Estimating-Multilevel-Models-using-SPSS--Stata--SA

EstimatingMultilevelModelsusingSPSS,Stata,SAS,andRJeremyJ.AlbrightandDaniM.MarinovaJuly14,20101Multileveldataarepervasiveinthesocialsciences.1Studentsmaybenestedwithinschools,voterswithindistricts,orworkerswithinrms,tonameafewexam-ples.Statisticalmethodsthatexplicitlytakeintoaccounthierarchicallystructureddatahavegainedpopularityinrecentyears,andtherenowexistseveralspecial-purposestatisticalprogramsdesignedspecicallyforestimatingmultilevelmodels(e.g.HLM,MLwiN).Inaddition,theincreasinguseofofmultilevelmodelsalsoknownashierarchicallinearandmixedeectsmodelshasledgeneralpurposepackagessuchasSPSS,Stata,SAS,andRtointroducetheirownproceduresforhandlingnesteddata.Nonetheless,researchersmayfacetwochallengeswhenattemptingtodeterminetheappropriatesyntaxforestimatingmultilevel/mixedmodelswithgeneralpurposesoftware.First,manyusersfromthesocialsciencescometomultilevelmodelingwithabackgroundinregressionmodels,whereasmuchofthesoftwaredocumenta-tionutilizesexamplesfromexperimentaldisciplines[duetothefactthatmultilevelmodelingmethodologyevolvedoutofANOVAmethodsforanalyzingexperimentswithrandomeects(Searle,Casella,andMcCulloch,1992)].Second,notationformultilevelmodelsisofteninconsistentacrossdisciplines(Ferron1997).ThepurposeofthisdocumentistodemonstratehowtoestimatemultilevelmodelsusingSPSS,StataSAS,andR.Itrstseekstoclarifythevocabularyofmultilevelmodelsbydeningwhatismeantbyxedeects,randomeects,andvariancecomponents.Itthencomparesthemodelbuildingnotationfrequentlyemployedinapplicationsfromthesocialscienceswiththemoregeneralmatrixnotationfound1Jeremywrotetheoriginaldocument.DaniwrotethesectiononRandrewrotepartsofthesectiononStata.2inmuchofthesoftwaredocumentation.Thesyntaxforcenteringvariablesandestimatingmultilevelmodelsisthenpresentedforeachpackage.1VocabularyofMixedandMultilevelModelsModelsformultileveldatahavedevelopedoutofmethodsforanalyzingexperi-mentswithrandomeects.Thusitisimportantforthoseinterestedinusinghierar-chicallinearmodelstohaveaminimalunderstandingofthelanguageexperimentalresearchersusetodierentiatebetweeneectsconsideredtoberandomorxed.Inanidealexperiment,theresearcherisinterestedinwhetherornotthepresenceorabsenceofonefactoraectsscoresonanoutcomevariable.2Doesaparticularpillreducecholesterolmorethanaplacebo?Canbehavioralmodicationreduceaparticularphobiabetterthanpsychoanalysisornotreatment?Thefactorsintheseexperimentsaresaidtobexedbecausethesame,xedlevelswouldbeincludedinreplicationsofthestudy(MaxwellandDelaney,pg.469).Thatis,theresearcherisonlyinterestedintheexactcategoriesofthefactorthatappearintheexperiment.Thetypicalmodelforaone-factorexperimentis:yij=+j+eij(1)wherethescoreonthedependentvariableforindividualiisequaltothegrandmean2Intheparlanceofexperiments,afactorisacategoricalvariable.Thetermcovariatereferstocontinuousindependentvariables.3ofthesample(),theeectofreceivingtreatmentj,andanindividualerrortermeij.Ingeneral,somekindofconstraintisplacedonthealphavalues,suchthattheysumtozeroandthemodelisidentied.Inaddition,itisassumedthattheerrorsareindependentandnormallydistributedwithconstantvariance.Insomeexperiments,however,aparticularfactormaynotbexedandperfectlyreplicableacrossexperiments.Instead,thedistinctcategoriespresentintheexper-imentrepresentarandomsamplefromalargerpopulation.Forexample,dierentnursesmayadministeranexperimentaldrugtosubjects.Usuallytheeectofaspecicnurseisnotoftheoreticalinterest,buttheresearcherwillwanttocontrolforthepossibilitythatanindependentcaregivereectispresentbeyondthexeddrugeectbeinginvestigated.Insuchcasestheresearchermayaddatermtocontrolfortherandomeect:yij=+j+k+()jk+eij(2)whererepresentstheeectofthekthleveloftherandomeect,andrepresentstheinteractionbetweentherandomandxedeects.Amodelthatcontainsonlyxedeectsandnorandomeects,suchasequation1,isknownasaxedeectsmodel.Onethatincludesonlyrandomeectsandnoxedeectsistermedarandomeectsmodel.Equation2isactuallyanexampleofamixedeectsmodelbecauseitcontainsbothrandomandxedeects.Whilethenotationinequation2fortherandomeectisthesameasforthexedeect(thatis,botharedenotedbysubscriptedGreekletters),animportantdierenceexistsinthetestsforthedrugandnursefactors.Forthexedeect,the4researcherisinterestedinonlythoselevelsincludedintheexperiment,andthenullhypothesisisthattherearenodierencesinthemeansofeachtreatmentgroup:H0:1=2=:::=jH1:j6=j0Fortherandomeectinthedrugexample,theresearcherisnotinterestedintheparticularnursespersebutinsteadwishestogeneralizeaboutthepotentialeectsofdrawingdierentnursesfromthelargerpopulation.Thenullhypothesisfortherandomeectisthereforethatitsvarianceisequaltozero:H0:2=0H1:20Theestimatedvarianceisknownasavariancecomponent,anditsestimationisanessentialstepinmixedeectsmodels.Oftentimesinexperimentalsettings,therandomeectsarenuisancesthatne-cessitatestatisticalcontrols.Intheaboveexample,theeectofthedrugwastheprimaryinterest,whereasthenursefactorwaspotentiallyconfoundingbuttheoreti-callyuninteresting.Itisnonethelessnecessarytoincludetherelevantran