On Autocorrelation in a Poisson Regression Model

OnAutocorrelationinaPoissonRegressionModelRichard.A.Davis1ColoradoStateUniversityWilliam.T.M.DunsmuirUniversityofNewSouthWalesandYingWangColoradoStateUniversityMay22,1998AbstractThispaperisconcernedwithdevelopingapracticalapproachtodiagnosingtheexistenceofalatentstochasticprocessinthemeanofaPoissonregressionmodel.First,arigorousderivationoftheasymptoticdistributionofstandardGLMestimatesisderivedforthecasethatanautocorrelatedlatentprocessispresent.Simpleformulaefortheeectofautocovarianceonstandarderrorsoftheregressioncoecientsarealsoprovided.Second,thepaperexaminestestsforthepresenceofalatentprocessandconsidersestimatesoftheautocovarianceofthelatentprocess.Methodsforadjustingfortheseverebiasinpreviouslyproposedestimatorarederivedandtheirbehaviourinvestigated.ApplicationsofthemethodstotimeseriesofmonthlypoliocountsintheU.S.anddailyasthmapresentationsatahospitalinSydneyareusedtoillustratetheresultsandmethods.1IntroductionInthispaperweareconcernedwithmodelsforatimeseriesofobservedcounts,fYt:t=1;:::;ngwhichhavemeanfunctionspeciedbyalinearpredictormodiedbya\latentpro-cess.SuchmodelshavebeenconsideredbyZeger(1988),Campbell(1994),BrannasandJohansson(1994)andChanandLedolter(1995)forexample.Therehasbeenconsiderableeortinrecentyearsdevotedtothedevelopmentofmethodstoecientlytalltheparametersinthesetypeofmodels.Howeverallofthesetechniquesrelyontheidenticationofasuitablemodelforthecorrelationstructureinthenoiselatentprocess.AsinlinearregressionwithcorrelatederrorsthereisaneedfordiagnostictechniqueswhichcanbeappliedtodecideifitisnecessarytoincludealatentprocessinthespecicationofthemeanofthePoissoncountsandifso,isthereanyevidenceofautocorrelationinsuchaprocess.Thispapertakesapracticalperspectivetothemodellingoftimeseriesofcountsand,tosomeextent,followsmodellingapproachesthathaveprovedsuccessfulinlinearregression.Tobepreciseweconsiderthesituationforwhichthereisanon-negativetimeseriestsuchthatYtjt;xtsP(tt);(1.1)1ThisresearchsupportedinpartbyNSFDMSgrantNo.9504596wheret=exTtinwhichxtisap-vectorofobservedregressorsand=(1;;p)Tisavectorofregressioncoecients.Therstcomponentofxtisassumedthroughouttobeunitysothattheregressioncomponentalwaysincludesaninterceptterm.Itisfurtherassumedthatftgisanon-negativestationarytimeserieswithE(t)=1,variance2,autocovariancefunction(ACVF)(h)=E[(t1)(t+h1)]andautocorrelationfunction(ACF)(h)=(h)=2.Theunitmeanconditionisimposedforidentiabilityreasons;otherwise,ifE(t)6=1,thenthemeancouldbeabsorbedintotheintercepttermofthelinearregression(see,forexample,ChanandLedolter,1995,p.247).TheabovespecicationofalatentprocessisthatsuggestedinZeger(1988)andusedinCampbell(1994).Analternative,usedinChanandLedolter(1995)isYtjt;xtsP(et+xTt);wheretisalatentprocesswhichisassumedtobeastationaryGaussianprocesswithmean,variance2andautocovariancefunctiongivenby(h)=E[(t)(t+h)].WithappropriatechoiceofmeanandvariancethisisthespecialcaseoftheZeger(1988)specicationinwhichthethavealog-normaldistribution.Inparticular,inordertosatisfytheidentiabilityrequirementthatE(t)=E[exp(t)]=1itisrequiredthattsN(2=2;2).Notethat,forthischoiceofmeanandvarianceinthelog-normaldistribution,(h)=exp((h))1forallh.Theregressorsxtmaydependonnandsoformatriangulararray.Aswillbemadeclearlaterthisisoftenrequiredinorderforthe\informationmatrixassociatedwiththeestimationproceduretodivergeasn!1.Forexampleifalineartimetrendisincludedinthemodel,thenitisnecessarytodividetimebythesamplesizen(e.g.xt=t=n)inordertoallowforthecasewherethetrendcoecientmightbenegative(inwhichcasethePoissonmeanwilleventuallybecomearbitrarilyclosetozero).Zeger’streatmentoftheabovemodelisbasedonaquasi-likelihoodapproachusedtocorrectforserialcorrelationinthelatentprocessftg.Assumptionsonthedistributionalpropertiesofthisprocessarenotexplicitlystatedbutformuchofhistreatmentthesearenotrequired.HoweverforthealternativespecicationintermsoftheftgprocesstherequirementthatthesebenormallydistributedismadequiteexplicitlyinthetreatmentinChanandLedolter(1995).Detectionofautocorrelationinthelatentprocessusingtheobservedcountprocess,Yt,isnotstraightforwardasweshowinSection2below.TypicallyuseoftheautocorrelationsforYtwillleadtounderestimatesofthetruesizeofautocorrelationinthelatentprocesst.ThisisnotedinZeger(1988).Also,inpractice,theregressiononxtwillneedtobeperformedbeforeattemptingtoestimatethisautocorrelation.Howeversuchestimationshouldadjustforautocorrelationinordertoarriveatcorrectinferences.Inordertotestfortheexistenceoflatentprocessandsubsequentlyidentifyitscorrelationstructure,itisnecessarytohaveaconsistentestimationprocedurefortheregressioncoecientvector.Anaturalandeasywaytocomputesuchestimatesisobtainedfromperformingastandardgeneralisedlinearmodel(GLM)analysis.InSection3,weestablishtheconsistencyandasymptoticnormalityoftheGLMestimates^whenastationaryautocorrelatedprocessispresentinthemeanofthePoissoncounts.Theseresultsareanalogoustolongstandingresultsforlinearregressionwithautocorrelatederrorsbutdierinthat,whiletheor