多水平统计模型 第3章.docx
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多水平统计模型 第3章.docx
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多水平统计模型第3章
Chapter3
Extensionstothebasicmultilevelmodel
3.1Complexvariancestructures
Inallthemodelsofchapter2wehaveassumedthatasinglevariancedescribestherandomvariationatlevel1.Atlevel2wehaveintroducedamorecomplexvariancestructure,asshowninfigure2.7,byallowingregressioncoefficientstovaryacrosslevel2units.Themodellingandinterpretationofthiscomplexvariation,however,wassolelyintermsofrandomlyvaryingcoefficients.Nowwelookathowwecanmodelthevariationexplicitlyasafunctionofexplanatoryvariablesandhowthiscangivesubstantivelyinterestinginterpretations.Weshallconsidermainlythelevel1variation,butthesameprinciplesapplytohigherlevels.Weshallalsointhischapterconsiderextensionsofthebasicmodeltoincludeconstraintsonparameters,unitweighting,standarderrorestimationandaggregatelevelanalyses.
IntheanalysisoftheJSPdatainchapter2wesawthatthelevel1residualvariationappearedtodecreasewithincreasing8-yearmathsscore.Wealsosawhowtheestimatedindividualschoollinesappearedtoconvergeathigh8-yearscores.Weconsiderfirstthegeneralproblemofmodellingthelevel1variation.
Sinceweshallnowconsiderseveralrandomvariablesateachlevelthenotationusedinchapter2needstobeextended.Fora2-levelmodelwecontinuetousethenotation
forthetotalvariationatlevels2and1andwewrite
(3.1)
wherethe
'sareexplanatoryvariables.Normally
refertotheconstant(=1)definingabasicorinterceptvariancetermateachlevel.
Forthreelevelmodelswewillusethenotation
whereireferstolevel1units,jtolevel2units,andktolevel3unitsandhindexestheexplanatoryvariablesandtheircoefficientswithineachlevel..
Onesimplemodelforthelevel1variationistomakeitalinearfunctionofasimpleexplanatoryvariable.Considerthefollowingextensionof(2.1)
(3.2)
sothatthelevel1contributiontotheoverallvarianceisthelinearfunctionof
Thisdeviceofconstrainingavarianceparametertobezerointhepresenceofanonzerocovarianceisusedtoobtaintherequiredvariancestructure.Thusitisonlythespecifiedfunctionsoftherandomparametersin(3.2)whichhaveaninterpretationintermsofthelevel1variancesoftheresponses
.Thiswillgenerallybethecasewherethecoefficientsarerandomatthesamelevelatwhichtheexplanatoryvariablesaredefined.Thusforexample,intheanalysesoftheJSPdatainchapter2,wecouldmodeltheaverageschool8-year-score,whichisalevel-2variable,asrandomatlevel2.Iftheresultingvarianceandcovariancearenon-zero,theinterpretationwillbethatthebetween-schoolvarianceisaquadraticfunctionofthe8-yearscorenamely
where
istheaverage8-yearscore.
Furthermore,wecanallowavarianceparametertobenegative,solongasthetotallevel1varianceremainspositivewithintherangeofthedataInchapter5wediscussmodellingthetotallevel1varianceasanonlinearfunctionofexplanatoryvariables,forexampleasanegativeexponentialfunctionwhichautomaticallyconstrainsthevariancetobepositive.
Whereacoefficientismaderandomatalevelhigherthanthatatwhichtheexplanatoryvariableitselfisdefined,thentheresultingvariance(andcovariance)canbeinterpretedasthebetween-higher-levelunitvarianceofthewithin-unitrelationshipdescribedbythecoefficient.Thisistheinterpretation,forexample,oftherandomcoefficientmodeloftable2.5wherethecoefficientofthestudent8-yearscorevariesrandomlyacrossschools.Inaddition,ofcourse,wehaveacomplexvariance(andcovariance)structureatthehigherlevel.
Themodel(3.2)doesnotconstraintheoveralllevel1contributiontothevarianceinanyway.Inparticular,itisquitepossibleforthelevel1varianceandhencethetotalresponsevariancetobecomenegative.Thisisclearlyinadmissibleandwillalsoleadtonumericalestimationproblems.Toovercomethiswecanconsiderelaboratingthemodelbyaddingaquadraticterm,mostsimplybyremovingthezeroconstraintonthevariance.Inchapter5weconsiderthealternativeofmodellingthevarianceasanonlinearfunctionofexplanatoryvariables.
Intable3.0weextendthemodeloftable2.5toincorporateasuchaquadraticfunctionforthelevel1variance.Ifweattempttofitalinearfunctionweindeedfindthatanegativetotalvarianceispredicted.
TheresultsfrommodelAshowasignificantcomplexlevel1variation(chisquaredwith2degreesoffreedom=123).Furthermore,thelevel2correlationbetweentheinterceptandslopeisnowreducedto-0.91andwithlittlechangeamongthefixedpartcoefficients.Thepredictedlevel1standarddeviationvariesfromabout9.0atthelowest8-yearscorevaluetoabout1.9atthehighest,reflectingtheimpressionfromthescatterplotinfigure2.1.
Table3.1JSPdatawithlevel1varianceaquadraticfunctionof8-yearscoremeasuredaboutthesamplemean.ModelAwithoriginalscale;modelsBandCwithNormalscoretransformof11-yearscore.
Parameter
Estimate(s.e.)
Estimate(s.e.)
Estimate(s.e.)
A
B
C
Fixed:
Constant
31.7
0.13
0.14
8-yearscore
0.58(0.029)
0.097(0.004)
0.096(0.004)
Gender(boys-girls)
-0.35(0.26)
-0.04(0.05)
-0.03(0.05)
Socialclass(NonMan-Man)
0.74(0.29)
0.16(0.06)
0.16(0.06)
Schoolmean8-yearscore
0.02(0.11)
-0.008(0.02)
8-yrscorexschoolmean8-yrscore
0.02(0.01)
0.0006(0.02)
Random:
Level2
2.84(0.88)
0.084(0.024)
0.086(0.024)
-0.17(0.07)
-0.0024(0.0015)
-0.0030(0.0015)
0.012(0.007)
0.00018(0.00016)
0.00021(0.00016)
Level1
16.5(1.02)
0.413(0.029)
0.412(0.022)
-0.90(0.02)
-0.0032(0.0017)
0.06(0.02)
0.0000093(0.00041)
Oneofthereasonsforthehighnegativecorrelationbetweentheinterceptandslopeattheschoollevelmaybeassociatedwiththefactthatthe11-yearscorehasa'ceiling'withathirdofthestudentshavingscoresof35ormoreoutof40.Astandardprocedurefordealingwithsuchskeweddistributionsistotransformthedata,forexampletonormality,andthisismostconvenientlydonebycomputingNormalscores;thatisbyassigningNormalorderstatisticstotherankedscores.TheresultsfromthisanalysisaregivenundermodelBintable3.1.Notethatthescalehaschangedsincetheresponseisnowastandardnormalvariablewithzeromeanandunitstandarddeviation.Wenowfindthatthereisnolongeranyappreciablecomplexvariationatlevel1;thechisquaredtestyieldsavalueof3.4on2degreesoffreedom.Noristhereanyeffectofthecompositionalvariableofmeanschool8-yearscore;thechisquaredtestforthetwofixedcoefficientsassociatedwiththisgiveavalueof0.2on2degreesoffreedom.ThereducedmodelisfittedasC.Theparametersassociatedwiththerandomslopeatlevel2remainsignificant(
=7.7,P=0.02)andthelevel2correlationisfurtherreducedto-0.71.Figure3.1showsthelevel1standardisedresidualsplottedagainstthepredictedvaluesfromwhichitisclearthatnowthevarianceismuchmorenearlyconstant.Thisexampledemonstratesthatinterpretationsmaybesensitivetothescaleonwhichvariablesaremeasured.Itistypicalofmanymeasurementsinthesocialsciencesthattheirscalesarearbitraryandwecanjustifynonlinear,butmonotone,orderpreserving,transformationsiftheyhelptosimplifythestatisticalmodelandtheinterpretation.
Figure3.1Level1standardisedresidualsbypredictedvaluesforanalysisCintable3.1
Wearenotlimitedtomakingthevarianceafunctionofasingleexplanatoryvariable,andwecanconsidergeneralfunctionsofthesecombined.Somemaybeabsentfromthefixedpartofthemodel,orequivalentlyhavetheirfixedcoefficientsconstrainedtozero.Atraditional,singlelevel,exampleis'regressionthroughtheorigin'inwhichthefixedintercepttermiszerowhilealevel1varianceassociatedwiththeinterceptisfitted.
Wecanconsideranyparticularfunctionofexplanatoryvariablesasthebasisformodellingthevariance.Onepossibilityistotakethefixedpartpredictedvalue
anddefinethelevel1randomtermas
assumingthepredictedvalueispositive,sothatthelevel1variancebecomes
thatisproportionaltothepredictedvalue;oftenknownasa'constantcoefficientofvariation'model.Otherfunctionsareclearlypossible,andasweshallseeinchapter7oftentherearenaturalchoicesassociatedwithdistributionalassumptionsmadeabouttheresponses.
3.1.1Variancesforsubgroupsdefinedatlevel1
Acommonexampleofcomplexvariationatlevel1iswherevariancesarespecificforsubgroups.Forexample,formanymeasurementstherearegenderorsocialclassdifferencesinthelevel1variation.Astraightforwardwaytomodelthissituationinthecaseofasinglesuchgroupingisbydefiningthefollowingversionof(3.2)foramodelwithdifferentvariancesforchildrenwithmanualandwithnon-manualsocialclassbackgrounds.
(3.3)
Table3.2JSPdatawithnormalscoreof11-yearmathsasresponse.Subscript1refersto8-yearmathsscore,2tomanualgroup,3tononmanualgroupand4toboys.
Parameter
Estimate(s.e.)
Estimate(s.e.)
Estimate(s.e.)
A
B
C
Fixed
Constant
0.13
0.13
0.13
8-yearscore
0.096(0.004)
0.096(0.004)
0.096(0.004)
Gender(boys-girls)
-0.03(0.05)
-0.03(0.05)
-0.03(0.05)
SocialClass(NonMan-Man)
0.16(0.05)
0.16(0.05)
0.16(0.05)
Random
level2
0.086(0.025)
0.086
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