A new variance estimator for parameters of semiparametric ge(3)

ResultsofthesecondsetofMonteCarloexperiments—basedonhypotheticaldata(situation2)—areshowninTable2.Theseresultsfurthersupportthesesamepatterns.

VARIANCEOFSEMIPARAMETRICGAMS

Table3.

Experiment/

Distribution

8 /Poisson

9/Poisson

10/Poisson

11/Poisson

12/Poisson

13/Normal

14/BinomialMonteCarloSimulationResults,ExpectedValuesCalculatedHypotheticalData253N,df100,6100,450,650,450,2100,2100,4 )bS Trueβ1aSE(βE oldc.031.044.031.056.076 .038.037.0100.0083.0134.0121.0119.0920.0116.0076.0072.0072.0111.0112.0904.0088CoveragecCoveraged 95%CI oldSE newd95%CI new89.4%82.4%71.6%92.2%93.3%94.7%79.9%.0100.0085.0133.0119.0119.0913.011395.1%93.4%94.4%94.8%94.9%94.9%94.4% EachexperimentislikeExperiment6inTable2,exceptfor:theerrordistribution,thenumberoftimepoints(N),andthedegreesoffreedom(df).

aValueofβ1usedforeachsetof1,000MonteCarloexperiments.

b 1foreachsetof1,000experiments.StandarderrorofβcMeanestimatedstandarderror,andcoverageof95%CIproducedbySASPROCGAM(3.1);95%CIcalculatedasthepointestimate+/ 1.96timestheestimatedstandarderror.

dMeanestimatedstandarderror,andcoverageof95%CIbasedonnewvarianceestimator;95%CIcalculatedasthepointestimate+/ 1.96timestheestimatedstandarderror.

Speci cally,theoldstandarderrorestimatesareconsistentlylowerthanthesamplestandarderroroftheestimatedβ’s,andthecoverageoftheassociatedcon denceintervalswasconsistentlylowerthanthenominal95%level.

InthethirdsetofMonteCarlosexperiments,thenewestimatorofthestandarderroryieldedresultsclosetothesimulatedstandarddeviationoftheestimatedβ,whenwereducedthenumberoftimepointsfrom100asinearlierexperimentstoeither100or500inthese.Inafewexperimentswiththelowernumberoftimepointsandifweusedmorethan2–4

differedslightly,butsigni cantlyfromdegreesoffreedomforeachspline,theaverageβ

thetrueβ(datanotshown).

5.EXAMPLE:APPLICATIONOFNEWMETHOD

WeappliedtheestimatortodatafromanongoingstudyofairpollutionandEDvisitsforcardiorespiratorydiseasesinAtlanta(Tolbertetal.2000),thedatausedherefromAugust1,1998,toJuly31,1999.OurrationaleforusingGAMsre ectstheirinherentappealdueinparttothesemiparametricnatureofthetimedependencyandconsequentrelaxationofassumptions,andthefrequentuseofthesemodelsintheairpollutionliterature.Wecalculatedthestandarderrorforparameterestimatesfromasemiparameticgeneralizedadditivemodel,withthePoissondistributionandloglink,executedusingPROCGAMinSAS,andcomparedthisestimatetothestandarderrorestimatedwithournewmethod.WechosethedegreesoffreedomforthesplinestobesimilartothoseweusedinparametricPoissonregression;Useofgeneralizedcross-validation,thedefaultapproachinSAS(SASInstitute2001),suggestedslightlyfewerdegreesoffreedom,butledtothesameresults.Weevaluatedtheassociationbetweennitrogendioxide(NO2)andERvisitsforallCVD,usingthethree-daymovingaverageofNO2,inpartduetoaprioriinterest,andcontrollingfor

254W.D.FLANDERS,M.KLEIN,ANDP.TOLBERT

time,formeantemperature,andfordewpointusingcubicsplineswith7,7,and5degreesoffreedom,respectively.Wealsocontrolledfordayoftheweekusingindicatorvariablesandthenumberofemergencyroomvisitsfornoncardiovasculardisease.Tosimplifycalculationsbyavoidingties,weaddedasmallrandomnumbertothetemperatureanddewpoint(whichdidnotchangetheestimateoftheparameteroritsstandarderror).Wefoundlittleevidenceofautocorrelationofresiduals(Durbin-Watson=2.155,p=.40bysimulation).InthemodelforCVDvisits,theparameterestimateforNO2was.020(rateratio=1.020)andthestandarderrorestimatedinPROCGAMwas.018.Thestandarderrorestimatedusingthenewestimatorwas.020,about10%largerthanthatobtainedusingSAS.Thisdifferenceisimportantforevaluatingthestabilityofresults,forintervalestimation,andwouldaffectanymeta-analysisthatusedthisresult.

6.DISCUSSION

OurresultsprovideevidencethatthevarianceestimatorinEquation(2.10)workswellwith nitesamples—atleastforthesituationsconsidered.Morework,however,needstobedonetoverifythatitsperfomanceremainsgoodunderotherconditions.OurresultsalsofurthersupportandareconsistentwiththeworkofKleinetal.(2002)andofRamseyetal.(2003)whoshowedthatthevarianceestimationproceduresusedincommerciallyavailableprogramscouldbeinadequate.Thesetendenciesofcommercialsoftwaretounderestimatevarianceshavebeenattributedtoconcurvityinthedata(Ramsayetal.2003),andtoin-adequatelinearapproximationsusedforthesmoothfunctions(Dominici,McDermott,andHastie2003).Recognitionoftheseandotherproblemshasmotivatedreanalysesofatleast20studiesofairpollutionandhealtheffectswiththeoverallconclusionthatuseofGAMs,implementedwiththefaultyvarianceestimator,wasassociatedwithsmallerstandarderrorsthanuseofgeneralizedlinearmodels(HealthEffectsInstitute2003).

Wehavepresentedandevaluatedavarianceestimatorforuptothreesplines,extendingthepreviousworkofHastieandTibshirani(1990)whopresentedexplicitresultsforasinglespline,andofFlanders,Klein,andTolbert(2003).Implementationofthisapproachformoresplinesisstraightforward;forexample,perhapsusingtherecursiveequationsgiven,butasthenumberofsplinesincreases,ofcourse,computationsbecomemoreandmoreonerous.OurresultalsoappliesdirectlytothecaseofonlyoneortwosplinesbysimplytakingS1and/orS2equalto0.

Ourargumentsdependheavilyontheassumptionofconsistencyofβandη,andconvergenceoftheback ttingalgorithmasarguedbyHastieandTibshirani(1990).Wehavenotinvestigatedperformanceofthevarianceestimatorwhenthatassumptionmightfail.ConditionsotherthanthosenotedbyHastieandTibshirani(1990)mayalsoleadtoconsistency.Inparticular,wemightalsoexpectconsistencyifthenumberof(say,time)pointsremains xed,buttheexpectedmeanincreasesforeachpoint,otherparametersremainconstant,andthemodeliscorrectlyspeci edwithjudiciouschoiceofdegreesoffreedomforthesplines.Moreworkonconsistency,notthefocusofthisarticle,remains.Forexample,theapplicationtotimeseriesshouldprobablybebasedonfutherspeci cationof

VARIANCEOFSEMIPARAMETRICGAMS255

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