分層模型分析與應(yīng)用五講ghlm

第五一第五一般化線性分層模型ld模型采用的結(jié)果變量可用離散變量不同測量層次結(jié)果的影響分析成為主流潛在變量已被植入分層模型分層模型的貝葉斯推斷廣泛普及和應(yīng)用HLM只描述連續(xù)型分布的層HLM只描述連續(xù)型分布的層-1結(jié)果基于模型誤差的正態(tài)分布統(tǒng)計推斷可在正態(tài)分布理論范圍內(nèi)進行若結(jié)果變量屬于離散型變量，推斷則更為復(fù)雜若結(jié)果變量屬于離散型變量，推斷則更為復(fù)雜二分類變量的結(jié)果(如結(jié)婚、離婚多分類定類測量結(jié)果如絕育、I服藥)定序次分類結(jié)果(如工作的低、中、高的滿意度計數(shù)數(shù)據(jù)(如某女性一年內(nèi)人工流產(chǎn)次數(shù)這些結(jié)果，不能再假定層-1是正態(tài)性的線性模型HLM的局限例解HLM的局限例解二分類結(jié)果變量Y的預(yù)測值在作為Y＝1的發(fā)生概率來理解時，必須處于(0,1)值域內(nèi)該限制使模型定義的效應(yīng)規(guī)模有意義須對預(yù)測值進行非線性轉(zhuǎn)換，如logit或probit轉(zhuǎn)換給定預(yù)測結(jié)果值條件下，層-1隨機效應(yīng)只能取兩值之一，不可能為正態(tài)分布層-1隨機效應(yīng)方差不–HLM的局HLM的局限例解計數(shù)數(shù)據(jù)Y的可能值是非負的整數(shù)0、1、2、……典型的正偏態(tài)分若0很少，Y*=log(1+Y或Y*=Y轉(zhuǎn)換可進行線性若事件發(fā)生頻數(shù)很少且存在大量的不能通過轉(zhuǎn)換來近似取得正態(tài)假定線性模型的預(yù)測值可能是負的模型系數(shù)的意義不能ConceptualandStatisticalConsiderasimplelevel-1ConceptualandStatisticalConsiderasimplelevel-10j1j(SES1ifinY0ifE[Yij|0j,1j]0j1j(SESthepersonspecificpredictedprobabilityProblemswiththisHLMassumesthelevel-1randomeffectrijtobewithconstantvariance.But,–Level-1ProblemswiththisHLMassumesthelevel-1randomeffectrijtobewithconstantvariance.But,–Level-1randomeffecthastwodiscrete?(SES] [01(SES)][ (SES01]01–Level-1varianceis?Var(Yij|0j,1j)ij(1ijTheTheLinearModelforijactuallymakesno–EachunitincreaseinSESshouldleadtoasmallerchangeinasijapproaches0orTherefore,wemovetoa“generalizedlinearratherthanastandardlinearmodelatlevel一般一般化線性模型這種方法可以適合許多類型的數(shù)據(jù)，包括連續(xù)型結(jié)果、分類型結(jié)果、以及計數(shù)數(shù)據(jù)。表兩層表兩層一般化線性模型的例子一般化線性模型可以擴展到包括多個層次嵌套的場合。一般化線性模型可以擴展到包括多個層次嵌套的場合。標(biāo)準(zhǔn)一般化線性模型便成為其中的層-1模型。對離散變量用最大似然估計(ML)做一階近似對離散變量用最大似然估計(ML)做一階近似–Stiratelli、基于ML開發(fā)軟件，可采用不同的離散型結(jié)果Goldstein(1991),但Reslow&Clayton（1993）、MIXOR軟件和SASProcMixed軟件高階拉普拉斯變換(RaudenbushYang&Yosef,Osgood,Marshall,1995）重復(fù)測量研結(jié)果為二分類變量(是否發(fā)生犯罪就業(yè)或失業(yè)作為時間協(xié)變量，其它變量不涉及時變層-1模型解釋個人犯罪的OR的logistic模型個體參數(shù)在層-2中也存在變學(xué)生流失的影響因素研究學(xué)生流失的影響因素研究兩層模型－學(xué)生層次和學(xué)校層次層-1模型logistic回歸早期流失的OR依賴于學(xué)生的背景特征Sampson,Raudenbush計數(shù)數(shù)據(jù)作為結(jié)果研究小區(qū)年輕人的初始抽齡Reardon&Buka,生存模型層-1模型的組成一個抽樣模型一個連接函數(shù)一個結(jié)構(gòu)模型HLM可視為GHLM的特例正態(tài)抽樣模型恒等連接函數(shù)線性結(jié)構(gòu)模型在保持層-1模型中所有隨機系數(shù)不變時的層-結(jié)果的期望值。在保持層-1期望值不變條件下，層-1結(jié)果被假設(shè)為從某種概率分布中產(chǎn)生。這種層-1概率分布就稱為層-1抽樣模型。層-1抽樣模型兩層GHLM抽樣模型表層-1抽樣模型兩層GHLM抽樣模型表示為是獨立正態(tài)–表示給定預(yù)測值 條件下，層-1結(jié)果布，且有期望值μij和同方差–層-1期望值和方差可2對層-1結(jié)果的期望值進行轉(zhuǎn)換，令其轉(zhuǎn)換結(jié)果對層-1結(jié)果的期望值進行轉(zhuǎn)換，令其轉(zhuǎn)換結(jié)果等于回歸系數(shù)的線性函數(shù)。這種轉(zhuǎn)換被稱為層-1連接函數(shù)。連接函數(shù)將層-1預(yù)測值μij進行保證預(yù)測結(jié)果限制于給定區(qū)間這種轉(zhuǎn)換的預(yù)測值標(biāo)注為有些連續(xù)變量無需轉(zhuǎn)換，即“恒等連接函數(shù)”與一個具與一個具有層-1系數(shù)的線性令層-1連接函數(shù)模型相等。這個線性模型就是層-1結(jié)構(gòu)模型。線性結(jié)構(gòu)模型為：層-1抽樣模型、層-1連接函數(shù)與層-1結(jié)構(gòu)模型的組合起來，即為通常HLM的層-1模型過離散指層過離散指層-1結(jié)果的變異性可能大于根據(jù)層-抽樣模型所期望的水平。二分二分類結(jié)果的兩層模型層-1抽樣模型定義Yij層-1抽樣模型定義Yijmij次試驗當(dāng)中“成功”的數(shù)量，?ij為–上式標(biāo)志Yij服從有mij次試驗、每一次試驗成功概率為?ij的二項(Binomial)分布–根據(jù)二項分布性質(zhì)，Yij的期望值和方差 層-2模型層-2模型量，具有多元正態(tài)分布，各分量的平均值為零、方差協(xié)方差矩陣為Ｔ。一個Bernoulli一個Bernoulli分布的例子：泰國學(xué)生留級研究TheThreeTheThreePartsoftheLevel-1ModelinSamplingmodel–describesthedistributionofthelevel-1observationsLinkfunction–transformsthemodel-predictedvaluestobewithintherequired–Structuralmodel–sameaswehaveseeninconventionalHLMmodels(Level2modelremainsthesameasinstandardHLM-multivariatenormalrandomeffects)Revisedlevel-1modelforbinaryLevel-1Sampling|ijRevisedlevel-1modelforbinaryLevel-1Sampling|ij~Bernoulli(ijE(Yij|ij)ij|ij)ij(1ijLevel-1Link 1Level-1Structural0j1j(SESHGLMforbinaryoutcomeLevel-1LinkexpOddsHGLMforbinaryoutcomeLevel-1LinkexpOdds 1 1log :``logodds(logitslogodds1Inprobabilityfromlogodds:ijexp(ij 1exp(ij ConversionLog<<ConversionLog<<10>>HLMasaSpecialCaseofLevel-1SamplingY|HLMasaSpecialCaseofLevel-1SamplingY|~2E(Y|)|)Level-1LinkLevel-1Structural0j1j(SESBinomialConsiderasimplelevel-1/mijBinomialConsiderasimplelevel-1/mij0j1j(SES)ijwhereYijisnumberofcoursefailureoutofcourses|0j,1j01j(SES) EthepersonspecificpredictedprobabilityofcourseProblemswiththistoHLMassumesthelevel-1randomeffectProblemswiththistoHLMassumesthelevel-1randomeffectwithconstantvariance.Level-1randomeffecthasonlydiscreteLevel-1varianceisij(1ij|0j,1mTheLinearModelforijactuallymakesnoEachunitincreaseinSESshouldleadtoasmallerchangeinijijapproaches0orRevisedlevel-1modelforbinomialLevel-1Sampling| ~BRevisedlevel-1modelforbinomialLevel-1Sampling| ~B(mij,ijisnumberofisprobabilityofE(Yij|ij)mijijVar(Yij|ij)mijij(1ijLevel-1Linklog 1Level-1Structural0j1j(SESCountConsiderasimplelevel-1CountConsiderasimplelevel-10j1j(SES)ij0,1,2,...numberofE[Yij|0j,1j]0j1j(SEStheperson-specificpredictedeventProblemswiththisProblemswiththistoHLMassumesthelevel-1randomeffectwithconstantvariance.Level-1randomeffectwillbeLevel-1varianceisVar(Yij|0j,1j)TheLinearModelforijactuallymakesnoEachunitincreaseinSESshouldleadtoasmallerchangeinijijapproachesRevisedlevel-1modelforcountLevel-1Revisedlevel-1modelforcountLevel-1Sampling|ij~Poisson(ijE(Yij|ij)ijLevel-1Link|ij) Level-1Structural0j1j(SESLevel-1modelforcountoutcome(variableLevel-1SamplingLevel-1modelforcountoutcome(variableLevel-1Sampling|ij~Poisson(mijijE(Yij|ij)mijijwheremijis|ij)mijLevel-1LinkLevel-1Structural0j1j(SESBernoulliExample:GradeBernoulliExample:GradeRetentionin–GraderepetitioninprimarygradesLevel-1–Sex,pre-primaryLevel-2–MeanProbabilityandLogOddsofProbabilityandLogOddsof(95%p.v.,ijPopulationAveragevs.Unit-Specificu0j,i.e.,PopulationAveragevs.Unit-Specificu0j,i.e.,u0jNotice:Foraschoolatthemedianprob.ofrepetitiondoesnotequalpopulationave.11ij(0)0.097[1Becauseprob.distributionispositivelyskewedMEAN>MEDIAN,whereasforMEAN=Unit-specificmodels(witharandomeffectforeachschoolàlaHLM)willnotdirectlyreproducepopulationavg.results.Resultofnon-linearlinkfunction.CharacteristicsCharacteristicsofUnit-SpecificDescribesaprocessatlevel1recurringwithineachWhatistheeffectofalevel-1predictorinagivenschoolholdingconstantthatschool’srandomHowdoesthiseffectvaryacrossHowdifferencesinlevel-2predictorsexplainvariabilityacrossgroupsinlevel-1coefficientsareintrinsicallyunit-specificquestions.CharacteristicsofPopulation-Average?CharacteristicsofPopulation-Average?Answerpopulationaverage?E.g.,Howdoesriskofgraderepetitiondependpre-primaryeducationacrossthe?Manypolicyconcernsmaybeaddressedpopulation-averageP-AVsU-Population-averageeffectwillP-AVsU-Population-averageeffectwilldifferfrom?isclosespecificestimates,especially000or1andis?Unit-specificmodelsmoredependentonassumptionsatlevelUnit-specific0j00u0| 11expUnit-specific0j00u0| 11exp 000011?Y| 0)10 100Population-average1E(Y)1 *0011?Y)11exp(?*)00ComparisonofUnit-SpecificandPopulation-AverageEffectsforThaiDataLevelLevelLogitLink:ComparisonofUnit-SpecificandPopulation-AverageEffectsforThaiDataLevelLevelLogitLink:LogitLink:Mean-----Pre-Primary---HGLMforHomicideHGLMforHomicideCountsinChicagoNeighborhoodsLevel-1SamplingYj|j~P(mj,jjishomiciderateper100,000peopleinneighborhoodmjisthepopulationsizeofneighborhoodj(inunitsofE(Yj|j)mjVar(Yj|j)mjLevel-1Linkjlog(jParameterParameterEstimationin“Penalizedquasi-likelihood”–requiresadoublyiterativealgorithm:microiterations–sameasstandardHLMiterations;continuetoconvergencealternatewithmacroiterations–dependentvariableandweightsrecomputed;thenbacktomicroiterationsParameterEstimationParameterEstimationinQuasi-penalizedlikelihoodisfastandconvergesreliablybutwilltendtounderestimateτespeciallyasτgetslarge.Therearebetterapproximations--LaplaceestimationinHLM6hasanerroroforderO(n-2)wherenisthetypicallevel1samplesize.Ifnissmall,however,shoulduseadaptiveGauss-Hermitequadratureorhigh-orderLaplace(whichweareintheprocessofaddingtoVersion7.)OrderedCategoricalOutcomeOrderedCategoricalOutcomeisordered–E.g.,Stronglydisagree,disagree,agree,stronglyagree;Never,sometimes,often,always.Examplequestionnaire–TeacherCommitment–“Ifyoucouldstartoveragainwouldyouchooseteachingasacareer?”–???1=“Yes”coded2=“Notsure”ProbabilityofrespondingincategorymPr(Rm)mProbabilityofrespondingincategorymPr(Rm)m1,...,MPr(R1)Pr("yes"Pr(R2)Pr("notsure"Pr(R1)Pr("no"ConvenienttousecumulativePr(R1)*112)Pr(RPr(R*123)3123ConceptualConceptualframefortheWeposittheexistenceofanunderlyingcontinuouslatentvariablethatdeterminesthecategorypeopleselect.Lowvaluesofthislatentvariableindicatelowlevelsofteachersatisfaction.SopeoplewhoareverylowwillsayPeopleinthemiddlewillsay“NotPeoplewhoareveryhighwillsay“Yes”So,M-1cumulativeprobabilitiesareofThecumulativePr(RmSo,M-1cumulativeprobabilitiesareofThecumulativePr(Rm)*mPr(Rm)1m*mwherem1,...,MThisbecomestheoutcomeinalogisticregression.“ProportionalOddsModel”mConsidertwocaseswhereConsidertwocaseswhereCase1:twopersonsdifferinX:Howdotheirpredictedvaluesdifferforagivencategory?mmXsom1m2(X1X2),whichdoesnotdependonCase2:Howdothepredictedvaluesofeachlogitdifferforanypersonacrosscategories?12so1i2i12,whichdoesnotdependonXorInsteadofoneInsteadofonethresholdforeachcategory,weshalluseaninterceptplusadifferenceforeachcategory.SoifM=3,wehave101201Xi301Xi12030ExtensiontoTwoLevel-1samplingmodelExtensiontoTwoLevel-1samplingmodelism*m)kLevel-1StructuralMQqjXLevel-1StructuralMQqjXqij0Level-2StructuralqjuqjHierarchicalGeneralizedHierarchicalGeneralizedLinearModelswithMultinomialCategoricalDataUseDataarenotDataviolateproportionaloddsPredictorshavedifferentassociationswiththeprobabilitiesofdifferentresponsesIfPr(Rij1) 2)Pr(Rij3)3ij1Level-1Samplingm,0DummySo,Level-1Samplingm,0DummySo,forthe3-category|mij)Var(Ymij|mij)mij(1mijCov(Ymij,Ym'ij)100100Level-1LinkForeachcategorym=1,…,M- m)Level-1LinkForeachcategorym=1,…,M- m)log log M)Misthe“referenceLevel-1Structuralmij0j(m)qj(m)XqijForM=3therewouldbetwolevel-1QQj(1)qj(1)XqijQj(2)qj(2)XqijLevel-2StructuralLevel-2Structuralq0(m)qj(m)uqj(m)whereq0,...,Post-secondaryDestinationsExamplePost-secondaryDestinationsExampleOutcome:20-year-oldswereaskedif–––Wereattendinga4-yearcollege(1)Wereattendinga2-yearcollegeDidnotattendpost-secondaryinstitutionLevel1:Grade-8mathachievement,race,sex,socio-economicstatus,familystructureLevel2:Catholicschoolattendance,otherprivateschoolattendanceComparisonofCoefficientsAcrossE.g.,RelativeoddsofCatholic-schoolandpublic-schoolComparisonofCoefficientsAcrossE.g.,RelativeoddsofCatholic-schoolandpublic-schoolstudentsattending4-yearschool(asopposedtonoschool))Pr(R1|Catholic)/Pr(R3|CatholicPr(R1|Public)/Pr(R3|PublicSimilarly,relativeoddsforattending2-yearschool(vs.no)Pr(R2|Catholic)/Pr(R3|CatholicPr(R2|Public)/Pr(R3|PublicAnd,relativeoddsforattending4-yearschoolvs.2-year)Pr(R1|Catholic)/Pr(R2|CatholicPr(R1|Public)/Pr(R2|Public與HLM有與HLM有關(guān)的一、如何匯報一、如何匯報的結(jié)果？定量研究文獻中，會有不同的回歸分析結(jié)果的方式傳達的信息才是較為完備、較為準(zhǔn)確的呢？常見的表達方法(1)第一種表達方法在實際研究中，常常用*(1)第一種表達方法在實際研究中，常常用*表示在水平下統(tǒng)計性顯著，用**表示在平統(tǒng)計性顯著，用因變量：收自變量***表示在水平統(tǒng)計父親受教育水平母親受教育水平 鞋的尺碼-請大家思考，這一回歸分析結(jié)果的表達存在什么問題？評例如母親受評例如母親受教育水平鞋的尺碼-那么，在這種情況下，父親教育水平有*，而鞋的尺碼沒有*，那父親教育水平對收入的影響是否比