混合效应模型 多水平模型(英)

IntroductiontoMixedModelsBIOST2086Lecture1Introduction•Inthebeginning,therewastheLinearModel(LM)Y=X¯+ee~(0,R)•Containsonlyfixedeffects,exceptforthemodelerrors•Errorsareassumedtobeuncorrelated/independent•However,thisassumptionmaybeviolatedandincertainsituationsneedstoberelaxedtoallowformorecomplicateddatastructures.•Ifweviolateindependence,wehavesomeinterdependence.Thisinterdependencecanbemodeleddirectly.ClusteredData•Cluster-correlateddataarisewhenthereisaclustered/groupedstructuretothedata–Anoutcomeismeasuredonceforeachsubject,andsubjectsbelongto(orare“nested”in)clusters,suchasfamilies,schools,orcenters.•Naturallyoccurringclusters:–Ex:familymembers(level1units)nestedwithinfamilies(level2units)–Ex:eyes,ears,lungs(level1units)nestedwithinpatients(level2units)•Designlevelclusters:–Ex:grouprandomizedclinicaltrials(RCT)–Ex:multicenterRCTInbothcases,patientsarelevel1unitsandgroups/centersarelevel2units.ClusteredData•Longitudinaldata:anoutcomeismeasuredforthesamesubjectrepeatedlyoveraperiodoftime.–Aspecialcaseofclustereddata.–Patientsarelevel2units,measurementsatdifferenttimesarelevel1units•Complexsurveydata:primarysamplingunits(suchascounties)arefirstsampled(level2units),andthenhouseholdsaresampledwithintheprimarysamplingunits(level1units).LevelsofClustering•Clustered/longitudinaldataaremoregenerallyknownashierarchicaldataor“multilevel”data.•Level1isthelowestormostgranularlevelofthedata,andwheretheoutcomevariableofinterestismeasured.•Levels2,3,…capturehigherlevelinformation–Previousexamplesallhave2levelsofclustering–Canhave3ormorelevelsofclustering:Longitudinalobservations(level1units)frompatients(level2units)nestedwithincenters(level3units)ofamulticenterRCTDataonchildren(level1)nestedwithinclassrooms(level2)thatarenestedwithinschools(level3)Two-LevelClusteredDataExample•Aresearchstudyineducationaimstoassesstheimpactofschooltype(publicvs.catholic)aswellasstudentgenderandstudentSESonstudent-levelmathachievementscores.Scoresaremeasuredonceforthestudentsintheschool.Two-LevelClusteredData(StudentsNestedwithinSchools)Student2Student3School1Student1School2…Student2...Student1Level2Level1Level1Variables:StudentAchievementScore,Gender,Student’sSES….Level2Variables:PublicorCatholicSchool…Two-LevelLongitudinalDataExample•Researchersarestudyingtheeffectofamother’svocabularyandthechild’sgenderonthechild’svocabularygrowth.LongitudinalData(VocabularyMeasuredOverTime)VocabMeasuredatTime2VocabMeasuredatTimen1Child1VocabMeasuredatTime1Child2……VocabMeasuredatTime2…VocabMeasuredatTime1Level2Level1Level1Variables(Time-Varying):ChildVocabularyCount,AgeateachmeasurementLevel2Variables(Time-Invariant):Mother’sVocabulary,Child’sGenderThree-LevelClusteredDataExample•Aresearchstudyineducationaimstoassesstheimpactofschool,classroom,andstudent-levelvariablesonstudentachievement.Three-LevelClusteredData(Studentsnestedinclassroomsnestedinschools)Level3Level1Level1Variables:StudentAchievementScore,Gender,Student’sSES…Level2Variables:Teacherexperience,Classsize…Level3Variables:Schoollocale(RuralorUrban),SchoolpercentlowincomeStudent2Studentn1Classroom1Student1Classroom2……Student2…Student1Level2School1…Three-LevelClustered-LongitudinalDataExample•Mathskillsaremeasuredforthesamestudenteachyearfromgrades1through6,withstudentsclusteredwithinschools.Thegoalistomodelhowstudentcharacteristics,suchasethnicityandgender,andschoolcharacteristics,suchasschoolsizeandpercentlow-incomestudents,affectthemathscoresofstudentsovertime.Three-LevelClustered-LongitudinalData(Mathscoresmeasuredovertimeforstudentsnestedinschools)Level2Level1Level1Variables(Time-Varying):Student’smathscore,GradeateachmeasurementLevel2Variables(Time-Invariant):Student’sEthnicity,Student’sGenderLevel3Variables(Time-Invariant):Schoolsize,EducationalInterventionatSchoolLevelMathScoreatGrade2MathScoreatGrade6Student1MathScoreatGrade1Student2……MathScoreatGrade3…MathScoreatGrade2School1…Level3MoreAboutClusteredData•Forclustereddata,itisimportanttoaccommodatethedependenceofthedataamongtheobservationsinthesamecluster.•Weexpectthemeasurementswithinaclusterwillbemoresimilarthanbetweenthedifferentclusters.•Thissimilarity(orcorrelation)violatestheindependenceassumptionofstandardregressionmodels.•Whatmodelsshallweusethen?MixedModels•Forexample,ifwehaveanumberofrepeatedmeasurementsmadeonthesamepatient,amixedmodelwillallowustospecifyapatternforthecorrelationofthesemeasurements.“MixedModels”arestatisticalmodelsthatcontainbothfactorswithfixedeffectsandfactorswithrandomeffects.ThreeTypesofMixedModels•Randomeffectsmodel(alsoknownasrandominterceptmodel)–Inthistypeofmixedmodel,certainvariables(effects)areallowedtoberandom(andthusarecalledrandomeffects).–Thismeansthatweassumethatthesevariableshavearisenfromaparticularprobabilitydistributionsotheybecomeanothersourceofrandomvariation.–Wecanspecifymorethanonevariableinourmodeltoberandom.Foreachrandomeffect,weareaddinganadditionalsourceofrandomvariation.ThreeTypesofMixedModels•Randomcoefficientsmodel(alsoknownasrandomslopemodel)–Hereacovariateisallowedtovaryra