信息与通信英文简化版方差分析参考资料ch10DesignofExperimentsandAnalysis.ppt

Chapter10,DesignofExperimentsandAnalysisofVariance,One-WayANOVAF-Test,TypesofRegressionModels,Experimental,Designs,One-WayAnova,CompletelyRandomized,RandomizedBlock,Two-WayAnova,Factorial,One-WayANOVAF-Test,1.TeststheEqualityof2orMore(p)PopulationMeans2.VariablesOneNominalScaledIndependentVariable2orMore(p)TreatmentLevelsorClassificationsOneIntervalorRatioScaledDependentVariable3.UsedtoAnalyzeCompletelyRandomizedExperimentalDesigns,One-WayANOVAF-TestAssumptions,1.Randomness&IndependenceofErrorsIndependentRandomSamplesareDrawnforeachcondition2.NormalityPopulations(foreachcondition)areNormallyDistributed3.HomogeneityofVariancePopulations(foreachcondition)haveEqualVariances,One-WayANOVAF-TestHypotheses,H0:1=2=3=.=pAllPopulationMeansareEqualNoTreatmentEffectHa:NotAlljAreEqualAtLeast1Pop.MeanisDifferentTreatmentEffectNOT12.p,One-WayANOVAF-TestHypotheses,H0:1=2=3=.=pAllPopulationMeansareEqualNoTreatmentEffectHa:NotAlljAreEqualAtLeast1Pop.MeanisDifferentTreatmentEffectNOT12.p,X,f(X),1,=,2,=,3,X,f(X),1,=,2,3,WhyVariances?,ObserveonesamplefromeachtreatmentgroupTheirmeansmaybeslightlydifferentHowdifferentisenoughtoconcludepopulationmeansaredifferent?DependsonvariabilitywithineachpopulationHighervarianceinpopulationhighervarianceinmeansStatisticaltestsareconductedbycomparingvariabilitybetweenmeanstovariabilitywithineachsample,TwoPossibleExperimentOutcomes,SametreatmentvariationDifferentrandomvariation,A,Cantrejectequalityofmeans!,Rejectequalityofmeans!,TwoMorePossibleExperimentOutcomes,SametreatmentvariationDifferentrandomvariation,A,B,DifferenttreatmentvariationSamerandomvariation,Cantrejectequalityofmeans!,Reject,Reject,1.Compares2TypesofVariationtoTestEqualityofMeans2.ComparisonBasisIsRatioofVariances3.IfTreatmentVariationIsSignificantlyGreaterThanRandomVariationthenMeansAreNotEqual4.VariationMeasuresAreObtainedbyPartitioningTotalVariation,One-WayANOVABasicIdea,One-WayANOVAPartitionsTotalVariation,One-WayANOVAPartitionsTotalVariation,Totalvariation,One-WayANOVAPartitionsTotalVariation,Variationduetotreatment,Totalvariation,One-WayANOVAPartitionsTotalVariation,Variationduetotreatment,Variationduetorandomsampling,Totalvariation,One-WayANOVAPartitionsTotalVariation,Variationduetotreatment,Variationduetorandomsampling,Totalvariation,SumofSquaresAmongSumofSquaresBetweenSumofSquaresTreatmentAmongGroupsVariation,One-WayANOVAPartitionsTotalVariation,Variationduetotreatment,Variationduetorandomsampling,Totalvariation,SumofSquaresWithinSumofSquaresError(SSE)WithinGroupsVariation,SumofSquaresAmongSumofSquaresBetweenSumofSquaresTreatment(SST)AmongGroupsVariation,TotalVariation,X,Group1,Group2,Group3,Response,X,TreatmentVariation,X,X3,X2,X1,Group1,Group2,Group3,Response,X,Random(Error)Variation,X2,X1,X3,Group1,Group2,Group3,Response,X,SS=SSE+SST,But,Thus,SS=SSE+SST,One-WayANOVAF-TestTestStatistic,1.TestStatisticF=MST/MSEMSTIsMeanSquareforTreatmentMSEIsMeanSquareforError2.DegreesofFreedom1=p-12=n-pp=#Populations,Groups,orLevelsn=TotalSampleSize,One-WayANOVASummaryTable,Sourceof,Variation,Degrees,of,Freedom,Sumof,Squares,Mean,Square,(Variance),F,Treatment,p-1,SST,MST=,SST/(p-1),MST,MSE,Error,n-p,SSE,MSE=,SSE/(n-p),Total,n-1,SS(Total)=,SST+SSE,TheFdistribution,TwoparametersincreasingeitheronedecreasesF-alpha(exceptforv2F-Betweengroups6109.714123054.857057.690.0007Withingroups54045.8255136397.395776-Total60155.5396138435.909707Bartlettstestforequalvariances:chi2(2)=7.1931Probchi2=0.027,One-WayANOVAF-TestThinkingChallenge,YoureatrainerforMicrosoftCorp.Isthereadifferenceinmeanlearningtimesof12peopleusing4differenttrainingmethods(=.05)?M1M2M3M41011131891682359925Usethefollowingtable.,1984-1994T/MakerCo.,SummaryTable(PartiallyCompleted),Sourceof,Variation,Degreesof,Freedom,Sumof,Squares,Mean,Square,(Variance),F,Treatment,(Methods),348,Error,80,Total,F,0,4.07,One-WayANOVAF-TestSolution*,H0:1=2=3=4Ha:NotAllEqual=.051=32=8CriticalValue(s):,TestStatistic:Decision:Conclusion:,Rejectat=.05,ThereIsEvidencePop.MeansAreDifferent,=.05,F,MST,MSE,116,10,11,6,.,SummaryTableSolution*,Sourceof,Variation,Degreesof,Freedom,Sumof,Squares,Mean,Square,(Variance),F,Treatment,(Methods),4-1=3,348,116,11.6,Error,12-4=8,80,10,Total,12-1=11,428,10.26:condition1vs.2,Two-samplettestwithequalvariances-Group|ObsMeanStd.Err.Std.Dev.95%Conf.Interval-+-1|5030.642.83343920.0354424.9459936.334012|4226.214293.6572923.7019518.8282433.60033-+-combined|9228.619572.27025321.775524.1099933.12914-+-diff|4.4257144.559216-4.63196313.48339-Degreesoffreedom:90Ho:mean(1)-mean(2)=diff=0Ha:diff0t=0.9707t=0.9707t=0.9707P|t|=0.3343Pt=0.1671,10.26condition2vs.3,Two-samplettestwithequalvariances-Group|ObsMeanStd.Err.Std.Dev.95%Conf.Interval-+-2|4226.214293.6572923.7019518.8282433.600333|4715.127662.29055415.7032510.5170119.73831-+-combined|8920.359552.17653320.5333716.0341524.68495-+-diff|11.086634.2207672.69739419.47586-Degreesoffreedom:87Ho:mean(2)-mean(3)=diff=0Ha:diff0t=2.6267t=2.6267t=2.6267P|t|=0.0102Pt=0.0051,MultipleComparisonsProblem,PAtleastoneofpintervalsfailstocontainthetruedifference=1PAllcintervalscontainthetruedifferences=1(1-alpha)calphaIfcomparingmanypairs,needgreaterconfidenceforanyoneofthemthanyouwouldforrejectingequalityofanyonepair,MultipleComparisonsProcedure,1.TellsWhichPopulationMeansAreSignificantlyDifferentExample:1=232.PostHocProcedureDoneAfterRejectionofEqualMeansinANOVAOutputFromManyStatisticalcomputerProgramsvariousversions(Tukey,Bonferroni,etc.),10.26MultipleComparisons,(Bonferroni)RowMean-|ColMean|12-+-2|-4.42571|0.872|3|-15.5123-11.0866|0.0010.029,RandomizedBlockDesign,TypesofRegressionModels,Experimental,Designs,One-WayAnova,CompletelyRandomized,RandomizedBlock,Two-WayAnova,Factorial,RandomizedBlockDesign,1.ExperimentalUnits(Subjects)AreAssignedRandomlytoBlocksBlocksareAssumedHomogeneous2.OneFactororIndependentVariableofInterest2orMoreTreatmentLevelsorClassifications3.OneBlockingFactor,RandomizedBlockDesign,RandomizedBlockF-Test,1.TeststheEqualityof2orMore(p)PopulationMeans2.VariablesOneNominalScaledIndependentVariable2orMore(p)TreatmentLevelsorClassificationsOneNominalScaledBlockingVariableOneIntervalorRatioScaledDependentVariable3.UsedwithRandomizedBlockDesigns,RandomizedBlockF-TestAssumptions,1.NormalityProbabilityDistributionofeachBlock-TreatmentcombinationisNormal2.HomogeneityofVarianceProbabilityDistributionsofallBlock-TreatmentcombinationshaveEqualVariances,RandomizedBlockF-TestHypotheses,H0:1=2=3=.=pAllPopulationMeansareEqualNoTreatmentEffectHa:NotAlljAreEqualAtLeast1Pop.MeanisDifferentTreatmentEffect12.pIsWrong,RandomizedBlockF-TestHypotheses,H0:1=2=3=.=pAllPopulationMeansareEqualNoTreatmentEffectHa:NotAlljAreEqualAtLeast1Pop.MeanisDifferentTreatmentEffect12.pIsWrong,X,f(X),1,=,2,=,3,X,f(X),1,=,2,3,TheFRatioforRandomizedBlockDesigns,SS=SSE+SSB+SST,RandomizedBlockF-TestTestStatistic,1.TestStatisticF=MST/MSEMSTIsMeanSquareforTreatmentMSEIsMeanSquareforError2.DegreesofFreedom1=p-12=nbp+1p=#Treatments,b=#Blocks,n=TotalSampleSize,RandomizedBlockF-TestCriticalValue,Ifmeansareequal,F=MST/MSE1.OnlyrejectlargeF!,AlwaysOne-Tail!,F,a,p,n,p,(,),1,0,RejectH,0,DoNot,RejectH,0,F,1984-1994T/MakerCo.,RandomizedBlockF-TestExample,Youwishtodeterminewhichoffourbrandsoftireshasthelongesttreadlife.Yourandomlyassignoneofeachbrand(A,B,C,andD)toatirelocationoneachof5cars.Atthe.05level,isthereadifferenceinmeantreadlife?,F,0,3.49,RandomizedBlockF-TestSolution,H0:1=2=3=4Ha:NotAllEqual=.051=32=12CriticalValue(s):,TestStatistic:Decision:Conclusion:,Rejectat=.05,ThereIsEvidencePop.MeansAreDifferent,=.05,F=11.9933,Exercise10.47,Whatisthepurposeofblockingonweeksinthisstudy?c.Arethemeannumberofwalkersdifferentamongthepromptingconditions?d.Whichpairwisemeansaresignificantlydifferent?e.Whatassumptionsarerequiredfortheanalysisincandd?,FactorialExperiments,TypesofRegressionModels,Experimental,Designs,One-WayAnova,CompletelyRandomized,RandomizedBlock,Two-WayAnova,Factorial,FactorialDesign,1.ExperimentalUnits(Subjects)AreAssignedRandomlytoTreatmentsSubjectsareAssumedHomogeneous2.TwoorMoreFactorsorIndependentVariablesEachHas2orMoreTreatments(Levels)3.AnalyzedbyTwo-WayANOVA,FactorialDesignExample,Treatment,Factor2(TrainingMethod),Factor,Levels,Level1,Level2,Level3,Level1,19hr.,20hr.,22hr.,Factor,1,(High),11hr.,17hr.,31hr.,(Motivation),Level2,27hr.,25hr.,31hr.,(Low),29hr.,30hr.,49hr.,AdvantagesofFactorialDesigns,1.SavesTime&Efforte.g.,CouldUseSeparateCompletelyRandomizedDesignsforEachVariable2.ControlsConfoundingEffectsbyPuttingOtherVariablesintoModel3.CanExploreInteractionBetweenVariables,Two-WayANOVA,TypesofRegressionModels,Experimental,Designs,One-WayAnova,CompletelyRandomized,RandomizedBlock,Two-WayAnova,Factorial,Two-WayANOVA,1.TeststheEqualityof2orMorePopulationMeansWhenSeveralIndependentVariablesAreUsed2.SameResultsasSeparateOne-WayANOVAonEachVariableButInteractionCanBeTested3.UsedtoAnalyzeFactorialDesigns,Two-WayANOVAAssumptions,1.NormalityPopulationsareNormallyDistributed2.HomogeneityofVariancePopulationshaveEqualVariances3.IndependenceofErrorsIndependentRandomSamplesareDrawn,Two-WayANOVADataTable,Xijk,LeveliFactorA,LeveljFactorB,Observationk,Factor,FactorB,A,1,2,.,b,1,X,111,X,121,.,X,1b1,X,112,X,122,.,X,1b2,2,X,211,X,221,.,X,2b1,X,212,X,222,.,X,2b2,:,:,:,:,:,a,X,a11,X,a21,.,X,ab1,X,a12,X,a22,.,X,ab2,Two-WayANOVANullHypotheses,1.NoDifferenceinMeansDuetoFactorAH0:1.=2.=.=a.2.NoDifferenceinMeansDuetoFactorBH0:.1.=.2.=.=.b.3.NoInteractionofFactorsA&BH0:ABij=0,TotalVariation,Two-WayANOVATotalVar