回歸分析面板分層模型與應(yīng)用

分層模型分析與應(yīng)第五 一般化線(xiàn)性分層模HLM只描述連續(xù)型分布的層-1二分類(lèi)變量的結(jié)果(如結(jié)婚、離婚 定序次分類(lèi)結(jié)果(如工作的低、中、高的滿(mǎn)意度計(jì)數(shù)數(shù)據(jù)(如某女性一年內(nèi)人 次數(shù)這些結(jié)果，不能再假定層-1是正態(tài)性的線(xiàn)性模于(0,1)值域內(nèi)該限須對(duì)預(yù)測(cè)值進(jìn)行非線(xiàn)性轉(zhuǎn)換，如logit或probit給定預(yù)測(cè)結(jié)果值條件下，層-1層-1隨機(jī)

Y的可能值是非負(fù)的整數(shù)0、1、2、…… 不能線(xiàn)性ConceptualandStatisticalConsiderasimplelevel-1

0j1j(SES)ijY

1ifretainedin 0if E[Yij|0j,1j]0j1j(SES)ij

specificpredictedprobabilityofProblemswiththisHLMassumesthelevel-1randomeffectrijtobe withconstantvariance.But,Level-1randomeffecthastwodiscrete

(SES)

[

(SES)]

0 1

0 1

[

1

(SES)ij0–Level-1varianceis0?Var(Yij|0j,1j)ij(1ij?TheLinearModelforijactuallymakesnoEachunitincreaseinSESshouldleadtoasmallerchangeinasijapproaches0orTherefore,wemovetoa“generalizedlinearmodel”ratherthanastandardlinearmodelatlevel1、分類(lèi)型結(jié)果、以及計(jì)表兩層一般化線(xiàn)性模型的例標(biāo)準(zhǔn)一般化線(xiàn)性模型便成為其中的層-1 ),Goldstein(1991),但Reslow&Clayton（1993）、 MIXOR軟件和SASProcMixed 斯變換(Raudenbush,Yang&Yosef, Osgood,結(jié)果為二分類(lèi)變量(是否發(fā) 就業(yè) 學(xué)生流失的影響因素研究(Rumberger, Sampson,Raudenbush計(jì)數(shù)數(shù)研究小區(qū)年輕人的初始Reardon&Buka,生存模兩層層-1一個(gè)抽一個(gè)結(jié)HLM可視為GHLM正態(tài)抽恒等連線(xiàn)性結(jié)在保持層-1模型中所有隨機(jī)系數(shù)不變時(shí)的層-結(jié)果的期望布，且有期望值μij和同方差σ2ηij等于回歸系數(shù)的線(xiàn)性函數(shù)。這種轉(zhuǎn)換被稱(chēng)連接

將層-1預(yù)測(cè)值μij進(jìn)行轉(zhuǎn)保證預(yù)測(cè)結(jié)果限制于給定這種轉(zhuǎn)換的預(yù)測(cè)值標(biāo)注為有些連續(xù)變量無(wú)需轉(zhuǎn)換，即“恒等連接函令層-1連接函數(shù)ηij與一個(gè)具有層-1系數(shù)的線(xiàn)性過(guò)離散指層-1結(jié)果的變異性可能大于根據(jù)層-抽樣模型所期望定義Yij為mij次試驗(yàn)當(dāng)中“成功”的數(shù)量，?ij為上式標(biāo)志Yij服從有mij次試驗(yàn)、每一次試驗(yàn)成功概率為?ij的二項(xiàng)(Binomial)分布根據(jù)二項(xiàng)分布性質(zhì)，Yij的期當(dāng)mij=1時(shí)，Yijuqj，q=0,…,Q，構(gòu)成了uj的一個(gè)向一個(gè)Bernoulli學(xué)生留TheThreePartsoftheLevel-1ModelinSamplingmodel–describesthedistributionofthelevel-1observationsLinkfunction–transformsthemodel-basedpredictedvaluestobewithintherequiredrangeStructuralmodel–sameaswehaveseeninconventionalHLMmodels(Level2modelremainsthesameasinstandardmultivariatenormalrandomRevisedlevel-1modelfor Yij|

E(Yij|ij)ij Var(Yij|ij)ij(1ijLevel-1Link

log 1ij 0j1j(SESHGLMfor eLevel-1LinkexpOdds

1 1

log :``logodds(logitslogodds1 Inprobabilityfromlogodds

exp(ij1exp(ij ConversionLog<<10>>HLMasaSpecialCaseof2Yij|ij~NID(ij,2

|

)ij

Var(Yij

)Level-1Linkij 0 1j(SES)ijBinomialConsiderasimplelevel-1Yij/ 0

1j(SES)ijwhereYijisnumberofcoursefailureoutofmijcoursesmEm

| ,0| ,

0

1j(SES

specificpredictedprobabilityof ProblemswiththisHLMassumesthelevel-1randomeffectrijtobe withconstantvariance.But,Level-1randomeffecthasonlydiscreteLevel-1varianceis

|0j,1j

ij(1ijm TheLinearModelforijactuallymakesnoEachunitincreaseinSESshouldleadtoasmallerchangeinijijapproaches0orRevisedlevel-1modelforRevisedlevel-1modelforeYij|

B(mij,ijmijisnumberoftrials,ijisprobabilityofE(Yij|ij)mijij Var(Yij|ij)mijij(1ijLevel-1Link

log 1ij Level-1Structural 0j1j(SESCountConsiderasimplelevel-1

0j1j(SES)ijYij0,1,2,...numberofdaysE[Yij|0j,1j]0j1j(SES

the -specificpredictedeventProblemsProblemswiththisHLMassumesthelevel-1randomeffectrijtobe withconstantvariance.But,Level-1randomeffectwillbeLevel-1varianceisVar(Yij|0j,1j)TheLinearModelforijactuallymakesnoEachunitincreaseinSESshouldleadtoasmallerchangeinijijapproachesRevisedRevisedlevel-1modelforeYij|

Poisson(ijE(Yij|ij)ij Var(Yij|ij)Level-1Link log 0j1j(SESLevel-1modelforLevel-1modelfore(variableYij|

Poisson(mijijE(Yij|ij)mijij Var(Yij|ij)wheremijisLevel-1Link

0j1j(SESBernoulliExample:GradeRetentioninGraderepetitioninprimarygradesLevel-1,pre-primaryLevel-2MeanProbabilityandLogOddsof95%p.v.,ij2.2295%p.v.,ij

PopulationAveragevs.Unit-SpecificNotice:ForaschoolPopulationAveragevs.Unit-Specific

u0j,i.e.,u0jprob.ofrepetitiondoesnotequalpopulationave.ij(0) [1ij(0)

[1

0.097Becauseprob.distributionispositivelyskewed

MEAN>MEDIAN,whereasforijMEAN= Unit-specificmodels(witharandomeffectforeachschoolàlaHLM)willnotdirectlyreproducepopulationavg.results.Resultofnon-linearlinkfunction.CharacteristicsofUnit-SpecificDescribesaprocessatlevel1recurringwithineachWhatistheeffectofalevel-1predictorinagivenschoolholdingconstantthatschool’srandomeffect?HowdoesthiseffectvaryacrossHowdifferencesinlevel-2predictorsex variabilityacrossgroupsinlevel-1coefficientsareintrinsicallyunit-specificquestions.CharacteristicsofPopulation-AverageAnswerpopulationaverageE.g.,Howdoesriskofgraderepetitiondependonpre-primaryeducationacrossthenation? concernsmaybeaddressedbypopulation-averagemodelsP-AVsU-Population-averageeffectwilldifferfromunit-specificestimates,especiallywhen iscloseto0or1andwhenP-AVsU-Unit-specificmodelsmoredependentonmodelassumptionsatlevel2.Unit-specificUnit-specific

0j00u0EY| 0 00

111

?(Yij

|u0

0)

1exp(?00

1

Population-averageE(Yij)? )

1exp( 00

.1400 1exp(? 00

1exp(1.72ComparisonComparisonofUnit-SpecificandPopulation-AverageEffectsforThaiDataLevelLevelLogitLink:LogitLink:--MeanPre-Primary---HGLMforHomicideHGLMforHomicideinChicagoYj|j~P(mj,jjishomiciderateper100,000peopleinneighborhoodmjisthepopulationsizeofneighborhoodj(inunitsofE(Yj|j)mj Var(Yj|j)mjLevel-1Link log(jParameterEstimationin“Penalizedquasi-likelihood”–requiresadoublyi tivealgorithm:microi tions–sameasstandardHLM tions;continuetoconvergencealternatewithmacroi tions–linearizeddependentvariableandweights backtomicroi ParameterEstimationinQuasi-penalizedlikelihoodisfastandconvergesreliablybutwilltendtounderestimateτespeciallyasτgetslarge.Therearebetterapproximations--La estimationinHLM6hasanerroroforderO(n-2)wherenisthetypicallevel1samplesize.Ifnissmall,however,shoulduseadaptiveGauss-Hermitequadratureorhigh-orderLa ce(whichweareintheprocessofaddingtoVersion7.)Ordered eisorderedE.g.,Stronglydisagree,disagree,agree,stronglyNever,sometimes,often,ExamplequestionnaireTeacherCommitment–“Ifyoucouldstartoveragainwouldchooseteachingasa1=“Yes”coded2=“NotProbabilityProbabilityofrespondingincategory Pr(Rm)m1,...,so1Pr(R1)Pr("yes" Pr(R2)Pr("notsure"1Pr(R1)Pr("no"ConvenientConvenienttousecumulativePr(R1) 2Pr(R2)2

12Pr(R

3) ConceptualframefortheWeposittheexistenceofanunderlyingcontinuouslatentvariablethatdeterminesthecategorypeopleselect.Lowvaluesofthislatentvariableindicatelowlevelsofteachersatisfaction.SopeoplewhoareverylowwillsayPeopleinthemiddlewillsay“NotPeoplewhoareveryhighwillsay“Yes”So,So,M-1cumulativeprobabilitiesareofThecumulative *

Pr(Rm)m log log m1* Pr(Rm)wherem1,...,M m esthe einalogisticregression.“ProportionalOddsModel” mXiConsidertwocaseswhereCase1:two sdifferinX:Howdotheirpredictedvaluesdifferforagivencategory?m1mm2mXsom1m2(X1X2),whichdoesnotdependonCase2:Howdothepredictedvaluesofeachlogitdifferforany acrosscategories?1i1Xi2i2Xiso1i2i12,whichdoesnotdependonXorInsteadofonethresholdforeachcategory,weshalluseaninterceptplusadifferenceforeachcategory.SoifM=3,wehave10201Xi

30 1202303ExtensionExtensiontoTwoLevel-1samplingmodelism m)m

k

Level-1Structural M 0jqjXqij Level-2Structuralqj q0qsWqsjuqjHierarchicalHierarchicalGeneralizedLinearModelswithMultinomialCategoricalDataUseDataarenotDataviolateproportionaloddsPredictorshavedifferentassociationswiththeprobabilitiesofdifferentresponsesIfPr(Rij1)Pr(Rij2)Pr(Rij3) 1

Level-1SamplingDummy Ymij1ifRijm,0So,forthe3-category100100E(Ymij|mij)Var(Ymij|mij)mij(1mijCov(Ymij,Ym'ij)Level-1LinkForeachcategorym=1,…,M-

Pr(Rijm)

log log

M) Misthe“referenceLevel-1StructuralQQmij0j(m)qj(m)XqijQForM=3therewouldbetwolevel-1Q1ij0j(1)qj(1)XqiqQ2ij 0j(2)qj(2)XqiQqqj(m) q0(m)qs(m)Wsjuqj(m)whereqPost-secondaryDestinationsExamplee:20-year-oldswereaskedifWereattendinga4-yearcollegeWereattendinga2-yearcollegeDidnotattendpost-secondaryinstitutionLevel1:Grade-8mathachievement,race, ,socio-economicstatus,familystructureLevel2:Catholicschoolattendance,otherschoolComparisonofCoefficientsAcrossE.g.,RelativeoddsofCat