Table Of ContentBAYESIANANALYSISOFCOMPETINGRISKSMODELS
By
CHEN-PINWANG
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL
OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT
OFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
1999
©Copyright1999
by
Chen-PinWang
Tomyfamily
ACKNOWLEDGMENTS
IwouldliketoexpressmydeepestgratitudetoDr.MalayGhosh,withoutwhom
thisworkwouldneverhavebeencompleted. Hewasalwayswillingtogivemeguid-
ance,knowledge,andfriendshipwhenitwasneeded. Iwouldalsoliketothankmy
committeemembers, Dr. RandolphCarter, Dr. AndrewRosalsky, Dr. MarkYang,
andDr.YumeiChen,forsupportingmyresearch. Additionally,thecollectiveknowl-
edgeandexperienceofmyfellowstudentsandmembersofthefacultyneverfailedto
inspireandencouragemeduringthecompletionofmywork. Iwouldliketoespecially
thankDr.RomanLittelforbeingwillingtohelpmebeyondstatistics.
Finally,Iwouldliketothankmyfamily,fortheirloveandsupport,bothfinancial
andemotional. IwouldespeciallyliketothankmybestfriendinGainesville, F.J.
Huang,whoputupwithmeandconstantlyencouragedmeduringthecompletionof
mydoctoralwork,withoutwhomthisworkwouldhaveneverbeenfinished.
IV
TABLEOFCONTENTS
ACKNOWLEDGMENTS
iv
LISTOFTABLES vii
LISTOFFIGURES viii
ABSTRACT ix
CHAPTERS
1 LITERATUREREVIEW 1
1.1 Introduction 1
1.2 CompetingRisks 2
1.3 BivariateLifetimeDistributions 5
1.4 NoninformativePriors 19
1.5 ResearchProposal 23
2 BAYESIANANALYSISOFSELECTEDBIVARIATE
EXPONENTIALMODELS 25
2.1 Introduction 25
2.2 Notation 26
2.3 NoninformativePriorModification 27
2.4 BayesianAnalysisforMarshall-OlkinBVE 29
2.5 BayesianAnalysisfortheACBVEModels 32
2.6 PriorPerformance 37
2.7 IdentifiableACBVEviaInformativePriors 48
3 BAYESIANANALYSISOFGENERALIZEDBIVARIATE
EXPONENTIALMODELS 51
3.1 Introduction 51
3.2 GeneralizedBVEModels 52
3.3 GeneralizationofACBVEModels 53
3.4 GeneralizationoftheMarshall-OlkinBVEModel 59
3.5 ApplicationtoaSamplewithCategoricalCovariates 62
v
3.6 SamplingSchemes 65
3.7 IllustratedExamples 67
4 GEOMETRICCOMPETINGRISKSMODELS 78
4.1 Introduction 78
4.2 GeneralizedGeometricModelsandLikelihoodFunctions 79
4.3 BayesianAnalysis 83
4.4 ApplicationtoaSamplewithCategoricalCovariates 88
4.5 DataAnalysis 90
5 SUMMARYANDFUTURERESEARCH 94
APPENDIX
PROOFSOFMATCHINGPROPERTIESOFnJVANDnj 95
REFERENCES 96
BIOGRAPHICALSKETCH 100
vi
LISTOFTABLES
Table Page
2.1 ReferencePriorsfortheMarshall-OlkinBVE 31
2.2 PosteriorDistributionsfortheMarshall-OlkinBVE 32
2.3 TheACBVEReparameterization 33
2.4 PosteriorDistributionsforAandpofACBVE’s 36
2.5 PosteriorDistributions(Moments)for0ofACBVE’s 36
2.6 FrequentistCoverageProbabilitiesforAunder7rjj,nuu,kj,kju 39
2.7 NumericalSimulationResultsofTable2.6 39
2.8 PosteriorEstimatesfor<j>ofACBVE’sunderBetaPriors 49
3.1 ProbabilityMatchingforA_1(l,0.5)'undernju,/kuu irj,Tru 68
,
3.2 ProbabilityMatchingforA_1(l,0.5,2)'underttju,ttuu,txj, 69
3.3 ECOGPosteriorEstimatesunderMainEffectModelandiruu 73
3.4 ECOGPosteriorEstimatesunderInteractionModelandnju,t^uu • 75
3.5 PosteriorEstimatesforContrastsofInterestundernuu(orkju) •••• 77
4.1 StochasticTransitionStatus 80
4.2 ComparisonofPosteriorEstimatesforP(A=1|T^=1) 90
vii
j 77
LISTOFFIGURES
Figure Page
2.1 TransformedQ-QPlotsfor(f)under^tj7 andn=10 42
2.2 TransformedQ-QPlotsfor4>undern 777,andn=50 43
2.3 TransformedQ-QPlotsfpr<f>under7tj777,andn=100 44
2.4 TransformedQ-QPlotsforpundernj,7 /,andn—10 45
2.5 TransformedQ-QPlotsforpunder7tj,7 /,andn=50 46
2.6 TransformedQ-QPlotsforpunderttj,777,andn—100 47
2.7 BetaPriorsandtheAssociatedPosteriorsof<pundern=10,50 50
3.1 R’sandPosteriorDensitiesof(3i,fa,fa,andfaunder7tjju 71
3.2 R'sandPosteriorDensitiesof71(j2,73,and74under7Tuu 72
3.3 Analysis1: Goodness-of-fitPlotsforTj’sandAj’sunderBVE 74
3.4 Analysis2: Goodness-of-fitPlotsforTj’sandAj’sunderBVE 76
4.1 BayesianGoodness-of-FitPlotforn^’s 91
4.2 SurvivalCurvesforTundertheACBVEandtheGEM 92
viii
AbstractofDissertationPresentedtotheGraduateSchool
oftheUniversityofFloridainPartialFulfillment
oftheRequirementsfortheDegreeof
DoctorofPhilosophy
BAYESIANANALYSISOFCOMPETINGRISKSMODELS
By
Chen-PinWang
August1999
Chairman: MalayGhosh
MajorDepartment: Statistics
Bivariate exponential (BVE) models have been widely used in the analysis of
competingrisksdatainvolvingtworiskcomponents. Forsuchanalysis,frequentist
approachoftenrunsintodifficultyduetononidentifiablelikelihood. Withanendto
overcomethenonindentifiability,recentliteraturehasbeengearedtowardsBayesian
analysiswithinformativepriors. However,systematicpriorelicitationisoftendiffi-
cult. ThisstudyfocusesinsteadonBayesiananalysiswithnoninformativepriors.
DuetothenatureofthecompetingrisksdataandBVEmodelstructure,quite
oftenitleadsalikelihoodfunctionwithanonregularFisherinformationmatrixim-
pedingthereby thecalculation ofstandard noninformative priorssuch as Jeffreys’
priorsandtheirvariants. Asaremedy,astagewisenoninformativepriorelicitation
strategyisproposed. Avarietyofnoninformativepriorsaredeveloped,andareused
fordataanalysis. Inaddition,thefrequentistprobabilitymatchingcriteriaareinves-
tigatedamongthenewlydevelopednoninformativepriors.
Oftenduetoresourcelimitationorothereconomicandpracticalreasons,individ-
ualsareonlyperiodicallyscreened. Theresultingcompetingrisksdataarenecessarily
discrete. Aclassofflexiblemodelsisintroducedtohandlesuchdiscretizeddata.
IX
CHAPTER
1
LITERATUREREVIEW
1.1 Introduction
Ofteninalife-testingsituation,failureofanindividualcanbeidentifiedasone
ormoreofs (s > 1) mutuallyexclusive, butpossiblydependent causesoffailure.
Inotherwords,eachindividualissubjecttosdistinctrisksreferredtoascompeting
risksthreateningitslife. Associatedwithcausei,thereisanonnegativeabsolutely
continuous random variable representingthe lifetime ofan individual when no
otherpotentialrisksarepresent. Supposethattheterminationtimeofanindividual
isdefinedasthetimetothefirstfailure. Thus,lifetimeofanindividualisgivenbyT=
min{T!,••-Ts}. Theavailableinformationisusuallygivenbythepair(T,/),where
/indicatesthecause(s) offailure. Thecompetingrisksconceptcanappropriately
be applied to many areas ofstudy, such as industrial reliability analysis, market
transactionanalysis,andclinicaltrialonpairedorgans.
OurmaingoalistouseBayesianmethodologyformakingstatisticalinferencein
competingrisksmodelstostudycertainlifetimefeaturesofinterestandthecovariate
effectsontheunderlyingsurvivalfunctions. Thefirststepisnecessarilymultivariate
lifetime modeling. Intheone-dimensionalcase, theWeibulldistribution hasbeen
consideredasthemostflexibleoneforlifetimemodelingsinceitaccommodatesthree
majortypesoffailurerates: agingtype,decayingtype,andconstanttype. Anatural
extensiontomultidimensionallifetimemodelingwouldbethemultivariateWeibull
model. Animportantspecialcaseisthemultivariateexponentialdistribution. When
the shape parameters are known, then one can make a power transformation on
1
