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Survival And Reliability Analysis Under Polya Tree Processes Priors

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Survival And Reliability Analysis Under Polya Tree Processes Priors
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english
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Huang, Lei
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University of Florida
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Ghosh, Malay
Committee Members:
Khare, Kshitij
Doss, John
Banerjee, Arunava

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Subjects / Keywords:
nonparametric -- survival
Statistics -- Dissertations, Academic -- UF
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Statistics thesis, Ph.D.
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theses   ( marcgt )
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Abstract:
The thesis consists of three components, which are related to survival and lifetime data analysis. We propose a Bayesian nonparametric approach to testing treatment effects and estimating regression coefficients in the Cox proportional hazards model by considering not only the rank statistics, but also the spacings between rank statistics. The Bayesian solution of our approach has a closed form even with censored data, which can be calculated extremely fast and has a ready interpretation. Simulation studies are carried out and the method is applied to analyze a real data set. Finally, we propose a nonparametric estimator for reliability, which degenerates to the well-known Mann Whitney U statistic in special cases.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
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by Lei Huang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Ghosh, Malay.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

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MISSING IMAGE

Material Information

Title:
Survival And Reliability Analysis Under Polya Tree Processes Priors
Physical Description:
1 online resource (78 p.)
Language:
english
Creator:
Huang, Lei
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Ghosh, Malay
Committee Members:
Khare, Kshitij
Doss, John
Banerjee, Arunava

Subjects

Subjects / Keywords:
nonparametric -- survival
Statistics -- Dissertations, Academic -- UF
Genre:
Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
The thesis consists of three components, which are related to survival and lifetime data analysis. We propose a Bayesian nonparametric approach to testing treatment effects and estimating regression coefficients in the Cox proportional hazards model by considering not only the rank statistics, but also the spacings between rank statistics. The Bayesian solution of our approach has a closed form even with censored data, which can be calculated extremely fast and has a ready interpretation. Simulation studies are carried out and the method is applied to analyze a real data set. Finally, we propose a nonparametric estimator for reliability, which degenerates to the well-known Mann Whitney U statistic in special cases.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Lei Huang.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Ghosh, Malay.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045745:00001


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SURVIVALANDRELIABILITYANALYSISUNDERPOLYATREEPROCESSESPRIORSByLEIHUANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013LeiHuang 2

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Idedicatethistomybelovedwifeandsupportiveparents. 3

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ACKNOWLEDGMENTS IwouldliketoexpressmyheartfeltgratitudetomyPh.Dadvisor,Dr.MalayGhosh.Heismorethanamentortome.Inthepastveyear,hehelpedmepickedthecorrectdirectionandwasalwaystheretohelp.TherewasatimewhenIconstantlycameacrossunsolvableproblemsandfeltdown.Dr.Ghoshalwayshasfaithonmeandencouragesmetokeeptrying.HealsospentquitealotoftimeonhelpingmecorrectmyEnglishwriting.Ihavelearnedmanyusefultechniquesonwritingtechnicalarticles.Icouldnothavedonethiswithouthiskindandgeneroushelp.Iwillforeverbethankfultomycommitteemembers,ProfessorsHaniDoss,KashitijKhareandArunavaBanerjee.Dr.Dossmadelargeeffortstomakemyscheduleworking.HisMCMCclassesdiscussmanyBayesiannonparametricsurvivalmodels,whichenlightensmanynewideasonmyresearch.Dr.Khareprovidesmanyinsightfulcommentsinmyproposal,whichhelpsmebetterdevelopthismethodology.Dr.Banerjeemadeagreatsuggestiononincludingasimplebuteffectiveexampletodemonstratethesuperiorityofourmethods.HealsomadealotofusefulcommentsinourweeklyseminarofDirichletprocess,makingitmucheasierformetogetintotheBayesiannonparametriceld.IwouldalsoliketothankmycolleaguesandfriendsintheStatisticsDepartment.Theyarealwayssupportiveandhelpful.Ienjoyeddiscussingproblemswiththem.Atlast,butnottheleast,Iwanttothankmyparentsandmybelovedwifefortheirunconditionalloveandsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 1.1Prologue .................................... 10 1.2HypothesisTestingUnderLehmannAlternatives .............. 10 1.3CoxProportionalHazardsModel ....................... 13 1.4Reliabiity .................................... 17 1.5PolyaTreeProcess ............................... 18 2HYPOTHESISTESTINGUNDERLEHMANNALTERNATIVES ......... 22 2.1Two-samplecase ................................ 22 2.1.1Two-SampleTestsWithFixed=0 ................. 22 2.1.1.1DerivationoftheBayesFactor ............... 22 2.1.1.2PropertiesOfTheBayesFactor .............. 24 2.1.2RealDataAnalysis ........................... 29 2.1.3RobustnessStudyThroughSimulations ............... 30 2.2One-sampleCase ............................... 32 2.2.1DerivationofBayesFactor ....................... 32 2.2.2AsymptoticResults ........................... 33 2.2.3SimulationStudies ........................... 36 3COXPROPORTIONALHAZARDSMODELUNDERPOLYATREEPROCESSPRIOR ........................................ 38 3.1LikelihoodFunction ............................... 38 3.1.1MarginalLikelihoodFunctionfor .................. 39 3.1.2EffectsofSpacings ........................... 42 3.2FullyBayesianAnalysis ............................ 43 3.2.0.1SimulationStudyOnConsistencyOfPosteriorMean ... 44 3.3RealDataAnalysis ............................... 44 4RELIABILITY ..................................... 47 5

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5CONCLUSIONANDDISCUSSION ......................... 50 APPENDIX APROOFOFTHEOREM2.1 ............................. 52 BPROOFOFTHEOREM3.1 ............................. 60 CPROOFOFTHEOREM4.1 ............................. 68 DPROOFOFCOROLLARY1 ............................. 72 REFERENCES ....................................... 73 BIOGRAPHICALSKETCH ................................ 77 6

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LISTOFTABLES Table page 1-1PartialLikelihoodFunctionEstimates ....................... 15 2-1OvarianCancer .................................... 29 2-2Coxproportionalhazardsmodel .......................... 30 7

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LISTOFFIGURES Figure page 1-1Twoexampleswithsamerankorderstatistics .................. 15 2-1OvarianCancer:Log(BF)fordifferent0 ..................... 30 2-2Log(BF)assamplesizeincreasesunderthealternativehypothesis ...... 31 2-3Log(BF)assamplesizeincreasesunderthenullhypothesis .......... 32 2-4Log(BF)growsdrasticallyassamplesizeincreasesunderthenullhypothesis 36 2-5Log(BF)decreasesdrasticallyassamplesizeincreasesunderthealternativehypothesis ...................................... 37 3-1EffectsofspacingsonMLE ............................. 43 3-2KernelDensityforPosteriorsof ......................... 44 3-3PosteriorMeansof'sasSampleSizeIncreases ................ 45 3-4KernelDensityforPosteriorsof's ........................ 46 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySURVIVALANDRELIABILITYANALYSISUNDERPOLYATREEPROCESSESPRIORSByLeiHuangAugust2013Chair:MalayGhoshMajor:StatisticsThethesisconsistsofthreecomponents,whicharerelatedtosurvivalandlifetimedataanalysis.WeproposeaBayesiannonparametricapproachtotestingtreatmenteffectsandestimatingregressioncoefcientsintheCoxproportionalhazardsmodelbyconsideringnotonlytherankstatistics,butalsothespacingsbetweenrankstatistics.TheBayesiansolutionofourapproachhasaclosedformevenwithcensoreddata,whichcanbecalculatedextremelyfastandhasareadyinterpretation.Simulationstudiesarecarriedoutandthemethodisappliedtoanalyzearealdataset.Finally,weproposeanonparametricestimatorforreliability,whichdegeneratestothewell-knownMannWhitneyUstatisticinspecialcases. 9

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CHAPTER1INTRODUCTION 1.1PrologueThisdissertationdealswiththreetopicsinBayesiannonparametricswithPolyatreepriors.First,wehaveconsideredPolyatreepriorsforhypothesistestingunderLehmmanalternatives.Second,wehaveusedPolyatreepriorsforthebaselinesurvivalfunctioninCoxproportionalhazardsmodelandhavecarriedoutsemi-parametricBayesianinferencefortheregressioncoefcients.Third,weusedPolyatreepriorsforestimationinstress-strengthmodel.Thetopicsaredescribedinmoredetailsbelow. 1.2HypothesisTestingUnderLehmannAlternativesLehmannalternatives,asthenamesuggests,wereintroducedby Lehmann ( 1953 )inthetwo-samplehypothesistestingcontext.Specically,oneconsiderstwoindependentsetsofrandomsamples,X1,,Xn1eachdistributedasF(x)andY1,,Yn2eachdistributedasH(x).OnetestsH0:F=HagainstthealternativesH1:H(x)=1)-275(f1)]TJ /F7 11.955 Tf 12.6 0 Td[(F(x)g,where>0and6=1.Lehmannconsideredthesetestsnotjustfortheirmathematicalsimplicity,butalsoforstraightforwardinterpretationofalternatives.Forinstance,whenisaninteger,H(x)isthedistributionfunctionoftheminimumofindependentrandomvariableseachhavingdistributionfunctionF.Moreover,thealternatives,ingeneral,introduceaverynaturalstochasticorderingofFandH.Inparticular,Hisstochasticallylarger(smaller)thanFwhen0<<1(>1).MuchofthecurrentliteratureonLehmannalternativesisrestrictedtorankordertests.Inparticular,thesealternativesareusedtocomparetheperformanceoflocallymostpowerfulranktestsagainstuniformlymostpowertfulranktestsforspecicalternatives.Thiswasdonein Lehmann ( 1953 )and Savage ( 1956 ).Sinceallthetestswetalkaboutisbasedonorderstatistics,withoutlossofgenerality,weassumethatX1,,Xn1andY1,,Yn2arebothordered.Inaddition,theranksofY'sinthecombinedsamplearedenotedbyE1,,En2.Thecompletesetofrankis,ofcourse, 10

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determinedbytheranksofY'salone. Lehmann ( 1953 )derived P(E1=e1,,En2=en2)=n2 0B@n1+n2n11CAn2Yj=1\(ej+j)]TJ /F7 11.955 Tf 11.95 0 Td[(j) \(ej)\(ej+1) \(ej+1+j)]TJ /F7 11.955 Tf 11.95 0 Td[(j)(1)UsingEquation( 1 ),onecancomputethepowerofvariousranktestsagainstthealternatives. Savage ( 1956 )dealtwithitdifferently.HeconsideredHL:F(x)=F0(x)1andH(x)=F0(x)2where2>1>0andF0(x)isacontinuouscumulativedistributionfunction.IfweputthesetwosamplestogetheranddenoteitbyV1,,Vn1+n2.ThusVk,k=1,,n1+n2areasampleF(x)ofsizen1+n2underthenullhypothesisH0:1=2andamixtureoftwosamplescomingfromdifferentdistributionsunderthealternativeHL:16=2.Withoutlossofgenerality,weassumethatVk'sareordered.Intheabsenceofties,V1<>:0,ifVk2X=fX1,,Xn1g1,ifVk2Y=fY1,,Yn2g(1)Then Savage ( 1956 )provedthatunderHL,theprobabilityofarankorderz1,,zn1+n2isgivenbyn1!n2!n11n22 Qn1+n2i=1(Pij=1[(1)]TJ /F7 11.955 Tf 11.96 0 Td[(zj)1+zj2]).Subsequently, Davies ( 1971 )showedasymptoticequivalenceoftheapproachesof Lehmann ( 1953 )and Savage ( 1956 ). Brooks ( 1974 )addressedtheproblemfromaBayesianperspective.Thepowerwasregardedasanunknownparameterinhismodel.Hebeganwiththejointdistributionofrankorderstatisticsasderivedby Savage ( 1956 ),andthenassignedanFdistributionpriortotocompletetheanalysis. 11

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Oneofthekeyfeaturesofallthesearticlesisthatthetestsarebasedonlyontherankswhichremaininvariantunderone-to-onetransformation.However,thisconsidersonlythepositions,butnotthemagnitudesofdifferencesoforderstatistics.Forexample,twoconsecutiveorderstatisticswithhighlydifferentmagnitudesdifferonlyby1intheirranks.Oneofthehighlightsofthepresentapproachisthatitisnotjusttheranks,butalsothespacingsoforderstatisticsthataretakenintoaccountinthetestingproblem.WealsotreatFasfullynonparametricandassignpriorstoF.Bayesiannonparametricmethodshavereceivedextensiveattentionrecentlybecauseoftheirexibility.Dirichletprocesspriors,introducedby Ferguson ( 1973 ),arethemostcommonlyusednonparametricpriors.Overtheyears,alargebodyoftheoryhasdevelopedforsuchpriors.However,theirscopeissomewhatlimitedduetothefactthatDirichletprocessselectswithprobability1onlydiscreteprobabilitymeasures.Forexample,inlifetestingandsurvivalanalysis,peopleusuallydealwithcontinuousrandomvariablesratherthandiscreteones.Therefore,itmaynotbeallthatreasonabletoemployDirichletprocesspriors,forexample,inthesurvivalcontext.Foryears,peopleareseekingotherapproachestoaccommodatecontinuousproblems.Forinstance, Kalbeisch ( 1978 )denesafamilyofrandomprobabilitiescalledtheGammaprocess. Hjort ( 1990 )discussedBetaprocessesinthecontextofsurvivalanalysis.Betaprocesspriorswerealsousedby DamienandWalker ( 2002 )fortestingtheeffectsoftwotreatments.ThemixtureofDirichletprocesspriors,introducedby Antoniak ( 1974 ),offersareasonablecompromisebetweenpurelyparametricandpurelynonparametricmethodsandareusedextensively.Additionally,theDirichletprocessmixturemodelintroducedby EscobarandWest ( 1995 )receivedgreatsuccessinBayesiannonparametrics.Incontrast,weareinterestedinanothernonparametricprior,namelyPolyatreepriors,originallyintroducedby Ferguson ( 1974 )asaspecialcaseofpriorswiththetailfreeproperty.ThesepriorsextendDirichletprocesspriorsandcanselectcontinuousdistributionsaswellwithpositiveprobability,andifnecessary, 12

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evenwithprobability1.Themainreasonforconsideringthesepriorsistheirgreatexibilitytoaccommodatediscrete,continuousormixeddistributionsbyappropriatechoiceofparameters. Lavine ( 1992 1994 )investigatedthebasicbutveryimportantpropertiesofPolyatreepriors.Sufcientconditionsforthesepriorstoassignprobability1tothesetofcontinuousdistributionsarediscussedin Mauldinetal. ( 1992 )(MSW)and Lavine ( 1992 ). ChenandHanson ( 2012 )proposeatestforH0:F=Hvs.H1F6=HthatusesaPolyatreepriorcenteringatadistributionthatestimatedfromaparametrict.SuchapproachesrequireMCMCmethodstocarryoutthecomputation.Comparedtotwo-sampleLehmannalternatives,theone-samplecaseislessinvestigated.Inone-sampletest,wewanttotestH0:X1,...,XnindependentlyidenticallydistributedasFversusH1:X1,...,XnindependentlyidenticallydistributedasF. MiuraandTsukahara ( 1993 )discussedtheestimationprobleminone-samplegeneralizedLehmannalternativemodel.HoweverthereiscurrentlynoestablishedmethodfortestingonesampleproblemsunderLehmannalternatives,whichwehaveaddressedinthisdissertation. 1.3CoxProportionalHazardsModel Cox ( 1972 )introducedtheproportionalhazardsmodelbyspecifyingthehazardrateattimetforanindividualwithcovariatevectorx.Thehazardrateh(tjx)isgivenby h(tjx)=h0(t)exp(xT),(1)whereh0(t)isthebaselinehazardfunction.Thismodelimpliesthattheratioofthehazardsfortwoindividualsisconstantovertimeprovidedthatthecovariatesstaythesameovertime.Theproportionalhazardsmodelhasbeenusedextensivelyforitsmathematicalsimplicityandeasyinterpretation.Clearly,LehmannalternativesarealsoveryusefulinsurvivalanalysissincethesurvivalfunctionisS(x)=1)]TJ /F7 11.955 Tf 12.23 0 Td[(F(x)underthedistributionfunctionF.Itfollowsthatthe 13

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survivalfunctioninCoxmodelisgivenbyS(tjx)=[S0(t)]exp(xT)Thenullhypothesiscorrespondstoazeroregressionvector.Anotherimportantapplicationwaspointedoutby Davies ( 1971 ).IftheXiandYjarelifetimesoftwosetsofsimilararticles,thehypothesisassertsthatthefailurerateofoneisaconstantmultipleofthefailurerateoftheother.Thus,atestofhypothesisunderLehmannalternativecouldbepotentiallyinterestingasitcanberegardedasagoodness-of-ttestforCoxmodel.AnotherinterestingtopicrelatedtoCoxmodelisestimationofregressioncoefcients.Supposeweobservecensoreddata(tk,k,xk),(k=1,,n),wheret1
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Toillustratetheeffectsofspacingsoforderstatistics,herewepresentasimpletoyexample.InbothscenariosinFigure 1-1 ,theblackdashedlinesdenotethesamecontrolgroup(xi=0),whiletheredlinesaresurvivalcurvesofthetreatmentgroup(xi=1). A BFigure1-1. Twoexampleswithsamerankorderstatistics Therankorderstatisticsinbothscenariosareexactlythesamebutitisnotthecaseforspacings.Therefore,weexpectthattheestimatesoftreatmenteffectsbasedonpartiallikelihoodfunctionarethesame.Table 1-1 liststheresults.Thisisnotquitereasonablebecauseapparentlyspacingsplayanveryimportantroleinbothcases.Inscenario 1-1A ,itisobviousthatthetreatmentgrouphasamuchlargermeansurvivaltimethanthatofthecontrolgroup.Hence,ideallywewouldliketotakethatspacingeffectintoaccountandendupwithamoresignicanttreatmentgroup.Itistheoppositeforscenario 1-1B .Therefore,weareproposingamethodtotakebothranksandspacingsoforderstatisticsintoaccountthroughBayesiannonparametricmethods. Table1-1. PartialLikelihoodFunctionEstimatescoefexp(coef)se(coef)P-valuelower.95upper.95 Treatment-1.10510.0.33120.70820.1190.082651.327 Overtheyears,alargebodyoftheoryhasdevelopedforDirichletpriors.Forexample, SusarlaandVanRyzin ( 1976 )derivedtheBayesestimatorofthesurvivalfunctionundertheDirichletprocesspriorand FergusonandPhadia ( 1979 )derived 15

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theposteriordistributionofthecumulativedistributionfunctionwithrightcensoreddata.However,theirscopeissomewhatlimitedduetothefactthatDirichletprocessselectswithprobability1onlydiscreteprobabilitymeasures.Forexample,insurvivalanalysis,peopleusuallydealwithcontinuousrandomvariablesratherthandiscreteones.Therefore,itmaynotbeallthatreasonabletoemployDirichletprocesspriors.Morerecently,peopletendtouseamixtureofnonparametricprocesspriorsinsteadofasingleoneduetodevelopmentsofcomputationpowerandMCMCmethods. Doss ( 1994 )and DossandHuffer ( 1998 )discusstheimplementationofmixtureofDirichletpriorsforF(t)=1)]TJ /F7 11.955 Tf 12.3 0 Td[(S(t)inthepresenceofrightcensoreddatausingGibbssampler.Ontheotherhand,mixturesofPolyatreepriorsarerelativelynew.Inparticular, HansonandJohnson ( 2002 )hasusedamixtureofPolyatreepriorsinCoxproportionalhazardsmodel. Hanson ( 2006 )and HansonandJara ( 2012 )appliedamixtureofnitePolyatreepriorsonavarietyofimportantsurvivalmodelsandcomparedtheresults.ComparedtoaparticularBayesiannonparametricprocessprior,themixturecontainsawiderclassofdistributionsandthemixtureparametersmoothensthepriortosomeextent,butitalsobringsincertaincomplexity.ApproximatecalculationsareusuallyemployedformixturesofPolyatreepriorsinthesensethatthePolyatreeistruncatedatanitenumberofsteps,duetotheextremecomplexityofthestructure.Ourinterestliesinthecasewhereanon-truncatedPolyatreeprocessisused.ThereasonwechoosePolyatreepriorsinsteadofmixturesofPolyatreesisthatweendupwithasimpleandexplicitexpressionofthemarginallikelihoodfunction,andthelikelihoodisexactinsteadofanapproximationasonegetsunderanitePolyatree. MuliereandWalker ( 1997 )discussedhowPolyatreepriorsmightbeusedinsurvivalanalysiswithoutcovariates.TheiroriginalideaofmakingthePolyatreepartitiondatadependentgreatlysimpliestheproblem.Wefurtherpushforwardtheirideatothecasewithcovariates.MarginallikelihoodsforregressioncoefcientsintheCoxproportionalhazardsmodelwithaBayesiannonparametricprioronthebaselinesurvivalfunctionhave 16

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beenstudiedforyears. KalbeischandPrentice ( 1973 )provedthat,withoutties,whenweuseaverydiffuseGammaprocessprioronthebaselinecumulativehazard,themarginallikelihoodisapproximatedbythepartiallikelihood.Thisresultisrestrictedtothecaseofcontinuousunivariatesurvivaldataandxedtime-constantcovariates. Sinhaetal. ( 2003 )generalizedtheseresultsbyestablishingBayesianjusticationofpartiallikelihoodsaslimitingcasesofmarginaldistributionsoftheregressionparametersunderthegroupeddatalikelihood,continuous-datalikelihoodwithtime-varyingcovariatesandregressioncoefcients.Weobtainsimilarresults.WederiveaclosedformforthemarginallikelihoodfunctionoftheregressioncoefcientsintheCoxproportionalhazardsmodelassumingthatthebaselinehazardfunctionisfromaPolyatreeprocess.Also,themarginallikelihoodresemblesthepartiallikelihoodfunctionwithsomenaturalheuristicinterpretationinalimitingcase. 1.4ReliabiityLastbutnottheleast,reliabilityanalysisalsoreceivesextensiveattentionforitsbroadapplicabilityinreal-worldproblem.Specically,weareinterestedinestimatingtheprobabilityP(X
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( 1945 )and MannandWhitney ( 1947 ).Laterresearchersmakevariousparametricassumptionsonthedistributions.WewanttoprovideasolutiontothisproblembyemployingPolyatreepriors.Reliabilityisanoldbutchallengingproblem.Itcapturedmanyresearchers'attentionbecauseitoriginatesfromreal-worldpractice. WolfeandHogg ( 1971 )arguedthatnumericalvaluesofP(X
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mixturesofnormalsmarginally.Theyconsideredbothdependentandindependentscenariosundernoninformativebenchmarkpriors. 1.5PolyaTreeProcessPolyatreeprocessisalargeclassofpriorsthatincludestheDirichletprocessasaspecialcase.ItprovidesaexiblewayforBayesiananalysisofnonparametricproblems.Thetreeisconstructedbysuccessivepartitioningofthesamplespace.UnliketheDirichletprocess,thepartitionplaysadeterministicroleinPolyatrees.Alsoalargecollectionofparametersmakesitpossibletoincorporateawiderangeofbeliefs.LetE=f0,1g,E0=;.LetEmbethem-foldproductEEEEandE=S10Em.Deneaseparatingbinarytreeofpartitionof,=fm,m=0,1,2,...g,suchthat0=.Also,0,1,formasequenceofpartitionssuchthatS10mgeneratesthemeasurablesetsandeveryB2m+1isobtainedbysplittingsomeB2mintotwosets.Degeneratesplitsarepermitted,i.e.someB2mcanbesplitintoBS;. Denition1. Foreachm,m=fB~m:~m=1,...,m2Emgisapartitionofsuchthatforall~m2E,B~m,0,B~m,1isapartitionofB~m.LetA=fa~m:~m2Egbeasetofnonnegativerealnumbersandy=fY~m:~m2Egbeacollectionofrandomvariables.Following Lavine ( 1992 ),wesayarandomprobabilitymeasurePonhaveaPolyatreedistributionwithparameter(,A),writtenPPT(,A),ifthefollowingconditionshold: 1. alltherandomvariablesinywithsubscriptsendingwith0areindependent,i.e.Y~m,0,forall~m2E,areindependent;Y~m,1=1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y~m,0 2. forevery~m2E,Y~m,0hasaBetadistributionwithparametersa~m,0anda~m,1; 3. foreverym=1,2,...andevery~m2E, P(B1,...,m)=(mYj=1;j=0Y1,...,j)mYj=1;j=1(1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y1,...,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0)=mYj=1Y1,...,j(1)WearegivinganalternativeformofP(B1,...,m)thanwhatisgivenin Lavine ( 1992 )byre-arrangingY~manddeningY~m,1=1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y~m,0.Bydoingthis,wecankeepgoodtrack 19

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onY~mintermsofacompactexpressionforP(B1,...,m)asin( 1 ).TheonlythingweneedtokeepinmindisthatY~m,0andY~m,1arenotindependent.ThereareseveralwellestablishedpropertiesofPolyatreeprocessthatweneedtouse.Theyarelistedbelow.FormorepropertiesofPolyatrees,thereadermayreferto Lavine ( 1992 ),and GhoshandRamamoorthi ( 2003 ). 1. Polyatreesareconjugate.IfPhasaPolyatreedistribution,andXjPP,thenPjXhasaPolyatreedistribution.Theposteriordistributionisupdatedinthefollowingmanner:forevery~msuchthatX2B~m,add1toa~m.Sometimeswhencensoreddatumisobserved,thatisweonlyobservethatXinsomesetI,thenforevery~msuchthatB~mI,add1toa~m.Generallyforcensoreddata,weonlyneedtoupdatenitelymanysteps. 2. SomePolyatreesassignprobability1tothesetofcontinuousdistributions.AbroadlyusedsufcientconditionforthiscanbefoundinTheorem3.3.7in GhoshandRamamoorthi ( 2003 ).Forexample,takea~m=c.m2. 3. IfwehaveaPolyatreewithpartitionsfB~m:~m2EgandparametersA,thepredictivedensityatx2B~misf(x)=limm!+1Pr(B~m) (B~m)=limm!+1Qmi=1a1,...,j a1,...,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,...,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1 (B~m) (1)where()istheLebesguemeasure. 4. APolyatreecanbeconstructedwithcenteringatanarbitrarydistribution.ThisisimportantbecausewhenweusePolyatreepriorsweusuallyhaveaguessoftheunderlyingdistribution.Thusbyproperconstruction,wecanmaketheexpectationofthetreecoincidewithourguess.Thereareactuallytwoclassicalwaystodothis.Suppose=R,saywewantaPolyatreetocenteratapre-specieddistributionfunctionG,thebaselinemeasureofthePolyatree.Thenwecanletthepartitionbesuchthattheelementsofmaretakenastheintervals[G)]TJ /F5 7.97 Tf 6.59 0 Td[(1(k=2m),G)]TJ /F5 7.97 Tf 6.59 0 Td[(1((k+1)=2m))fork=0,1,...,2m)]TJ /F4 11.955 Tf 11.01 0 Td[(1,withtheobviousinterpretationforG)]TJ /F5 7.97 Tf 6.58 0 Td[(1(0)andG)]TJ /F5 7.97 Tf 6.58 0 Td[(1(1).WewillrefertothisasMethod1.Theotherapproachistomakethepartitiondata-dependent,asmentionedin MuliereandWalker ( 1997 ).Thiswouldmaketheupdatingforcensoreddataextremelysimple.Supposewespecifyanumberofpointsx1<...
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B11=[x2,+1),...,B1,...,1| {z }n=[xn,+1).Thenweneedtheparametersa~mtosatisfya1,...,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0 a1,...,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1=G(B1,...,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0) G(B1,...,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1)anda~mgrowsquicklyenoughtoensurethecontinuityproperty.Hereandlater,anyotherunspeciedsubintervalsaregeneratedbysplittingtheirparentintervalsintotwoequalpartswithrespecttotheGmeasure.Forexample,B0=(0,x1),andB00=(0,xx)andB01=[xx,x1)areobtainedbyspecifyingaxx2(0,x1)suchthatG(B00)=G(B01)=G(B0)=2.ThiswereferasMethod2.Wewilltakea~m=c.m2toensurecontinuouspriors.Method2involvessomeextrabenets.First,itiseasytoseethattheexpectationofthisPolyatreeisalsoG.Second,asseenby( 1 ),itassignsprobability1tothesetofcontinuousprobabilitymeasures.OurpartitionandparametersdifferfromMethod1onlybynitelymanyterms.Thusifwecalculatethelimitin( 1 )foranyx2R+,thelimitexistsandisnite.WecarriedoutthecalculationsbasedonthepartitionsdescribedinMethod2.However,aswewillseeinlatersections,theresultsonlydependonnitelymanyparameters,a1,...,1| {z }kanda1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0fork=1,...n.ThecalculationswillgothroughaslongasPolyatreesselectcontinuousdistributionswithprobability1anda1,...,mgrowstoinnityasm!+1. 21

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CHAPTER2HYPOTHESISTESTINGUNDERLEHMANNALTERNATIVES 2.1Two-samplecase 2.1.1Two-SampleTestsWithFixed=0Werstconsiderthetwo-samplecasewithxed=0.AlthoughtherearesomeestablishedPolyatreesbasedonnonparametrichypothesistestsintwosamplecase,suchas ChenandHanson ( 2012 ), Holmesetal. ( 2009 )and MaandWong ( 2012 ),theyfocusprimarilyontestingthedifferenceofthedistributionsofthetwosamples.We,ontheotherhand,aregoingtolimitourattentiontotestingLehmannalternativesexclusively.Supposewehavetwosamplesofsizesn1andn2,drawnfromdistributionsFandHrespectively.Specically,wewilldiscussthecasewhenH(x)=1)-93(f1)]TJ /F7 11.955 Tf 10.41 0 Td[(F(x)g.LetX1,,Xn1andY1,,Yn2bethetwosamples.WeputthesetwosamplestogetheranddenoteitbyV1,,Vn1+n2.ThusVk,k=1,,n1+n2areasampleF(x)ofsizen1+n2underthenullhypothesisH0:=1andamixtureoftwosamplescomingfromdifferentdistributionsunderthealternativeH1:=0,where0issomexedknownrealnumbernotequalto1.Withoutlossofgenerality,weassumethatVk'sareordered.Whennotieoccurs,wehaveV1<>:0,ifVk2X=fX1,,Xn1g1,ifVk2Y=fY1,,Yn2g(2)Itmaybenotedthat(Vk,Zk)isequivalentto(Xk,Yk)becauseeachisuniquelydeterminedbytheother.Inaddition,weconsiderthecensoringcasebecausethelatterismostcommonlyencounteredinthesurvivalcontextandwewouldliketheresultstoapplytothosecasesaswell.Letd1,,dn1+n2bethecensoringindicatorscorrespondingtoV1,,Vn1+n2. 22

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2.1.1.1DerivationoftheBayesFactorSupposethataPolyatreepriorisappliedtoF(x).UndertheBayesianhypothesistestingparadigm,theBayesfactorisawidelyusedtool. KassandRaftery ( 1995 )reviewedandsummarizedtheusesoftheBayesfactorforsomescienticapplications.Bydenition,theBayesfactorisBF01=posteriorodds priorodds=P(H0jV1,...,Vn1+n2) P(H1jV1,...,Vn1+n2) (H0) (H1)Let(H0)=p0=1)]TJ /F3 11.955 Tf 11.96 0 Td[((H1),where0
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ofthetestisasfollowsBF01=1 Pn1+n2i=1diZi0\(a1+n1+n2)\(a0+a1+n1+n20) \(a0+a1+n1+n2)\(a1+n20)n1+n2)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yi=1[\(a1,...,1| {z }i,1+n1+n2)]TJ /F7 11.955 Tf 11.95 0 Td[(i) \(a1,...,1| {z }i,1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)ti+1)\(a1,...,1| {z }i,0+a1,...,1| {z }i,1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1+di) \(a1,...,1| {z }i,0+a1,...,1| {z }i,1+n1+n2)]TJ /F7 11.955 Tf 11.95 0 Td[(i+di)] (2)whereti=Pn1+n2j=iZj.Theproofofthetheoremistechnical,andisprovidedinappendix.NotethatBF01dependson0,wewillwriteitasafunctionof0,namely,BF01(0)IntuitivelytishowshowmanyobservationsonthetailofthemixedsamplearecomingfromsampleYattimeVi.ThusifF(x)>H(x),wewouldexpectlargerti'sbecausesampleYisstochasticallylargerthansampleX.Notethatknowingtk,k=1,,n1+n2impliesfullknowledgeofZk,k=1,,n1+n2sincethereisaone-to-onecorrespondencebetweentk'sandZk's.ItmaybenotedthattheexpressionoftheBayesfactordoesnotdependonm.Second,theBayesfactorinvolvesonlyanitenumberofparameters,a1,...,1| {z }kanda1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0,fork=1,...,n1+n2.Were-parameterize 8>>>>>><>>>>>>:k=a1,...,1| {z }k+a1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0rk=G([vk,+1)) G([vk)]TJ /F9 5.978 Tf 5.76 0 Td[(1,+1))=a1,...,1| {z }k a1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,...,1| {z }k(2)wherewehaveusedtheassumptionthata1,...,1| {z }k/G([vk,+1))anda1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0/G([vk)]TJ /F5 7.97 Tf 6.59 0 Td[(1,vk)). 24

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ThustheBayesfactorsimpliestoBF01(0)=1 Pn1+n2i=1diZi0\(1r1+n1+n2)\(1+n1+n20) \(1+n1+n2)\(1r1+n1+n20)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yi=1\(i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i)\(i+1+n1+n2)]TJ /F7 11.955 Tf 11.95 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1+di) \(i+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+di)\(i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)ti+1) 2.1.1.2PropertiesOfTheBayesFactorMonotonicityoftheBayesFactorNotethatthedataappearintheexpressionoftheBayesfactorthroughri'sandti's.Basically,ti'scorrespondtotheeffectsofrankorderstatisticsandri'sexplaintheeffectofthespacingsoforderstatistics.Fixti,thenBF01(0)isafunctionofri.Fori=1,..,n)]TJ /F4 11.955 Tf 11.96 0 Td[(1,letqi+1(ri+1)=log(\(i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i)))]TJ /F7 11.955 Tf 11.95 0 Td[(log(\(i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.95 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1))Thederivativeisd dri+1qi+1(ri+1)= (i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i))]TJ /F3 11.955 Tf 11.95 0 Td[( (i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.95 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1))where ()isDigammafunction.Hence,qi+1(ri+1)isdecreasingif0>1andincreasingif0<0,because ()isstrictlyincreasingin(0,+1).Thesameresultholdsforthersttermlog(\(1r1+n1+n2)))]TJ /F7 11.955 Tf 12.19 0 Td[(log(\(1r1+n1+n20)).Thismakessensebecauselargeri'simplythatdataarehighlyclustered,whichismoreplausibletohappenunderthealternativehypothesisasH(x)=1)-300(f1)]TJ /F7 11.955 Tf 12.89 0 Td[(F(x)g0ismorelikelytoproduceaclusteredsamplewhen0>1thanF(x)does.Thefactthatincreasingri'slowerBF01correctlygivesanedgetothealternativehypothesis.Oppositeisthecaseforsmallerri's.Theanalogousstatementisalsotruewhen0<1.SimilarcalculationsshowthemonotonicityoftheBayesfactoronti'swhenri'sarexed.ItturnsoutthatBF01(0)isincreasingif0>1anddecreasingif0<1.Thisresultisreasonablesincewhen0>1,F(x)
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likelytoobservelargeti's(moreYsamplesintherighttail)underthealternativehypothesisthanunderthenullhypothesis.Therefore,largeti'sleadtolargeBF01,whichfavorsthenullhypothesis.Analogously,when0<1,F(x)>H(x),i.e.,sampleXisstochasticallysmallerthansampleY.Largeti'saremorelikelytooccurunderthealternativehypothesis.ThemonotonedecreasingpropertyofBF01inti'scorrectlysupportsthisconclusion.EffectsofSpacingsTheproposedtestwithPolyatreepriorsnotonlyconsiderstherankorderstatistics,butalsotakesintoaccountthespacingsoforderstatistics.Thisclearlyenhancestherobustnessofthetest.Bydoingthis,theeffectoftheorderingisreducediftwoobservationsarefairlyclose.Thisisdesirablesincethedifferenceoftwoconsecutiveobservationshasasignicantimpactwhentheyarefarapart.Ontheotherhand,iftwoobservationsarefairlyclose,thentheorderingmightnotbethatcrucial.Thusarobusttestshouldsatisfythecriterionthatreversingtheorderoftwoadjacentobservationsinthecombinedsampleneednotmakelargedifferencewhenthetwoobservationsfromthetwogroupsareveryclose.Infact,supposeforsomek02f2,,n1+n2g,Vk0)]TJ /F5 7.97 Tf 6.59 0 Td[(1andVk0areveryclose(naturallydk0)]TJ /F5 7.97 Tf 6.59 0 Td[(1=dk0=1).IfVk0)]TJ /F5 7.97 Tf 6.58 0 Td[(1andVk0arebothcomingfromsampleXorsampleY,exchangingthepositionsofthemwouldnotaffecttheresultingBayesfactoratall.Inthatcase,rk0=G([vk0,+1)) G([vk0)]TJ /F5 7.97 Tf 6.59 0 Td[(1,+1))1.andhence\(k0rk0+n1+n2+1)]TJ /F7 11.955 Tf 11.96 0 Td[(k0)\(k0+n1+n2+1)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)tk0+dk0)]TJ /F5 7.97 Tf 6.58 0 Td[(1) \(k0+n1+n2+1)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+dk0)]TJ /F5 7.97 Tf 6.59 0 Td[(1)\(k0rk0+n1+n2+1)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)tk0) (2)k0+n1+n2+1)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)tk0 k0+n1+n2+1)]TJ /F7 11.955 Tf 11.95 0 Td[(k0 (2) 26

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Inaddition,itisclearthatreversingtheorderofVk0)]TJ /F5 7.97 Tf 6.59 0 Td[(1andVk0willleaveallti'sintactexceptfortk0.Hence,iftheorderischanged,thenewBayesfactordiffersfromtheoriginaloneonlybyoneterm.Itisstraightforwardthattnewk0=toldk01.Theplus(minus)signcorrespondstothecasewhereVk0)]TJ /F5 7.97 Tf 6.59 0 Td[(12Y(X)andVk02X(Y).TocomputethenewBayesfactor,( 2 )isreplacedbyk0+n1+n2+1)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)tnewk0 k0+n1+n2+1)]TJ /F7 11.955 Tf 11.95 0 Td[(k0Therefore,BF01(0)new=k0+n1+n2+1)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)tnewk0 k0+n1+n2+1)]TJ /F7 11.955 Tf 11.95 0 Td[(k0+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)toldk0BF01(0)old=(10)]TJ /F4 11.955 Tf 11.96 0 Td[(1 k0+n1+n2+1)]TJ /F7 11.955 Tf 11.96 0 Td[(k0+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)toldk0)BF01(0)oldWecallj0)]TJ /F5 7.97 Tf 6.58 0 Td[(1 k0+n1+n2+1)]TJ /F8 7.97 Tf 6.59 0 Td[(k0+(0)]TJ /F5 7.97 Tf 6.59 0 Td[(1)toldk0jtheexpansionrate.TheexibilityofchoosingparametersenablesonetotestH0vsH1accordingtoone'sneeds.Forinstance,let1==n1+n2,theexpansionrategoesupask0increases,whichimpliesthattheorderstatisticsplayamoreimportantroleinthetailthanatthebeginning.Ontheotherhand,ifweletk0growfastwithk0,theexpansionratecouldbeadecreasingfunctionofk0.Thischoiceofparametersrendersatestmoresensitiveinthebeginning.Lastbutnotleast,theexpansionrategrowswhen0increasesif0>1or0!0+if0<1.Thismeansthattheeffectoforderisenlargedwhenthedistanceofthealternativefromthenullisincreased.Generalselectionofk0=k20ensuresthattheexpansionrateissmallwhenk0islarge;Whenk0issmall,withreasonablesamplesizes,n1+n2)]TJ /F7 11.955 Tf 12.31 0 Td[(k0isreasonablylarge,andtheexpansionratedoesnotexpandorshrinktheoriginalBayesfactoroverwhelmingly.AsymptoticResultsAnotherinterestingpointistheasymptoticbehavioroftheBayesfactorwhenthepowerofthealternativehypothesis,0approachesitslimit.Weillustratethecasewhen0!1(0>1).Thepowerof0intheexpressionofBF01(0)isthenegativeof 27

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Pn1+n2i=1diZi,whichisthenumberofuncensoredobservationsinsampleY.WheneverthereexistsoneuncensoredobservationinsampleY,thereisacorrespondingdi6=0forsomei21,,n1+n2.Inotherwords,foranyisuchthatdiZi6=0,wehavedi6=0,di=1.ThusonetermintheexpressionofBayesfactorassociatedwiththisparticulardiisgivenby\(i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i)\(i+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1+di) \(i+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+di)\(i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)ti+1)=const.\(i+1+n1+n2)]TJ /F7 11.955 Tf 11.95 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1+di) \(i+1ri+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)ti+1)>const.\(i+1+n1+n2)]TJ /F7 11.955 Tf 11.95 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1+di) \(i+1+n1+n2)]TJ /F7 11.955 Tf 11.95 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1)=const.fi+1+n1+n2)]TJ /F7 11.955 Tf 11.96 0 Td[(i+(0)]TJ /F4 11.955 Tf 11.95 0 Td[(1)ti+1g (2)whichisO(0)forti+16=0.AllthesetermsoffsetthepowerofnegativePn1+n2i=1diZi.Thus,itisstraightforwardthatifthereexistsatleastoneuncencoredobservationsinsampleXandtheindicatorfunctionin( 2 )isnonzero,BF01!1.Thatsaid,aparticularlyinterestingcaseiswhenalluncensoredobservationsinsampleYareclusteredcloselyandtheyareuniformlysmallerthananyobservationsinsampleX.Inthiscase,ti'sattaintheirminimums,denotedbytmini's,where 8><>:t1=n2,t2=n2)]TJ /F4 11.955 Tf 11.96 0 Td[(1,,tn2=1ti=0,fori=n2+1,,n1+n2(2)Also,ri1fori=2,,n2.OnemayshowthatinthiscaseBF01!0,whichiswhatweexpectedsinceas0!1,thealternativedistributionHessentiallydegeneratesattheinmumofthesupportofF.ImpactofPriorParametersLast,butnottheleast,itisofsomeinteresttodiscusstheparametersweuseinthePolyatreepriors.TechnicallyspecifyingaPolyatreepriorrequiresspecicationof 28

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innitelymanyparameters,a~m,orkandrk,fork=1,2,.ItiswellestablishedthattheposteriordistributionofaPolyatreeisupdatedbyreplacinga~mbya~m+1wheneveranobservationsatisfyingx2B~missampled.Thusifoneletsk=a1,...,1| {z }k+a1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0=k2,oneobservationleadstoamoredifferentposteriorwhena~missmall.Thatis,anxhaslargeimpactwhena~missmall.Ourpartitionisessentiallybasedonorderstatistics,puttinglargerobservationsinahigherlevelofthePolyatree.Hence,farthertheobservationisfromtheorigin,higherthelevelthatitbelongsto.Thisleadstobiggerk,andaccordinglyitcausessmallerimpacttothePolyatree.Therefore,thesetestsareinsensitiveinthetails.Onecanalterthesituationbyassigninga~maccordingly.Forinstance,onemayuseacommonvalueforallcorrespondingk's,whichleadstoatestequallysensitiveonR+.Furthermore,aslongasonerestrictsassignmentofk'stoanitenumber,itisnotgoingtoaffectthecontinuitypropertyofthePolyatreepriorsbecauseintrinsicallyoneonlymodiesanitenumberofparameters. 2.1.2RealDataAnalysisWetaketheovariancancerdatasetintheSurvivalpackageofRsoftwareasarealdataanalysisexample.Thedatasetwasoriginallyreportedby J.H.EdmunsonandKvols ( 1979 ),andwaslateronanalyzedinanumberofliteratures,suchas Collett ( 2003 ).Thestudyincludedn=26patientswithadvancedovariancarcinoma(stagesIIIBandIV).Treatmentofpatientsusingeithercyclophosphamidealone(1g/m2)orcyclophosphamide(500mg/m2)plusadriamycin(40mg/m2)byivinjectionevery3weekseachproducedpartialimprovementinapproximatelyonethirdofthepatients.Theobjectiveofthetrialwastoseeiftheproposedtwotreatmentdifferentiatesinprolongingthetimeofsurvival. Table2-1. OvarianCancerTreatmentSurvivalTime(Days) Treatment159,115,156,268,329,431,448+,477+,638,803+,855+,1040+,1106+Treatment2353,365,377+,421+,464,475,563,744+,769+,770+,1129+,1206+,1227+ 29

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Asanillustration,wewillusetheTreatment1groupasthebaselinegroup.AsimpleCoxproportionalhazardsmodelregressiongivesestimated^=0.55.Thisnumberiscalculatedjustforapowerthatisreasonabletotestagainstintheexample.Inreality,canbepre-speciedbyphysiciansorresearchersaccordingtomedicalinterests. Table2-2. Coxproportionalhazardsmodelcoefexp(coef)se(coef)P-valuelower.95upper.95 Treatment2-0.59640.55080.58700.310.17431.74 ThemaximumlikelihoodestimatesofunderlyingWeibulldistributionparameterare(0.947,980.4).Thelargescaleparameterisduetothefactthatonethirdsofallpatientsinthestudyshowedimprovement.WeassignaWeibull(0.947,980.4)distributionasthebaselineofPolyatreejustforillustrationpurpose.SupposewewanttotestH0:=1vsH1:=0=0.55.Bypluggingin,weendupwithlog(BF01(0.55))=)]TJ /F4 11.955 Tf 9.29 0 Td[(0.604.Basedonthecriteriadescribedin KassandRaftery ( 1995 ),itshowslittleevidenceagainstH0.Ifwerepeatthetestmanytimeswithdifferent0valuesrangingfrom0to10usingthesamedataset,weobtainacurveasshowninFigure 2-1 Figure2-1. OvarianCancer:Log(BF)fordifferent0 Thelowerreddottedlineisatlevel)]TJ /F4 11.955 Tf 9.29 0 Td[(3,whichshowstheregionforthestrongevidenceagainstH0withrespectto KassandRaftery ( 1995 )criteria.Hence,forthisparticulardataset,thereappearstobenoevidenceagainstH0atany. 30

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2.1.3RobustnessStudyThroughSimulationsSimulationstudiesarecarriedouttoinvestigatetherobustnessofmis-speciyingthecenterofthePolyatreepriorprocessofthetests.Therobustnessofmis-specieddistributionisanothervaluableadvantageofourapproach.Theresultsofmanyparametricmethodsdependheavilyonthedistributionassumptions.Weintendtoalleviatethisdependencebyassigningaprioronalargefamilyofdistributionsratherthansomeparticularone.SincePolyatreeprocesscoverssolargeasetofdistributions,itispossiblethatitmightcapturethetruedistributioneventhoughthecenter(akeyparameter)oftheprocessisincorrectlyspecied,thusyieldingrobustresults.Toillustratethis,wegenerateXsamplefromaWeibull(3,12)distributionandthisdistributionofsampleYischosenaccordinglytoH1. ABaselineMeasure:TrueandEst.WeibullDistributionwith0=1.5 BBaselineMeasure:TrueandEst.WeibullDistributionwith0=0.5Figure2-2. Log(BF)assamplesizeincreasesunderthealternativehypothesis Figure 2-2 showshowlog(BF01(0))decresesassamplesizen1andn2increasewhen0issettobe1.5and0.5respectively.Ineachgraph,threeBayesfactorsarecalculated,onewithtrueWeibulldistributionsasthecenterofPolyatreeprocessprior(asshownbyblackcircles),onewithGammadistributionwithestimatedparametersbasedonsampleXasthecenter(asshownbyredtriangles)andonewithNormaldistributionwithestimatedparameters(asshownbygreendiamonds).Twoconclusionscanbemadebasedonthesimulationresults.First,thetestisconsistentinthesense 31

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thattheBayesfactordecreasestoassamplesizegrows.Second,thetestisrobusttomis-speciedcentersofthePolyatreeprocesspriorssincethetwocurveswithGammaandNormaldistributionsdonotdepartawayfromtheonewithtruedistribution.Figure 2-3 demonstratesthepatternsoflog(BF01(0))whentestingH1with0=1.5and=0.5underthenullhypothesis.Anagously,thesimulationsshowstheconsistencyofthetestaswellastherobustnessagainstmis-speciedcenterdistributionsforPolyatreeprocesspriors. ABaselineMeasure:TrueandEst.WeibullDistributionwith0=1.5 BBaselineMeasure:TrueandEst.WeibullDistributionwith0=0.5Figure2-3. Log(BF)assamplesizeincreasesunderthenullhypothesis 2.2One-sampleCaseNowweconsidertheone-samplecase.SupposewewanttotestH0:X1,,XnF(x)independentlyagainstH1:X1,,XnH(x)whereH(x)isdenedasinSection3,withcensoringindicatorsd1,,dn. 2.2.1DerivationofBayesFactorThenexttheoremgivesanexactformoftheBayesfactorofthistest. Theorem2.2. SupposethepartitionandparametersofPolyatreeareasgiveninSection2,andGisastrictlyincreasingbaselinemeasure.ThentheBayesfactorofthe 32

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testisasfollowsBF01=1 Pni=1di\(a1+n)\(a0+a1+n) \(a0+a1+n)\(a1+n)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yi=1\(a1,...,1| {z }i,1+n)]TJ /F7 11.955 Tf 11.95 0 Td[(i)\(a1,...,1| {z }i,0+a1,...,1| {z }i,1+(n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)+di) \(a1,...,1| {z }i,0+a1,...,1| {z }i,1+n)]TJ /F7 11.955 Tf 11.96 0 Td[(i+di)\(a1,...,1| {z }i,1+(n)]TJ /F7 11.955 Tf 11.95 0 Td[(i)) (2)TheproofofTheorem2isessentiallysimilartothatofTheorem1.ThenumeratoroftheBayesfactorstaysthesame.Oneonlyneedstore-calculatethebottompartanalogously.Again,itmaybenotedthattheexpressionoftheBayesfactordoesnotdependonm.Secondly,theBayesfactorinvolvesonlyanitenumberofparameters,a1,...,1| {z }kanda1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0,fork=1,...,n.Thusbytheassumptionthata1,...,1| {z }k/G([xk,+1))anda1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0/G([xk)]TJ /F5 7.97 Tf 6.58 0 Td[(1,xk)),wemayre-parameterize 8>>>>>><>>>>>>:k=a1,...,1| {z }k+a1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0rk=G([xk,+1)) G([xk)]TJ /F9 5.978 Tf 5.76 0 Td[(1,+1))=a1,...,1| {z }k a1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,...,1| {z }k(2)ThuswecansimplifytheformulaforBayesfactortoBF01=1 Pni=1di\(1r1+n)\(1+n) \(1+n)\(1r1+n)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yi=1\(i+1ri+1+n)]TJ /F7 11.955 Tf 11.95 0 Td[(i)\(i+1+(n)]TJ /F7 11.955 Tf 11.95 0 Td[(i)+di) \(i+1+n)]TJ /F7 11.955 Tf 11.95 0 Td[(i+di)\(i+1ri+1+(n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)) 2.2.2AsymptoticResultsForsimplicity,wewilldiscusstheasymptoticpropertiesofBF01onlywithoutcensoring,i.e.,d1==dn=1.Fori=1,..,n)]TJ /F4 11.955 Tf 11.96 0 Td[(1,letqi+1(ri+1)=log(\(i+1ri+1+n)]TJ /F7 11.955 Tf 11.95 0 Td[(i)))]TJ /F7 11.955 Tf 11.96 0 Td[(log(\(i+1ri+1+(n)]TJ /F7 11.955 Tf 11.96 0 Td[(i))) 33

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Thederivativeisd dri+1qi+1(ri+1)= (\(i+1ri+1+n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)))]TJ /F3 11.955 Tf 11.95 0 Td[( (i+1ri+1+(n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)))where ()isDigammafunction.Hence,qi+1(ri+1)isdecreasingif>1andincreasingif<0,because ()isstrictlyincreasingin(0,+1).Thesameresultholdsforlog(\(1r1+n)))]TJ /F7 11.955 Tf 11.95 0 Td[(log(\(1r1+n)).Inaddition,ri2(0,1)fori=1,...,n,andricannotbe0or1sinceourdenitionofaPolyatreeensuresthatPiscontinuouswithprobability1.Thustheprobabilitythattherearetiesinthedatais0.Now,say>1.ThenthelowerboundandupperboundofBF01areachievedwhenri!1andri!0,respectively.SincetheGammafunctioniscontinuous,thelowerbound(LB)andupperbound(UB)forBF01aregivenbyLB=1 nn)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yi=1i+1+(n)]TJ /F7 11.955 Tf 11.95 0 Td[(i) i+1+n)]TJ /F7 11.955 Tf 11.96 0 Td[(i!0UB=1 n\(n)\(1+n) \(n)\(1+n)n)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yi=1\(n)]TJ /F7 11.955 Tf 11.95 0 Td[(i)\(i+1+(n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)+1) \((n)]TJ /F7 11.955 Tf 11.96 0 Td[(i))\(i+1+n)]TJ /F7 11.955 Tf 11.96 0 Td[(i+1)!+1asn!+1.UBisprovedbythestandardStirlingapproximation.ThisresultjustiestheconsistencyoftheBayesfactortosomeextent.Infact,nomatterwhatfunctionwespecifyasG,tomakeri!0,thedatahavetobefairlyspreadoutonR+;Incontrast,datahavetobeclusteredtogethertomakeri!1.Thus,thespacingsofdatadeterminetheBayesfactor.SupposewearetestingthatX1,...,XnindependentlyidenticallydistributedassomespecicF0=1)]TJ /F7 11.955 Tf 12.46 0 Td[(S0against1)]TJ /F7 11.955 Tf 12.46 0 Td[(S20.IffisthedensityofF0,thenthedensityof1)]TJ /F7 11.955 Tf 12.8 0 Td[(F20is2S0f.Whendataarewidelyspread,i.e.,thereareplentyofobservationsinthetail,thentheusuallikelihoodratioteststatisticsforoneoftheseparticularobservationsisf(x) 2S0(x)f(x)=1 2S0(x) 34

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isverylargebecausethesurvivalfunctiongetssmallinthetail.Hence,wetendtoacceptH0.Oppositeisthecasefor<1.Ontheotherhand,wemayxthedatathatwehaveobserved,i.e.riinthiscase,andinvestigatethebehavioroftheBayesfactorasvaries.Intuitively,whenapproachestheboundary,either0or+1,thedivergenceofthenullandalternativehypothesesbecomesbigger.Thenwewouldexpectthediscernibilityofthenullfromthealternativetoincrease.IntermsoftheBayesfactor,weanticipatethatBF01behavesinawaysuchthatitprovidesmoreevidenceinfavorofthenullorthealternativeasthecasemaybe.ConsideringBF01asafunctionof,BF01=const.k()wherek()=1 n\(1+n) \(1r1+n)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yi=1\(i+1+(n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)+1) \(i+1ri+1+(n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)).Bysomealgebra,as!+1, 8>><>>:k()!+1if(1)]TJ /F7 11.955 Tf 11.96 0 Td[(r1)+n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Pj=1[1+k+1(1)]TJ /F7 11.955 Tf 11.95 0 Td[(rk+1)])]TJ /F7 11.955 Tf 11.96 0 Td[(n>0k()!0if(1)]TJ /F7 11.955 Tf 11.96 0 Td[(r1)+n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Pj=1[1+k+1(1)]TJ /F7 11.955 Tf 11.95 0 Td[(rk+1)])]TJ /F7 11.955 Tf 11.96 0 Td[(n<0(2)Writec(r)=(1)]TJ /F7 11.955 Tf 11.96 0 Td[(r1)+n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xj=1f1+k+1(1)]TJ /F7 11.955 Tf 11.95 0 Td[(rk+1)g)]TJ /F7 11.955 Tf 20.59 0 Td[(nAccordingly,BF01givesedgetothenullhypothesiswhenc(r)>0,andviceversaforc(r)<0.Notingthatwhenislarge,foranysurvivalfunctionS(x),S(x)isalmost0exceptforaverynarrowregionneartheorigin.Hence,underthealternative,weexpecttoobservecloselyclusteredobservations,whichmeansthatrkiscloseto1,fork=1,...,n.Inthiscase,(1)]TJ /F7 11.955 Tf 12.47 0 Td[(r1)+Pn)]TJ /F5 7.97 Tf 6.59 0 Td[(1j=1f1+k+1(1)]TJ /F7 11.955 Tf 11.95 0 Td[(rk+1)gisapproximatelyequalton)]TJ /F4 11.955 Tf 12.4 0 Td[(1,whichamountstoc(r)<0,leadingtotheconclusionthatBF01!0.Hence,theBayesfactorsupportsthealternative.Onthecontrary,underH0,theobservations 35

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aremuchmorelikelytobespreadthanunderH1.Itiseasilyseenthataslongassomeoftheserk'sarenotcloseto1,c(r)isgoingtobepositive,whichleadstothefactthatBF01!+1.Thecasewhenc(r)=0needsfurthercalculationbasedonrealizationsofrk's.Thesituationwhen!0+issomewhatdifferent.Aftersomecalculations,itturnsoutthatifthereexistsatleastonerk>0,k()!+1.Ifrk=0,fork=1,...,n,thenk()goestosomepositivefunctiondependingonn,andthisfunctiongoesto+1veryquicklyasngrowsbig.Theinterpretationforthisresultisalsoreasonable.Asapproaches0fromtherighthandside,S(x)isalmostatuntilxisextremelyfarfromtheorigin.Thus,inthelimit,thealternativeisgivingapointmassto+1,whichcannothappeninreality.Hence,asexpected,theBayesfactorincreasesto+1,supportingthenull,inthiscase.Inpractice,weneedthecutoffpointssuchthatwecanmakedecisionsbasedonBF01.However,exactcriticalvalueisnotavailablebecauseofthecomplexityofthedistributionofBF01underH0.Nevertheless,itispossibletogeneratesamplesfromPolyatrees.Thus,peoplecangeneratesufcientlymanysamplesfromthespecicPolyatreeandndtheempiricalcriticalvalueofthetest. 2.2.3SimulationStudiesWerunsimulationstoinvestigateasymptoticbehaviorsofthederivedBayesfactorunderthenullandalternativehypotheses.Ideally,theBayesfactorshouldhavetheconsistencyproperty,namely,itgrowstoinnityunderthenullhypothesisandshrinkstozerounderthealternativehypothesisasthesamplesizeincreases.ItturnsouttobethecaseasshowninFigure 2-4 .Inthissimulation,samplesfromaWeibull(3,12)distributionsaredrawnconstantlyandcorrespondingBayesfactorsarecalculatedandplotted.ThechoiceofaWeibulldistributionismotivatedfromthefactthatithasrelativelysimpleformunderthealternativehypothesisanditiswidelyusedinthesurvivalcontext.Theinitialsample 36

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ABaselineMeasure:TrueWeibullDistri-bution BBaselineMeasure:EstimatedWeibullDistributionFigure2-4. Log(BF)growsdrasticallyassamplesizeincreasesunderthenullhypothesis sizeissettobe5andgrowsto500,wheretheBayesfactorsarereasonablylargeenoughtomeetourillustrationpurpose.Thepowerofthealternative,issettobe0.5,1.5,2.0respectively.Figure 2-4A demonstratesthebehavioroflog(BF01)whenaPolyatreepriorcenteringatthetruedistributionisgiven;Figure 2-4B illustrateshowlog(BF01)growswhenaPolyatreepriorcenteringattheWeibulldistribution,whoseparametersareestimatesbasedonthesample,isusedincomputation.Itisworthmentioningthatthelinerepresenting=2(blue)hasalargerslopethanthelinerepresenting=1.5(red)does.Itmeansthatonemaybeabletodiscriminatethealternativefromthenullhypothesissoonerwhen=2ascomparedto=1.5,andthisisdesirable.ResultswhenPolyatreepriorsarecenteredatdistributionsotherthantheWeibullaresimilar,andareomitted.Ontheotherhand,theBayesfactordecreasessharplyunderthealternativehypothesis.Forexample,ifF(x)=1)]TJ /F7 11.955 Tf 13.12 0 Td[(S(x)isaWeibull(,!)distribution,thenG(x)=1)]TJ /F4 11.955 Tf 12.99 0 Td[([S(x)]isdistributedasWeibull(,!=1=).Therefore,samplesfromaWeibull(3,12=1=3)aredrawnandBayesfactorsarecalculatedwithaPolyatreepriorcenteringatWeibull(3,12).TheresultisshowninFigure 2-5 .Notethatthelinerepresenting=2(blue)goesdownfasterthanthelinerepresenting=1.5(red). 37

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Figure2-5. Log(BF)decreasesdrasticallyassamplesizeincreasesunderthealternativehypothesis 38

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CHAPTER3COXPROPORTIONALHAZARDSMODELUNDERPOLYATREEPROCESSPRIOR 3.1LikelihoodFunctionNowweconsiderthecasewithcensoringinsurvivaldataanalysis.Thedataisoftheform(tk,k,xk),wherexkisthesetofcovariatesassociatedwiththekthsubject,andkisthecensoringindicator,with=1beingthecasethataneventhappens.AssumingCoxproportionalhazardsmodelasin( 1 ),itfollowsthatthesurvivalfunctionisS(tjx)=[S0(t)]exp(xT) KalbeischandPrentice ( 1973 )derivedthemarginallikelihoodfunctionsoftheCoxmodelwithandwithoutties.Later, Kalbeisch ( 1978 )usedGammaprocess(GP)forthebaselineandderivedthemarginallikelihoodfunctionofthecoefcients. DykstraandLaud ( 1981 )specifyanextendedGammaprocessprioronthehazardrateitself.InthissectionwepresenttheresultsverysimilartoKalbeisch'swork.InsteadofputtingaGammaprocesspriorforthebaseline,weuseaPolyatreeprior.ThePolyatreepriorappearstobemoreexiblethanGammaprocessprior.Toseethis,supposeP1GP(a(t),)andP2PT(,a).Supposet1
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However,thisisnotthecaseinPolyatree.Bychoosingproperparameters,itispossibletomakeE[P2((t1,t2))]=E[P2((t2,t3))]andVar[P2((t1,t2))]6=Var[P2((t2,t3))].Thisisusefulbecauseinsomecases,duetosomereason,theresearchermightnotbesureaboutthedistributionovercertaininterval,andwantstoletthedistributionhavearelativelylargevarianceinthatinterval.Polyatreecandealwiththissituation.Forexample,supposeahorribleearthquaketookplaceatt2andaftershockskeptcomingduringt2tot3.Thismightbringinalotofcensoreddataintroducingbias.Onemightwanttoletthevarianceofthebaselinedistributionbebiggerduring(t2,t3)thanthatduring(t1,t2).APolyatreepriorcanaccommodatesituationsofthistype. 3.1.1MarginalLikelihoodFunctionforThroughoutthissection,tiesarenotconsidered.Weassumethatthecensoringoccursinnitesimallybeforeaneventhappens.Namely,ifweobserveacensoreddatumt,thecontributionofthisdatumtolikelihoodisPr([t,+1)).Incaseoneneedstodealwiththeoppositesituation,oneshouldletthepartitionbeB1,...,1| {z }i=(Xi,+1)andredothecomputations.NowwesupposethatthebaselinedistributionF0=1)]TJ /F7 11.955 Tf 12.44 0 Td[(S0isdrawnfromaPolyatreeprocess.Ourinterestistoestimate.Forthetimebeing,wearenotconsideringthisprobleminafullyBayesianframework.Thatis,nopriorisgivenforandthusndingtheposteriorofisnotourprimarygoal.ThereasonisthatthestructureofPolyatreeissocomplicated,thatevenifweusethesimplestpriorfor,theposteriorisnotavailableanalytically.Hence,weconneoutinterestinthemarginallikelihoodofandseekingMaximumLikelihoodEstimator(MLE)throughoutthissubsection,whichisalsotheposteriormodeunderauniformpriorfor.Thefollowingtheoremgivestheexactformofthemarginallikelihoodfunctionof. 40

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Theorem3.1. Assumethemodel( 1 )andnotiesoccurindata.SupposethebaselinedistributionisdistributedasPolyatree.ThentheexactlikelihoodfunctionofcoefcientsisE[Lm]=nYk=1f1 (B1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(k)gkn)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yk=1fa1,...,1| {z }k,0gkfa1,...,1| {z }n,0 a1,...,1| {z }n,0+a1,...,1| {z }n,1gnnYk=1fa1,...,1| {z }k,00 a1,...,1| {z }k,00+a1,...,1| {z }k,01a1,...,1| {z }k,000 a1,...,1| {z }k,000+a1,...,1| {z }k,001a1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(k a1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(k+a1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1gkfnYk=1exp(xTk)kg\(a0+a1)\(a1+Pnj=1exp(xTj)) \(a0+a1+Pnj=1exp(xTj))\(a1)fn)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yk=1\(a1,...,1| {z }k,0+a1,...,1| {z }k,1)\(a1,...,1| {z }k,1+Pnj=k+1exp(xTj)) \(a1,...,1| {z }k,0+a1,...,1| {z }k,1+Pnj=k+1exp(xTj)+k)\(a1,...,1| {z }k,1)g (3)Notethatthetermsinvolvingareindependentofm.TogettheMLEof,weonlyneedtomaximizethefollowingfunctionwithrespecttoL()=fnYk=1exp(xTk)kg\(a1+Pnj=1exp(xTj)) \(a0+a1+Pnj=1exp(xTj))fn)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yk=1\(a1,...,1| {z }k,1+Pnj=k+1exp(xTj)) \(a1,...,1| {z }k,0+a1,...,1| {z }k,1+Pnj=k+1exp(xTj)+k)g (3)ItworthspointingoutthattheexactlikelihoodfunctionofisjustL()multipliedbyaconstant,i.e.Lik()=c.L().Ifwere-parameterizeasfollows, 41

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8>>>>>><>>>>>>:k=a1,...,1| {z }k+a1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0rk=G([vk,+1)) G([vk)]TJ /F9 5.978 Tf 5.76 0 Td[(1,+1))=a1,...,1| {z }k a1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,...,1| {z }k(3)thusL()reducestoL()=fnYk=1exp(xTk)kg\(1r1+Pnj=1exp(xTj)) \(1+Pnj=1exp(xTj))fn)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yk=1\(k+1rk+1+Pnj=k+1exp(xTj)) \(k+1+Pnj=k+1exp(xTj)+k)g (3)Thisresultissimple,andiseasiertoworkwiththan( 3 ).Moreover,when1=...=n=0,L()reducestoL()=fnYk=1exp(xTk)kgfn)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yk=1\(Pnj=k+1exp(xTj)) \(Pnj=k+1exp(xTj)+k)g=n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yk=1;k=1exp(xTk) Pnj=k+1exp(xTj)whichresemblesthepartiallikelihoodfunctionfortheCoxproportionalhazardsmodel.TheexactpartiallikelihoodfunctionisPL()=nYk=1;k=1exp(xTk) Pnj=kexp(xTj)=n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yk=1;k=1exp(xTk) Pnj=kexp(xTj)Theyonlydifferinthatexp(xTk)isnotincludedinthedenominatorofeachtermintheproduct.IfonethinksofthemotivationofpartiallikelihoodfunctionheuristicallyasPr(theparticularindiv.diesattkjonedeathattk)=h(tkjxk) Pnj=kh(tjjxj)=exp(xTk) Pnj=kexp(xTj), 42

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thenthelimitingcaseofourlikelihoodfunctionisnothingbutconsideringtheconditionaloddsinsteadofconditionalprobabilities,namely,Odds(theparticularindiv.diesattkjonedeathattk)=h(tkjxk) Pnj=kh(tjjxj) 1)]TJ /F8 7.97 Tf 22.51 5.7 Td[(h(tkjxk) Pnj=kh(tjjxj)=exp(xTk) Pnj=k+1exp(xTj) 3.1.2EffectsofSpacingsInthissubsection,weinvestigatetheeffectsofspacingsoforderstatisticsontheMLE.Forsimplicity,weconsideronlyone-dimensionalcovariateinthissub-section.Withoutlossofgenerality,assumethatthecovariateisnon-negative.Weconsideroneatatimethecomponenttermsin( 3 ).Fork=1,,n)]TJ /F4 11.955 Tf 11.96 0 Td[(1,weconsiderh(rk+1)=log(\(k+1rk+1+nXj=k+1exp(xj))))]TJ /F7 11.955 Tf 11.95 0 Td[(log(\(k+1+nXj=k+1exp(xj)+k)).Leth(rk+1)=[ (k+1rk+1+nXj=k+1exp(xj)))]TJ /F3 11.955 Tf 11.55 0 Td[( (k+1+nXj=k+1exp(xj)+k)](nXj=k+1exp(xj)xj),where ()isDigammafunction.Hence,h(rk+1)isincreasinginrk+1because ()isstrictlyincreasingin(0,+1)andPnj=k+1exp(xj)xj>0unlessallcovariatesaretrivially0.Thisisalsotrueforh(r1)whereh(r1)isdenedintheobviousway.ThustheMLEisobtainedbysolvingthetargetfunction h()=n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Xk=0h(rk+1)=0.(3)Notethath()isanincreasingfunctionofrk,k=1,,n.Hencelargerrk'sresultinlargerh(),andaccordinglylarger's,i.e.largertreatmenteffects,asshowninFigure 3-1 .Thefallingtendencyofh()iscrucialbecauseotherwiseweendupwithexactlytheoppositeconclusion.ThistendencyisguaranteedbythedenitionofMLE,because 43

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Figure3-1. EffectsofspacingsonMLE anascendingtrendwouldsuggestthattherootof( 3 )isalocalminimuminsteadofmaximum. 3.2FullyBayesianAnalysisFullyBayesiansetupisavailableinthisframeworkandisdiscussedinthissub-section.Assumethatweputagenericprior()on.Itiswellknownthattheposteriorisgivenby(jdata)/Lik()()/L()().SincewehaveexplicitexpressionforL(),namely,thelikelihoodfunctionuptoanunknownconstant,itisstraightforwardtousestandardMetropolis-Hastingsalgorithmtosimulatetheposteriordistributionof.Herewerevisitthetoyexample(Figure 1-1 )proposedinSection2.Arelativelyun-informativeN(0,100)priorisassignedfor.Figure 3-2 showsthekerneldensityestimatesoftheposteriorsofusingMCMCalgorithm.ComparedtothepartiallikelihoodestimatePL=)]TJ /F4 11.955 Tf 9.29 0 Td[(1.1051inTable 1-1 ,theposteriormeanofinscenario 1-1A isevaluatedat)]TJ /F4 11.955 Tf 9.3 0 Td[(2.44,showingasignofmuchlargertreatmenteffect;theposteriormeanofinscenario 1-1B isevaluatedat)]TJ /F4 11.955 Tf 9.3 0 Td[(0.79,showingasmallertreatmenteffect.Anotherinterestingresultisthatthehighest 44

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AMean=-2.44,95%HPD=(-3.38,-1.57) BMean=-0.79,95%HPD=(-1.53,-0.003)Figure3-2. KernelDensityforPosteriorsof posteriordensity(HPD)intervalforscenario 1-1A is()]TJ /F4 11.955 Tf 9.3 0 Td[(3.38,)]TJ /F4 11.955 Tf 9.3 0 Td[(1.57),whichisclearlyfarawayfrom0,identifyingastatisticalsignicanttreatmenteffect.Incontrast,theHPDintervalforscenario 1-1B is()]TJ /F4 11.955 Tf 9.3 0 Td[(1.53,)]TJ /F4 11.955 Tf 9.29 0 Td[(0.003),whichbarelyexcludes0,makingthesignicanceoftreatmenteffectlessconvincing. 3.2.0.1SimulationStudyOnConsistencyOfPosteriorMeanTostudytheasymptoticpropertiesoftheestimation,wecarriedoutaseriesofsimulationstudies.WesetthebaselinesurvivaldistributiontobeaWeibull(3,12).ObservationsarecensoredbyanindependentExp(50)distribution.Weassumethatthereare2covariates.Thetrueregressioncoefcientsaresettobe0.5and)]TJ /F4 11.955 Tf 9.3 0 Td[(5respectively.Werandomlygeneratecensoredsamplewithsizesfrom15to150.Throughoutallsamplesizes,anun-informativeprior,N(0,100)isassignedtothetwocoefcients.Foreachsamplesize,aMCMCalgorithmisusedtocomputetheposteriormean.TheposteriormeansareplottedinFigure 3-3 .Asonecanseeinthesimulations,theposteriormeansoftheregressioncoefcientsconvergetothetruevaluesquicklyandstaystableeversince. 3.3RealDataAnalysisWetaketheovariancancerdatasetintheSurvivalpackageofRsoftwareasarealdataanalysisexample.ThedescriptionofthedatasetisgiveninSection2.1.2and 45

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A BFigure3-3. PosteriorMeansof'sasSampleSizeIncreases Table 2-1 .Therearethreecovariatesotherthantreatments,whicharepatients'ages(inyears),presenceofresidualdiseaseaswellasECOGperformancestatus(1isbetter,see J.H.EdmunsonandKvols ( 1979 )).Thefollowingtableprovidesaglanceofthedatasetgroupedbytreatments.Asanillustration,wewillapplythefullyBayesianparametricmodeldescribedby DellaportasandSmith ( 1993 ).ThelikelihoodundertheWeibullmodelisgivenby L(,jdata)=fnYj=1t)]TJ /F5 7.97 Tf 6.58 0 Td[(1jexp(xj)gfn+mYj=1exp[)]TJ /F7 11.955 Tf 9.3 0 Td[(tjexp(xj)]g.(3)Weassumethat1,2,3N(0,100)independently,where1,2,3areregressioncoefcientscorrepondingtotreatment,ageandresidualdiseaserespectively.DuetothefactthatforECOGscore,1isbetterthan2,weassignalognormal(0,100)priorto4.ThroughGibbssamplerinWinbugs,theposteriordensitiesaregiveninFigure 3.3 .Nowweproceedwithourproposedmethod.Arelativelyun-informativepriorisused.Thesamepriorsforareused.StandardMetropoliswithinGibbssampleralgorithmcanbeappliedandthekerneldensityestimatesofresultingposteriordistributionsaredisplayedinFigure 3-4 .Asimplecomparisonoftheresultsshowsthatourmethodleadstoatreatmenteffectwithlargermagnitude.Besides,ourmethodsuccessfullyidentiesapositive 46

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AMean=-1.88,95%HPD=(-2.908,-0.578) BMean=0.039,95%HPD=(0.0099,-0.071) CMean=0.416,95%HPD=(-0.514,1.379) DMean=0.031,95%HPD=(0,0.170)Figure3-4. KernelDensityforPosteriorsof's ageeffectsinceitmakesmoresensethatolderpeoplearemorelikelytohavelargerhazards. 47

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CHAPTER4RELIABILITYMuchoftheliteraturediscussedinChapter1makesspecicandstrongassumptionsonthedistributionsofXandY,whichgreatlylimitstheapplicabilityoftheresults.Ideally,wewouldliketoweakenthedistributionassumptions,butstillbeabletoincorporatethepriorinformationintotheestimation.Bayesiannonparametricmethodsprovideapropersolutionforseveralreasons.Nonparametricpriorsgenerallycoverafairlylargerangeofdistributions,butcouldstillbeabletoconcentrateonsomepre-specieddistributiondependingonthechoiceoftheprecisionparameter. Ferguson ( 1973 )gaveanestimatorforP(X
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istheMann-Whitneystatistics.Itworthmentioningthat^P(X
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whereLi'saredenedasfollows L1=a1+n0 a0+a0+nb1+m1 b0+b1+mL2=a0+n0 a0+a1+n0a00+n00 a00+a01+n0b0+m0 b0+b1+mb01+m01 b00+b01+m0+a1+n1 a0+a1+n0a10+n10 a10+a11+n1b1+m1 b0+b1+mb11+m11 b10+b11+m1Lk=P1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1=0or1[a1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+n1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0 a1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+n1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1b1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1+m1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1 b1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+b1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1+m1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1Qk)]TJ /F5 7.97 Tf 6.59 0 Td[(1j=1a1,,j+n1,,j a1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1+n1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1] (4) withn1,,k=#fXi:Xi2B1,,k,i=1,,ngandm1,,k=#fYi:Yi2B1,,k,i=1,,mg.ThefollowingcorollaryrevealstherelationshipbetweenourestimatorandtheMann-Whitneyestimator. Corollary1. Ifonetakesa1,,k=b1,,k!0forall1,,kandallk,whichstandsfornopriorinformationonthedistributionfunctionisavailable,namelynobeliefonthecenterofthePolyatreeprocess, ^P(X
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CHAPTER5CONCLUSIONANDDISCUSSIONSofartheresultsonBayesfactorsworkonlyincasewithoutties.However,inreality,theremightbesituationswheretiesoccur.Forexample,somemassivedestructiondisasters,likeearthquakes,mightcausemultipleexperimentsbeingcensoredatthesametime.Basically,therearetwocasesforoccurrencesofties,tiesofcensoreddataandtiesofeventtimes.Fortherstcasewhenmultipleobservationsarecensoredatthesametime,ourcalculationsworkperfectly.However,thisisnotthecaseforthesecondscenario.Theproblemiscausedfortworeasons.First,thelimit( 1 )maynotexistorisinnityforsomex2R+becausethedensitydoesnotnecessarilyexistwhentheunderlyingdistributionisnotabsolutelycontinuous.Second,whenmultipleeventsareobservedatthesametime,weknowthattheunderlyingdistributionmustbenon-continuous.Inthiscase,tokeepthepriorreasonable,weshouldnotconsiderPolyatreepriorswhichgiveprobability1tothesetofcontinuousdistributions.WeneedtoassignparameterstoPolyatreesaccordingly,whetheritgivespositiveprobabilitytothesetofdiscretedistributions,ortothesetofpartlydiscrete,andpartlycontinuousdistributions.TakingintoaccountthefactthatDirichletProcessisaspecialcaseofPolyatree,onewouldliketoutilizeDirichletprocessincaseswheretiesoccurbecausetheprobabilityofobservingtieswithDirichletprocesspriorsgoesto1assamplesizegoesup.Wedidcalculatethelikelihoodfunctionunderthisassumption.Butnoasymptoticpropertiescouldbedevelopedinthiscase,sincethelikelihoodfunctiondependsnotonlyonthenumberoftiedobservations,butalsoonthelocationonR+oftheoccurrencesofties.Thisisaseriousproblemwhensamplesizeincreases,sincethenumberofpossibletiedsituationsgrowstoinnityveryquickly.However,thisdoesnotmeanthatwecannotdoanythingwhentiesoccur.Wecanborrowsomeideasfromhowpeopledealwithtiesinpartiallikelihoodfunction. 51

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Supposeobserveddataset(tk,k,xk),k=1,2,,n(withoutlossofgenerality,assumingdataaresortedbytk)hastiedeventtimestk0=tk0+1=t.Assumethattheunderlyingdistributionoftimesiscontinuous.Inthiscase,weobservetiesbecauseofmeasurementerror.Indeed,tk06=tk0+1,buttheyaresoclosetoeachothersuchthatwedonotseethedifference.Hence,inrealitytherearetwoequallylikelypossibilities,i.e.tk0>tk0+1andtk0
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APPENDIXAPROOFOFTHEOREM2.1WehaveX1,,Xn1jPF(x)andY1,,Yn2jPH(x)=1)-231(f1)]TJ /F7 11.955 Tf 12.06 0 Td[(F(x)gandwanttotestH0:=1vsH1:>1(or<1).wherePPT(G)istheprobabilitymeasureinducedbyF(x),and>0.Forsimplicity,letPbetheprobabilitymeasureinducedbyH(x).FirstweputthesetwosamplestogetheranddenotethembyV1,,Vn1+n2anddeneZ1,,Zn1+n2asdescribedinChapter2.Letn=n1+n2.Incasethatthereisnocensoring,takem>n,suchthatatlevelm,V1,...,Vnareseparatedindifferentintervals.Forv2[0,+1),let~m(v)=1,...,msuchthatv2B1,...,m.Inaddition,withappropriateparameters,Piscontinuouswithprobability1.Thuswithoutlossofgenerality,assumeV1<...
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P(B~m+jP).Atmthlevelofthetree,givenP,asimpleexpressionisprovidedby,P(B~m+jP)=fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1jP)0(n)+Pr(B~mjP)g.Therefore,P(B~mjP)=P(B~m+jP))]TJ /F4 11.955 Tf 13.24 2.65 Td[(P((B~m+))]TJ /F7 11.955 Tf 11.96 0 Td[(B~mjP)=fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1jP)0(n)+Pr(B~mjP)g)-222(fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1jP)0(n)g (A)wherethe-signintheprobabilitymeansexclusion.NowusingthesecondorderTaylorExpansionforfunctionh(t)=t,h(t+))]TJ /F7 11.955 Tf 11.96 0 Td[(h(t)=t)]TJ /F5 7.97 Tf 6.59 0 Td[(1+()]TJ /F4 11.955 Tf 11.96 0 Td[(1)(t+))]TJ /F5 7.97 Tf 6.59 0 Td[(22where2(0,).ItfollowsthatP(B~mjP)=Pr(B~mjP)fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1jP)0(n)g)]TJ /F5 7.97 Tf 6.58 0 Td[(1+()]TJ /F4 11.955 Tf 11.95 0 Td[(1)Pr(B~mjP)2fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1jP)0(n)+g)]TJ /F5 7.97 Tf 6.58 0 Td[(2where2(0,Pr(B~mjP)).Forsimplicity,writeWm(x)=mXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1jP)0(n)Wmdependsonvbecause1,...,mdependonv.NowweareinplacetocalculatetheconditionaljointpdfofY1,,Yn2underH1.fm(y1,...,yn2jP)=Qn2i=1P(B~mijP) Qn2i=1(B~mi)=NUM Qn2i=1(B~mi) 54

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whereNUMisthenumeratorofthefraction,namelyNUM=n2Yi=1P(B~mijP)=Qn2i=1Pr(B~mijP)Qn2i=1fWm(Yi))]TJ /F5 7.97 Tf 6.59 0 Td[(1+()]TJ /F4 11.955 Tf 11.96 0 Td[(1)[Wm(Yi)+i])]TJ /F5 7.97 Tf 6.58 0 Td[(2Pr(B~mijP)g Qn2i=1(B~mi)Again,theexactmarginaljointpdfisfoundbylettingm!+1andthentakingexpectedvalue.Notethat limm!+1E[NUM Qn2i=1(B~mi)]=limm!+1E[Qn2i=1Pr(B~mijP)Qn2i=1fWm(Yi))]TJ /F5 7.97 Tf 6.59 0 Td[(1g Qn2i=1(B~mi)](A)Indeed,ifwewritetheproductsinthenumeratorassummation,wehaven2Yi=1[Wm(Yi))]TJ /F5 7.97 Tf 6.58 0 Td[(1+()]TJ /F4 11.955 Tf 11.95 0 Td[(1)fWm(Yi)+ig)]TJ /F5 7.97 Tf 6.59 0 Td[(2Pr(B~mijP)]=XS[Yj2SWm(Xj))]TJ /F5 7.97 Tf 6.58 0 Td[(1Yk2Sc()]TJ /F4 11.955 Tf 11.96 0 Td[(1)fWm(Yk)+kg)]TJ /F5 7.97 Tf 6.58 0 Td[(2Pr(B~mkjP)] (A)where=f1,...,ng,andthesummationistakenforall(properandimproper)subsetsS.However,ifjScj1,i.e.thereexistsk02Sc,thenj[Yj2SWm(Yj))]TJ /F5 7.97 Tf 6.58 0 Td[(1Yk2Sc()]TJ /F4 11.955 Tf 11.96 0 Td[(1)fWm(Yk)+kg)]TJ /F5 7.97 Tf 6.58 0 Td[(2Pr(B~mkjP)]jjn()]TJ /F4 11.955 Tf 11.96 0 Td[(1)jScjPr(B~mk0jP)jTheaboveinequalityusesthefactthat0Wm(Y)Wm(Y)+Wm(Y)+Pr(B~mkjP)mXn=1Pr(B1,...,njP)=Pr([0,+1)jP)=1. 55

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Hence,theexpectationofthecorrespondingterminsummationsatisesE[jQn2i=1Pr(B~mijP)[Qj2SWm(Yj))]TJ /F5 7.97 Tf 6.59 0 Td[(1Qk2Sc()]TJ /F4 11.955 Tf 11.95 0 Td[(1)fWm(Yk)+kg)]TJ /F5 7.97 Tf 6.59 0 Td[(2Pr(B~mkjP)] Qn2i=1(B~mi)j]E[jQn2i=1Pr(B~mijP)n()]TJ /F4 11.955 Tf 11.96 0 Td[(1)jScjPr(B~mk0jP) Qn2i=1(B~mi)j]=const.E[fQi6=k0Pr(B~mijP)gPr(B~mk0jP)2 Qn2i=1(B~mi)] (A)Comparing( B )to( A ),itfollowsthatE[fQi6=k0Pr(B~mijP)gPr(B~mk0jP)2 Qn2i=1(B~mi)]=E[fm(y1,...,yn2jP)]mYj=1ak01,...,k0j+nk01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1+1 ak01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0+ak01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+nk01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1+1wherenk01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1=]fj:Yj2Bk01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1g.Whenm>n,wespecifytheparametersasa1,...,,m)]TJ /F9 5.978 Tf 5.76 0 Td[(10=a1,...,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1=m2whichimpliesthatwhenmislarge,ak01,...,k0j+nk01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1+1 ak01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0+ak01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+nk01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1+1!1 2.AndE[fm(y1,...,yn2jP)]isnite.Therefore,E[fQi6=k0Pr(B~mijP)gPr(B~mk0jP)2 Qn2i=1(B~mi)]!0asm!+1.SoallthetermswithjScj1in( B )eventuallygoesto0.TheonlytermleftiswhenjScj=0,i.e.S=,whichcompletestheprooftotheclaim. 56

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Furthermore,theconditionaljointpdfofV1,,VnunderH1isgivenbyfm(v1,...,vnjP)=fm(x1,,xn1jP)fm(y1,,yn2jP)=Qn1i=1P(B~mijP) Qn1i=1(B~mi)Qn2i=1Pr(B~mijP)Qn2i=1fWm(Yi))]TJ /F5 7.97 Tf 6.59 0 Td[(1g Qn2i=1(B~mi)=Qni=1Pr(B~mijP)Qn2i=1fWm(Yi))]TJ /F5 7.97 Tf 6.59 0 Td[(1g Qni=1(B~mi)NowwecomputetheBayesfactor.Beforewedothat,letusgureoutwhat~m(vi)is.Bythemechanismofpartition,form>n,andi=1,...,n,viisanendpointatleveliofthetree.Beforetheithlevel,viliesintherightsubintervaleverytimethecurrentintervalsplitsintotwo;afterithlevel,viwouldbealwaysintheleftsubintervalgeneratedbysplittingthecurrentintervalthatcontainsvi.Thus,~m(vi)=1,...,1| {z }i,0,...,0| {z }m)]TJ /F8 7.97 Tf 6.59 0 Td[(iClearly,Pr(B~mijP)=Y1Y11...Y1,...,1| {z }iY1,...,1| {z }i,0...Y1,...,1| {z }i,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(i.AndbydenitionofWm(v),Wm(v1)=Y1,Wm(v2)=Y1Y11,,Wm(vn)=Y1Y11...Y1,...,1| {z }n.Intuitively,Wmisthesurvivalfunctionatlevelm.TheseleadtonYi=1Pr(B~mijP)=fY1Y10Y100...Y1,0,...,0| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(1gfY1Y11Y110Y1100...Y11,0,...,0| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(2gfY1Y11...Y1,...,1| {z }nY1,...,1| {z }n,0Y1,...,1| {z }n,00...Y1,...,1| {z }n,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(ng (A) 57

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Also,nYi=1Pr(B~mijP)n2Yi=1fWm(Yi))]TJ /F5 7.97 Tf 6.59 0 Td[(1g=nYi=1fPr(B~mijP)g1)]TJ /F8 7.97 Tf 6.59 0 Td[(ZifPr(B~mijP)Wm(Vi))]TJ /F5 7.97 Tf 6.59 0 Td[(1gZi (A)=n2Yn1+n21Yn)]TJ /F5 7.97 Tf 6.58 0 Td[(1+()]TJ /F5 7.97 Tf 6.59 0 Td[(1)t211...Yn+1)]TJ /F8 7.97 Tf 6.58 0 Td[(k+()]TJ /F5 7.97 Tf 6.59 0 Td[(1)tk1,...,1| {z }k...Y1+()]TJ /F5 7.97 Tf 6.58 0 Td[(1)tn1,...,1| {z }nfY10Y100...Y1,0,...,0| {z }n)]TJ /F9 5.978 Tf 5.75 0 Td[(1gfY110Y1100...Y11,0,...,0| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(2gfY1,...,1| {z }n,0Y1,...,1| {z }n,00...Y1,...,1| {z }n,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(ng (A)Wheretk=Pni=kZievaluatestheorderingofXsampleandYsample.ItturnsoutthatalltheseY'sarenotindependentbecause Y1,...,j,1=1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y1,...,j,0(A)forallj=1,...,nandall(1,...,j).Takingintoaccountthatthedenominatorsin( A )and( B )arethesame,theBayesfactorreducesto BF01=limm!+1E[Qni=1Pr(B~mijP)] E[Qni=1Pr(B~mijP)Qn2i=1fWm(Yi))]TJ /F5 7.97 Tf 6.58 0 Td[(1g](A)Combining( B ),( A )and( B ),andcancelingtheindependentcommontermsin( A ),weendupwithBF01=1 n2E1 E2 (A) 58

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whereE1=E[(1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y0)nY10(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y10)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1...Y1,...,1| {z }i,0(1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y1,...,1| {z }i,0)n)]TJ /F8 7.97 Tf 6.59 0 Td[(i...Y1,...,1| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y1,...,1| {z }n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0)]E2=E[(1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y0)n1+n2Y10(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y10)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1+()]TJ /F5 7.97 Tf 6.59 0 Td[(1)t2...Y1,...,1| {z }i,0(1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y1,...,1| {z }i,0)n)]TJ /F8 7.97 Tf 6.58 0 Td[(i+()]TJ /F5 7.97 Tf 6.59 0 Td[(1)tiY1,...,1| {z }n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y1,...,1| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0)1+()]TJ /F5 7.97 Tf 6.59 0 Td[(1)tn]AlltheY'sappearingintheaboveequationareindependentwithBetadistributions,thatis,Y1,...,1| {z }i,0independentlyBeta(a1,...,1| {z }i,0,a1,...,1| {z }i,1)fori=1,...,n.HencesimplecalculationsofmomentsofBetadistributionsyieldthenalresultBF01=1 n2\(a1+n)\(a0+a1+n1+n2) \(a0+a1+n)\(a1+n1+n2)n)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yi=1\(a1,...,1| {z }i,1+n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)\(a1,...,1| {z }i,0+a1,...,1| {z }i,1+()]TJ /F4 11.955 Tf 11.95 0 Td[(1)ti+1+1) \(a1,...,1| {z }i,0+a1,...,1| {z }i,1+n)]TJ /F7 11.955 Tf 11.95 0 Td[(i+1)\(a1,...,1| {z }i,1+()]TJ /F4 11.955 Tf 11.95 0 Td[(1)ti+1) (A)Whendataarecensored,simplyreplacethetermPr(B~mijP) (B~mi)orP(B~mijP) (B~mi)byPr(B~iijP)orP(B~iijP)respectivelydependingonwhethertheobservationiscomingformsampleXorY.Analogouscalculationleadsto BF01=1 n2Ec1 Ec2(A) 59

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whereEc1=E[(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y0)nYd110(1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y10)n)]TJ /F5 7.97 Tf 6.58 0 Td[(1...Ydi1,...,1| {z }i,0(1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y1,...,1| {z }i,0)n)]TJ /F8 7.97 Tf 6.58 0 Td[(i...Ydn)]TJ /F9 5.978 Tf 5.75 0 Td[(11,...,1| {z }n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y1,...,1| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0)]Ec2=E[(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y0)n1+n2Yd110(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y10)n)]TJ /F5 7.97 Tf 6.58 0 Td[(1+()]TJ /F5 7.97 Tf 6.59 0 Td[(1)t2...Ydi1,...,1| {z }i,0(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y1,...,1| {z }i,0)n)]TJ /F8 7.97 Tf 6.59 0 Td[(i+()]TJ /F5 7.97 Tf 6.59 0 Td[(1)ti+1Y1,...,1| {z }n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y1,...,1| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0)1+()]TJ /F5 7.97 Tf 6.59 0 Td[(1)tn],andn2=Pi=1ndiZiisthetotalnumberofuncensoredobservationsinsampleY.IntegrationyieldstheBayesfactorfordatawithcensoringasfollowsBF01=1 n2\(a1+n)\(a0+a1+n1+n2) \(a0+a1+n)\(a1+n1+n2)n)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yi=1\(a1,...,1| {z }i,1+n)]TJ /F7 11.955 Tf 11.96 0 Td[(i)\(a1,...,1| {z }i,0+a1,...,1| {z }i,1+()]TJ /F4 11.955 Tf 11.95 0 Td[(1)ti+1+di) \(a1,...,1| {z }i,0+a1,...,1| {z }i,1+n)]TJ /F7 11.955 Tf 11.95 0 Td[(i+di)\(a1,...,1| {z }i,1+()]TJ /F4 11.955 Tf 11.96 0 Td[(1)ti+1) (A) 60

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APPENDIXBPROOFOFTHEOREM3.1Takem>n,suchthatatlevelm,t1,...,tnareseparatedindifferentintervals.Fort2[0,+1),let~m(t)=1,...,msuchthatt2B1,...,m.Inaddition,withappropriateparameters,Piscontinuouswithprobability1.Thuswithoutlossofgenerality,assumet1<...
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providedby,Pk(B~m+jP)=fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1jP)0(n)+Pr(B~mjP)gk.Therefore,Pk(B~mjP)=Pk(B~m+jP))]TJ /F7 11.955 Tf 11.96 0 Td[(Pk((B~m+))]TJ /F7 11.955 Tf 11.96 0 Td[(B~mjP)=fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1jP)0(n)+Pr(B~mjP)gk)-222(fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1jP)0(n)gk (B)wherethe-signintheprobabilitymeansexclusion.NowusingthesecondorderTaylorExpansionforfunctionh(t)=t,h(t+))]TJ /F7 11.955 Tf 11.96 0 Td[(h(t)=t)]TJ /F5 7.97 Tf 6.59 0 Td[(1+()]TJ /F4 11.955 Tf 11.96 0 Td[(1)(t+))]TJ /F5 7.97 Tf 6.59 0 Td[(22where2(0,).ItfollowsthatPk(B~mjP)=kPr(B~mjP)fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1jP)0(n)gk)]TJ /F5 7.97 Tf 6.59 0 Td[(1+k(k)]TJ /F4 11.955 Tf 11.95 0 Td[(1)Pr(B~mjP)2fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1jP)0(n)+gk)]TJ /F5 7.97 Tf 6.59 0 Td[(2where2(0,Pr(B~mjP)).Forsimplicity,writeWm(t)=mXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1jP)0(n)Wmdependsontbecause1,...,mdependont. 62

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Nowweareinplacetocalculateconditionaljointpdfoft1,,tnatmthlevelofthePolyatree,whichisdenotedbyfm(t1,...,tnjP).fm(t1,...,tnjP)=Qni=1Pi(B~mijP) Qni=1(B~mi)=Qni=1Pr(B~mijP)Qni=1fiWm(ti)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1+i(i)]TJ /F4 11.955 Tf 11.96 0 Td[(1)[Wm(ti)+i]i)]TJ /F5 7.97 Tf 6.59 0 Td[(2Pr(B~mijP)g Qni=1(B~mi)=NUM Qni=1(B~mi)whereNUMisthenumeratorofthefraction.Theexactmarginaljointpdfisfoundbylettingm!+1andthentakingexpectedvaluewithregardtoPolyatreeprocess.Weclaimthat limm!+1E[NUM Qni=1(B~mi)]=limm!+1E[Qni=1Pr(B~mijP)Qni=1fiWm(ti)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1g Qni=1(B~mi)](B)Indeed,ifwewritetheproductsinthenumeratorassummation,wehavenYi=1[iWm(ti)i)]TJ /F5 7.97 Tf 6.58 0 Td[(1+i(i)]TJ /F4 11.955 Tf 11.95 0 Td[(1)fWm(ti)+igi)]TJ /F5 7.97 Tf 6.58 0 Td[(2Pr(B~mijP)]=XS[Yj2SiWm(tj)i)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yk2Sci(i)]TJ /F4 11.955 Tf 11.96 0 Td[(1)fWm(tk)+kgi)]TJ /F5 7.97 Tf 6.58 0 Td[(2Pr(B~mkjP)] (B)where=f1,...,ng,andthesummationistakenforall(properandimproper)subsetsS.However,ifjScj1,i.e.thereexistsk02Sc,thenj[Yj2SiWm(tj)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yk2Sci(i)]TJ /F4 11.955 Tf 11.95 0 Td[(1)fWm(tk)+kgi)]TJ /F5 7.97 Tf 6.59 0 Td[(2Pr(B~mkjP)]j.jn()]TJ /F4 11.955 Tf 11.96 0 Td[(1)jScjPr(B~mk0jP)jwhere=maxi=1,,ni.Theaboveinequalityusesthefactthat0Wm(t)Wm(t)+Wm(t)+Pr(B~mkjP)mXn=1Pr(B1,...,njP)=Pr([0,+1)jP)=1. 63

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Hence,theexpectationofthecorrespondingterminsummationsatisesE[jQni=1Pr(B~mijP)[Qj2SiWm(tj)i)]TJ /F5 7.97 Tf 6.59 0 Td[(1Qk2Sci(i)]TJ /F4 11.955 Tf 11.95 0 Td[(1)fWm(tk)+kgi)]TJ /F5 7.97 Tf 6.59 0 Td[(2Pr(B~mkjP)] Qni=1(B~mi)j]E[jQni=1Pr(B~mijP)n()]TJ /F4 11.955 Tf 11.95 0 Td[(1)jScjPr(B~mk0jP) Qni=1(B~mi)j]=const.E[fQi6=k0Pr(B~mijP)gPr(B~mk0jP)2 Qni=1(B~mi)] (B)Comparing( B )to( B ),itfollowsthatE[fQi6=k0Pr(B~mijP)gPr(B~mk0jP)2 Qni=1(B~mi)]=E[fm(x1,...,xnjP)]mYj=1ak01,...,k0j+nk01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1+1 ak01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+ak01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+nk01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1+1wherenk01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1=]fj:Xj2Bk01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1g.Whenm>n,wespecifytheparametersasa1,...,,m)]TJ /F9 5.978 Tf 5.76 0 Td[(10=a1,...,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1=m2whichimpliesthatwhenmislarge,ak01,...,k0j+nk01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1+1 ak01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0+ak01,...,k0j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+nk01,...,k0j)]TJ /F9 5.978 Tf 5.75 0 Td[(1+1!1 2,andE[fm(x1,...,xnjP)]isnite.Therefore,E[fQi6=k0Pr(B~mijP)gPr(B~mk0jP)2 Qni=1(B~mi)]!0asm!+1.SoallthetermswithjScj1in( B )eventuallygoesto0.TheonlytermleftiswhenjScj=0,i.e.S=,whichcompletestheprooftotheclaim.Asforthecensoreddata,theircontributiontothelikelihoodfunctionissimpler.Thecontributionof(tk,dk=0)atmthlevelofPolyatreeisjust Pk(B~m+jP)=fmXn=1Pr(B1,...,n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1jP)0(n)+Pr(B~mjP)gk=Wm(tk)k(B) 64

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Beforeweputtheresultsfrom( B )and( B )together,letusgureoutwhat~m(ti)is.Bythemechanismofpartition,form>n,andi=1,...,n,tiisanendpointatleveliofthetree.Beforetheithlevel,tiliesintherightsubintervaleverytimethecurrentintervalsplitsintotwo;afterithlevel,tiwouldbealwaysintheleftsubintervalgeneratedbysplittingthecurrentintervalthatcontainsti.Thus,~m(ti)=1,...,1| {z }i,0,...,0| {z }m)]TJ /F8 7.97 Tf 6.59 0 Td[(iClearly,Pr(B~mijP)=Y1Y11...Y1,...,1| {z }iY1,...,1| {z }i,0...Y1,...,1| {z }i,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(i.AndbydenitionofWm(t), Wm(t1)=Y1,Wm(t2)=Y1Y11,,Wm(tn)=Y1Y11...Y1,...,1| {z }n. (B) TheseleadtonYi=1Pr(B~mijP)=fY1Y10Y100...Y1,0,...,0| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(1gfY1Y11Y110Y1100...Y11,0,...,0| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(2gfY1Y11...Y1,...,1| {z }nY1,...,1| {z }n,0Y1,...,1| {z }n,00...Y1,...,1| {z }n,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(ng (B)ItturnsoutthatalltheseY'sarenotindependentbecause Y1,...,j,1=1)]TJ /F7 11.955 Tf 11.95 0 Td[(Y1,...,j,0(B)forallj=1,...,nandall(1,...,j).Combiningtheresultsfrom( B ),( B ),( B )and( B )yieldsthatthecontribution,atlevelm,of(k,tk,k)tothelikelihoodisasymptoticallyequalto 65

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Lk,m=Pr(B~mkjP)kWm(tk)k)]TJ /F5 7.97 Tf 6.58 0 Td[(1 (B~mk)=Y1Y11...Y1,...,1| {z }kY1,...,1| {z }k,0...Y1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(kk(Y1Y11...Y1,...,1| {z }k)k)]TJ /F5 7.97 Tf 6.59 0 Td[(1 (B1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(k)=Y1,...,1| {z }k,0...Y1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(kk(Y1Y11...Y1,...,1| {z }k)k (B1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(k) (B)When=0,tkisacensoredtime.Thusthecontributiontothelikelihood,givenP,isLk,m=Wm(Xk)k=fY1Y11...Y1,...,1| {z }kgkCombiningtheprevioustwoequations,thecontributionof(k,tk,k)tothelikelihoodisLk,m=(Y1Y11...Y1,...,1| {z }k)kfkY1,...,1| {z }k,0...Y1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(k (B1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(k)gkThusthetruelikelihoodfunctionhastheformL=E[limm!+1nYk=1Lk,m]Again,ByLavine(1992),thedensityexistsandisnite.Bydominatedconvergencetheorem,wecanexchangetheorderoftheexpectationandthelimit. 66

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DeneLm=nYk=1Lk.m=YPni=1i1YPni=2i11YPni=ki1,...,1| {z }kYn1,...,1| {z }nnYj=1fY1,...,1| {z }j,0Y1,...,1| {z }i,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(i (B1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(k)gj=[nYk=1kk (B1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(k)k](1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y0)Pni=1i(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y10)Pni=2iY111(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y1,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0)Pni=kiYk)]TJ /F9 5.978 Tf 5.76 0 Td[(11,...,1| {z }k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0(1)]TJ /F7 11.955 Tf 11.96 0 Td[(Y1,...,1| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0)nYn)]TJ /F9 5.978 Tf 5.75 0 Td[(11,...,1| {z }n)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0fY100Y1000Y1,0,...,0| {z }m)]TJ /F9 5.978 Tf 5.75 0 Td[(1g1fY1100Y11000Y11,0,...,0| {z }m)]TJ /F9 5.978 Tf 5.75 0 Td[(2g2fY1,...,1| {z }n,0Y1,...,1| {z }n,00Y1,...,1| {z }n,000Y1,...,1| {z }n,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(ngnWithallY'sappearedinlastequationbeingindependentlybetadistributed,takingexpectationleadsto 67

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E[Lm]=nYk=1f1 (B1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(k)gkn)]TJ /F5 7.97 Tf 6.58 0 Td[(1Yk=1fa1,...,1| {z }k,0gkfa1,...,1| {z }n,0 a1,...,1| {z }n,0+a1,...,1| {z }n,1gnnYk=1fa1,...,1| {z }k,00 a1,...,1| {z }k,00+a1,...,1| {z }k,01a1,...,1| {z }k,000 a1,...,1| {z }k,000+a1,...,1| {z }k,001a1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(k a1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.75 0 Td[(k+a1,...,1| {z }k,0,...,0| {z }m)]TJ /F15 5.978 Tf 5.76 0 Td[(k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1gk 68

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APPENDIXCPROOFOFTHEOREM4.1Foraxedintegerk,letFk(x)betheapproximatecumulativedistributionfunctionofPT(F0)truncatedatthekthlevelofthetree.Thusforanyx2R,thereexist1,,k2f0,1gsuchthatx2B1,,k.Itfollowsthat Fklow(x)Fk(x)Fklow(x)+P(B1,,k),(C)where Fklow(x)=kXi=1P(B1,,i)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0)1(i),(C)with1()beingdenedasanindicatorfunctionof1.Itisstraightforwardthat E[Fklow(x)]E[Fk(x)]E[Fklow(x)+P(B1,,k)].(C)Theleftandrighthandsideof( C )canbeexpressedexplicitlysince P(B1,,i)=iYj=1a1,,j a1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1(C)Whendatabecomeavailable,Fk(x)isreplacedbyFk(xjdata),whichisboundedaccordinglywith P(B1,,ijdata)=iYj=1a1,,j+n1,,j a1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+n1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1.(C)Notethat( C )usetherelationn1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1=n1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+n1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1implicitly.Inaddition,thedensityofGexistsbyassumption,whoseexpectationisgivenby E[gk(x)]=1 (B1,,k)[kYj=1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1],(C)where()istheLebesguemeasure.Thusby( C ), I1Z+1Fk(x)dGk(x)=Z+1Fk(x)gk(x)dxI2,(C) 69

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whereI1=Z+1E[Fklow(x)]gk(x)dxI2=Z+1E[Fklow(x)+P(B1,,k)]gk(x)dx.ItisstraightforwardtocheckthatI2)]TJ /F7 11.955 Tf 11.95 0 Td[(I1!0becauseaskincreases,I2)]TJ /F7 11.955 Tf 11.96 0 Td[(I1=Z+1P(B1,,k)[kYj=1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1]dx=X1,,k=0or1P(B1,,k)[kYj=1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1]!0. (C)Theequation( C )holdsbecausekYj=1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1!0. (C)foranycombinationof1,,kandP1,,kP(B1,,k)=1.Combining( C )and( C )andthefactthatI2isnitesinceitisaprobability,wehavelimk!1I1=limk!1I2=limk!1Z+1Fk(x)dGk(x)=^P(X
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Thereare2ktermstobeaddedtogetherin( C ).Weproceedcalculationsin( C )bygroupingthe2kintervalsatlevelkinto2k)]TJ /F5 7.97 Tf 6.59 0 Td[(1groupsbysameparents.Specically,foranyxedcombinationof1,,k)]TJ /F5 7.97 Tf 6.58 0 Td[(1,B1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0andB1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1willbegroupedtogethertocarryoutthecalculationbecausetheycomefromthesameparent,whichisB1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1.Thatsaid,forsimplicityletk=0andFF=E[kXi=1P(B1,,i)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0)1(i)]=E[k)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xi=1P(B1,,i)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0)1(i)+P(B1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0)1(k)]=E[k)]TJ /F5 7.97 Tf 6.58 0 Td[(1Xi=1P(B1,,i)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0)1(i)] (C)denotingtheexpectedprobabilityofallsubsetstotheleftofB1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0.Nowifweaddthetwotermscorrespondingto(1,,k)]TJ /F5 7.97 Tf 6.59 0 Td[(1,0)and(1,,k)]TJ /F5 7.97 Tf 6.59 0 Td[(1,1),denotedbyter(1,,k)]TJ /F5 7.97 Tf 6.58 0 Td[(1,0)andter(1,,k)]TJ /F5 7.97 Tf 6.58 0 Td[(1,1)respectively,itfollowster(1,,k)]TJ /F5 7.97 Tf 6.59 0 Td[(1,0)+ter(1,,k)]TJ /F5 7.97 Tf 6.59 0 Td[(1,1)=FF[kYj=1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1]+[FF+fkYj=1a1,,j+n1,,j a1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1+n1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1g][kYj=1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1]=FF[k)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yj=1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1]+a1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0+n1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0 a1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+n1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1b1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+m1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1 b1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0+b1,,k)]TJ /F9 5.978 Tf 5.75 0 Td[(1,1+m1,,k)]TJ /F9 5.978 Tf 5.76 0 Td[(1k)]TJ /F5 7.97 Tf 6.59 0 Td[(1Yj=1a1,,j+n1,,j a1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+a1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+n1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1b1,,j+m1,,j b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0+b1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1+m1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1. (C)Notethat( C )holdsforanycombinationof1,,k)]TJ /F5 7.97 Tf 6.59 0 Td[(1.Thuswhenwetakethesummationin( C ),wecanusetheresultsin( C )foreachofthe2k)]TJ /F5 7.97 Tf 6.59 0 Td[(1groups.Thereforeweendupwithasummationofthelasttermin( C )overallpossible 71

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combinationsof1,,k)]TJ /F5 7.97 Tf 6.58 0 Td[(1,whichisthedenitionofLkin( 4 ).ThesummationoftheleftoverFFtermsisnothingbuttheanalogouscalculationatlevelk)]TJ /F4 11.955 Tf 12.57 0 Td[(1.Thusbyrecursion, I1=L1+L2++Lk.(C)Therefore, ^P(X
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APPENDIXDPROOFOFCOROLLARY1Takea1,,k=b1,,k=0forall1,,k=0or1andallk2N.Firstxk,bydirectlypluggingin( C ),I1=n0 nm1 m+n0 nn00 n0m0 mm01 m0+n1 nn10 n1m1 mm11 m1+=n0m1 mn+n00m01 mn+n10m11 mn+=1 mnkXj=1[X1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1n1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,0m1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1] (D)Whenkislargeenoughsuchthatatlevelk,eachsubintervalcontainsatmostoneobservationfromsampleXandsampleYeach,thereareonlynitelymanytermsin( D )thatarenonzero.Somecarefulthoughtson( D )leadstotheconclusionthatkPj=1[P1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1n1,,j)]TJ /F9 5.978 Tf 5.75 0 Td[(1,0m1,,j)]TJ /F9 5.978 Tf 5.76 0 Td[(1,1]isjustanotherwaytocountPi=1,,n;j=1,,mI(Xi
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BIOGRAPHICALSKETCH LeiHuangreceiveshisPh.DdegreeunderthementorshipofMalayGhosh,aninternationallyrenownedexpertofBayesianstatistics,fromtheDepartmentofStatisticsatUniversityofFloridain2013.HewillbejoiningNovartisCo.asaseniorbiostatistician.Broadly,hismainresearchinterestsincludeBayesiannonparametricmethodsforhypothesistestingandsurvivalanalysis,especiallywithPolyatreeprocessespriors.LeireceivedtheStatisticsFacultyAwardandtheWilliamMendenhallAwardinrecognitionofbeingchosenastheoutstandingstudentinthedepartment.In2008,LeigraduatedfromBeijingNormalUniversityinChinawithhisB.S.instatistics.Heenjoysplayingtennisandlisteningtomusicinhisrecreationaltime. 78