ANOVA英文版).ppt

AnalysisofVariance,2,ANalysisOfVAriance(ANOVA),ANOVAisastatisticalanalysistechniqueinwhichthetotalvariationisdividedintomeaningfulsourcesofvariation.RecallfromMeasurementSystemAnalysis,3,AnalysisofVariance(ANOVA),ANOVAisoftenusedforstudyingtherelationshipbetweenaresponsevariable(Y)andoneormoreexplanatoryorpredictorvariables(Xs).Thepredictorvariablesarealsocalledfactorsortreatments.Whiletheresponseisquantitative,thepredictorsmaybeeitherquantitativeorqualitative.However,quantitativepredictorsareanalyzedasiftheyarequalitative(orcategorical).,4,AnalysisofVariance(ANOVA),Tostudytheeffectofoneormorefactorsonaresponse,eachfactorhavingtwoormorelevels.ANOVAcanbeusedtodeterminethestatisticalsignificanceofeffectscalculatethecomponentsofvarianceestimatethecontributiontovariationbyeachidentifiedsourceestimatetheunderlyingnoisewithintheprocess,5,ANOVAPrinciple,N(1,1),N(2,2),N(3,3),N(4,4),6,ANOVA-Terminology,yijisthejthobservationwhenthefactorisatitsithlevel.,y14,y23,y31,y42,7,ANOVA-Terminology,yiistheexpectedresponsewhenthefactorisatitsithlevel.,_,8,ANOVA-Terminology,yistheoverallmeanofallobservationsacrossallfactorlevels.,9,ANOVA-Terminology,i=yiyistheeffectofthefactoratitsithlevel.,_,10,ANOVA-Terminology,ij=yijyiistheresidualorerrorofthejthobservationattheithlevelofthefactor.,_,11,ANOVA-Terminology,TotalDeviation=ExplainedDeviation+UnexplainedDeviation,12,ANOVAtable,1)DFTotal=DFFactor+DFErrorDFError=DFTotalDFFactor=(N1)(k1)=Nk2)SSTotal=SSFactor+SSError,13,ANOVATable,TotalDeviation=ExplainedDeviation+UnexplainedDeviation,SSTotal=SSExplained+SSUnexplained=SSFactor+SSError,14,HypothesisTesting,H0:1=2=kallgroupmeansareequalHa:ijforsomeijatleastonepairofgroupmeansisnotequalANOVAverifiesthenullhypothesisbycomparingthevariancebetweenthegroupsagainstthevariationwithinagroupmean:Thenullhypothesisisrejectedif.,15,HypothesisTesting,Ingeneral,Aswillbeseeninfollowingexamples,theactualformulationofF*isdependentonthetypeofANOVAmodel.,16,ANOVA-Assumptions,Theobservationsaremutuallyindependent.StatNonparametricsRunsTestThekgroupsexhibithomogeneityofvariance.i.e.1=2=kStatANOVATestforEqualVariancesTheresidualsarenormallydistributed.i.e.eNID(0,)StatBasicStatisticsNormalityTest,17,ANOVA-Assumptions,Independance,Homoscedasticity,Normality,18,ANOVA-Models,FixedEffects(ANOVAI)ModelfactorlevelsintheexperimentarespecificallychosentheconclusionisrelevantonlyforthechosenlevelsandmaynotbeextendedtosimilartreatmentsthatwerenotconsideredRandomEffects(ANOVAII)ModelfactorlevelsarerandomlydrawnfromalargepopulationoftreatmentstheconclusionisvalidfortheentirepopulationoftreatmentsMixedEffects(ANOVAIII)Modeloneormorefactorsarefixedfactorswhileothersarerandomfactors,19,ANOVA-Models,AkWayANOVAmodelinvolvesthestudyofkfactors.kWayANOVAI:kfactors;allfactorsoffixedeffectkWayANOVAII:kfactors;allfactorsofrandomeffect(WillnotbediscussedinGBcourse)kWayANOVAIII:kfactors;somefactorsoffixedeffect(WillnotbediscussedinGBcourse)othersofrandomeffect,20,One-WayANOVA-IModel,whereYijisthevalueoftheresponsevariableinthejthtrialfortheithfactorlevelortreatment.istheoverallmeanforallobservationsi=i.istheeffectoftheithtreatments.t.ijistherandomerrorcomponents.t.Thehypothesismaynowberephrased:H0:1=2=r=0Ha:notalliequal0,21,One-WayANOVA-IModel,22,Example,Anengineerwanttoevaluatetheeffectofthepercentageofthestrengtheneroftheadhesivetothebondingforce(breakageforceoftheassy)ofabondedactuator.Theengineerdecidestoinvestigatefourlevelsofpercentageofthestrengthener(ie,5%,10%15%&20%).Sixsamplesweretestedateachpercentagelevel.Thedataisstoredin“4.ANOVA.MTW”,23,Example,StatANOVAOne-way,24,Example,SessionWindowsOne-wayANOVA:ForceversusStrengtherAnalysisofVarianceforForceSourceDFSSMSFPStrength3382.79127.6019.610.000Error20130.176.51Total23512.96Individual95%CIsForMeanBasedonPooledStDevLevelNMeanStDev-+-+-+-+-5610.0002.828(-*-)10615.6672.805(-*-)15617.0001.789(-*-)20621.1672.639(-*-)-+-+-+-+-PooledStDev=2.55110.015.020.025.0,25,Example,26,Example,27,Example,StatRegressionResidualPlots,28,Example,29,One-WayANOVA-IIModel,whereYijisthevalueoftheresponsevariableinthejthtrialfortheithfactorlevelortreatment.istheoverallmeanforallobservationsi=i.istheeffectoftheithfactorlevels.t.ijistherandomerrorcomponents.t.Thehypothesismaynowberephrased:H0:=0Ha:0,30,One-WayANOVA-IIModel,ThereisnodifferenceincalculationoftheF-StatisticbetweenaOne-WayANOVA-IModelandaOne-WayANOVA-IIModel.Thedifferenceliesintheconclusion.,31,Example,Anengineerisinterestedinwindingmachinevariabilityinpeelstrengthofacoil.Theengineerselectsfourwindingmachinesatrandomanddeterminesthepeelstrengthoffoursampleschosenatrandomfromeachwindingmachine.,32,Example,StatANOVAGeneralLinearModel,33,Example,SessionWindowsGeneralLinearModel:PeelStrength