A new variance estimator for parameters of semiparametric ge(2)

where

V1 I S1S3–S1(I S3)(I S2S3) S2(I S3) S1I S3 (I S3)I S2S)3S2(I S3)

=I S2S1–S2(I S1)(I S3S1) S3(I S1) S2I S1 (I S1)[I S3S1] S3(I S1) =I S3S2 S3(I S2)[I S1S2]

S1(I S2)] S3I S2 (I S2)[I S1S2] S1(I S2),=(2.8)V2V3

i 1,f 1,i,f 2,i,f 3,i,andη.andZcisexpressedintermsofβ

Equation(2.8)isanexplicitformformultiplesmoothingsplines(intermsoftheindividualsmoothersS1,S2,andS3),oftheresultgivenbyHastieandTibshirani(1990)of

=[XtA(I S)X] (XtA(I S))Zc.BytakingS=(I V1 V2 V3),onetheform:β

seesthattheone-stepupdateforβinEquation(2.8)isconsistentwithHastieandTibshirani(1990)whogaveexplicitresultsforasinglespline,butnotedthatthesameformoftheequationwouldholdformultiplesplines.Anexplicitformformultiplesmoothingsplines(intermsoftheindividualsmoothers)canbeobtainedbyapplyingtherecursiveequationdescribedintheAppendix.

Becausetheestimatesconvergeinprobabilitybyassumption,wecanimaginestarting

istheprocessatthetruevalueηt.Then,withalargesamplesize,theone-stepestimatorβ

250W.D.FLANDERS,M.KLEIN,ANDP.TOLBERT

givenby:

=XtA(I V1 V2 V3)X XtA(I V1 V2 V3)Zt,β(2.9)

whereZt=ηt+A 1u—allparametersandexpressionsnowevaluatedattheirtruevalues.Thus,followingtheargumentsofbyMcCullaghandNelder(1989),theone-stepestimatorisalinearfunctionoftheobservations,andsubsequentupdatesshouldbenegligibleforalargesamplesize(asymptotically).

Thus,theasymptoticvarianceisgivenby:

)var(β

where

W=≈W var(ZT) Wt t W=XtA(I V1 V2 V3)XXA(I V1 V2 V3).

(2.10)

Furthermore,wecanestimatevar(ZT)byA 1andW,withallparametersevaluatedattheestimatedvalues.

3.SIMULATIONS

Toevaluatetheperformanceofthisvarianceestimator(Equation(2.10))in nitesamplesizes,weperformedMonteCarlosimulationsintwodifferentsetsofsituations—onebasedonrealdata,theotherusinghypotheticaldata.Inthe rstsituation,weuseddatafromourongoingstudyofemergencydepartment(ED)visitsforcardiorespiratorydiseasesandairpollutioninAtlanta(Tolbertetal.2000).TheoutcomevariablewaseitherdailyEDvisitsforalltypesofcardiovasculardisease(CVD)orforasthmafromAugust1,1998,toJuly31,1999.Weanalyzeddatafordailynitrogendioxide(NO2)andseparatelyfordailyparticulatematter(PM10),resultinginatotaloffourexperimentalconditionsbasedonrealdata(Table

1).WecontrolledforotherEDvisits,temperature,dewpoint,dayoftheweek,andtimeusingsmoothingsplinesinaGAMmodel:

EYi|Xi,Z1i,...,ZJi

=exp(α+βXi+f1(Z1i)+f2(Z2i)+f3(Z3i)),fori=1,2,...,n,(3.1)whereXti=(airpollutantondayi,allEDvisitsondayi,ITue,IWed,IThu,IFri,ISat,ISun),ITue ISunareindicatorsfordayi;Z1i=numberofdayssincethestartofthestudy,Z2i=themeantemperaturefordayi,andZ3i=themeandewpointondayi.Toavoidtiesanddivisionby0,weaddedasmallrandomnumber(mean0,variance.01)toeachtemperatureanddewpoint.Wethen tthisGAMtotheobserveddata,using14degreesoffreedomforthetimespline,correspondingapproximatelytomonthlyknots;and,5eachfortemperatureanddewpointsplines.(Weused7degreesoffreedomfortheCVDoutcome,correspondingapproximatelytoseasonalknots).Wethensavedthesemodelpredictedvalues,andusedtheminthesimulationsastheexpectedvalues,andgeneratedindependentlyforeachday,

VARIANCEOFSEMIPARAMETRICGAMS251

Table1.MonteCarloSimulationResults,ExpectedValuesCalculatedfromActualEDvisitsandAir

Pollutants,inAtlanta

CoveragedCoveragee

95%CI oldSE newe95%CI new

90.7%

93.4%

89.4%

90.2%.0123.0345.0226.038395.3%95.5%95.7%94.5% )cS OutcomePollutantTrueβ1bSE(βE olddCVDCVDAsthmaAsthma

a

bPM10NO2PM10NO2.011.006.040.014.0124.0336.0219.0378.0111.0313.0187.0314CVD(Asthma):Emergencyroomvisitsforcardiovasculardisease(asthma),usedtodetermineexpecteddailycountsValueofβ1usedtodetermineexpectedvalueofY,foreachsetof1,000MonteCarloexperiments.c 1foreachsetof1,000experiments.StandarderrorofβdMeanestimatedstandarderror,andcoverageof95%CIproducedbySASPROCGAM(3.1);95%CIcalculatedasthepointestimate+/ 1.96timestheestimatedstandarderror.

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

Poissonrandomvariables.Weanalyzedthisrandomlygeneratedseries,by ttingtheGAMmodel(Equation(3.1)),andsavingthestandarderrorgeneratedbySASandthatcalculatedusingEquation(2.10).Werepeatedthisprocess1,000timesforeachoutcome-airpollutantcombination.Wefocusonβ1,thelograteratiofortheassociationofEDvisitswiththeairpollutant.

Thesecondsetofsituationsusedwassimilartothatjustdescribed,exceptthathypo-thetical,simulateddatareplacedtheobserveddatausedabovetogeneratethedailyexpectedcountsofEDvisits.Speci cally,fori=1,2,...200wegenerated(hypothetical)exposureandcovariates:εi,x2,i,t2,i,t3,iasindependent,standardGaussianvariables;x1,i=εi+x2,iandt1,i=i.Wethende nedtheexpectedvalueofYias:

EYi|x1i,x2i,t1i,t2i,t2i

1132=expβ0+β1x1 β2x2+cos(t1/5)+sin(t2/5)+(t3 t3+3/4t3)/200.22

(3.2)

Wechoseβ0=3,4,or5andβ1=.0or.04,andβ2=.1,resultinginatotalofsixadditionalexperimentalconditions(Table2).Werandomlygenerated200preliminaryvaluesofYiasindependentPoissonvariableswithmeangivenbyEquation(3.2).Sothatthemodelwouldbecorrectlyspeci ed,wethen taGAMwithasmoothingsplinesusingsixdegreesoffreedomeachfort1,t2,andt3tothesepreliminaryobservationsandusedthemodel-predictedvaluesastheexpectedvaluestogeneratethe200observationsYiusedforeachMonteCarloexperiment.Weadjustedthesmoothingparameters(λ1,λ2,λ3)tocorrespondtothesixdegreesoffreedomthatwespeci edforuseinPROCGAM.(Thatis,weretainedthesamex1,x2,t1,t2,andt3foreachexperiment,butusedthepredictedvaluesfromthemodel ttothepreliminarydataastheexpectedvaluesforall1,000MonteCarloexperimentsineachset.Thisprocedureshouldensurethattheexpectedvalueswereinthespaceofpossible ts.Becauseofthisprocess,the“truevalue”ofβ1foreachsimulation

252

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