利用Excel进行统计分析-Chapter11-Analysis of Variance.ppt

Chap11 1 StatisticsforManagersUsingMicrosoft Excel5thEdition Chapter11AnalysisofVariance Chap11 2 LearningObjectives Inthischapter youlearn ThebasicconceptsofexperimentaldesignHowtousetheone wayanalysisofvariancetotestfordifferencesamongthemeansofseveralgroupsHowtousethetwo wayanalysisofvarianceandinterprettheinteraction Chap11 3 ChapterOverview AnalysisofVariance ANOVA F test Tukey Kramertest One WayANOVA Two WayANOVA InteractionEffects Chap11 4 GeneralANOVASetting InvestigatorcontrolsoneormorefactorsofinterestEachfactorcontainstwoormorelevelsLevelscanbenumericalorcategoricalDifferentlevelsproducedifferentgroupsThinkofthegroupsaspopulationsObserveeffectsonthedependentvariableArethegroupsthesame Experimentaldesign theplanusedtocollectthedata Chap11 5 CompletelyRandomizedDesign Experimentalunits subjects areassignedrandomlytothedifferentlevels groups SubjectsareassumedhomogeneousOnlyonefactororindependentvariableWithtwoormorelevels groups Analyzedbyone factoranalysisofvariance one wayANOVA Chap11 6 One WayAnalysisofVariance EvaluatethedifferenceamongthemeansofthreeormoregroupsExamples Accidentratesfor1st 2nd and3rdshiftExpectedmileageforfivebrandsoftiresAssumptionsPopulationsarenormallydistributedPopulationshaveequalvariancesSamplesarerandomlyandindependentlydrawn Chap11 7 Hypotheses One WayANOVA Allpopulationmeansareequali e notreatmenteffect novariationinmeansamonggroups Atleastonepopulationmeanisdifferenti e thereisatreatment groups effectDoesnotmeanthatallpopulationmeansaredifferent atleastoneofthemeansisdifferentfromtheothers Chap11 8 Hypotheses One WayANOVA AllMeansarethesame TheNullHypothesisisTrue NoGroupEffect Chap11 9 Hypotheses One WayANOVA Atleastonemeanisdifferent TheNullHypothesisisNOTtrue TreatmentEffectispresent or Chap11 10 PartitioningtheVariation Totalvariationcanbesplitintotwoparts SST TotalSumofSquares Totalvariation SSA SumofSquaresAmongGroups Among groupvariation SSW SumofSquaresWithinGroups Within groupvariation SST SSA SSW Chap11 11 PartitioningtheVariation TotalVariation theaggregatedispersionoftheindividualdatavaluesaroundtheoverall grand meanofallfactorlevels SST Within GroupVariation dispersionthatexistsamongthedatavalueswithintheparticularfactorlevels SSW Among GroupVariation dispersionbetweenthefactorsamplemeans SSA SST SSA SSW Chap11 12 PartitioningtheVariation AmongGroupVariation SSA WithinGroupVariation SSW TotalVariation SST Chap11 13 TheTotalSumofSquares Where SST Totalsumofsquaresc numberofgroupsnj numberofvaluesingroupjXij ithvaluefromgroupjX grandmean meanofalldatavalues SST SSA SSW Chap11 14 TheTotalSumofSquares Chap11 15 Among GroupVariation Where SSA Sumofsquaresamonggroupsc numberofgroupsnj samplesizefromgroupjXj samplemeanfromgroupjX grandmean meanofalldatavalues SST SSA SSW Chap11 16 Among GroupVariation 1 2 c Chap11 17 Within GroupVariation Where SSW Sumofsquareswithingroupsc numberofgroupsnj samplesizefromgroupjXj samplemeanfromgroupjXij ithvalueingroupj SST SSA SSW Chap11 18 Within GroupVariation Chap11 19 ObtainingtheMeanSquares MeanSquaresAmong MeanSquaresWithin MeanSquaresWithin Chap11 20 One WayANOVATable c numberofgroupsn sumofthesamplesizesfromallgroupsdf degreesoffreedom Chap11 21 One WayANOVATestStatistic TeststatisticMSAismeansquaresamongvariancesMSWismeansquareswithinvariancesDegreesoffreedomdf1 c 1 c numberofgroups df2 n c n sumofallsamplesizes H0 1 2 cH1 Atleasttwopopulationmeansaredifferent Chap11 22 One WayANOVATestStatistic TheFstatisticistheratiooftheamongvariancetothewithinvarianceTheratiomustalwaysbepositivedf1 c 1willtypicallybesmalldf2 n cwilltypicallybelarge DecisionRule RejectH0ifF FU otherwisedonotrejectH0 0 05 RejectH0 DonotrejectH0 FU Chap11 23 One WayANOVAExample Club1Club2Club3254234200263218222241235197237227206251216204 Youwanttoseeifthreedifferentgolfclubsyielddifferentdistances Yourandomlyselectfivemeasurementsfromtrialsonanautomateddrivingmachineforeachclub Atthe 05significancelevel isthereadifferenceinmeandistance Chap11 24 One WayANOVAExample 270260250240230220210200190 Distance Club 123 Club1Club2Club3254234200263218222241235197237227206251216204 Chap11 25 One WayANOVAExample X1 249 2X2 226 0X3 205 8X 227 0 n1 5n2 5n3 5n 15c 3 SSA 5 249 2 227 2 5 226 227 2 5 205 8 227 2 4716 4 SSW 254 249 2 2 263 249 2 2 204 205 8 2 1119 6 Chap11 26 One WayANOVAExample MSA 4716 4 3 1 2358 2 MSW 1119 6 15 3 93 3 F 25 275 0 05 FU 3 89 RejectH0 DonotrejectH0 CriticalValue FU 3 89 Chap11 27 One WayANOVAExample H0 1 2 3H1 jnotallequal 05df1 2df2 12 Decision RejectH0at 0 05 Conclusion Thereisevidencethatatleastone jdiffersfromtherest Chap11 28 One WayANOVAinExcel EXCEL SelectTools DataAnalysis ANOVA singlefactor Chap11 29 TheTukey KramerProcedure Tellswhichpopulationmeansaresignificantlydifferente g 1 2 3DoneafterrejectionofequalmeansinANOVAAllowspair wisecomparisonsCompareabsolutemeandifferenceswithcriticalrange x 1 2 3 Chap11 30 Tukey KramerCriticalRange where QU ValuefromStudentizedRangeDistributionwithcandn cdegreesoffreedomforthedesiredlevelof seeappendixE 9table MSW MeanSquareWithinnjandnj Samplesizesfromgroupsjandj Chap11 31 TheTukey KramerProcedure 1 Computeabsolutemeandifferences Club1Club2Club3254234200263218222241235197237227206251216204 2 FindtheQUvaluefromthetableinappendixE 9withc 3and n c 15 3 12degreesoffreedomforthedesiredlevelof 05usedhere Chap11 32 TheTukey KramerProcedure 5 Alloftheabsolutemeandifferencesaregreaterthancriticalrange Thereforethereisasignificantdifferencebetweeneachpairofmeansatthe5 levelofsignificance 3 ComputeCriticalRange 4 Compare Chap11 33 ANOVAAssumptions RandomnessandIndependenceSelectrandomsamplesfromthecgroups orrandomlyassignthelevels NormalityThesamplevaluesfromeachgrouparefromanormalpopulationHomogeneityofVarianceCanbetestedwithLevene sTest Chap11 34 ANOVAAssumptionsLevene sTest Teststheassumptionthatthevariancesofeachgroupareequal First definethenullandalternativehypotheses H0 21 22 2cH1 Notall 2jareequalSecond computetheabsolutevalueofthedifferencebetweeneachvalueandthemedianofeachgroup Third performaone wayANOVAontheseabsolutedifferences Chap11 35 Two WayANOVA ExaminestheeffectofTwofactorsofinterestonthedependentvariablee g PercentcarbonationandlinespeedonsoftdrinkbottlingprocessInteractionbetweenthedifferentlevelsofthesetwofactorse g Doestheeffectofoneparticularcarbonationleveldependonwhichlevelthelinespeedisset Chap11 36 Two WayANOVA AssumptionsPopulationsarenormallydistributedPopulationshaveequalvariancesIndependentrandomsamplesareselected Chap11 37 Two WayANOVASourcesofVariation TwoFactorsofinterest AandBr numberoflevelsoffactorAc numberoflevelsoffactorBn numberofreplicationsforeachcelln totalnumberofobservationsinallcells n rcn Xijk valueofthekthobservationoflevelioffactorAandleveljoffactorB Chap11 38 Two WayANOVASourcesofVariation SSTTotalVariation SSAFactorAVariation SSBFactorBVariation SSABVariationduetointeractionbetweenAandB SSERandomvariation Error DegreesofFreedom r 1 c 1 r 1 c 1 rc n 1 n 1 SST SSA SSB SSAB SSE Chap11 39 Two WayANOVAEquations TotalVariation FactorAVariation FactorBVariation Chap11 40 Two WayANOVAEquations InteractionVariation SumofSquaresError Chap11 41 Two WayANOVAEquations r numberoflevelsoffactorAc numberoflevelsoffactorBn numberofreplicationsineachcell Chap11 42 Two WayANOVAEquations r numberoflevelsoffactorAc numberoflevelsoffactorBn numberofreplicationsineachcell Chap11 43 Two WayANOVAEquations Chap11 44 Two WayANOVA TheFTestStatistic FTestforFactorBEffect FTestforInteractionEffect H0 1 2 r H1 Notall i areequal H0 theinteractionofAandBisequaltozeroH1 interactionofAandBisn tzero FTestforFactorAEffect H0 1 2 c H1 Notall j areequal RejectH0ifF F