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1、7HierarchicalModeling7.1IntroductionInthischapter,weillustratetheuseofRtosummarizeanexchangeablehierarchicalmodel.Webeginbygivingabriefintroductiontohierarchicalmodeling.Thenweconsiderthesimultaneousestimationofthetruemortal-ityratesfromhearttransplantsforalargenu
2、mberofhospitals.Someoftheindividualestimatedmortalityratesarebasedonlimiteddata,anditmaybedesirabletocombinetheindividualratesinsomewaytoobtainmoreaccurateestimates.Wedescribeatwo-stagemodel,amixtureofgammadistributions,torepresentpriorbeliefsthatthetruemortalityr
3、atesareexchangeable.WedescribetheuseofRtosimulatefromtheposteriordistribution.Wefirstusecontourgraphsandsimulationtolearnabouttheposteriordistributionofthehyperparameters.Oncewesimulatehyperparameters,wecansimulatefromtheposteriordistributionsofthetruemortalityrate
4、sfromgammadistribu-tions.Weconcludebyillustratinghowthesimulationofthejointposteriorcanbeusedtoperformdifferenttypesofinferencesinthehearttransplantapplication.7.2ThreeExamplesInmanystatisticalproblems,weareinterestedinlearningaboutmanypa-rametersthatareconnectedin
5、someway.Toillustrate,considerthefollowingthreeproblemsdescribedinthischapterandthechapterstofollow.1.SimultaneousestimationofhospitalmortalityratesInthemainexampleofthischapter,oneisinterestedinlearningaboutthemortalityratesduetohearttransplantsurgeryfor94hospital
6、s.Eachhospitalhasatruemortalityrateλi,andsoonewishestosimultaneouslyestimatethe94ratesλ1,...,λ94.Itisreasonabletobelieveapriorithatthetrueratesaresimilarinsize,whichimpliesadependencestructureJ.Albert,BayesianComputationwithR,UseR,DOI10.1007/978-0-387-92298-07,©Sp
7、ringerScience+BusinessMedia,LLC20091547HierarchicalModelingbetweentheparameters.Ifoneistoldsomeinformationaboutaparticularhospital’struerate,thatinformationwouldlikelyaffectone’sbeliefaboutthelocationofasecondhospital’srate.2.EstimatingcollegegradepointaveragesInan
8、exampleinChapter10,admissionspeopleataparticularuniversitycollectatableofmeansoffreshmangradepointaverages(GPA)organizedbythestudent’shighschoolrankandh
