.&01 522 -.546 0.348

516 1.094 1.599 458 0.827 1.5615 30 674 0.558 1.609 752 0.366 1.118 669 0.L80 1.026 674 -. 199 0.576 674 -. 584 0.282

660 0.523 1.104 601 0.286 1.0295 50 801 0.254 1.559 850 0.184 0.869 756 0.105 0.863 801 -. 193, 0.541 801 -. 615 Y. 2325 787 0.261 0.850 722 0.150 0.858

75 893 0.128 0.817 915 0.112 0.728 827 0.014 0.747 893 -.189 0.535 893 -.628 0. 21.-878 0.153 0.715 814 0.("28 0.744

5 100 892 0.033 0.683 913 0.020) ).628 83 -. 064 0.666 892 -. 206 0.494 892 -. 644 0.185879 0.055 0.615 821 -. 050 0.663

TABLE 3BIAS AND STANDARD DEVtAT[ON(SO) OF ESTIMATORS OF SYSTEM RELLABILITY AT T-.9065

MAXIMUM WEIGHTED METHOD OFLIKELIHOOD LEAST SQUARES LEAST SQUARES MOMENTS BERGER'S METHOD

AN 14 BIAS D M IAS ..SD M SEAS - i M S --- ai S -AS SDo --- BIAS -- SD... .; ....... -... ... ... - e 0 ----....-----2 -13 769 -. 1 .479-04 .5672.02089 70 0.037 .0 5 70 0.04 .0463

766 -.006 .0577 715 0.002 .05862 30 916 -.005 .0473 953 0.002 .0424 877 0.010 .0434 916 0.037 .0357 916 0.069 .0335

912 0.001 .0426 957 0.009 .04342 50 979 -.001 .0372 999 0.003 .0349 956 0.010 .0359 979 0.035 .0300 979 0.071 .0233

976 0.003 .0348 952 0.010 .0359r., 2 75 996 0.000 .0290 998 0.004 .0274 974 0.010 .0292 996 o.034 .0244 996 0.02 .0238

996 0.003 .0274 972 0.010 .02932 100 999 0.011 .0243 100 0.004 .0234 999 0.010 .0248 999 0.034 .0223 999 0.075 .0216

999 0.004 .0233 989 0.010 .0249T 15 642 -.018 .0815 753 -.010 .0767 653 -.Q06 .0748 643 0.031 .0661 643 0.067 .0616

636 -.015 .0764 573 -.008 .07513 30 809 -.007 .0577 870 -.003 .0Z52 769 0.002 .0551 810 0.024 .0490 809 0.062 .0471

804 -.006 .0544 731 0.001 .05503 50 916 -.003 .0429 935 -.001 .0412 864 0.005 .0431 916 0.022 .0366 916 0.066 .0340

912 -.002 .0411 851 0.005 .04323 75 963 -.001 .0372 977 0.000 .0356 925 0.007 .0374 963 0.021 .0327 963 0.067 .0313

959 0.000 .0357 923 0.007 .0375. 100 9788 0.001 .0309 989 0.002 .0299 956 0.008 .0311 978 0.019 .0267 978 0.067 .0261

978 0.002 .0299 952 0.008. .03125 15 520 -.02.9 .1011 665 -.024 .0967 558 -.020 .0977 52Z 0.012 .0926 522 0.054 .0921

516 -.030 .0968 458 -.022 .0968 -5" 30 674 -.022 .0691 752 -.020 .0674 669 -.013 .0651 674 0.001 .0613 674 0.045 .060e

66') -.027 .0665 601 -.016 .0652550 901 -. 006 .0545 850 -. 006 0530 756 -. 002 .0532 901 0.010 .0494 801 0.055 o09

787 -.008 .0528 722 -.002 .05355 75 893 -. 005 .0442 915 -. 005 0436 827 -. 002 .0431 893 0.007 .0406 893 0.051 .038o

.979 -. ()06 .0430 814 -. 002 .0432S 100 892 -. 01)2 .0375 913 --. ))1 .0392 831 0.003 .0380 892 0.008 .0353 892 0,052 .0.345

879 -. u02 0390 821 0.003 .0381

.Z

13

Acknowledgment

This work was supported by the U.S. Air Forc Office of Scientific Research under conact

AFOSR-82-030.

iw.

414

Bibliography

Bain, L J. (1978). Statistical Analysis of Reliability and Life-Testing Models. New York:

Marcel Dekker, Inc.

S Barlow, R. E. and Campo, R. (1975). Total Time on Test Processes and Applications to Failure

Data Analysis. Raliability and Fault Tree Anlysis. (Barlow, Fussell, and Singpurwalla,

I eds.). SIAM, Philadelphia, 451-481.

S Basu, A. P. and Klein, J. P. (1982). Some Recent Results in Competing Risk Theory. Suvival

Anasis. (Crowley and Johnson, eds.). The Institute of MathematicalStatistics, 216-227.

Clayton, D. G. (1978). A Model for Association in Bivariate Life Tables and Its Application in

Epidemiological Studies of Familiar Tendency in Chronic Disease Incidence. Hiometia 6,

141-151.

b Hui, S. L. and Berger, J. (1983). Empirical Bayes Estimation of Rates. Jounalft

Amerantatiatial Association L 753-760.

Hutchinson, T.P. (1981). Compound Gamma Bivariate Distributions. Metrika8.263-271.

Klein, . P. and Moeschberger, M. L (1984). Consequences of Departure from Independence in

I Exponential Series Systems. Techometrics 26, 277-284.

, Lindley D. V. and Singpurwalla, N. D. (1985). Multivariate Distributions for the Reliability of a

System of Components Sharing a Common Environment. George Washington University

Technical Report.

Oakes, D. (1982). A Model for Association in Bivariate Survival Data. Journal of the Royal

Statistical Society B 44 414-422.

Takahasi, K. (1965). Note on the multivariate Burr's Distribution. Annals of the Institute of

Stati~Ltal themaics.Tokx. 17,257-260.

..... .

A d

* Appendix E

H

~~~TTiir- 1 a a4F:toAr

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BEFORE COMPLE'TING FORM

I. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

4. TITLE (and Subtitle) 5. TYPE OF RErPORT & PERIOD COVEREDOOn Dependent Competing Risks 10/1/84 to 10/31/85

3 6. PERFORMING O'1G. REPORT NUMBER

714837/7632657. AUTHOR(e) 6. CONTRACT OR GRANT NUMBER(s)

John P. Klein and Sukhoon Lee Grant No.AFOSR 82-0307 "

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

AREA & WORK UNIT NUMBERS

The Ohio State University Research Foundation1314 Kinnear Rd., Columbus, Ohio 43212 92

If. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

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16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited

a17. DISTRIBUTION STATEMENT (of the abetract entered in Block 20, if different from Report)

IS. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if neceesary nd identify by block number)

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SECURITY CLASSIFICATION OF THIS PAGE (When Date Enteefd

% %, % ,,%.

On Dependent Competing Risks

byJohn P. Klein and Sukhoon Lee

Department of StatisticsThe Ohio State University

Columbus. Ohio 43210 USA

1. Introduction

The problem of competing rLsks arises naturally in a number of contexts.namely the modeling of series systems in reliability, the problem of estimationwith censored data and the analysis of physical or biological systems with

Vmultiple failure modes. A common, untestable assumption is usually made thatthe potential failure times for each risk are statistically independent (seeBasu and Klein (1982)). Hoeschberger and Klein (1984) show that an investiga-tion may be appreciably misled in modeling series systems reliability and inestimating component parameters by incorrectly assuming independent componentlifetimes. In this paper we model dependence between components through acommon environmental effect on each component. Such a dependence structurehas also been suggested by Oakes (1982). Lindleyand Singpurvalla (1985), andHutchinson (1981).

2. The Model

Consider a two-component system. Suppose that under ideal, controlledconditions, as encountered in the testing stage, the times to failure of thetwo components are X0 and Y'. Under these condittons. XOY ate independent

with marginal survival functions F and G0. Now suppose t e two componentsare linked into a system and exposea to the environment. The effect of theenvironment is to select a random factor, Z, from a distribution, H(z). which

z zchanges the marginal survival functions of the two components to F0

Z GO0

respectively. A value of Z less than one means that component reliability isimproved. while a value greater than one implies a joint degradation. In thesequel we assume that X0 and Y follow a Weibull distribution with parameters

, and That is, Fo(x) - exp(-kx x). The resulting joint

reliability of the two components' lifetimes. (X.Y). in the operating environ-

ment is F(x.y) - E(exp(-Z(A x x + Y Z. F(x,y) is positive quadrant dependent.

Also E(X) E(XE(Z'L ); V(X) - E(Xo)E(Z2/"'x) -(E(X(Z-lcx)) 2 ; and

Cov(XY) - E(X0)E(Y0 ) Cov(Z" G/ x. z- 1 /6Y). The correlation is always positive

and is bounded above by (l + 1/a) /r( + 2/a) whena - a - a. Explicit.

though lengthy. formula for the moments of (X,Y), system aJd component

. reliability, and system components can be obtained vhen Z is assumed to beeither a uniform or gammar random variables. The gamma case leads tobivariate Burr distributions.

•3. Estimation

The estimation of model pramer.ters is carried out under the assumption

that Z has a gamma distribution with density h(s)m exp(-ba). Since the

S paramacers 4,,, Ay are not identifiable when only data from either a series or

parallel system is available, we incorporate sample information obtainedindependently on each component under test conditions. Maximum likelihoodand method of moments estimators are obtained and their properties are studiedby Monte-Carlo methods since no closed form maximum likelihood estimates areavailable.

Acknowledgement

This research was supported by contract AFOSR820307 for the U.S. AirForce Office of Scientific Research.

BIBLIOGRAPHY

1. Basu, A. P. and Klein, J. P. (1982). "Some Recent Results in Competing

Risk Theory," in Survival Analysis, Crowley and Johnson, eds., IMS, 216-

271.

2. Hutchinson, T. P. (1981). "Compound Gamma BivarLate Distributions,"

Metrika, 28, 263-271.

3. Lindley,D. V. and Singpurwalla, N. D. (1985). "Multivariate Distributionsfor the Reliability of a System of Components Sharing a Common Environ-ment," Proceedings of the International Conference on Reliability andQuality Control (to appear).

4. Hoeschberger, M. L. and Klein, J. P. (1984). "Consequences of Departurefrom Independence in Exponential Series Systems," Technometrics. 26,277-284.

5. Oakes, D. (1982). "A Model for Association in Bivariate Survival Data,"Journal of the Royal Statistical Society B, 44, 414-422.

Resume

Un modile pour les systimesd'pendancs dans l'analyse de la fiabilite estexamine. Le modele suppose que, sous les conditions ideales, les cemps desurvie des constituanca du sysceme ont des distributions Weibull ind;pendantes.Sous des conditions d'operation un facteur excfrieur aleacoLre affecce chaque€onatituant simultandmnt ean multipliant son caux de hasard par une quanticEal acoire. Lea propriltes de ce mod'tle ec l'estimacion des paramecres dumodble sont considdrds, I partir des exemples concrets du laboratoire ct dela pratique.

6,.

i

FILMEF

SDTIC