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1、ComputationalMethodsforMultilevelModellingDouglasM.BatesJoseC.Pinheiro′DepartmentofStatisticsBellLaboratoriesUniversityofWisconsin–MadisonLucentTechnologiesAbstractAmultilevelmixed-effectsmodelhasrandomeffectsateachofseveralnestedlevelsofgroupingofth
2、eobservedresponses.Wemayusethese,forexample,whenmodellingobservationstakenovertimeonstudentswhoaregroupedintoclassesthataregroupedintoschoolsthataregroupedintodistricts.Ifeachofthedistri-butionsoftherandomeffectsisGaussianandifthedisturbancetermatthel
3、owestlevelofgroupingisalsoGaussianitisstraightforwardtode?nealikelihoodforthe?xedeffectsandtheparametersde?ningtherandomeffectsdistribution.Weshowthatbyexpressingtherandomeffectsdistributionintermsofrelativepreci-sionfactorsandusingmatrixdecomposition
4、s,thislikelihoodcanbepro?ledandcanbecompactlyexpressed.Thesamedecompositionsproviderapidevaluationofthepro?ledlog-restricted-likelihoodforREMLestimation.Theconditionaldistributionoftherandomeffectsgiventhedatacanbederivedfromthedecomposedmatrices.From
5、thisacompactandrapidlyevaluatedexpres-sionfortheEMiterationscanbederived.Reasonablestartingestimatesfortherelativeprecisionfactorscanbederivedfromthedesignalone.Thesestartinges-timates,re?nedbyamoderatenumberofEMiterations,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
