A new variance estimator for parameters of semiparametric ge

ANewVarianceEstimatorforParametersofSemiparametricGeneralizedAdditiveModels

W.DanaFLANDERS,MitchKLEIN,andPaigeTOLBERT

Generalizedadditivemodels(GAMs)havebecomepopularintheairpollutionepi-demiologyliterature.Twoproblems,recentlysurfaced,concernimplementationofthesesemiparametricmodels.The rstproblem,easilycorrected,waslaxityofthedefaultconver-gencecriteria.Theother,notedindependentlybyKlein,Flanders,andTolbert,andRamsay,Burnett,andKrewskiconcernedvarianceestimatesproducedbycommerciallyavailablesoftware.Insimulations,theywereasmuchas50%toosmall.Wederiveanexpressionforavarianceestimatorfortheparametriccomponentofgeneralizedadditivemodelsthatcanincludeuptothreesmoothingsplines,andshowhowthestandarderror(SE)ingMonteCarloexperiments,weevaluatedperformanceoftheestimatorin nitesamples.TheestimatorperformedwellinMonteCarloexperiments,inthesituationsconsidered.However,ingdatafromourstudyofairpollutionandcardiovasculardisease,thestandarderrorestimatedusingthenewmethodwasabout10%to20%largerthanthebiased,commerciallyavailablestandarderrorestimate.

KeyWords:Epidemiologicmethods;Generalizedadditivemodels;Semiparametricmod-els;Variance.

1.INTRODUCTION

Generalizedadditivemodels(GAMs),arelativelynewapproachtononparametricorsemiparametricsmoothinganddataanalysis(HastieandTibshirani1990),havebecomewidelyused,particularlyintimeseriesanalysesofacutehealtheffectsofairpollution.Insemiparametricmodels,thefocusofthisarticle,themeanofthedependentvariableismodeledasaparametric,linearfunctionofsomepredictorsplusasumoffunctionsofotherpredictors,whichinsomeapplicationsmaybeconfoundersornuisancefactors.Theformofthefunctionusedfortheseotherpredictorsisquitegeneral,hencethetermsemiparametric.W.DanaFlandersisProfessor,RollinsSchoolofPublicHealth,DepartmentofEpidemiology,EmoryUniversity,1518CliftonRoad,Atlanta,GA30327(E-mail: anders@sph.emory.edu).MitchKleinisAssistantProfessor,andPaigeTolbertisAssociateProfessor,RollinsSchoolofPublicHealth,DepartmentofEpidemiology andDepartmentofEnvironmentalandOccupationalHealth,EmoryUniversity,RollinsSchoolofPublicHealth,1518CliftonRoad,Atlanta,GA30327.

©2005AmericanStatisticalAssociationandtheInternationalBiometricSociety

JournalofAgricultural,Biological,andEnvironmentalStatistics,Volume10,Number2,Pages246–257DOI:10.1198/108571105X47010

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VARIANCEOFSEMIPARAMETRICGAMS247

SchwartzproposedapplicationofGAMStotimeseriesstudiesassessingtheassociationofairpollutionwithmortalityorotheroutcomemeasuresin1994(Schwartz1994a),andinitiallypresentedGAMmodelsasasensitivityanalysisaugmentingaparametricapproach(Schwartz1994b).Intheinterveningyears,GAMshavegainedwidespreadpopularityforuseinthesetypesoftimeseriesstudies(e.g.,Borja-Aburtaetal.1998;Michelozzietal.1998;Burnettetal.1999;Conceicaoetal.2001;Moolgavkar2000;Pope,Hill,andVillegas1999;Sametetal.2000).

GAMscangenerallybe tusingS-PlusorusingPROCGAMinSAS(SAS2001).Asdiscussedinthefollowing,themodelscanbe tusingaback ttingalgorithm.HastieandTibshirani(1990)discussedconditionsthatassureconvergenceofthisapproach.Twoproblemshaverecentlysurfaced,however,concerningimplementationofthesemodels.The rstproblem,easilycorrected,wasthatthedefaultconvergencecriteriawerenotadequatelystrict(Dominici,McDermott,Zeger,andSamet2002;Katsouyannietal.2002).TheotherproblemconcernsthevarianceestimatesproducedbytheseprogramsfortheparametriccomponentofthesemiparameticGAMs.TheproblemwasnotedindependentlybyKlein,Flanders,andTolbert(2002)andbyRamsay,Burnett,andKrewski(2003).Theyshowedthatthevarianceestimatescouldbeasmuchas50%lowerthanthesimulatedvarianceinsomeofthesituationsconsidered.Thisarticleaddressesthesecondproblembyderivingarelativelyeasilyimplementablevarianceestimatorforthesemodels.

Oneofthespeci cproblemsthatmotivatedthisworkarethenumerouspublishedorongoingstudiesoftheassociationsbetweenhealthoutcomes,suchasrespiratorydisease,andairpollution(e.g.,Borja-Aburtaetal.1998;Michelozzietal.1998;Burnettetal.1999;Conceicaoetal.2001;Moolgavkar2000;Pope,Hill,andVillegas1999;Sametetal.2000;Tolbertetal.2000).Someofthestudiespublishedbyothershaveusedgeneralizedadditivemodelstoassesstheassociationbetweenairpollutionanddisease,butanappropriatevarianceestimatorhasbeenunavailable.

Thepurposeofthisarticleisthree-fold.First,wepresentanasymptoticvarianceesti-matorfortheparametriccomponentofGAMsemiparametricmodels,providinganexplicitformulationforuptothreesplines.Second,weempiricallyevaluatetheperformanceofthisestimatorin nitesamplesusingMonteCarlosimulationsandbasethesesimulationsonactualdatafromanongoingstudyofairpollution.Finally,weapplytheestimatortoanongoingstudyofairpollutionandemergencydepartmentvisits.Weillustratethat,inthisstudy,thevarianceweestimatediffersfromthecorrespondingestimatesproducedbycommerciallyavailablesoftware.

2.METHODS

Inthesemiparametricsituationsofinteresthere,thegeneralizedadditivemodelisgivenby:

E(Yi|Xi,Z1i,...,ZJi)=g–1(ηi)=g 1(α+βXi+f1(Z1i)

+···+fJ(ZJi)),i=1,2,...,n,(2.1)

248W.D.FLANDERS,M.KLEIN,ANDP.TOLBERT

whereYiisthenumberofeventsfortheithobservation;gisastrictlymonotonelinkfunction;ηi=α+βXi+f1(Z1i)+···+fJ(ZJi);βis(p×1)parameterofinteresttobeestimated;Xiisa(1×p)vectorofpredictors;Zjiisthevalueofthejthcovariatefortheithobservation;andfj(Zji)isanarbitrary(smoothing)functionwithcontinuoussecondderivatives,forj=1toJ.(Here,welimitconsiderationtoJ≤3,butresultsshouldextendinananalogouswayforJ>3.)WeassumeherethattheYigivenXi,andZ1iareindependentwithaPoissondistributionwhosemeanisgivenbyEquation(2.1).ThePoissondistributionistypicallyusedinapplicationsinairpollutionepidemiology.However,resultsholdwithobviousmodi cationsforotherdistributionsintheexponentialfamily(HastieandTibshirani1990).

WederiveanexplicitexpressionforthevarianceestimatorforaclassofestimatorsofβinthemodelgivenbyEquation(2.1),estimatedbypenalizedlikelihood(HastieandTibshirani1990).Thatis,onemaximizes

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