多水平统计模型 第8章.docx

多水平统计模型 第8章.docx

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多水平统计模型第8章

Chapter8

Multilevelcrossclassifications

8.1Randomcrossclassifications

Inpreviouschapterswehaveconsideredonlydatawheretheunitshaveapurelyhierarchicalornestedstructure.Inmanycases,however,aunitmaybeclassifiedalongmorethanonedimension.Anexampleisstudentsclassifiedbothbytheschooltheyattendandbytheneighbourhoodwheretheylive.Wecanrepresentthisdiagramaticallyasfollowsforthreeschoolsandfourneighbourhoodswithbetweenoneandsixstudentsperschool/neighbourhoodcell.Thecrossclassificationisatlevel2withstudentsatlevel1.

School1

School2

School3

Neighbourhood1

xxxx

xx

x

Neighbourhood2

x

xxxxxx

xxx

Neighbourhood3

xx

x

xxxx

Neighbourhood4

xxx

xx

xx

Figure8.1Arandomcrossclassificationatlevel2

Anotherexampleisinarepeatedmeasuresstudywherechildrenaremeasuredbydifferentratersatdifferentoccasions.Ifeachchildhasitsownsetofratersnotsharedwithotherchildrenthenthecrossclassificationisatlevel1,occasionsbyraters,nestedwithinchildrenatlevel2.Thiscanberepresenteddiagramaticallyasfollowsforthreechildrenwithupto7measurementoccasionsanduptothreeratersperchild.

Weseethatthecrossclassificationtakesplaceentirelywithinthelevel2units.Wenotethat,bydefinition,alevel1crossclassificationhasonlyoneunitpercell.Wecan,however,alsoviewsuchacrossclassificationasaspecialcaseofalevel2crossclassificationwith,atmost,asinglelevel1unitpercell.Itseemsappropriatetoviewsuchcasesaslevel1crossclassificationsonlywherethesubstantivecontextdeterminesthatthereisatmostoneunitpercell（seesection8.6）.

Child1

Child2

Child3

Occasion:

1234567

12346

147

Rater1

xxxxx

Rater2

xxxxx

Rater3

xxxxx

Rater4

xxxxx

Rater5

xxx

Rater6

x

Figure8.2Arandomcrossclassificationatlevel1.

Ifnowthesamesetofratersisinvolvedwithallthechildrenthecrossingisatlevel2ascanbeseeninthefollowingdiagramwiththreeratersandthreechildrenanduptofiveoccasions.

Child1

Child2

Child3

Occasion:

1234

12

12345

Rater1

xxx

x

x

Rater2

x

xx

Rater3

x

xx

Figure8.3Arandomcrossclassificationatlevel2.

Figure8.3isformallythesamestructureasFigure8.1withthelevel1variancebeingthatbetweenoccasions.

Thesebasiccrossclassificationsoccurcommonlywhenasimplehierarchicalstructurebreaksdowninpractice.Consider,forexample,arepeatedmeasuresdesignwhichfollowsasampleofstudentsovertime,sayonceayear,withinasetofclassesforasingleschool.Weassumefirstthateachclassgroupistakenbythesameteacher.Thehierarchicalstructureisthenathreelevelonewithoccasionsgroupedwithinstudentswhoaregroupedwithinclasses.Ifwehadseveralschoolsthenschoolswouldconstitutethelevel4units.Suppose,however,thatstudentschangeclassesduringthecourseofthestudy.Forthreestudents,threeclassesanduptothreeoccasionswemighthavethefollowingpatterninFigure8.4..

Student1

Student2

Student3

Occasion:

123

12

123

Class/teacher1

xx

x

x

Class/teacher2

x

Class/teacher3

x

xx

Figure8.4Studentschangingclasses/teachers.

FormallythisisthesamestructureasFigure8.3,thatisacrossclassificationatlevel2forclassesbystudents.Suchdesignswilloccuralsoinpanelorlongitudinalstudiesofindividualswhomovefromonelocalitytoanother,orworkerswhochangetheirplaceofemployment.Ifwenowincludeschoolsthesewillbeclassifiedaslevel3units,butifstudentsalsochangeschoolsduringthecourseofthestudythenweobtainalevel3crossclassificationofstudentsbyschoolswithclassesnestedatlevel2withinschoolsandoccasionsasthelevel1units.Thestudentshavemovedfrombeingcrossedwithclassestobeingcrossedwithschools.Notethatsincestudentsarecrossedatlevel3withschoolstheyarealsoautomaticallycrossedwithanyunitsnestedwithinschoolsandwedonotneedseparatelytospecifythecrossingofclasseswithstudents.

Supposenowthat,insteadofthesameteacherstakingtheclassesthroughoutthestudy,theclassesaretakenbyacompletelynewsetofteacherseveryyearandwherenewgroupingsofstudentsareformedeachyeartoo.SuchastructurewithfourdifferentteachersattwooccasionsforthreestudentsisgiveninFigure8.5.

Student1

Student2

Student3

Occasion:

12

12

12

Teacher1

1

x

x

Teacher2

x

Teacher3

2

x

x

Teacher4

x

Figure8.5.Studentschangingteachersandgroups

Thisisnowacrossclassificationofteachersbystudentsatlevel2withoccasionasthelevel1unit.Wenotethatmostofthecellsareemptyandthatthereisatmostonelevel1unitpercellsothatnoindependentbetweenoccasionvariancecanbeestimatedaspointedoutabove.Infactwecanalsoviewthisasalevel1crossclassificationofteachersbystudents,withmissingdata,andoccasioncanbemodelledinthefixedpart,forexampleusingapolynomialfunctionofage.Raudenbush（1993）givesanexampleofsuchadesign,andprovidesdetailsofanEMestimationprocedurefor2-level2-waycrossclassificationswithworkedexamples.

WecanhaveadesignwhichisamixtureofthosegivenbyFigure8.4andFigure8.5wheresometeachersareretainedandsomearenewateachoccasion.Inthiscasewewouldhaveacrossclassificationofteachersbystudentsatlevel2wheresomeoftheteachersonlyhadobservationsatoneoccasion.Moregenerally,wecanhaveanunbalanceddesignwhereeachteacherispresentatavariablenumberofoccasions.Otherexamplesofsuchdesignsoccurinpanelstudiesofhouseholdswhere,overtime,somehouseholdssplitupandformnewhouseholds.Thetotalsetofallhouseholdsiscrossedwithindividualatlevel2withoccasionatlevel1.Thehouseholdswhichremainintactformorethanoneoccasionprovidetheinformationforestimatinglevel1variation.

Occasion2

Teacher1

Teacher2

Teacher3

Teacher1

xxxxx

x

xx

Occasion1

Teacher2

xx

xxxx

Teacher3

x

xxx

xxxx

Figure8.6.Teacherscrossclassifiedbythemselvesattwooccasions

Withtwooccasionswherewehavethesameteachersorintactgroupswecanformulateanalternativecrossclassificationdesignwhichmaybemoreappropriateinsomecases.Insteadofcrossclassifyingstudentsbyteachersweconsidercrossclassifyingthesetofallteachersatthefirstoccasionbythesamesetatthesecondoccasion,asfollows.

Wehave22studentswhoarenestedwithinthecrossclassificationofteachersateachoccasion.ThedifferencebetweenthisdesignandthatinFigure8.4isanalogoustothedifferencebetweenatwo-occasionlongitudinaldesignwhereasecondoccasionmeasurementisregressedonafirstoccasionmeasurementandthetwo-occasionrepeatedmeasuresdesignwhereameasurementisrelatedtoageortime.InFigure8.6weareconcernedwiththecontributionfromeachoccasiontothevariationin,say,ameasurementmadeatoccasion2.InFigure8.4ontheotherhand,althoughwecouldfitaseparatebetweenteachervarianceforeachoccasion,theresponsevariableisessentiallythesameonemeasuredateachoccasion.DesignssuchasthatofFigure8.6areusefulwhere,forexample,measurementsaremadeonthesamesetofstudentsandschoolsatthestartandendofschooling,asinschooleffectivenessstudies,andwherestudentscanmovebetweenschools.Insuchcaseswemayalsowishtointroducea‘weight’toreflectthetimespentineachschool,andweshalldiscussthisbelow.

Wenowsetoutthestructureofthesebasicmodelsandthengoontoconsiderextensionsandspecialcasesofinterest.

8.2Abasiccrossclassifiedmodel

Goldstein（1987a）setsoutthegeneralstructureofamodelwithbothhierarchicalandcrossclassifiedstructuresandRasbashandGoldstein（1994）providefurtherelaborations.WeconsiderfirstthesimplemodelofFigure8.1withvariancecomponentsatlevel2andasinglevariancetermatlevel1.

Weshallrefertothetwoclassificationsatlevel2usingthesubscripts

andingeneralparentheseswillgroupclassificationsatthesamelevel.Wewritethemodelas

（8.1）

Thecovariancestructureatlevel2canbewritteninthefollowingform

（8.2）

Notethatifthereisnomorethanoneunitpercell,thenmodel（8.1）isstillvalidandcanbeusedtospecifyalevel1crossclassificationasdefinedinSection8.1.

Thusthelevel2varianceisthesumoftheseparateclassificationvariances,thecovariancefortwolevel1unitsinthesameclassificationisequaltothevarianceforthatclassificationandthecovariancefortwolevel1unitswhichdonotshareeitherclassificationiszero.Ifwehaveamodelwhererandomcoefficientsareincludedforeitherorbothclassifications,thenanalogousstructuresareobtained.Wecanalsoaddfurtherwaysofclassificationwithobviousextensionstothecovariancestructure.

Appendix8.1showshowcrossclassifiedmodelscanbespecifiedandestimatedefficientlyusingapurelyhierarchicalformulationandwecansummarisetheprocedureusingthesimplemodelof8.1.Wespecifyoneoftheclassifications,mostefficientlytheonewiththelargernumberofunits,asastandardhierarchicallevel2classification.Fortheotherclassificationwedefineadummy（0,1）variableforeachunitwhichisoneiftheobservationbelongstothatunitandzeroifnot.Thenwespecifythateachofthesedummyvari