Computation Method for Mixed Model

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1、ComputationalMethodsforMultilevelModellingDouglasM.BatesJoseC.Pinheiro´DepartmentofStatisticsBellLaboratoriesUniversityofWisconsin–MadisonLucentTechnologiesAbstractAmultilevelmixed-effectsmodelhasrandomeffectsateachofseveralnestedlevelsofgroupingofth

2、eobservedresponses.Wemayusethese,forexample,whenmodellingobservationstakenovertimeonstudentswhoaregroupedintoclassesthataregroupedintoschoolsthataregroupedintodistricts.Ifeachofthedistri-butionsoftherandomeffectsisGaussianandifthedisturbancetermatthel

3、owestlevelofgroupingisalsoGaussianitisstraightforwardtodefinealikelihoodforthefixedeffectsandtheparametersdefiningtherandomeffectsdistribution.Weshowthatbyexpressingtherandomeffectsdistributionintermsofrelativepreci-sionfactorsandusingmatrixdecomposition

4、s,thislikelihoodcanbeprofiledandcanbecompactlyexpressed.Thesamedecompositionsproviderapidevaluationoftheprofiledlog-restricted-likelihoodforREMLestimation.Theconditionaldistributionoftherandomeffectsgiventhedatacanbederivedfromthedecomposedmatrices.From

5、thisacompactandrapidlyevaluatedexpres-sionfortheEMiterationscanbederived.Reasonablestartingestimatesfortherelativeprecisionfactorscanbederivedfromthedesignalone.Thesestartinges-timates,refinedbyamoderatenumberofEMiterations,provideexcellentstartingvalu

6、esforaNewton-Raphsonorquasi-Newtonoptimizationofthelog-likelihoodorthelog-restricted-likelihood.Themethodswedescribeextendeasilytomodelswithnon-sphericaldistributionsforthewithin-grouperrorsandtononlinearmul-tilevelmodels.Keywordsandphrases:mixed-effe

7、ctsmodels,EMalgorithm,maximumlikeli-hood,restrictedmaximumlikelihood1IntroductionWeconsidercomputationalmethodsforGaussianmultilevelmixed-effectsmodelsasdescribed,forexample,inLongford(1993)orGoldstein(1995).Thesemodelsareusedwithdatawheretheindividua

8、lobservationsaregroupedatoneormorehierarchicallev-els.Forexample,wemaywishtomodelobservationsonstudentswhoaregroupedintoThisresearchwassupportedbytheNationalScienceFoundationthroughgrantDMS-9704349.1classesthataregroupedintoschoolsthataregrou