Spatially correlated allocation models for count d

SpatiallycorrelatedallocationmodelsforcountdataPeterJ.GreenUniversityofBristol,UK.SylviaRichardsonyImperialCollege,London.9August2000AbstractSpatialheterogeneityofcountdataonararephenomenonoccurscommonlyinmanydomainsofapplication,inparticularlyindiseasemapping.Wepresentnewmethodologytoanalysesuchdata,basedonahierarchicalallocationmodel.WeassumethatthecountsfollowaPoissonmodelatthelowestlevelofthehierarchy,andintroduceanitemixturemodelforthePoissonratesatthenextlevel.Thenoveltyliesintheallocationmodeltothemixturecomponents,whichfollowsaspatiallycorrelatedprocess,thePottsmodel,andintreatingthenumberofcomponentsofthespatialmixtureasunknown.InferenceisperformedinaBayesianframeworkusingreversiblejumpMCMC.ThemodelintroducedcanbeviewedasaBayesiansemiparametricapproachtospecifyingexiblespatialdistributioninhierarchicalmodels.Itcouldalsobeusedincontextswherethespatialmixturesubgroupsarethemselvesofinterest,asinhealthcaremonitoring.Performanceofthemodelandcomparisonwithanalternativewell-knownMarkovrandomeldmodelspecicationforthePoissonratesaredemonstratedonsyntheticdatasets.Wefoundthatourallocationmodelavoidstheproblemofoversmoothingincaseswheretheunderlyingratesexhibitdiscontinuities,whilegivingequallygoodresultsincasesofsmoothgradient-likeorhighlyautocorrelatedrates.ThemethodologyisillustratedonepidemiologicalapplicationstodataonrarediseaseandhealthoutcomeinFrance.Somekeywords:Allocation,Bayesianhierarchicalmodel,Diseasemapping,Finitemixturedis-tributions,Heterogeneity,MarkovchainMonteCarlo,Poissonmixtures,Pottsmodel,Reversiblejumpalgorithms,Semiparametricmodel,Split/mergemoves.1IntroductionWeconsiderthemodellingofspatialheterogeneityforcountdataonararephenomenon,observedinapre-denedsetofareas.Thisisageneralset-upwhicharisesinmanydomainsofapplication,forexampleinecologyoragriculturalscience.Themotivatingsituationthatwehaveinmindthroughout,thatofobserveddiseasecountswithfewobservedeventsineacharea,belongstothedomainofepidemiology.Therearemanyreasonsforsuspectingheterogeneityintheunderlyingeventrateandwantingtocharacteriseit.Indiseasemapping,forexample,itishardlyplausiblethatalltherelevantfactorsactingontheunderlyingdiseaseriskcanbeidentiedormeasuredatthearealevel.Thusthereremainsresidualheterogeneity,whichislikelytohaveaspatialstructureDepartmentofMathematics,UniversityofBristol,BristolBS81TW,UK.Email:[e-mail]yandPublicHealth,ImperialCollegeSchoolofMedicine,NorfolkPlace,London,W21PG.Email:[e-mail]ogeneityalsoarisesnaturallyinagriculturaleldtrialsandotherapplications.Itisinterestingtocharacterisethisspatialstructurefurther,asdiscoveryofeitherlocaldiscon-tinuitiesorsmoothgradientscanbeexploitedforfurtherstudyoraction.Inepidemiology,currentaetiologicalhypothesesmadeattheindividuallevelcanbeusefullyconfrontedwiththeiraggre-gatedcounterparts,keepinginmindthedelicateissueofecologicalbias(GreenlandandRobins,1994).Thesuspicionofalocalexcessindiseaseoccurrenceorthehighlightingofgeographicalinequalitiesinmedicaltreatmentareimportantpublichealthconcernsthatcanbeaddressedbyananalysisofthespatialheterogeneity.Notethat,mostoften,weareinanobservationalframeworkwherethereislittleornocontroloverthesourcesofvariability.Ouraiminthispaperisthustoproposeanewclassofspatialmodelsfortheheterogeneityofcountdataandtodemonstrateitsexibility.Thisproblemhasusuallybeenaddressedinahierarchicalframework.WeshalldothesameinthedevelopmentofourBayesianapproach,andspecicallyconsideraPoissonmodelatthelowestlevelofthehierarchy:yiPoisson(iEi)independentlyfori=1;2;:::;n(1)whereyidenotestheobservedcountofdiseaseincidencesordeathsinareai,Eiistheexpectedcountbasedonpopulationsize,adjustedforageandsex,etc.,andiisanarea-specicrelativeriskvariable,themainobjectofourinference.Weusethesimpleterm\risktorefertothefiginfuture.Thequestionwhicharisesnextisthatofthechoiceofstructureforthejointdistributionofthefi;i=1;2;:::;ngatthenextlevelofthehierarchy,achoicewhichcanbeinuentialontheeectivesmoothingofthePoissonnoise.IntheseminalworkofBesag,YorkandMollie(1991)andClaytonandBernardinelli(1992),andsubsequentpapers,loglinearGaussianmodelsforthefigwerepostulatedusingaconditionalformulationthatincludedaspatialautoregressivecomponentbasedoncontiguityinanundirectedgraph.Thisapproachhasbeencommonlyadoptedinrecentworkindiseasemapping,andhashelpedtohighlightmanyinterestingfeaturesofthegeographicaldistributionofsomerarediseases.AlternativeformulationsalsoinvolvingamultivariateGaussianmodelfortheflogigwithspatiallyparametrisedcovariancematrixhavealsobeendiscussed(WakeeldandMorris,1999).Inbothcases,theparameterscharacterisingthespatialdependenceareconstantacrosstheentirestudyregion.Whenusingtheseparametricmodels,thereisthepotentialriskofover-smoothingandmaskingoflocaldiscontinuities,duetotheglobaleectoftheparameters.Thisdicultycanbeaddressedbythedevelopmentofclusteringmodels(Knorr-HeldandRaer,2000,DenisonandHolmes,2000),thatprov