Chapter9DesignofExperimentsandAnalysisofVarianceContents1. ElementsofaDesignedExperiment2. TheCompletelyRandomizedDesign:SingleFactor3. MultipleComparisonsofMeans4. TheRandomizedBlockDesign5. FactorialExperiments:TwoFactorsWhereWe’reGoingDiscusscriticalelementsinthedesignofasamplingexperimentLearnhowtosetupthreeexperimentaldesignsforcomparingmorethantwopopulationmeans:completelyrandomized,randomizedblock,andfactorialdesignsShowhowtoanalyzedatacollectedfromadesignedexperimentusingatechniquecalledananalysisofvariance(ANOVA)Presentafollow-upanalysistoanANOVA:Rankingmeans9.1ElementsofaDesignedExperimentResponseVariableTheresponsevariableisthevariableofinteresttobemeasuredintheexperiment.Wealsorefertotheresponseasthedependentvariable.Typically,theresponse/dependentvariableisquantitativeinnature.FactorsFactorsarethosevariableswhoseeffectontheresponseisofinteresttotheexperimenter.Quantitativefactorsaremeasuredonanumericalscale,whereasqualitativefactorsarethosethatarenot(naturally)measuredonanumericalscale.Factorsarealsoreferredtoasindependentvariables.FactorLevelsandTreatmentsFactorlevelsarethevaluesofthefactorusedintheexperiment.Thetreatmentsofanexperimentarethefactor-levelcombinationsused.ExperimentalUnitAnexperimentalunitistheobjectonwhichtheresponseandfactorsareobservedormeasured.DesignedandObservationalExperimentAdesignedstudyisoneforwhichtheanalystcontrolsthespecificationofthetreatmentsandthemethodofassigningtheexperimentalunitstoeachtreatment.Anobservationalstudyisoneforwhichtheanalystsimplyobservesthetreatmentsandtheresponseonasampleofexperimentalunits.ExperimentalProcessOverviewDesignedexperiment:ProcessandterminologyExamplesofExperimentsThirtystoresarerandomlyassigned1

of4(levels)storedisplays(independentvariable)toseetheeffectonsales(dependentvariable).Twohundredconsumersarerandomlyassigned1of3(levels)brandsofjuice(independentvariable)tostudyreaction(dependentvariable).9.2TheCompletelyRandomizedDesign:SingleFactorCompletelyRandomizedDesignAcompletelyrandomizeddesignisadesigninwhichtheexperimentalunitsarerandomlyassignedtothektreatmentsorinwhichindependentrandomsamplesofexperimentalunitsareselectedforeachtreatment.CompletelyRandomizedDesignExperimentalunits(subjects)areassignedrandomlytotreatmentSubjectsareassumedhomogeneousOnefactororindependentvariableTwoormoretreatmentlevelsorclassificationsAnalyzedbyone-wayAnalysisofVariance(ANOVA)Example:RandomizedDesign-BottledWaterBrandsStudyFactors(BottledWaterBrands)FactorLevels(Treatments)BrandABrandBBrandCExperimentalUnits5randomconsumers5randomconsumers5randomconsumersDependentVariable(Response):TastePreferenceScale,1-102,5,8,6,66,7,9,2,58,4,4,6,7ANOVAF-TestTeststheequalityoftwoormore(k)populationmeansVariablesOnenominalscaledindependentvariableTwoormore(k)treatmentlevelsor

classificationsOneintervalorratioscaleddependent

variableUsedtoanalyzecompletelyrandomizedexperimentaldesignsANOVA

PartitionsTotalVariationSumofSquaresWithinSumofSquaresErrorWithinGroupsVariationSumofSquaresAmongSumofSquaresBetweenSumofSquaresTreatmentAmongGroupsVariationTotalvariationVariationduetotreatmentVariationduetorandomsamplingTotalVariationxGroup1Group2Group3Response,xTreatmentVariationx3xx2x1Group1Group2Group3Response,xRandom(Error)Variationx2x1x3Group1Group2Group3Response,xANOVAF-Test

TestStatisticTestStatistic

F=MST/MSEMSTisMeanSquareforTreatmentMSEisMeanSquareforErrorDegreesofFreedom

1=k–1numeratordegreesoffreedom

2=n–kdenominatordegreesoffreedomk=Numberofgroupsn=TotalsamplesizeANOVASummaryTableSourceof

VariationDegrees

of

FreedomSumof

SquaresMean

Square

(Variance)FTreatmentk–1SSTMST=

SST/(k–1)MSTMSEErrorn–kSSEMSE=

SSE/(n–k)Totaln–1SS(Total)=

SST+SSEANOVAF-TestCriticalValue

Ifmeansareequal,F=MST/MSE

≈1.OnlyrejectlargeF!Alwaysaone-sidedtail!F(α;k–1,n–k)0RejectH0DoNotRejectH0FANOVAF-TesttoComparekTreatmentMeans:CompletelyRandomizedDesignH0:μ1=μ2=…=μk

Ha:AtleasttwotreatmentmeansdifferTestStatistic:Rejectionregion:F>F

,p-value:P(F>Fc)whereF

isbasedon(k–1)numeratordegreesoffreedom(associatedwithMST)and(n–k)denominatordegreesoffreedom(associatedwithMSE).ConditionsRequiredforaValidANOVAF-test:

CompletelyRandomizedDesign

1. Thesamplesarerandomlyselectedinanindependentmannerfromthektreatmentpopulations.(Thiscanbeaccomplishedbyrandomlyassigningtheexperimentalunitstothetreatments.)2. Allksampledpopulationshavedistributionsthatareapproximatelynormal.3. Thekpopulationvariancesareequal

(i.e.,ANOVAF-TestHypothesesH0:

1=

2=

3=...=

kAllpopulationmeans

areequalNotreatmenteffectHa:NotAll

iAreEqualAtleast2populationmeansaredifferentTreatmenteffect

iswrong

xf(x)

1=

2=

3123xf(x)

=

WhyVariances?SametreatmentvariationDifferentrandomvariationPossibletoconcludemeansareequal!Pop1Pop2Pop3Pop4Pop6Pop5VariancesWITHINdifferAPop1Pop2Pop3Pop4Pop6Pop5VariancesAMONGdifferBDifferenttreatmentvariationSamerandomvariationANOVABasicIdeaComparestwotypesofvariationtotestequalityofmeansComparisonbasisisratioofvariancesIftreatmentvariationissignificantlygreaterthanrandomvariationthenmeansarenotequalVariationmeasuresareobtainedby‘partitioning’totalvariationWhatDoYouDoWhentheAssumptionsAreNotSatisfiedforanANOVAforaCompletelyRandomizedDesign?Answer:UseanonparametricstatisticalmethodsuchastheKruskal-WallisH-test.StepsforConductinganANOVAforaCompletelyRandomizedDesignBesurethedesignistrulycompletelyrandomized,withindependentrandomsamplesforeachtreatment.Checktheassumptionsofnormalityandequalvariances.StepsforConductinganANOVAforaCompletelyRandomizedDesign(cont)CreateanANOVAsummarytablethatspecifiesthevariabilityattributabletotreatmentsanderror,makingsurethatitleadstothecalculationoftheF-statisticfortestingthenullhypothesisthatthetreatmentmeansareequalinthepopulation.Useastatisticalsoftwareprogramtoobtainthenumericalresults.Ifnosuchpackageisavailable,usethecalculationformulasinAppendixC.StepsforConductinganANOVAforaCompletelyRandomizedDesign(cont)IftheF-testleadstotheconclusionthatthemeansdiffer,Conductamultiplecomparisonsprocedureforasmanyofthepairsofmeansasyouwishtocompare.Usetheresultstosummarizethestatisticallysignificantdifferencesamongthetreatmentmeans.

Ifdesired,formconfidenceintervalsforoneormoreindividualtreatmentmeans.StepsforConductinganANOVAforaCompletelyRandomizedDesign(cont)IftheF-testleadstothenonrejectionofthenullhypothesisthatthetreatmentmeansareequal,considerthefollowingpossibilities:Thetreatmentmeansareequal–thatis,thenullhypothesisistrue.StepsforConductinganANOVAforaCompletelyRandomizedDesign(cont)Thetreatmentmeansreallydiffer,butotherimportantfactorsaffectingtheresponsearenotaccountedforbythecompletelyrandomizeddesign.Thesefactorsinflatethesamplingvariability,asmeasuredbyMSE,resultinginsmallervaluesoftheF-statistic.Eitherincreasethesamplesizeforeachtreatmentoruseadifferentexperimentaldesignthataccountsfortheotherfactorsaffectingtheresponse.StepsforConductinganANOVAforaCompletelyRandomizedDesign(cont)Note:BecarefulnottoautomaticallyconcludethatthetreatmentmeansareequalbecausethepossibilityofaTypeIIerrormustbeconsideredifyouacceptH0.Example:ProductionF-TestAsproductionmanager,youwanttoseeifthreefillingmachineshavedifferentmeanfillingtimes.Youassign15similarlytrainedandexperiencedworkers,5permachine,tothemachines.Atthe0.05

levelofsignificance,isthereadifferenceinmeanfillingtimes?

Mach1 Mach2

Mach3

25.40 23.40 20.00

26.31 21.80 22.20

24.10 23.50 19.75

23.74 22.75 20.60

25.10 21.60 20.40Example:ProductionF-Test(cont)H0:Ha:

=

1=

2=

F03.89

=0.05

1

=

2=

3Notallequal0.052

12CriticalValue(s):Recallthatk

=numberofgroupsandn

=totalsamplesize.Sowehavek=3andn=15forthisexample.Then

1

=k–1numeratordegreesoffreedom=2

2

=n–kdenominatordegreesoffreedom=12.SetuptheF-test:Example:ProductionF-Test(cont)ANOVASummaryTableTreatment

(Machines)3–1=247.164023.582025.60Error15–3=1211.05320.9211Total15–1=1458.2172Sourceof

VariationDegrees

of

FreedomSumof

SquaresMean

Square

(Variance)FExample:ProductionF-Test(cont)H0:Ha:

=

1=

2=

CriticalValue(s):TestStatistic:Decision:Conclusion:F03.89

=0.05

1=

2=

3Notallequal0.052

12Rejectat

=0.05ThereisevidencepopulationmeansaredifferentFMSTMSE

2358200.921125.6.9.3MultipleComparisonsofMeansDeterminingtheNumberofPairwiseComparisonsofTreatmentMeansIngeneral,iftherearektreatmentmeans,therearepairsofmeansthatcanbecompared.ErrorRatesForasinglecomparisonoftwomeansinadesignedexperiment,theprobabilityofmakingaTypeIerror(i.e.,theprobabilityofconcludingthatadifferenceinthemeansexists,giventhatthemeansarethesame)iscalledacomparisonwiseerrorrate

(CER).Formultiplecomparisonsofmeansinadesignedexperiment,theprobabilityofmakingatleastoneTypeIerror(i.e.,theprobabilityofconcludingthatatleastonedifferenceinmeansexists,giventhattheareallthesame)iscalledanexperimentwiseerrorrate(EER).TukeyMethodsTukey(1949)developedhismultiplecomparisonsmethodinANOVAspecificallyforpairwisecomparisonswhenthesamplesizesofthetreatmentsareequal.BonferroniMethodTheBonferronimethod(seeMiller,1981),liketheTukeyprocedure,canbeappliedwhenpairwisecomparisonsareofinterest;however,Bonferroni’smethoddoesnotrequireequalsamplesizes.SchefféMethodScheffé(1953)developedamoregeneralprocedureforcomparingallpossiblelinearcombinationsoftreatmentmeans(calledcontrasts).Consequently,whenmakingpairwisecomparisons,theconfidenceintervalsproducedbyScheffé’smethodwillgenerallybewiderthantheTukeyorBonferroniconfidenceintervals.TukeyProcedurexf(x)m1=m2m32Groupings Tellswhichpopulation

meansaresignificantly

differentExample:μ1=μ2

≠

μ3

PosthocprocedureDoneafterrejectionof

equalmeansinANOVA Outputfrommany

statisticalcomputer

programsGuidelines9.4TheRandomizedBlockDesignRandomizedBlockDesignTherandomizedblockdesignconsistsofatwo-stepprocedure:1. Matchedsetsofexperimentalunits,calledblocks,areformed,eachblockconsistingofkexperimentalunits(wherekisthenumberoftreatments).Thebblocksshouldconsistofexperimentalunitsthatareassimilaraspossible.2. Oneexperimentalunitfromeachblockisrandomlyassignedtoeachtreatment,resultinginatotalofn

=bkresponses.RandomizedBlockDesign

TotalVariationPartitioningANOVAF-TesttoComparekTreatmentMeans:RandomizedBlockDesignH0:μ1=μ2=…=μk

Ha:AtleasttwotreatmentmeansdifferTestStatistic:Rejectionregion:F>F

,p-value:P(F>Fc)whereFa

isbasedon(k–1)numeratordegreesoffreedomand(n–b–k+1)denominatordegreesoffreedom.ConditionsRequiredforaValidANOVAF-test:

RandomizedBlockDesign

Thebblocksarerandomlyselected,andallktreatmentsareapplied(inrandomorder)toeachblock.Thedistributionsofobservationscorrespondingtoallbkblock-treatmentcombinationsareapproximatelynormal.Allbkblock-treatmentdistributionshaveequalvariances.RandomizedBlockDesign

F-TestTestStatisticTestStatistic

F=MST/MSEMSTisMeanSquareforTreatmentMSEisMeanSquareforErrorDegreesofFreedom

1=k–1(numerator)

2=n–k–b+1(denominator)k=Numberofgroupsn=Totalsamplesizeb=NumberofblocksExample:RandomizedBlockDesignF-TestAproductionmanagerwantstoseeifthreeassemblymethodshavedifferentmeanassemblytimes(inminutes).Fiveemployeeswereselectedatrandomandassignedtouseeachassemblymethod.Atthe0.05

levelofsignificance,isthereadifferenceinmeanassemblytimes?Employee

Method1

Method2

Method3 1 5.4 3.6 4.0

2 4.1 3.8 2.9

3 6.1 5.6 4.3

4 3.6 2.3 2.6

5 5.3 4.7 3.4Example:RandomizedBlockDesignF-Test(cont)H0:Ha:

=

1=

2=F04.46

=0.05

1=

2=

3Notallequal0.052

8CriticalValue(s):Wehavek=3andn=5.Definethetest:Example:RandomizedBlockDesignF-Test(cont)Treatment

(Methods)3–1=25.432.7112.9Error15–3–

5+1

=81.68.21Total15–1=1417.8Sourceof

VariationDegrees

of

FreedomSumof

SquaresMean

Square

(Variance)FBlock

(Employee)5–1=410.692.6712.7Example:RandomizedBlockDesignF-Test(cont)H0:Ha:

=

1=

2=CriticalValue(s):TestStatistic:Decision:Conclusion:F04.46

=0.05

1=

2=

3Notallequal0.052

8Rejectat

=0.05ThereisevidencepopulationmeansaredifferentFMSTMSE

2.710.2112.9StepsforConductinganANOVAforaRandomizedBlockDesign1. Besurethedesignconsistsofblocks(preferably,blocksofhomogeneousexperimentalunits)andthateachtreatmentisrandomlyassignedtooneexperimentalunitineachblock.2. Ifpossible,checktheassumptionsofnormalityandequalvariancesforallblock-treatmentcombinations.[Note:Thismaybedifficulttodobecausethedesignwilllikelyhaveonlyoneobservationforeachblock-treatmentcombination.]StepsforConductinganANOVAforaRandomizedBlockDesign(cont)3. CreateanANOVAsummarytablethatspecifiesthevariabilityattributabletoTreatments,Blocks,andError,whichleadstothecalculationoftheF-statistictotestthenullhypothesisthatthetreatmentmeansareequalinthepopulation.UseastatisticalsoftwarepackageorthecalculationformulasinAppendixCtoobtainthenecessarynumericalingredients.StepsforConductinganANOVAforaRandomizedBlockDesign(cont)4. IftheF-testleadstotheconclusionthatthemeansdiffer,usetheBonferroni,Tukey,orsimilarproceduretoconductmultiplecomparisonsofasmanyofthepairsofmeansasyouwish.Usetheresultstosummarizethestatisticallysignificantdifferencesamongthetreatmentmeans.Rememberthat,ingeneral,therandomizedblockdesigncannotbeusedtoformconfidenceintervalsforindividualtreatmentmeans.StepsforConductinganANOVAforaRandomizedBlockDesign(cont)5.IftheF-testleadstothenonrejectionofthenullhypothesisthatthetreatmentmeansareequal,considerthefollowingpossibilities:a.Thetreatmentmeansareequal–thatis,thenullhypothesisistrue.StepsforConductinganANOVAforaRandomizedBlockDesign(cont)b.Thetreatmentmeansreallydiffer,butotherimportantfactorsaffectingtheresponsearenotaccountedforbytherandomizedblockdesign.Thesefactorsinflatethesamplingvariability,asmeasuredbyMSE,resultinginsmallervaluesoftheF-statistic.Eitherincreasethesamplesizeforeachtreatmentorconductanexperimentthataccountsfortheotherfactorsaffectingtheresponse.DonotautomaticallyreachtheformerconclusionbecausethepossibilityofaTypeIIerrormustbeconsideredifyouacceptH0.StepsforConductinganANOVAforaRandomizedBlockDesign(cont)6. Ifdesired,conducttheF-testofthenullhypothesisthattheblockmeansareequal.Rejectionofthishypothesislendsstatisticalsupporttousingtherandomizedblockdesign.StepsforConductinganANOVAforaRandomizedBlockDesign(cont)Note:Itisoftendifficulttocheckwhethertheassumptionsforarandomizedblockdesignaresatisfied.Thereisusuallyonlyoneobservationforeachblock-treatmentcombination.Whenyoufeeltheseassumptionsarelikelytobeviolated,anonparametricprocedureisadvisable.WhatDoYouDoWhentheAssumptionsAreNotSatisfiedforanANOVAforaCompletelyRandomizedDesign?Answer:UseanonparametricstatisticalmethodsuchastheFriedmanFrtest.9.5FactorialExperiments:

TwoFactorsFactorialDesignAcompletefactorialexperimentisoneinwhicheveryfactor-levelcombinationisemployed,thatis,thenumberoftreatmentsintheexperimentequalsthetotalnumberoffactor-levelcombinations.Alsoreferredtoasatwo-wayclassification.FactorialDesignTodeterminethenatureofthetreatmenteffect,ifany,ontheresponseinafactorialexperiment,weneedtobreakthetreatmentvariabilityintothreecomponents:InteractionbetweenFactorsAandB,MainEffectofFactorA,andMainEffectofFactorB.FactorialDesign(cont)TheFactorInteractioncomponentisusedtotestwhetherthefactorscombinetoaffecttheresponse,whiletheFactorMainEffectcomponentsareusedtodeterminewhetherthefactorsseparatelyaffecttheresponse.FactorialDesignExperimentalunits(subjects)areassignedrandomlytotreatmentsSubjectsareassumedhomogeneousTwoormorefactorsorindependentvariablesEachhastwoormoretreatments(levels)Analyzedbytwo-wayANOVAProcedureforAnalysisofTwo-FactorFactorialExperimentPartitiontheTotalSumofSquaresintotheTreatmentsandErrorcomponents.UseeitherastatisticalsoftwarepackageorthecalculationformulasinAppendixCtoaccomplishthepartitioning.ProcedureforAnalysisofTwo-FactorFactorialExperiment(cont)UsetheF-ratioofMeanSquareforTreatmentstoMeanSquareforErrortotestthenullhypothesisthatthetreatmentmeansareequal.Ifthetestresultsinnonrejectionofthenullhypothesis,considerrefiningtheexperimentbyincreasingthenumberofreplicationsorintroducingotherfactors.Alsoconsiderthepossibilitythattheresponseisunrelatedtothetwofactors.Ifthetestresultsinrejectionofthenullhypothesis,thenproceedtostep3.ProcedureforAnalysisofTwo-FactorFactorialExperiment(cont)PartitiontheTreatmentsSumofSquaresintotheMainEffectandInteractionSumofSquares.UseeitherastatisticalsoftwarepackageorthecalculationformulasinAppendixCtoaccomplishthepartitioning.ProcedureforAnalysisofTwo-FactorFactorialExperiment(cont)TestthenullhypothesisthatfactorsAandBdonotinteracttoaffecttheresponsebycomputingtheF-ratiooftheMeanSquareforInteractiontotheMeanSquareforError.Ifthetestresultsinnonrejectionofthenullhypothesis,proceedtostep5.Ifthetestresultsinrejectionofthenullhypothesis,concludethatthetwofactorsinteracttoaffectthemeanresponse.Thenproceedtostep6a.ProcedureforAnalysisofTwo-FactorFactorialExperiment(cont)ConducttestsoftwonullhypothesesthatthemeanresponseisthesameateachleveloffactorAandfactorB.ComputetwoF-ratiosbycomparingtheMeanSquareforeachFactorMainEffecttotheMeanSquareforError.Ifoneorbothtestsresultinrejectionofthenullhypothesis,concludethatthefactoraffectsthemeanresponse.Proceedtostep6b.ProcedureforAnalysisofTwo-FactorFactorialExperiment(cont)Ifbothtestsresultinnonrejection,anapparentcontradictionhasoccurred.Althoughthetreatmentmeansapparentlydiffer(step2test),theinteraction(step4)andmaineffect(step5)testshavenotsupportedthatresult.Furtherexperimentationisadvised.ProcedureforAnalysisofTwo-FactorFactorialExperiment(cont)Comparethemeans:Ifthetestforinteraction(step4)issignificant,useamultiplecomparisonsproceduretocompareanyorallpairsofthetreatmentmeans.Ifthetestforoneorbothmaineffects(step5)issignificant,useamultiplecomparisonsproceduretocomparethepairsofmeanscorrespondingtothelevelsofthesignificantfactor(s).PartitioningtheTotalSumofSquaresPartitioningtheTotalSumofSquaresforatwo-factorfactorialANOVATestsConductedforFactorialExperiments:CompletelyRandomizedDesign,rReplicatesperTreatmentTestforTreatmentMeansH0:NodifferenceamongtheabtreatmentmeansHa:AtleasttwotreatmentmeansdifferTestStatistic:Rejectionregion:F>F

p-value:P(F>Fc)whereFisbasedon(ab–1)numeratorand

(n–ab)denominatordegreesoffreedom

[Note:n=abr.]ANOVATestsConductedforFactorialExperiments:CompletelyRandomizedDesign,rReplicatesperTreatment(cont)TestforFactorInteractionH0:FactorsAandBdonotinteracttoaffectthe

responsemeanHa:FactorsAandBdointeracttoaffecttheresponse

meanTestStatistic:Rejectionregion:F>F

p-value:P(F>Fc)whereFisbasedon(a–1)(b–1)numeratorand

(n–ab)denominatordegreesoffreedomANOVATestsConductedforFactorialExperiments:CompletelyRandomizedDesign,rReplicatesperTreatment(cont)TestforMainEffectofFactorAH0:NodifferenceamongtheameanlevelsoffactorA

Ha:AtleasttwofactorAmeanlevelsdifferTestStatistic:Rejectionregion:F>F

,p-value:P(F>Fc)whereFisbasedon(a–1)numeratorand

(n–ab)denominatordegreesoffreedomANOVATestsConductedforFactorialExperiments:CompletelyRandomizedDesign,rReplicatesperTreatment(cont)TestforMainEffectofFactorBH0:NodifferenceamongthebmeanlevelsoffactorB

Ha:AtleasttwofactorBmeanlevelsdifferTestStatistic:Rejectionregion:F>F

,p-value:P(F>Fc)WhereFisbasedon(b–1)numeratorand

(n–ab)denominatordegreesoffreedomConditionsRequiredforValid

F-testsinFactorialExperimentsTheresponsedistributionforeachfactor-levelcombination(treatment)isnormal.Theresponsevarianceisconstantforalltreatments.Randomandindependentsamplesofexperimentalunitsareassociatedwitheachtreatment.ANOVADataTableSchemaxijkLeveliFactorALeveljFactorBObservationkFactorFactorBA12...b1x111x121...x1b1x112x122...x1b22x211x221...x2b1x212x222...x2b2:::::axa11xa21...xab1xa12xa22...xab2TreatmentExample:FactorialDesignFactor2(TrainingMethod)Factor

LevelsLevel1Self-PacedLevel2Class-roomLevel3On-lineVirtualLevel115hr.

10hr.

22hr.

Factor1

(Motivation)(High)11hr.

12hr.

17hr.

Level227hr.

15hr.

31hr.

(Low)29hr.

17hr.

49hr.

TreatmentAdvantages

ofFactorialDesignsSavestimeandefforte.g.,Coulduseseparatecompletely

randomizeddesignsforeachvariableControlsconfoundingeffectsbyputtingothervariablesintomodelCanexploreinteractionbetweenvariablesTwo-WayANOVATeststheequalityoftwoormorepopulationwhenseveralindependentvariablesareusedSameresultsasseparateone-wayANOVAoneachvariableNointeractioncanbetestedUsedtoanalyzefactorialdesignsInteractionOccurswheneffectsofonefactorvaryaccordingtolevelsofotherfactorWhensignificant,interpretationofmaineffects(AandB)iscomplicatedCanbedetectedIndatatable,patternofcellmeansinonerow

differsfromanotherrowIngraphofcellmeans,linescrossGraphsofInteractionEffectsofmotivation(highorlow)andtrainingmethod(A,B,C)onmeanlearningtimeInteractionNoInteractionAverageResponseABCHighLowAverageResponseABCHighLowTwo-WayANOVA

TotalVariationPartitioningTwo-WayANOVA

SummaryTableSourceof

VariationDegreesof

FreedomSumof

SquaresMean

SquareFA

(Row)a–1SS(A)MS(A)MS(A)MSEB

(Column)b–1SS(B)MS(B)MS(B)MSEAB

(Interaction)(a–1)(b–1)SS(AB)MS(AB)MS(AB)MSEErrorn–abSSEMSETotaln–1SS(Total)SameasotherdesignsExample:TreatmentMeansF-Test(1of4)HumanResourceswantstodetermineiftrainingtimeisdifferentbasedonmotivationlevelandtrainingmethod.ConducttheappropriateANOVAtests.Useα=0.05

foreachtest.TrainingMethodFactor

LevelsSelf–

pacedClassroomOn-lineVirtual15hr.10hr.22hr.MotivationHigh11hr.12hr.17hr.27hr.15hr.31hr.Low29hr.17hr.49hr.Example:TreatmentMeansF-Test(2of4)H0:Ha:

=

1=

2=CriticalValue(s):F04.39

=0.05

The6treatment

meansareequalAtleast2differ0.055

6Example:TreatmentMeansF-Test(3of4)Sourceof

VariationDegreesof

FreedomSumof

SquaresMean

SquareFModel51201.8240.35Error6188.531.42Corrected

Total7.65111390.3Two-WayANOVASummaryTableExample:TreatmentMeansF-Test(4of4)H0:Ha:

=

1=

2=CriticalValue(s):F04.39

=0.05

The6treatment

meansareequalAtleast2differ0.055

6TestStatistic:Decision:Conclusion:Rejectat

=0.05ThereisevidencepopulationmeansaredifferentExample:FactorInteractionF-Test(1of3)H0:Ha:

=

1=

2=CriticalValue(s):F05.14

=0.05ThefactorsdonotinteractThefactorsinteract0.052

6Example:FactorInteractionF-Test(2of3)Sourceof

VariationDegreesof

FreedomSumof

SquaresMean

SquareFA

(Row)1546.75546.75B

(Colu