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1、Class5,Page9Class5:ANOVA(AnalysisofVariance)andF-testsI.WhatisANOVAWhatisANOVA?ANOVAistheshortnamefortheAnalysisofVariance.TheessenceofANOVAistodecomposethetotalvarianceofthedependentvariableintotwoadditivecomponents,oneforthestructuralpart,andtheotherfo
2、rthestochasticpart,ofaregression.Todaywearegoingtoexaminetheeasiestcase.II.ANOVA:AnIntroductionLetthemodelbe.Assumingxisacolumnvector(oflengthp)ofindependentvariablevaluesfortheith'observation,.Thenisthepredictedvalue.sumofsquarestotal:because.Thisisalwa
3、ystruebyOLS.=SSE+SSRImportant:thetotalvarianceofthedependentvariableisdecomposedintotwoadditiveparts:SSE,whichisduetoerrors,andSSR,whichisduetoregression.Geometricinterpretation:[blackboard]DecompositionofVarianceIfwetreatXasarandomvariable,wecandecompos
4、etotalvariancetothebetween-groupportionandthewithin-groupportioninanypopulation:Class5,Page9Prove:(bytheassumptionthat,forallpossiblek.)TheANOVAtableistoestimatethethreequantitiesofequation(1)fromthesample.Asthesamplesizegetslargerandlarger,theANOVAtable
5、willapproachtheequationcloserandcloser.Inasample,decompositionofestimatedvarianceisnotstrictlytrue.Wethusneedtoseparatelydecomposesumsofsquaresanddegreesoffreedom.IsANOVAamisnomer?III.ANOVAinMatrixIwilltrytogiveasimpliedrepresentationofANOVAasfollows:(be
6、cause)(inyourtextbook,monsterlook)SSE=e'eClass5,Page9(because,asalways)(inyourtextbook,monsterlook)IV.ANOVATableSOURCESSDFMSFwithRegressionSSRDF(R)MSRMSR/MSEDF(R)ErrorSSEDF(E)MSEDF(E)TotalSSTDF(T)Letususearealexample.Assumethatwehavearegressionestimatedt
7、obey=-1.70+0.840xANOVATableSOURCESSDFMSFwithRegression6.4416.446.44/0.19=33.891,18Error3.40180.19Total9.8419Weknow,,,,.IfweknowthatDFforSST=19,whatisn?n=20=201.71.7+0.840.84509.12-21.70.84100-125.0Class5,Page9=6.44SSE=SST-SSR=9.84-6.44=3.40DF(Degreesoffr
8、eedom):demonstration.Note:discountingtheinterceptwhencalculatingSST.MS=SS/DFp=0.000[askstudents].Whatdoesthep-valuesay?V.F-TestsF-testsaremoregeneralthant-tests,t-testscanbeseenasaspecialcaseofF-tests.Ifyouhavedifficultywi
