多水平统计模型 第8章.docx
- 文档编号:11984373
- 上传时间:2023-04-16
- 格式:DOCX
- 页数:17
- 大小:193.23KB
多水平统计模型 第8章.docx
《多水平统计模型 第8章.docx》由会员分享,可在线阅读,更多相关《多水平统计模型 第8章.docx(17页珍藏版)》请在冰豆网上搜索。
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
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 多水平统计模型 第8章 水平 统计 模型