ANOVA Test: Definition, Types, Examples, SPSS - Statistics ...

A one way ANOVA is used to compare two means from two independent (unrelated) groups using the F-distribution. The null hypothesis for the test is that the two ... Skiptocontent StatisticsHowTo Menu FacebookTweet StatisticsDefinitions>ANOVA Contents: TheANOVATest OneWayANOVA HowtoRunaOneWayANOVAinSPSS TwoWayANOVA WhatisMANOVA? WhatisFactorialANOVA? HowtorunanANOVA ANOVAvs.TTest RepeatedMeasuresANOVA RepeatedMeasuresANOVAinSPSS:Steps Sphericity RelatedArticles TheANOVATest WatchthevideoforanintroductiontoANOVA. WhatisANOVA?WatchthisvideoonYouTube. Can’tseethevideo?Clickhere. AnANOVAtestisawaytofindoutifsurveyorexperimentresultsaresignificant.Inotherwords,theyhelpyoutofigureoutifyouneedtorejectthenullhypothesisoracceptthealternatehypothesis. Basically,you’retestinggroupstoseeifthere’sadifferencebetweenthem.Examplesofwhenyoumightwanttotestdifferentgroups: Agroupofpsychiatricpatientsaretryingthreedifferenttherapies:counseling,medicationandbiofeedback.Youwanttoseeifonetherapyisbetterthantheothers. Amanufacturerhastwodifferentprocessestomakelightbulbs.Theywanttoknowifoneprocessisbetterthantheother. Studentsfromdifferentcollegestakethesameexam.Youwanttoseeifonecollegeoutperformstheother. WhatDoes“One-Way”or“Two-WayMean? One-wayortwo-wayreferstothenumberofindependentvariables(IVs)inyourAnalysisofVariancetest. One-wayhasoneindependentvariable(with2levels).Forexample:brandofcereal, Two-wayhastwoindependentvariables(itcanhavemultiplelevels).Forexample:brandofcereal,calories. Whatare“Groups”or“Levels”? Groupsorlevelsaredifferentgroupswithinthesameindependentvariable.Intheaboveexample,yourlevelsfor“brandofcereal”mightbeLuckyCharms,RaisinBran,Cornflakes—atotalofthreelevels.Yourlevelsfor“Calories”mightbe:sweetened,unsweetened—atotaloftwolevels. Let’ssayyouarestudyingifanalcoholicsupportgroupandindividualcounselingcombinedisthemosteffectivetreatmentforloweringalcoholconsumption.Youmightsplitthestudyparticipantsintothreegroupsorlevels: Medicationonly, Medicationandcounseling, Counselingonly. Yourdependentvariablewouldbethenumberofalcoholicbeveragesconsumedperday. Ifyourgroupsorlevelshaveahierarchicalstructure(eachlevelhasuniquesubgroups),thenuseanestedANOVAfortheanalysis. WhatDoes“Replication”Mean? It’swhetheryouarereplicating(i.e.duplicating)yourtest(s)withmultiplegroups.WithatwowayANOVAwithreplication,youhavetwogroupsandindividualswithinthatgrouparedoingmorethanonething(i.e.twogroupsofstudentsfromtwocollegestakingtwotests).Ifyouonlyhaveonegrouptakingtwotests,youwouldusewithoutreplication. TypesofTests. Therearetwomaintypes:one-wayandtwo-way.Two-waytestscanbewithorwithoutreplication. One-wayANOVAbetweengroups:usedwhenyouwanttotesttwogroupstoseeifthere’sadifferencebetweenthem. TwowayANOVAwithoutreplication:usedwhenyouhaveonegroupandyou’redouble-testingthatsamegroup.Forexample,you’retestingonesetofindividualsbeforeandaftertheytakeamedicationtoseeifitworksornot. TwowayANOVAwithreplication:Twogroups,andthemembersofthosegroupsaredoingmorethanonething.Forexample,twogroupsofpatientsfromdifferenthospitalstryingtwodifferenttherapies. BacktoTop OneWayANOVA AonewayANOVAisusedtocomparetwomeansfromtwoindependent(unrelated)groupsusingtheF-distribution.Thenullhypothesisforthetestisthatthetwomeansareequal.Therefore,asignificantresultmeansthatthetwomeansareunequal. ExamplesofwhentouseaonewayANOVA Situation1:Youhaveagroupofindividualsrandomlysplitintosmallergroupsandcompletingdifferenttasks.Forexample,youmightbestudyingtheeffectsofteaonweightlossandformthreegroups:greentea,blacktea,andnotea. Situation2:Similartosituation1,butinthiscasetheindividualsaresplitintogroupsbasedonanattributetheypossess.Forexample,youmightbestudyinglegstrengthofpeopleaccordingtoweight.Youcouldsplitparticipantsintoweightcategories(obese,overweightandnormal)andmeasuretheirlegstrengthonaweightmachine. LimitationsoftheOneWayANOVA AonewayANOVAwilltellyouthatatleasttwogroupsweredifferentfromeachother.Butitwon’ttellyouwhichgroupsweredifferent.Ifyourtestreturnsasignificantf-statistic,youmayneedtorunanadhoctest(liketheLeastSignificantDifferencetest)totellyouexactlywhichgroupshadadifferenceinmeans. BacktoTop HowtorunaOneWayANOVAinSPSS WatchthevideoforinstructionsonhowtorunaonewayANOVAinSPSSforbetweengroups: HowtoRunaOneWayANOVAinSPSSWatchthisvideoonYouTube. Can’tseethevideo?Clickhere. TwoWayANOVA ATwoWayANOVAisanextensionoftheOneWayANOVA.WithaOneWay,youhaveoneindependentvariableaffectingadependentvariable.WithaTwoWayANOVA,therearetwoindependents.UseatwowayANOVAwhenyouhaveonemeasurementvariable(i.e.aquantitativevariable)andtwonominalvariables.Inotherwords,ifyourexperimenthasaquantitativeoutcomeandyouhavetwocategoricalexplanatoryvariables,atwowayANOVAisappropriate. Forexample,youmightwanttofindoutifthereisaninteractionbetweenincomeandgenderforanxietylevelatjobinterviews.Theanxietylevelistheoutcome,orthevariablethatcanbemeasured.GenderandIncomearethetwocategoricalvariables.Thesecategoricalvariablesarealsotheindependentvariables,whicharecalledfactorsinaTwoWayANOVA. Thefactorscanbesplitintolevels.Intheaboveexample,incomelevelcouldbesplitintothreelevels:low,middleandhighincome.Gendercouldbesplitintothreelevels:male,female,andtransgender.Treatmentgroupsareallpossiblecombinationsofthefactors.Inthisexampletherewouldbe3x3=9treatmentgroups. MainEffectandInteractionEffect TheresultsfromaTwoWayANOVAwillcalculateamaineffectandaninteractioneffect.ThemaineffectissimilartoaOneWayANOVA:eachfactor’seffectisconsideredseparately.Withtheinteractioneffect,allfactorsareconsideredatthesametime.Interactioneffectsbetweenfactorsareeasiertotestifthereismorethanoneobservationineachcell.Fortheaboveexample,multiplestressscorescouldbeenteredintocells.Ifyoudoentermultipleobservationsintocells,thenumberineachcellmustbeequal. Twonullhypothesesaretestedifyouareplacingoneobservationineachcell.Forthisexample,thosehypotheseswouldbe: H01:Alltheincomegroupshaveequalmeanstress. H02:Allthegendergroupshaveequalmeanstress. Formultipleobservationsincells,youwouldalsobetestingathirdhypothesis: H03:Thefactorsareindependentortheinteractioneffectdoesnotexist. AnF-statisticiscomputedforeachhypothesisyouaretesting. AssumptionsforTwoWayANOVA Thepopulationmustbeclosetoanormaldistribution. Samplesmustbeindependent. Populationvariancesmustbeequal(i.e.homoscedastic). Groupsmusthaveequalsamplesizes. BacktoTop WhatisMANOVA? MANOVAisjustanANOVAwithseveraldependentvariables.It’ssimilartomanyothertestsandexperimentsinthatit’spurposeistofindoutiftheresponsevariable(i.e.yourdependentvariable)ischangedbymanipulatingtheindependentvariable.Thetesthelpstoanswermanyresearchquestions,including: Dochangestotheindependentvariableshavestatisticallysignificanteffectsondependentvariables? Whataretheinteractionsamongdependentvariables? Whataretheinteractionsamongindependentvariables? MANOVAExample Supposeyouwantedtofindoutifadifferenceintextbooksaffectedstudents’scoresinmathandscience.Improvementsinmathandsciencemeansthattherearetwodependentvariables,soaMANOVAisappropriate. AnANOVAwillgiveyouasingle(univariate)f-valuewhileaMANOVAwillgiveyouamultivariateFvalue.MANOVAteststhemultipledependentvariablesbycreatingnew,artificial,dependentvariablesthatmaximizegroupdifferences.Thesenewdependentvariablesarelinearcombinationsofthemeasureddependentvariables. InterpretingtheMANOVAresults IfthemultivariateFvalueindicatesthetestisstatisticallysignificant,thismeansthatsomethingissignificant.Intheaboveexample,youwouldnotknowifmathscoreshaveimproved,sciencescoreshaveimproved(orboth).Onceyouhaveasignificantresult,youwouldthenhavetolookateachindividualcomponent(theunivariateFtests)toseewhichdependentvariable(s)contributedtothestatisticallysignificantresult. AdvantagesandDisadvantagesofMANOVAvs.ANOVA Advantages MANOVAenablesyoutotestmultipledependentvariables. MANOVAcanprotectagainstTypeIerrors. Disadvantages MANOVAismanytimesmorecomplicatedthanANOVA,makingitachallengetoseewhichindependentvariablesareaffectingdependentvariables. Onedegreeoffreedomislostwiththeadditionofeachnewvariable. Thedependentvariablesshouldbeuncorrelatedasmuchaspossible.Iftheyarecorrelated,thelossindegreesoffreedommeansthatthereisn’tmuchadvantagesinincludingmorethanonedependentvariableonthetest. Reference: (SFSU) BacktoTop WhatisFactorialANOVA? AfactorialANOVAisanAnalysisofVariancetestwithmorethanoneindependentvariable,or“factor“.ItcanalsorefertomorethanoneLevelofIndependentVariable.Forexample,anexperimentwithatreatmentgroupandacontrolgrouphasonefactor(thetreatment)buttwolevels(thetreatmentandthecontrol).Theterms“two-way”and“three-way”refertothenumberoffactorsorthenumberoflevelsinyourtest.Four-wayANOVAandabovearerarelyusedbecausetheresultsofthetestarecomplexanddifficulttointerpret. Atwo-wayANOVAhastwofactors(independentvariables)andonedependentvariable.Forexample,timespentstudyingandpriorknowledgearefactorsthataffecthowwellyoudoonatest. Athree-wayANOVAhasthreefactors(independentvariables)andonedependentvariable.Forexample,timespentstudying,priorknowledge,andhoursofsleeparefactorsthataffecthowwellyoudoonatest FactorialANOVAisanefficientwayofconductingatest.Insteadofperformingaseriesofexperimentswhereyoutestoneindependentvariableagainstonedependentvariable,youcantestallindependentvariablesatthesametime. Variability Inaone-wayANOVA,variabilityisduetothedifferencesbetweengroupsandthedifferenceswithingroups.InfactorialANOVA,eachlevelandfactorarepairedupwitheachother(“crossed”).Thishelpsyoutoseewhatinteractionsaregoingonbetweenthelevelsandfactors.Ifthereisaninteractionthenthedifferencesinonefactordependonthedifferencesinanother. Let’ssayyouwererunningatwo-wayANOVAtotestmale/femaleperformanceonafinalexam.Thesubjectshadeitherhad4,6,or8hoursofsleep. IV1:SEX(Male/Female) IV2:SLEEP(4/6/8) DV:FinalExamScore Atwo-wayfactorialANOVAwouldhelpyouanswerthefollowingquestions: Issexamaineffect?Inotherwords,domenandwomendiffersignificantlyontheirexamperformance? Issleepamaineffect?Inotherwords,dopeoplewhohavehad4,6,or8hoursofsleepdiffersignificantlyintheirperformance? Isthereasignificantinteractionbetweenfactors?Inotherwords,howdohoursofsleepandsexinteractwithregardstoexamperformance? Cananydifferencesinsexandexamperformancebefoundinthedifferentlevelsofsleep? AssumptionsofFactorialANOVA Normality:thedependentvariableisnormallydistributed. Independence:Observationsandgroupsareindependentfromeachother. EqualityofVariance:thepopulationvariancesareequalacrossfactors/levels. HowtorunanANOVA Thesetestsareverytime-consumingbyhand.Innearlyeverycaseyou’llwanttousesoftware.Forexample,severaloptionsareavailableinExcel: TwowayANOVAinExcelwithreplicationandwithoutreplication. OnewayANOVAinExcel2013. RunningthetestinExcel. ANOVAtestsinstatisticspackagesarerunonparametricdata.Ifyouhaverankorordereddata,you’llwanttorunanon-parametricANOVA(usuallyfoundunderadifferentheadinginthesoftware,like“nonparametrictests“). Steps Itisunlikelyyou’llwanttodothistestbyhand,butifyoumust,thesearethestepsyou’llwanttotake: Findthemeanforeachofthegroups. Findtheoverallmean(themeanofthegroupscombined). FindtheWithinGroupVariation;thetotaldeviationofeachmember’sscorefromtheGroupMean. FindtheBetweenGroupVariation:thedeviationofeachGroupMeanfromtheOverallMean. FindtheFstatistic:theratioofBetweenGroupVariationtoWithinGroupVariation. ANOVAvs.TTest AStudent’st-testwilltellyouifthereisasignificantvariationbetweengroups.At-testcomparesmeans,whiletheANOVAcomparesvariancesbetweenpopulations. Youcouldtechnicallyperformaseriesoft-testsonyourdata.However,asthegroupsgrowinnumber,youmayendupwithalotofpaircomparisonsthatyouneedtorun.ANOVAwillgiveyouasinglenumber(thef-statistic)andonep-valuetohelpyousupportorrejectthenullhypothesis. BacktoTop RepeatedMeasures(WithinSubjects)ANOVA ArepeatedmeasuresANOVAisalmostthesameasone-wayANOVA,withonemaindifference:youtestrelatedgroups,notindependentones. It’scalledRepeatedMeasuresbecausethesamegroupofparticipantsisbeingmeasuredoverandoveragain.Forexample,youcouldbestudyingthecholesterollevelsofthesamegroupofpatientsat1,3,and6monthsafterchangingtheirdiet.Forthisexample,theindependentvariableis“time”andthedependentvariableis“cholesterol.”Theindependentvariableisusuallycalledthewithin-subjectsfactor. RepeatedmeasuresANOVAissimilartoasimplemultivariatedesign.Inbothtests,thesameparticipantsaremeasuredoverandover.However,withrepeatedmeasuresthesamecharacteristicismeasuredwithadifferentcondition.Forexample,bloodpressureismeasuredoverthecondition“time”.Forsimplemultivariatedesignitisthecharacteristicthatchanges.Forexample,youcouldmeasurebloodpressure,heartrateandrespirationrateovertime. ReasonstouseRepeatedMeasuresANOVA Whenyoucollectdatafromthesameparticipantsoveraperiodoftime,individualdifferences(asourceofbetweengroupdifferences)arereducedoreliminated. Testingismorepowerfulbecausethesamplesizeisn’tdividedbetweengroups. Thetestcanbeeconomical,asyou’reusingthesameparticipants. AssumptionsforRepeatedMeasuresANOVA TheresultsfromyourrepeatedmeasuresANOVAwillbevalidonlyifthefollowingassumptionshaven’tbeenviolated: Theremustbeoneindependentvariableandonedependentvariable. Thedependentvariablemustbeacontinuousvariable,onanintervalscaleoraratioscale. Theindependentvariablemustbecategorical,eitheronthenominalscaleorordinalscale. Ideally,levelsofdependencebetweenpairsofgroupsisequal(“sphericity”).Correctionsarepossibleifthisassumptionisviolated. OneWayRepeatedMeasuresANOVAinSPSS:Steps WatchthevideofortheSteps: One-WayRepeatedMeasuresANOVAinSPSSWatchthisvideoonYouTube. Step1:Click“Analyze”,thenhoverover“GeneralLinearModel.”Click“RepeatedMeasures.” Step2:Replacethe“factor1”namewithsomethingthatrepresentsyourindependentvariable.Forexample,youcouldput“age”or“time.” Step3:Enterthe“NumberofLevels.”Thisishowmanytimesthedependentvariablehasbeenmeasured.Forexample,ifyoutookmeasurementseveryweekforatotalof4weeks,thisnumberwouldbe4. Step4:Clickthe“Add”buttonandthengiveyourdependentvariableaname. Step5:Clickthe“Add”button.ARepeatedMeasuresDefineboxwillpopup.Clickthe“Define”button. Step6:Usethearrowkeystomoveyourvariablesfromthelefttotherightsothatyourscreenlookssimilartotheimagebelow: Step7:Click“Plots”andusethearrowkeystotransferthefactorfromtheleftboxontotheHorizontalAxisbox. Step8:Click“Add”andthenclick“Continue”atthebottomofthewindow. Step9:Click“Options”,thentransferyourfactorsfromtheleftboxtotheDisplayMeansforboxontheright. Step10:Clickthefollowingcheckboxes: Comparemaineffects. DescriptiveStatistics. EstimatesofEffectSize. Step11:Select“Bonferroni”fromthedropdownmenuunderConfidenceIntervalAdjustment. Step12:Click“Continue”andthenclick“OK”torunthetest. BacktoTop Sphericity Instatistics,sphericity(ε)referstoMauchly’ssphericitytest,whichwasdevelopedin1940byJohnW.Mauchly,whoco-developedthefirstgeneral-purposeelectroniccomputer. Definition SphericityisusedasanassumptioninrepeatedmeasuresANOVA.Theassumptionstatesthatthevariancesofthedifferencesbetweenallpossiblegrouppairsareequal.Ifyourdataviolatesthisassumption,itcanresultinanincreaseinaTypeIerror(theincorrectrejectionofthenullhypothesis). It’sverycommonforrepeatedmeasuresANOVAtoresultinaviolationoftheassumption.Iftheassumptionhasbeenviolated,correctionshavebeendevelopedthatcanavoidincreasesinthetypeIerrorrate.ThecorrectionisappliedtothedegreesoffreedomintheF-distribution. Mauchly’sSphericityTest Mauchly’stestforsphericitycanberuninthemajorityofstatisticalsoftware,whereittendstobethedefaulttestforsphericity.Mauchly’stestisidealformid-sizesamples.Itmayfailtodetectsphericityinsmallsamplesanditmayover-detectinlargesamples. Ifthetestreturnsasmallp-value(p≤.05),thisisanindicationthatyourdatahasviolatedtheassumption.ThefollowingpictureofSPSSoutputforANOVAshowsthatthesignificance“sig”attachedtoMauchly’sis.274.Thismeansthattheassumptionhasnotbeenviolatedforthissetofdata. Image:UVM.EDU Youwouldreporttheaboveresultas“Mauchly’sTestindicatedthattheassumptionofsphericityhadnotbeenviolated,χ2(2)=2.588,p=.274.” Ifyourtestreturnedasmallp-value,youshouldapplyacorrection,usuallyeitherthe: Greehouse-Geissercorrection. Huynh-Feldtcorrection. Whenε≤0.75(oryoudon’tknowwhatthevalueforthestatisticis),usetheGreenhouse-Geissercorrection. Whenε>.75,usetheHuynh-Feldtcorrection. BacktoTop RelatedArticles Grandmean ANOVAvsRegression References Blokdyk,B.(2018).AdHocTesting.5STARCooks Miller,R.G.BeyondANOVA:BasicsofAppliedStatistics.BocaRaton,FL:Chapman&Hall,1997 Image:UVM.RetrievedDecember4,2020from:https://www.uvm.edu/~dhowell/gradstat/psych341/lectures/RepeatedMeasures/repeated1.html CITETHISAS:StephanieGlen."ANOVATest:Definition,Types,Examples,SPSS"FromStatisticsHowTo.com:ElementaryStatisticsfortherestofus!https://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/anova/ --------------------------------------------------------------------------- Needhelpwithahomeworkortestquestion?WithCheggStudy,youcangetstep-by-stepsolutionstoyourquestionsfromanexpertinthefield.Yourfirst30minuteswithaCheggtutorisfree! Comments?Needtopostacorrection?PleaseContactUs. 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