1、纵向数据非参数混合效应模型的一个局部不变估计J.Sys.Sci.&Math.Scis.27(1)(2007,2),102112(DepartmentofBiostatisticsandComputationalBiology,UniversityofRochesterMedicalCenter,Rochester,NY14642,USA)(710072)(Lin(2000)()Carroll,2000).LinCarrollGEE()LinCarroll(CV),MR(2000)62G08,62H12,62J99Shi,WeissTaylor1RiceWu2(NPMEmodel)yi(t)=s
2、i(t)+i(t)=(t)+vi(t)+i(t),i=1,2,n,(1)si(t)=(t)+vi(t)(t)vi(t)i(t)yi(t)vi(t)i(t)vi(t)0,(s,t)i(t)0,2(t)123,4*NIH(AI562247,AI059773)2006-09-19.(70471057)1103LinCarroll5GEE()n i=122Gip(t)Ti(t)Kih(t)Vi1(t)Kih(t)Yii(t)=0,11Yi=(Yi1,Yini)TiKih(t)=diagKh(tijt),Gip(t)K(h)ni2ji=E(Yi),j1Vi=SRiS,12ij=Si(tijt),Kh()
3、=h,K()h(1)GTGTp(tij),i=diagp(tij),1,h(1)()Ri5.iiGEERi()(1)n niKh(tijt LC(t)=i=1 )yij2(tij) j=1n niK,h(tijt)i=1j=12(tij) 2()2(t)=varyi(t)nii5GEEvi(t)6,7).)NPME9“”.().109KLME3LinCarroll(52.1LMEtij,j=1,2,niLinRi8(KLME)10.).(1)Carroll(5410427yi(tij)=si(tij)+i(tij)=(tij)+vi(tij)+i(tij),i=1,2,n;j=1,2,ni.(
4、2)yij=yi(tij)ij=i(tij).(t)si(t).(t)si(t)t0,(t)vi(t)tij1111,(tij),vi(tij)bitijt0t0(2)(LME)tijt0yij=+bi+ij,(3)i=(i1,i2,ini)T(0,Ri)2Ri=E(iTi)=diag(ti1),2(tini),bi(0,2)t),Kh(tinit),Kh()=K(h)hyi=(yi1,yi2,yini)T,Kih=diagKh(ti1LME2,(3),biRibi12ni=1(yi1i1ibi)niT1Ri(yib21i1ibi)+i+log|2|+log|Ri|,2(4)b2i21i1ni
5、=1(yi1i1ibi)(5)T12KihRiKih(yi121b21i1ibi)+i+log|2|+log|Ri|,2(5)2Kih.1().(5)12,13.Ri2,nn 1 TT =1iih1i1iihyi,i=1i=1 bi=21Ti=1,2,n,iih(yi1i),(6)11122ih=KihiKih,1=K12T2211Kiiihiih+Ri.(7)1105(t)n vi(t)KLME=1 (t)=i ni=1Bi(t)1+Ai(t)Ai(t)1+Ai(t),v i(t)= (t)Bi(t)Ai(t),1+Ai(t)i=1,2,n,(8)ni Kh(tijt),Ai(t)=(t)2
6、(t)ijj=12ni Kh(tijt)yijBi(t)=(t).2(t)ijj=12(9)(REML)Ri212,14.LME(8):22Kh(tijt)yij=Kh(tijt)(+bi)+ij,11i=1,2,n;j=1,2,ni,(10)Kh(tijt)yij12Kh(tijt)12(10)(10),LME(8)SplusLMESASPROCMIXED2.2KLME(10),()(t)vi(t)si(t)=(t)+vi(t).vi(t)(t)RiceSilverman15(SCV)(t),(PCV)vi(t).(t)().PCV(t)16,17PCVSCVPCV1)(t)SCVSCVvi
7、(t). scv(t)v scvj,i(t);2)v pvc,i(t);3)PCV(t)vi(t) pcv(t) scv(t),v scv,i(t),s i(t)= scv(t)+v pcv,i(t).106273(t)KLMEni,niA12341, 67tij,i=1,2,n;j=1,2,niidf();tff(t)=0,f (t)(t)=Eyi(t)t (t)(s,t)=covvi(s),vi(t)st2(t)=vari(t)tK K(u)udu=0V(K)= K(u)du= K(u)u2du=C10,K2(u)du;1432,n1h=C1(nn)5,n 11n=nn.ini=O(n);
8、3.1Ai=1 (t)(8)h2 2 (t)f (t) (t)+1+o(1),bias (t)=2f(t) 12(t)2(t)1 var (t)=+V(K)+o+,nnnhf(t)n(nnh)minini,n,2(t)=2(t)+2(t).3.13.2h=o(nn)5,1 (t)A7ni=O(n),141,ninin3.173.2A16nim,h=O(n5)1n (t)h2 2 (t)f (t) (t)+,bias (t)6f(t)2(t)+2(t)var (t),n2nmhf(t)52LC1Population Estimates with KLME and LC methods5AN4R3
9、2050100150Time1KLME()LC()(t)4AIDSAIDS(ACTG)ACTG31548HIV-1HIVRNA()t=0,2,7,10,14,21,28,56,84,1683918,19.KLMELCLC=hKLME,SCV=39.96hhKLME,PCV=33.01.1KLMEKLMEKLME2461044AIDS5LME51071108275GEEKLME(5)(5)5KMLESubject 15+4ANR32050100150TimeSubject 4454+AN3R21050100150Time2KLMESubject 10.55.4AN.4R5+.30.35.2050
10、100150TimeSubject 46.55.40.4ANR5.30.35.2050100150Time()()13.1R,R=E(R)+O1pvar2(R).(A.1)nihniKh(tijt)ni 1j=12(tij)=f(t)2(t)1+Opnih,Opi. (t)ni(tijt2)(tyij(t) (t)=( nKht)+wij)i(t)j=1,i=1 niKj=1h2(t(ijtt)ij)Awi(t)= n1+Ai(t)i(t)A,i(t)i=11+Ai(t)Ai(t)Bi(t)(9)(A.2)Ani2(t)f(t) 1i(t)=2(t)1+Opnih,w1+Oi(t)=p(1nn
11、),Op(n11i)=Op(n). (t)=(t)+n1 nZ 1i(t)1+Op,i=1nniKh(tijt2)(tyij(t)Zij)i(t)=j=1 niKh(2t(ijtt)j=1ij)(A.2)A7,(),niZ=1i(t)Kh(tijt)yij(t) f(t) 1n1+Oipj=12(tij)2( =Zi(t)f(t) 1 t)nih2(t)1+Opnih.109(A.1)(A.2)Taylor11027EZi(t)|Dh2 2 (t)f (t) (t)+1+O(h2).=2f(t)ni nini2 Kh(tijt)Kh(tilt)(tij,til) Kh(tijt)2(tij)+
12、varZi(t)|D=2(t)2(t)4(tij)ijilj=1l=1j=1 12(t)f(t)V(K)+o+h2=(t,t)+h220(t,t)B(K)+2(t)=f(t)2(t) nihf1 ( t)(nih)2(t)(t,t)+nihf(t)V(K)+o(nih),20(,)h2(n1ih).E1 n h2 (t)+2 (t)f (t) 1+o(1),nZi(t)=i=12f(t)nvar1nZ=(t,t)+ 1i(t)2(t)V(K)+onh).i=1nnhf(t)(n3.13.2(8) (t)n n mwij(t)yiji1+BAi(t)i(t)=1n=i=1 j=1n mi=11+
13、AAi(t)i(t)w,ij(t)i=1j=12(t)wijK2h(t)=(2h)1,j=1,2,m;i=1,2,n.1+2(t) (ttij)t)mijKh(tiltl=12(t)il)f(t)t,tij,j=1,2,ma)tij=tj=1,2,mb)tij0=t1j0mA1i)hmin1jm|tijt|Kh(tijt)=0,j=1,2,m,K1,1.2(t)(2h)wij(t)=K2h(tij01+2(t) (tt)ij)mKh(tilt)1+0=0,j=1,2,m,l=12(til)nih301111ii)h0,Kh(tijt)=0h0j=1,2,j01,j0+1,2,m.mj=1Kh(
14、tijt)=K(0)h.(2h)wij0(t)1=(2h)wij(t)0=wij(t)12(t)h1K(0)2(t)2h1K(0)+(t)2(t)10,=0,j=1,2,j01,j0+1,2,m.01+2(t)h1K(0)2(t)wij(t)=Uh(tijt)1+op(1),Uh(u)=U(uh)h,Un m (t)=i=1j=1mn Uh(tijt)yijUh(tijt)1+op(1).i=1j=1 (t)(t)131ShiM,WeissREandTaylorJMG.AnanalysisofpediatricCD4countsforacquiredimmunedeciencysyndrome
15、usingexiblerandomcurves.AppliedStatist.,1996,45:15163.2RiceJAandWuCO.Nonparametricmixedeectsmodelsforunequallysamplednoisycurves.Biometrics,2001,57:253259.3WangY.Mixed-eectssmoothingsplineANOVA.J.R.Statist.Soc.B,1998,60:159174.4ZhangD,LinX,RazJandSowersM.Semiparametricstochasticmixedmodelsforlongitu
16、dinaldata.J.Am.Statist.Assoc.,1998,93:710719.5LinXandCarrollRJ.Nonparametricfunctionestimationforclustereddatawhenthepredictorismeasuredwithout/witherror.J.Am.Statist.Assoc.,2000,95:52034.6HooverDR,RiceJA,WuCOandYangLP.Nonparametricsmoothingestimatesoftime-varyingcoecientmodelswithlongitudinaldata.B
17、iometrika,1998,85:80922.7WuCO,ChiangCTandHooverDR.Asymptoticcondenceregionsforkernelsmoothingofavarying-coecientmodelwithlongitudinaldata.J.Am.Statist.Assoc.,1998,93:1388402.8JonesMC.Donotweightforheteroscedasticityinnonparametricregression.Aus.J.Statist.,1993,35:8992.9WandMPandJonesMC.KernelSmoothi
18、ng.London:ChapmanandHall.1995.10LairdNMandWareJH.Randomeectsmodelsforlongitudinaldata.Biometrics1982,38:96374.11FanJandGijbelsI.LocalPolynomialModelinganditsApplications,London:ChapmanandHall.199612DavidianMandGiltinanDM.NonlinearModelsforRepeatedMeasurementData.London:ChapmanandHall.1995.1122713Har
19、villeD.Maximumlikelihoodapproachestovariancecomponentestimationandtorelatedproblems.J.Am.Statist.Assoc.,1977,72:32040.14VoneshEFandChinchilliVM.LinearandNonlinearModelsfortheAnalysisofRepeatedMea-surements.NewYork:MarcelDekker,Inc.1996.15RiceJAandSilvermanBW.Estimatingthemeanandcovariancestructureno
20、nparametricallywhenthedataarecurves.J.R.Statist.Soc.B.,1991,53:23343.16AltmanNS.Kernelsmoothingofdatawithcorrelatederrors.J.Am.Statist.Assoc.,1990,85:749759.17HartJD.Kernelregressionestimationwithtimeserieserrors.J.R.Statist.Soc.B,1991,53:173187.18LedermanMM,ConnickE,LandayA.Immunologicresponsesasso
21、ciatedwith12weeksofcom-binationantiretroviraltherapyconsistingofzidovudine,lamivudineandritonavir:resultsofAIDSClinicalTrialsGroupProtocol315.J.Infect.Diseases,1998,178:70-9.19WuH,KuritzkesDR,McClernon.Characterizationofviraldynamicsinhumanimmunodeciencyvirustype1-infectedpatientstreatedwithcombinat
22、ionantiretroviraltherapy:relationshipstohostfactors,cellularrestoration,andvirologicendpoints.J.Infect.Diseases,1999,179:19.ALOCALCONSTANTESTIMATORFORNONPARAMETRICMIXED-EFFECTSMODELSWITHLONGITUDINALDATALiangHua(DepartmentofBiostatisticsandComputationalBiology,UniversityofRochesterMedicalCenter,Roche
23、ster,NY14642)ShiYimin(DepartmentofAppliedMathematics,NorthwesternPolytechnicalUniversity,XiAn710072)AbstractNonparametrickernelregressionmethodshavebeenproposedforlongitudinaldataanalysisrecently(LinandCarroll,2000).Acontroversialquestioniswhetherthecorre-lationamonglongitudinaldatashouldbeconsidere
24、dinthenonparametrickernelregression.LinandCarroll(2000)haveshownthatthekernelestimatorbasedonworking-independence(ignoringthecorrelation)ismost(asymptotically)ecientinaclassofkernelGEEestima-tors.Inthispaperweproposeadierentclassofkernelestimatorsbasedonthemixed-eectsmodelapproachthatincorporatesthe
25、correlationstructureoflongitudinaldatanaturallyandeciently.WeshowthatourestimatorachievesthesameasymptoticeciencyasLinandCar-rollsestimator,butperformsbetterinnitesamples.Thenonparametriccurveestimatesforbothpopulationandindividualsubjects(clusters)canbereadilyobtainedfromtheproposedmethod.Thesegoodpropertiesoftheproposedestimatoraswellaseasyimplementationareattractivetopractitioners.KeywordsCross-validation(CV),kernelregression,mixed-eectsmodels,nonparamet-ricregression,relativeeciency.
