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BAYESIANSTATISTICS 9,

J.M.Bernar do,M.J. Bayarri,J.O.Berger, A.P.Dawid,

D.Heckerman, A.F.M.Smith andM.West (Eds.)

c?OxfordUniversityPress,2010

BayesianVariableSelectionfo rRandom

InterceptModeling ofGaussianand

non-GaussianData DepartmentofAppliedStatistics andEc onometrics,JohannesKeplerUniversit¨atLinz, Austria

1.INTRODUCTION

Thepaper considersBayesianv ariableselectionforrandomin terceptmodelsboth forGaussian andnon-Gaussiandata.F orGaussian datathemodelreads y it =x it i it it N

0,σ

2 ,(1) whereyitarerepeated responsesobservedfor Nunits(e.g. subjects)i=1,...,Non Tioccasionst=1,...,Ti.xitisthe(1×d)design matrixforanunkno wnregression coe ffi cientα=(α1,...,α d ofdim ensiond,including theoverallin tercept.For each unit,βiisasubjectsp ecificdeviati onfromtheov erallintercept. Fore ffi cientestimationitisnecessary tospecifythedi stributionofheterogene- ityp(β1,...,βN).Asusual weassum ethat β1,...,βN|θareindep endentgiven arandomh yperparameter θwithpriorp(θ).Marginall y,therandomintercepts β1,...,βNaredependen tandp(β1,...,βN)actsa smoothi ngpri orwhichtiesthe randomintercepts togetherandencouragesshrinkage ofβitowardtheoverall inter- ceptby "borrowingstrength"from observationsofothersubjects.Averypopular choiceisthefollo wingstandardrandominterceptm odel: i |Q≂N(0,Q),Q≂G -1 (c 0 ,C 0 ),(2) whichisbasedonassumingconditional normalityof therandomin tercept. Severalpapersdealwith theissueofspeci fyingalternative smoothing priors p(β1,...,βN),because misspecifyi ngthisdistributionmayleadtoinefficient,and forrandomi nterceptmo delfornon-Gaussiandata,eventoinconsisten testimation oftheregressi oncoe ffi cientα,seee. g.Neuhauset al.(1992).Recently ,Kom´arek andLesa ff re(2008)suggested tousefinite mixtureof normal priorsforp(β i |θ)to handlethisissue. Inthepresentpaper wealsodeviatefromthe commonl yused normalprior(2)and considermoregeneralpriors.Ho wever, inaddition tocorrect estimationofα,ourfo cuswil lbeonBa yesianvariableselection. TheBay esianvariableselectionapproachiscomm onlyappliedtoastandard regressionmodelwhere β i isequalto0i n(1) forall unitsandaim satseparating

2S.Fr ¨uhwirth-SchnatterandH.Wagner

non-zeroregression coe ffi cientsαj?=0from zeroregressionco efficientsαj=0. Bycho osinganappropriatepriorp(α),it ispossible toshrinksomeco efficientsαr toward0andidentify inthisw ayrel evantcoe ffi cients.Commonshrinkage priors arespik e-and-slabpriors(MitchellandBeauchamp, 1988;GeorgeandMcCulloch,

1993,1997;Ish waranand Rao,2005),whereaspikeat 0(either aDiracmeasure

oradensi tywi thverysmallvariance)is combinedinthesl abwithadensity with largevariance.Al ternatively,unimodalshrinkage priorshavebeenappliedlikethe doubleexponential orLaplacepriorleadingtotheBay esianLasso(P arkandCasella,

2008)orthe moregeneral normal-gam maprior(Gri

ffi nandBro wn,2010); seealso

Fahrmeiretal.(2010)forarecen treview.

Subsequentlyweconsidervariabl eselectionfortherandom interceptm odel(1). Althoughthisalsoconcernsα,we willfocuson variableselectionfortherandom e ff ectswhic h,todate,hasbeendiscussedonlyb yafew papers.F ollowing Kinney andDunson(2007), Fr¨ uhwirth-Schnatter andT¨uchler(2008),andT¨uchler(2008) wecouldconsider variableselectionfor therandominterceptmo delasaproblemof varianceselection.Underpri or(2),forinstance,asinglebinary indicatorδcouldbe introducedwhereδ=0corresp ondstoQ=0,whi leδ=1all owsQtobe different from0.Thi sim plicitlyimpli esvariableselectionfortherandomintercept,because settingδ=0forces allβ i tobe zero,whileforδ=1all randomin terceptsβ 1 N areall owedbedi ff erentfrom0. Inthepresen tpaper weareinterested inaslightlym oregeneralvariableselection problemforrandome ff ects.Rather thandiscriminatingas abov ebetweenamodel whereall randome ff ectsarezero andam odelwhere allrandome ff ectsaredi ff erent from0,i tmi ghtbeof interesttomakeunit-specificselectionof randome ff ectsin ordertoi dentify unitswhichare"average"inthesensethat theydonot deviate fromtheo verall mean,i.e.β i =0,and unitswhi ch deviatesignificantlyfromthe "average",i.e.β i ?=0. Inanalogy tovariableselection instandardregression model,wewillshowthat individualshrinkagefortherandome ff ectscanb eachi evedthroughappropriatese- lectionofthepriorp(β i |θ)ofthe randomeffects.For instance,ifp(β i |Q)is aLaplace ratherthana normalpri orasin (2)witha randomhyperparameterQ,we obtain aBay esianLassorandome ff ectsmodels wherethesmoothingadditionall yallo ws individualshrinkageoftherandomin tercepttoward0forspecific units.How ever, asfora standardregression modelto omuc hshrinkagetakes placeforthenon-zero randome ff ectsunder theLaplaceprior. Forthi sreasonw einv estigatealternative shrinkage-smoothingpriorsfortherandominterceptmo dellikethespik e-and-slab randome ff ectsmo delwhichiscloselyrelatedtothe finitemixturesofrandom e ff ects modelinvestigated byFr¨uhwirth-Schnatteretal.(2004)andKom´arekand Lesa ff re (2008).

2.VAR IABLESELECTIONINRANDOMINTERCEPTMODELS THROUGH

SMOOTHINGPRIORS

ff ectsapproach couldbeappli ed,meaningthateachunit specificparameterβ i istreatedjustasan- otherregression coe ffi cientandthehi ghdimensional parameterα 1 N isestimatedfrom alargeregressionmodelwithoutan yrandomin tercept: y it =x it it it N

0,σ

2 .(3)

Wecouldthenp erformvariableselection forα

inthelargeregression model(3), inwhichcase abinaryvariableselectionindicatorδ i isintroduced foreachrandom BayesianVariable SelectionforRandom InterceptModels3 e ff ectβiindividually.Thisappearstobethesolutiontothevariable selection problemaddressedinthein troduction, howev er,vari ableselectionin(3)isnot entirelystandard:first,thedimension ofα growswiththen umberNofunits; second,ani nformationimbal ancebetweenthe regressioncoe ffi cientsαjandthe randomintercepts β i ispresent,because thenumber ofobservationsis N i=1 T i for j ,butonl yT i forβ i .This makeitdi ffi culttochoose thepriorp(α ).Under a(Dirac)-spi ke-and-slabpriorforp(α ),fori nstance,aprior hastobec hosenfor allnon-zerocoe ffi cientsinα .Anasym ptotically optimalchoiceinastandard regressionmodelis Zellner'sg-prior,however, theinformationimbalancebetween j andβ i makeitimpossi bletocho oseavalueforgwhichissuitablefor allnon-zero elementsofα Theinform ationimbalancesuggeststochoosethe priorfortheregressionco- e ffi cientsindependentlyfromthe priorfortherandomintercepts,i.e.p(α p(α)p(β 1 N ).Vari ableselectionforβ i inthelargeregression model(3) isthen controlledthroughthechoiceofp(β 1 N )whic hisexactlythesameproblem as choosingthesmoothingintheoriginal randomintercept model(1).Thismotivated ustouse commonshri nkage priorsinBayesian variableselectionassmoothing priors intherandomin terceptmodel andtostudy howthischoicee ff ectsshrinkage for therandomi ntercept. Practicallyallpriorshav eahierarc hicalrepresentationwhere i i andβ j j areindep endentandp(ψ i |θ)depends onahyperparameter θ.The goalis toidentifyc hoicesof p(ψ i |θ)whic hleadtostrongshrinkage ifmanyrandom interceptsareclosetozero, butintro ducelittlebias,if allunitsare heterogeneous.

Notethatthe marginal distribution

p(βi|θ)= p(βi|ψi)p(ψi|θ)dψi isnon-Gaussianandthat thejointdensit yp(β 1 N )is smoothingpriorin the standardsenseonly ,ifat leastsomecomponents oftheh yperparameter θareran- dom.

3.VAR IABLESELECTIONINRANDOMINTERCEPTMODELS USING

SHRINKAGESMOOTHINGPRIORS

Thissubsectiondealswith unimodalnon-Gaussianshrink agepriorswhic hputa lot ofprior masscloseto0, buthaveheavy tails.Suchaprior encouragesshrinkage of insignificantrandome ff ectsto ward0and,thesametime, allowsthat theremaining randome ff ectsm aydeviateconsiderablyfrom0.F orsuch aprior,theposterior modeofp(βi|yi,θ)is typicallyto 0withpositiveprobability.Wecall sucha prior anon-Gaussian shrinkageprior.

3.1.Non-GaussianShrinkage Priors

ChoosingtheinvertedG ammapri orψ

i |ν,Q≂G -1 (ν,Q)leads totheStudent-t randomintercept modelwhere i |ν,Q≂t 2ν (0,Q/ν).(5)

4S.Fr ¨uhwirth-SchnatterandH.Wagner

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