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PROPHET StatGuide: Do your data violate one way ANOVA assumptions? If the populations from which data to be analyzed by a oneway analysis of variance (ANOVA) were sampled violate one or more of the oneway ANOVA test assumptions, the results of the analysis may be incorrect or misleading. For example, if the assumption of independence is violated, then the oneway ANOVA is simply not appropriate, although another test (perhaps a blocked oneway ANOVA ) may be appropriate. If the assumption of normality is violated, or outliers are present, then the oneway ANOVA may not be the most powerful test available, and this could mean the difference between detecting a true difference among the population means or not. A nonparametric test or employing a transformation may result in a more powerful test. A potentially more damaging assumption violation occurs when the population variances are unequal , especially if the sample sizes are not approximately equal (unbalanced ). Often, the effect of an assumption violation on the oneway ANOVA result depends on the extent of the violation (such as how unequal the population variances are, or how heavytailed one or another population distribution is). Some small violations may have little practical effect on the analysis, while other violations may render the oneway ANOVA result uselessly incorrect or uninterpretable. In particular, small or unbalanced sample sizes can increase vulnerability to assumption violations. Potential assumption violations include: Implicit factors : lack of independence within a sample Lack of independence : lack of independence between samples Outliers : apparent nonnormality by a few data points Nonnormality : nonnormality of entire samples Unequal population variances Patterns in plots of data : detecting violation assumptions graphically Special problems with small sample sizes Special problems with unbalanced sample sizes Multiple comparisons : effects of assumption violations on multiple comparison tests Implicit factors: A lack of independence within a sample is often caused by the existence of an implicit factor in the data. For example, values collected over time may be serially correlated (here time is the implicit factor). If the data are in a particular order, consider the possibility of dependence. (If the row order of the data reflect the order in which the data were collected, an index plot of the data [data value plotted against row number] can reveal patterns in the plot that could suggest possible time effects.) Lack of independence: Whether the samples are independent of each other is generally determined by the structure of the experiment from which they arise. Obviously correlated samples, such as a set of observations over time on the same subjects, are not independent, and such data would be more appropriately tested by a oneway blocked ANOVA or a repeated measures ANOVA. If you are unsure whether your samples are independent, you may wish to consult a statistician or someone who is knowledgeable about the data collection scheme you are using.

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  • 3/13/2015 PROPHETStatGuide:DoyourdataviolateonewayANOVAassumptions?

    http://www.basic.northwestern.edu/statguidefiles/oneway_anova_ass_viol.html 1/5

    PROPHETStatGuide:DoyourdataviolateonewayANOVAassumptions?

    Ifthepopulationsfromwhichdatatobeanalyzedbyaonewayanalysisofvariance(ANOVA)weresampledviolateoneormoreoftheonewayANOVAtestassumptions,theresultsoftheanalysismaybeincorrectormisleading.Forexample,iftheassumptionofindependenceisviolated,thentheonewayANOVAissimplynotappropriate,althoughanothertest(perhapsablockedonewayANOVA)maybeappropriate.Iftheassumptionofnormalityisviolated,oroutliersarepresent,thentheonewayANOVAmaynotbethemostpowerfultestavailable,andthiscouldmeanthedifferencebetweendetectingatruedifferenceamongthepopulationmeansornot.Anonparametrictestoremployingatransformationmayresultinamorepowerfultest.Apotentiallymoredamagingassumptionviolationoccurswhenthepopulationvariancesareunequal,especiallyifthesamplesizesarenotapproximatelyequal(unbalanced).Often,theeffectofanassumptionviolationontheonewayANOVAresultdependsontheextentoftheviolation(suchashowunequalthepopulationvariancesare,orhowheavytailedoneoranotherpopulationdistributionis).Somesmallviolationsmayhavelittlepracticaleffectontheanalysis,whileotherviolationsmayrendertheonewayANOVAresultuselesslyincorrectoruninterpretable.Inparticular,smallorunbalancedsamplesizescanincreasevulnerabilitytoassumptionviolations.

    Potentialassumptionviolationsinclude:

    Implicitfactors:lackofindependencewithinasampleLackofindependence:lackofindependencebetweensamplesOutliers:apparentnonnormalitybyafewdatapointsNonnormality:nonnormalityofentiresamplesUnequalpopulationvariancesPatternsinplotsofdata:detectingviolationassumptionsgraphicallySpecialproblemswithsmallsamplesizesSpecialproblemswithunbalancedsamplesizesMultiplecomparisons:effectsofassumptionviolationsonmultiplecomparisontests

    Implicitfactors:Alackofindependencewithinasampleisoftencausedbytheexistenceofanimplicitfactorinthedata.Forexample,valuescollectedovertimemaybeseriallycorrelated(heretimeistheimplicitfactor).Ifthedataareinaparticularorder,considerthepossibilityofdependence.(Iftheroworderofthedatareflecttheorderinwhichthedatawerecollected,anindexplotofthedata[datavalueplottedagainstrownumber]canrevealpatternsintheplotthatcouldsuggestpossibletimeeffects.)

    Lackofindependence:Whetherthesamplesareindependentofeachotherisgenerallydeterminedbythestructureoftheexperimentfromwhichtheyarise.Obviouslycorrelatedsamples,suchasasetofobservationsovertimeonthesamesubjects,arenotindependent,andsuchdatawouldbemoreappropriatelytestedbyaonewayblockedANOVAorarepeatedmeasuresANOVA.Ifyouareunsurewhetheryoursamplesareindependent,youmaywishtoconsultastatisticianorsomeonewhoisknowledgeableaboutthedatacollectionschemeyouareusing.

  • 3/13/2015 PROPHETStatGuide:DoyourdataviolateonewayANOVAassumptions?

    http://www.basic.northwestern.edu/statguidefiles/oneway_anova_ass_viol.html 2/5

    Outliers:Valuesmaynotbeidenticallydistributedbecauseofthepresenceofoutliers.Outliersareanomalousvaluesinthedata.Outlierstendtoincreasetheestimateofsamplevariance,thusdecreasingthecalculatedFstatisticfortheANOVAandloweringthechanceofrejectingthenullhypothesis.Theymaybeduetorecordingerrors,whichmaybecorrectable,ortheymaybeduetothesamplenotbeingentirelyfromthesamepopulation.Apparentoutliersmayalsobeduetothevaluesbeingfromthesame,butnonnormal,population.Theboxplotandnormalprobabilityplot(normalQQplot)maysuggestthepresenceofoutliersinthedata.

    TheFstatisticisbasedonthesamplemeansandthesamplevariances,eachofwhichissensitivetooutliers.(Inotherwords,neitherthesamplemeannorthesamplevarianceisresistanttooutliers,andthus,neitheristheFstatistic.)Inparticular,alargeoutliercaninflatetheoverallvariance,decreasingtheFstatisticandthusperhapseliminatingasignificantdifference.Anonparametrictestmaybeamorepowerfultestinsuchasituation.Ifyoufindoutliersinyourdatathatarenotduetocorrectableerrors,youmaywishtoconsultastatisticianastohowtoproceed.

    Nonnormality:Thevaluesinasamplemayindeedbefromthesamepopulation,butnotfromanormalone.Signsofnonnormalityareskewness(lackofsymmetry)orlighttailednessorheavytailedness.Theboxplot,histogram,andnormalprobabilityplot(normalQQplot),alongwiththenormalitytest,canprovideinformationonthenormalityofthepopulationdistribution.However,ifthereareonlyasmallnumberofdatapoints,nonnormalitycanbehardtodetect.Ifthereareagreatmanydatapoints,thenormalitytestmaydetectstatisticallysignificantbuttrivialdeparturesfromnormalitythatwillhavenorealeffectontheFstatistic.

    Fordatasampledfromanormaldistribution,normalprobabilityplotsshouldapproximatestraightlines,andboxplotsshouldbesymmetric(medianandmeantogether,inthemiddleofthebox)withnooutliers.

    TheonewayANOVA'sFtestwillnotbemuchaffectedevenifthepopulationdistributionsareskewed,buttheFtestcanbesensitivetopopulationskewnessifthesamplesizesareseriouslyunbalanced.Ifthesamplesizesarenotunbalanced,theFtestwillnotbeseriouslyaffectedbylighttailednessorheavytailedness,unlessthesamplesizesaresmall(lessthan5),orthedeparturefromnormalityisextreme(kurtosislessthan1orgreaterthan2).

    Robuststatisticaltestsoperatewellacrossawidevarietyofdistributions.Atestcanberobustforvalidity,meaningthatitprovidesPvaluesclosetothetrueonesinthepresenceof(slight)departuresfromitsassumptions.Itmayalsoberobustforefficiency,meaningthatitmaintainsitsstatisticalpower(theprobabilitythatatrueviolationofthenullhypothesiswillbedetectedbythetest)inthepresenceofthosedepartures.TheonewayANOVA'sFtestisrobustforvalidityagainstnonnormality,butitmaynotbethemostpowerfultestavailableforagivennonnormaldistribution,althoughitisthemostpowerfultestavailablewhenitstestassumptionsaremet.Inthecaseofnonnormality,anonparametrictestoremployingatransformationmayresultinamorepowerfultest.

    Unequalpopulationvariances:Theinequalityofthepopulationvariancescanbeassessedbyexaminationoftherelativesizeofthesamplevariances,eitherinformally(includinggraphically),orbyarobustvariancetestsuchasLevene'stest.(Bartlett'stestisevenmoresensitivetononnormalitythantheonewayANOVA'sFtest,andthusshouldnotbeusedforsuchtesting.)Theeffectofinequalityofvariancesismitigatedwhenthesamplesizesareequal:TheFtestisfairlyrobustagainstinequalityofvariancesifthe

  • 3/13/2015 PROPHETStatGuide:DoyourdataviolateonewayANOVAassumptions?

    http://www.basic.northwestern.edu/statguidefiles/oneway_anova_ass_viol.html 3/5

    samplesizesareequal,althoughthechanceincreasesofincorrectlyreportingasignificantdifferenceinthemeanswhennoneexists.Thischanceofincorrectlyrejectingthenullhypothesisisgreaterwhenthepopulationvariancesareverydifferentfromeachother,particularlyifthereisonesamplevarianceverymuchlargerthantheothers.

    Theeffectofinequalityofthevariancesismostseverewhenthesamplesizesareunequal.Ifthelargersamplesareassociatedwiththepopulationswiththelargervariances,thentheFstatisticwilltendtobesmallerthanitshouldbe,reducingthechancethatthetestwillcorrectlyidentifyasignificantdifferencebetweenthemeans(i.e.,makingthetestconservative).Ontheotherhand,ifthesmallersamplesareassociatedwiththepopulationswiththelargervariances,thentheFstatisticwilltendtobegreaterthanitshouldbe,increasingtheriskofincorrectlyreportingasignificantdifferenceinthemeanswhennoneexists.Thischanceofincorrectlyrejectingthenullhypothesisinthecaseofunbalancedsamplesizescanbesubstantialevenwhenthepopulationvariancesarenotverydifferentfromeachother.

    Althoughtheeffectofunbalancedsamplesizesandunequalpopulationvariancesincreasesforsmallersamplesizes,itdoesnotdecreasesubstantiallyifthesamplesizesareincreasedwithoutchangingthelackofbalanceinthesamplesizes.Forthisreason,andbecauseequalsamplesizesmitigatetheeffectofunequalpopulationvariances,thebestcourseistokeepthesamplesizesasequalaspossible.

    Ifbothnonnormalityandunequalvariancesarepresent,employingatransformationmaybepreferable.AnonparametrictestliketheKruskalWallisteststillassumesthatthepopulationvariancesarecomparable.

    Patternsinplotofdata:Theplotofeachsample'svaluesagainstitsmean(oritssampleID)willconsistofvertical"stacks"ofdatapoints,onestackforeachuniquesamplemeanvalue.Iftheassumptionsforthesamples'populationdistributionsarecorrect,thestacksshouldbeaboutthesamelength.Outliersmayappearasanomalouspointsinthegraph.

    Afanpatternliketheprofileofamegaphone,withanoticeableflareeithertotherightortotheleftasshowninthepicture(oneormoreofthe"stacks"ofdatapointsismuchlongerthantheothers),suggeststhatthevarianceinthevaluesincreasesinthedirectionthefanpatternwidens(usuallyasthesamplemeanincreases),andthisinturnsuggeststhatatransformationmaybeneeded.

    Sidebysideboxplotsofthesamplescanalsoreveallackofhomogeneityofvariancesifsomeboxplotsaremuchlongerthanothers,andrevealsuspectedoutliers.

  • 3/13/2015 PROPHETStatGuide:DoyourdataviolateonewayANOVAassumptions?

    http://www.basic.northwestern.edu/statguidefiles/oneway_anova_ass_viol.html 4/5

    Specialproblemswithsmallsamplesizes:Ifoneormorethesamplesizesissmall,itmaybedifficulttodetectassumptionviolations.Withsmallsamples,violationassumptionssuchasnonnormalityorinequalityofvariancesaredifficulttodetectevenwhentheyarepresent.Also,withsmallsamplesize(s)theonewayANOVA'sFtestofferslessprotectionagainstviolationofassumptions.

    Evenifnoneofthetestassumptionsareviolated,aonewayANOVAwithsmallsamplesizesmaynothavesufficientpowertodetectanysignificantdifferenceamongthesamples,evenifthemeansareinfactdifferent.Thepowerdependsontheerrorvariance,theselectedsignificance(alpha)levelofthetest,andthesamplesize.Powerdecreasesasthevarianceincreases,decreasesasthesignificancelevelisdecreased(i.e.,asthetestismademorestringent),andincreasesasthesamplesizeincreases.Withverysmallsamples,evensamplesfrompopulationswithverydifferentmeansmaynotproduceasignificantonewayANOVAFteststatisticunlessthesamplevarianceissmall.IfastatisticalsignificancetestwithsmallsamplesizesproducesasurprisinglynonsignificantPvalue,thenalackofpowermaybethereason.Thebesttimetoavoidsuchproblemsisinthedesignstageofanexperiment,whenappropriateminimumsamplesizescanbedetermined,perhapsinconsultationwithastatistician,beforedatacollectionbegins.

    Specialproblemswithunbalancedsamplesizes:TheonewayANOVAtestisnottoosensitivetoinequalityofvariancesifthesamplesizesareequal.Ifthesamplesizesarenotapproximatelyequal,andespeciallyifthelargersamplevariancesareassociatedwiththesmallersamplesizes,thenthecalculatedFstatisticmaybedominatedbythesamplevariancesforthelargersamples,sothatthetestislesslikelytocorrectlyidentifysignificantdifferencesinthemeansifthelargersamplesareassociatedwiththelargerpopulationvariances,andmorelikelytoreportnonexistentdifferencesinthemeansifthesmallersamplesareassociatedwiththelargerpopulationvariances.Unbalancedsamplesizesalsoincreaseanyeffectduetononnormality,andrequireadjustmentstobemadeincalculatingmultiplecomparisonstests.

    Multiplecomparisons:Ingeneral,themultiplecomparisonstestswillberobustinthosesituationswhentheonewayANOVA'sFtestisrobust,andwillbesubjecttothesamepotentialproblemswithunequalvariances,particularlywhenthesamplesizesareunequal.AswiththeonewayANOVAitself,thebestprotectionagainsttheeffectsofpossibleassumptionviolationsistoemployequalsamplesizes.Unequalvariancesmaymakeindividualcomparisonsofmeansinaccurate,becausethemultiplecomparisontechniquesrelyonapooledestimateforthevariance,basedontheassumptionthatthesamplevariancesareequal.

    Ideally,thesamplesizeswillbeequalforallpairwisemultiplecomparisontests.Whentheyarenot,anadjustmentmustbemadetothecalculations.TheTukeyKrameradjustment(basedonthe

  • 3/13/2015 PROPHETStatGuide:DoyourdataviolateonewayANOVAassumptions?

    http://www.basic.northwestern.edu/statguidefiles/oneway_anova_ass_viol.html 5/5

    harmonicmeanofeachpair'ssamplesizes),whichProphetuses,maybeconservative(thatis,itmaybelesslikelytoflagmeansasdifferentthanthenominalsignificancelevelwouldsuggest),butingeneralperformswell.Analternativeprocedureistousetheharmonicmeanofallthesamplesizesforallthepairwisecomparisons.ThishasthedisadvantagethattheactualsignificancelevelofthetestismoreoftendifferentfromthenominalsignificancelevelthanisthecasewiththeTukeyKrameradjustmentworse,theactualsignificancelevelofthetestmaybegreaterthanthenominalsignificancelevel,meaningthatthetestismorelikelytoincorrectlyflagameandifferenceassignificant.

    Examinetheglossary.

    DoakeywordsearchofPROPHETStatGuide.

    BacktoStatGuideonewayANOVApage.

    BacktoStatGuidehomepage.

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