常用质量统计方法ANOVA

ANOVAHypothesisTestingMethods2Attheendofthismoduleyouwillbeableto:ExplaintheprinciplesofmultipleparametertestingDescribethegeneralmethodformultipleparametertestingPerformanAnalysisofVariancefor1,2or3factorsandinterprettheresultsModuleObjectives3Weneverknowthetruepopulationparameters,butwecanusesamplestatisticstoestimatethem.OnetypeofanalysisiscalledAnalysisofVariance(ANOVA).»Allowscomparisonoftwoormoreprocess’sor’s.Wecanteststatisticallywhetherthesesamplesrepresentasinglepopulation,orifthe’sor’saredifferent.TheOUTPUTvariable(KPOV)isgenerallymeasuredonacontinuousscale(Yield,Temperature,Volts,%Impurities,etc...)TheINPUTvariables(KPIV’s)areknownasFACTORS.InANOVA,theLEVELSoftheFACTORSaretreatedascategoricalinnatureeventhoughtheymaynotbe.Whenthereisonlyonefactor,thetypeofanalysisusediscalled“One-WayANOVA.”For2factors,theanalysisiscalled“Two-WayANOVA.And“n”factorsentail“n-WayANOVA.”CharacteristicsAboutMultipleParameterTesting4ANOVA—ApplicationTostudytheeffectofoneormorefactorsonaresponse,eachfactorhavingtwoormorelevels.ANOVAcanbeusedtodeterminethestatisticalsignificanceofeffectscalculatethecomponentsofvarianceestimatethecontributiontovariationbyeachidentifiedsourceestimatetheunderlyingnoisewithintheprocess5Arethelevelsindependent(differentor),ordotheyallcomefromadistributionwiththesamemeanandstandarddeviationµ1µ2µ3Ho:pop1=pop2=pop3=...Ha:atleasttwoaredifferentHo:pop1=pop2=pop3=...Ha:atleasttwoaredifferentTheBasicANOVAQuestion6Step1:StatethePracticalProblemStep2:Dotheassumptionsforthemodelhold?•Responsemeansareindependentandnormallydistributed•Populationvariancesareequalacrossalllevelsofthefactor»Runahomogeneityofvarianceanalysis--byfactorlevel--first!Step3:StatetheNullandAlternateHypothesesStep4:ConstructtheANOVATableStep5:Dotheassumptionsfortheerrorshold(residualanalysis)?•ErrorsofthemodelareindependentandnormallydistributedStep6:InterprettheP-Value(ortheF-statistic)forthefactoreffect•P-Value0.05,thenREJECTHo•Otherwise,operateasifthenullhypothesisistrueStep7:CalculateepsilonsquaredforthetreatmentanderrortermsStep8:TranslatethestatisticalconclusionintoprocesstermsGeneralMethod7Arethemeansindependentandnormallydistributed»Randomizerunsduringtheexperiment»Ensureadequatesamplesizes»Runanormalitytestonthedatabylevel•Minitab:StatBasicStatsNormalityTestPopulationvariancesareequalforeachfactorlevel(runahomogeneityofvarianceanalysisfirst)•ForHo:pop1=pop2=pop3=pop4=...Ha:atleasttwoaredifferentNote:Theassumptionofequalvariancesgenerallyholds,especiallyifyourtestisBALANCED(same#ofobservationsineachlevel).Ifvariancesarenotequal,andatransformationdoesnotsucceed,thennon-parametrictestsarerequired.Step2:DotheAssumptionsfortheModelHold?8Ho:1=2=3=4=5Ha:AtleastonekisdifferentTheHypothesesinGraphicalForm(One-Way)Level12345Step3:StatetheHypotheses9SOURCESSdfMSTestStatisticBetweenSStreatmentg-1MStreatment=SStreatment/(g-1)F=MStreatment/MSerrorWithinSSerrorN-gMSerror=SSerror/(N-g)TotalSStotalN-1TestStatisticistheF-test=SignaltoNoiseRatio2g1in1jiijg1i2ig1in1j2ijxxxxnxxSSTotalSSTreatmentSSErrorWhere:g=numberofsubgroupsn=numberofreadingspersubgroupThisanalysisdeterminesifthedifferencesbetweentheaverageofthelevelsisgreaterthancouldreasonablybeexpectedfromthevariationthatoccurswithineachlevelTotalvariationBetweengroupvariationWithingroupvariationExercise:ANOVATableWorksheetStep4:ConstructtheANOVATable10SOURCESSdfMSTestStatisticBetweenSStreatmentg-1MStreatment=SStreatment/(g-1)F=MStreatment/MSerrorWithinSSerrorN-gMSerror=SSerror/(N-g)TotalSStotalN-1TestStatisticistheF-test=SignaltoNoiseRatioWhere:g=numberofsubgroupsn=numberofreadingspersubgroupOne-WayAnalysisofVarianceAnalysisofVarianceforTimeSourceDFSSMSFPOperator3149.549.84.350.016Error20229.211.5Total23378.6What’simportanttheprobabilitythattheOperatorvariationinmeanscouldhavehappenedbychance.Step4:ConstructtheANOVATable11Themeasureoftheoverallwithin-meanvariabilityequalsthesquarerootofthemeansquareerror(MSerror),calledtheRootMeanSquareError,orRMSE.Thisnumbercanoftenbecomparedtopreviouslyobservedwithin-groupstandarddeviationstoverifythattheresultsoftheANOVAareconsistentwithhistoricalobservations.Aninconsistencyusuallyindicatesasourceofvariability(suchasaninteraction)thatisnotbeingaccountedforintheANOVAmodel.RMSE:RootMeanSquareError12Step5:Dotheassumptionsfortheerrorshold(residualanalysis)?•Errorsofthemodelareindependentandnormallydistributed»Randomizerunsduringtheexperiment»Ensureadequatesamplesize»Plothistogramoferrorterms»Runanormalitycheckonerrorterms»Ploterroragainstrunorder(I-Chart)»PloterroragainstmodelfitStep6:InterprettheP-Value(ortheF-statistic)forthefactoreffect•P-Value0.05,thenREJECTHo.•Otherwise,operateasifthenullhypothesisistrue.ResidualAnalysisSteps5-613Step7:CalculateepsilonsquaredforthetreatmentanderrortermsStep8:TranslatethestatisticalconclusionintoprocesstermsEpsilon-Squareisacontroversialstatistic.Itprovidesagoodguidelineofthepracticalsignificanceoftheeffect.Epsilon-Squaredisameasureoftheamountofvariati