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I. IntroductionII. GraphingDistributionsIII. SummarizingDistributionsIV. DescribingBivariateDataV. ProbabilityVI. ResearchDesignVII. NormalDistributionVIII. AdvancedGraphsIX. SamplingDistributionsX. EstimationXI. LogicofHypothesisTestingXII. TestsofMeansXIII. PowerXIV. Regression

XV. AnalysisofVariance1. Contents

Standard

2. IntroductionStandardVideo

3. ANOVADesignsStandardVideo

4. OneFactorANOVAStandardVideo

5. OneWayDemoStandard

6. MultiFactorBetweenSubjectsStandardVideo

7. UnequalnStandardVideo

8. TestsSupplementingStandardVideo

9. WithinSubjectsStandard

10. PowerofWithinSubjectsDesignsDemoStandard

11. StatisticalLiteracyStandard

12. ExercisesStandard

XVI. TransformationsXVII. ChiSquareXVIII. DistributionFreeTestsXIX. EffectSizeXX. CaseStudiesXXI. CalculatorsXXII. Glossary

Chapter: 15.AnalysisofVariance Section: WithinSubjects

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WithinSubjectsANOVA

Author(s)DavidM.Lane

PrerequisitesDesigns,IntroductiontoANOVA,ANOVADesigns,MultiFactorANOVA,DifferenceBetweenTwoMeans(CorrelatedPairs)

LearningObjectives1. Defineawithinsubjectsfactor2. Explainwhyawithinsubjectsdesigncanbeexpectedtohavemorepower

thanabetweensubjectsdesign3. BeabletocreatetheSourceanddfcolumnsofanANOVAsummarytable

aonewaywithinsubjectsdesign4. Explainerrorintermsofinteraction5. Discusstheproblemofcarryovereffects6. BeabletocreatetheSourceanddfcolumnsofanANOVAsummarytable

adesignwithonebetweensubjectsandonewithinsubjectsvariable7. Definesphericity8. Describetheconsequencesofviolatingtheassumptionofsphericity9. Discusscoursesofactionthatcanbetakenifsphericityisviolated

Withinsubjectsfactorsinvolvecomparisonsofthesamesubjectsunderdifferentconditions.Forexample,inthe"ADHDTreatment"study,eachchild'sperformancewasmeasuredfourtimes,onceafterbeingoneachoffourdrugdosesforaweek.Therefore,eachsubject'sperformancewasmeasuredateachofthefourlevelsofthefactor"Dose."Notethedifferencefrombetweensubjectsfactorsforwhicheachsubject'sperformanceismeasuredonlyonceandthecomparisonsareamongdifferentgroupsofsubjects.Awithinsubjectsfactorissometimesreferredtoasarepeatedmeasuresfactorsincerepeatedmeasurementsaretakenoneachsubject.Anexperimentaldesigninwhichtheindependentvariableisawithinsubjectsfactoriscalledawithinsubjectsdesign

AdvantageofWithinSubjectsDesigns

ONEFACTORDESIGNSLet'sconsiderhowtoanalyzethedatafromthe"ADHDTreatment"casestudyThesedataconsistofthescoresof24childrenwithADHDonadelayofgratification(DOG)task.Eachchildwastestedunderfourdosagelevels.Fornow,wewillbeconcernedonlywithtestingthedifferencebetweenthemeanintheplacebocondition(thelowestdosage,D0)andthemeaninthehighestdosagecondition(D60).Thedetailsofthecomputationsarerelativelyunimportantsincetheyarealmostuniversallydonebycomputers.ThereforewejumprighttotheANOVASummarytableshowninTable1.

Table1.ANOVASummaryTable.Source df SSQ MS F pSubjects 23 5781.98 251.39

Dosage 1 295.02 295.02 10.38 0.004

Error 23 653.48 28.41

Total 47 6730.48

Thefirstsourceofvariation,"Subjects,"referstothedifferencesamongsubjects.Ifallthesubjectshadexactlythesamemean(acrossthetwodosages),thenthesumofsquaresforsubjectswouldbezerothemoresubjectsdifferfromeachother,thelargerthesumofsquaressubjects.

Dosagereferstothedifferencesbetweenthetwodosagelevels.Ifthemeansforthetwodosagelevelswereequal,thesumofsquareswouldbezero.Thelargerthedifferencebetweenmeans,thelargerthesumofsquares.

Theerrorreflectsthedegreetowhichtheeffectofdosageisdifferentfor

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differentsubjects.Ifsubjectsallrespondedverysimilarlytothedrug,thentheerrorwouldbeverylow.Forexample,ifallsubjectsperformedmoderatelybetterwiththehighdosethantheydidwiththeplacebo,thentheerrorwouldbelow.Ontheotherhand,ifsomesubjectsdidbetterwiththeplacebowhileothersdidbetterwiththehighdose,thentheerrorwouldbehigh.Itshouldmakeintuitivesensethatthelessconsistenttheeffectofdosage,thelargerthedosageeffectwouldhavetobeinordertobesignificant.ThedegreetowhichtheeffectofdosagediffersdependingonthesubjectistheSubjectsxDosageinteraction.Recallthataninteractionoccurswhentheeffectofonevariablediffersdependingonthelevelofanothervariable.Inthiscase,thesizeoftheerrortermistheextenttowhichtheeffectofthevariable"Dosage"differsdependingonthelevelofthevariable"Subjects."Notethateachsubjectisadifferentlevelofthevariable"Subjects."

OtherportionsofthesummarytablehavethesamemeaningasinbetweensubjectsANOVA.TheFfordosageisthemeansquarefordosagedividedbythemeansquareerror.Forthesedata,theFissignificantwithp=0.004.NoticethatthisFtestisequivalenttothettestforcorrelatedpairs,withF=t2.

Table2showstheANOVASummaryTablewhenallfourdosesareincludedintheanalysis.Sincetherearenowfourdosagelevelsratherthantwo,thedffordosageisthreeratherthanone.SincetheerroristheSubjectsxDosageinteraction,thedfforerroristhedffor"Subjects"(23)timesthedfforDosage(3)andisequalto69.

Table2.ANOVASummaryTable.Source df SSQ MS F pSubjects 23 9065.49 394.15

Dosage 3 557.61 185.87 5.18 0.003

Error 69 2476.64 35.89

Total 95 12099.74

CARRYOVEREFFECTSOftenperforminginoneconditionaffectsperformanceinasubsequentconditioninsuchawayastomakeawithinsubjectsdesignimpractical.Forexample,consideranexperimentwithtwoconditions.Inbothconditionssubjectsarepresentedwithpairsofwords.InConditionA,subjectsareaskedtojudgewhetherthewordshavesimilarmeaningwhereasinConditionB,subjectsareaskedtojudgewhethertheysoundsimilar.Inbothconditions,subjectsaregivenasurprisememorytestattheendofthepresentation.IfConditionwereawithinsubjectsvariable,thentherewouldbenosurpriseafterthesecondpresentationanditislikelythatthesubjectswouldhavebeentryingtomemorizethewords.

Notallcarryovereffectscausesuchseriousproblems.Forexample,ifsubjectsgetfatiguedbyperformingatask,thentheywouldbeexpectedtodoworseonthesecondconditiontheywerein.However,aslongastheorderofpresentationiscounterbalancedsothathalfofthesubjectsareinConditionAfirstandConditionBsecond,thefatigueeffectitselfwouldnotinvalidatetheresults,althoughitwouldaddnoiseandreducepower.ThecarryovereffectissymmetricinthathavingConditionAfirstaffectsperformanceinConditionBtothesamedegreethathavingConditionBfirstaffectsperformanceinConditionA.

Asymmetriccarryovereffectscausemoreseriousproblems.Forexample,supposeperformanceinConditionBweremuchbetterifprecededbyConditionA,whereasperformanceinConditionAwasapproximatelythesameregardlessofwhetheritwasprecededbyConditionB.Withthiskindofcarryovereffect,itisprobablybettertouseabetweensubjectsdesign.

ONEBETWEENANDONEWITHINSUBJECTSFACTORInthe"StroopInterference"casestudy,subjectsperformedthreetasks:namingcolors,readingcolorwords,andnamingtheinkcolorofcolorwords.Someofthesubjectsweremalesandsomewerefemales.Therefore,thisdesignhadtwofactors:genderandtask.TheANOVASummaryTableforthisdesignisshowninTable3.

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Table3.ANOVASummaryTableforStroopExperiment.

Source df SSQ MS F pGender 1 83.32 83.32 1.99 0.165

Error 45 1880.56 41.79

Task 2 9525.97 4762.99 228.06

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CONSEQUENCESOFVIOLATINGTHEASSUMPTIONOFSPHERICITYAlthoughANOVAisrobusttomostviolationsofitsassumptions,theassumptionofsphericityisanexception:ViolatingtheassumptionofsphericityleadstoasubstantialincreaseintheTypeIerrorrate.Moreover,thisassumptionisrarelymetinpractice.Althoughviolationsofthisassumptionhadatonetimereceivedlittleattention,thecurrentconsensusofdataanalystsisthatitisnolongerconsideredacceptabletoignorethem.

ApproachestoDealingwithViolationsofSphericityIfaneffectishighlysignificant,thereisaconservativetestthatcanbeusedtoprotectagainstaninflatedTypeIerrorrate.Thistestconsistsofadjustingthedegreesoffreedomforallwithinsubjectsvariablesasfollows:Thedegreesoffreedomnumeratoranddenominatoraredividedbythenumberofscorespersubjectminusone.ConsidertheeffectofTaskshowninTable3.Therearethreescorespersubjectandthereforethedegreesoffreedomshouldbedividedbytwo.Theadjusteddegreesoffreedomare:

(2)(1/2)=1forthenumeratorand(90)(1/2)=45forthedenominator

TheprobabilityvalueisobtainedusingtheFprobabilitycalculatorwiththenewdegreesoffreedomparameters.TheprobabilityofanFof228.06orlargerwith1and45degreesoffreedomislessthan0.001.Therefore,thereisnoneedtoworryabouttheassumptionviolationinthiscase.

PossibleviolationofsphericitydoesmakeadifferenceintheinterpretationoftheanalysisshowninTable2.TheprobabilityvalueofanFof5.18with1and23degreesoffreedomis0.032,avaluethatwouldleadtoamorecautiousconclusionthanthepvalueof0.003showninTable2.

Thecorrectiondescribedaboveisveryconservativeandshouldonlybeusedwhen,asinTable3,theprobabilityvalueisverylow.Abettercorrection,butonethatisverycomplicatedtocalculate,istomultiplythedegreesoffreedombyaquantitycalled(theGreekletterepsilon).Therearetwomethodsofcalculating.ThecorrectioncalledtheHuynhFeldt(orHF)isslightlypreferredtotheonecalledtheGreenhouseGeisser(orGG),althoughbothworkwell.TheGGcorrectionisgenerallyconsideredalittletooconservative.

Afinalmethodfordealingwithviolationsofsphericityistouseamultivariateapproachtowithinsubjectsvariables.Thismethodhasmuchtorecommendit,butitisbeyondthescopeofthistext.

Question1outof5.Whichofthefollowingrepresentwithinsubjectsvariables?

Age:Subjectsoffourdifferentageswereusedintheexperiment.

Trial:Eachsubjecthadthreetrialsonthetaskandtheirscorewasrecordedforeachtrial.

Dose:Eachsubjectwastestedundereachoffivedosagelevels.

Day:Eachsubjectwastestedonceadayforfourdays.

Intensity:Eachsubjectwasrandomlyassignedtooneoffiveintensitylevels.

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