One-Way Independent ANOVA - Discovering · PDF fileOne-Way Independent ANOVA There goes my hero . Watch him as he goes (to hospital) Children wearing superhero costumes are
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There goes my hero …. Watch him as he goes (to hospital) Childrenwearing superhero costumesaremore likely toharm themselvesbecauseof theunrealistic impressionofinvincibility that these costumes could create: For example, childrenhave reported tohospitalwith severe injuriesbecauseof trying ‘to initiate flightwithouthavingplanned for landingstrategies’ (Davies,Surridge,Hole,&Munro-Davies,2007).Icanrelatetotheimaginedpowerthatacostumebestowsuponyou;evennow,IhavebeenknowntodressupasFisherbydonningabeardandglassesandtrailingagoataroundonaleadinthehopethatitmightmakememoreknowledgeableaboutstatistics.Imaginewehaddataabouttheseverityofinjury(onascalefrom0,noinjury,to100,death)forchildrenreportingtotheemergencycentreathospitalsandinformationonwhichsuperherocostumetheywerewearing(hero):Spiderman,superman,thehulkorateenagemutantninjaturtle.TheDataareinTable1andtherearedescriptivestatisticsinOutput1.Theresearcherhypothesized:
1 Some of youmight take issuewith this because you probably think of the hulk as a fancy bit of CGI that leapsskyscrapers.However,the ‘proper’hulk,that is, theonethatwasonTVduringmychildhoodinthe late1970s(seeYouTube)wasinfactarealmanwithbigmusclespaintedgreen.Makenomistake,hewaswayscarierthananyCGI,buthedidnotjumpoverskyscrapers.
Figure1 showshowwewouldapplyRule1 to theSuperheroexample.We’re told thatwewant to compare flyingsuperheroes(i.e.SupermanandSpiderman)againstnon-flyingones(theHulkandNinjaTurtles)inthefirstinstance.Thatwillbecontrast1.However,becauseeachofthesechunksismadeupoftwogroups(e.g.,theflyingsuperheroeschunkcomprisesbothchildrenwearingSpidermanandthosewearingSupermancostumes),weneedasecondandthirdcontrastthatbreakseachofthesechunksdownintotheirconstituentparts.
Effect Sizes: Cohen’s d Wediscussedearlierinthemodulethatitcanbeusefulnotjusttorelyonsignificancetestingbutalsotoquantifytheeffectsinwhichwe’reinterested.Whenlookingatdifferencesbetweenmeans,ausefulmeasureofeffectsizeisCohen’sd.Thisstatisticisveryeasytounderstandbecauseitisthedifferencebetweentwomeansdividedbysomeestimateofthestandarddeviationofthosemeans:
Planned Comparisons Using SPSS Clickon toaccessthedialogueboxinFigure2,whichhastwosections.Thefirstsectionisforspecifyingtrendanalyses. If youwant to test for trends in the data then tick the box labelledPolynomial and select the degree ofpolynomialyouwouldlike.TheSuperherodatahasfourgroupsandsothehighestdegreeoftrendtherecanbeisacubic trend (see Field, 2013 Chapter 11). We predicted that the injuries will decrease in this order: Superman >Spiderman>Hulk>NinjaTurtle.Thiscouldbealineartrend,orpossiblyquadratic(acurveddescendingtrend)butnotcubic(becausewe’renotpredictingthatinjuriesgodownandthenup.
Itisimportantfromthepointofviewoftrendanalysisthatwehavecodedthegroupingvariableinameaningfulorder.Todetectameaningfultrend,weneedtohavecodedthegroupsintheorderinwhichweexpectthemeaninjuriestodescend;thatis,Superman,Spiderman,Hulk,NinjaTurtle.WehavedonethisbycodingtheSupermangroupwiththelowestvalue1,Spidermanwiththenextlargestvalue(2),theHulkwiththenextlargestvalue(3),andtheNinjaTurtlegroupwiththelargestcodingvalueof4.Ifwecodedthegroupsdifferently,thiswouldinfluencebothwhetheratrendisdetected,and ifby chancea trend isdetectedwhether it ismeaningful. For the superherowepredictatmostaquadratictrend(seeabove),soselectthepolynomialoption( ),andthenselectaquadraticdegreebyclickingon andthenselectingQuadratic(thedrop-downlistshouldnowsay )—seeFigure3.IfaquadratictrendisselectedSPSSwilltestforbothlinearandquadratictrends.
Wewillspecifycontrast1first.Itisimportanttomakesurethatyouenterthecorrectweightingforeachgroup,soyoushouldrememberthatthefirstweightthatyouentershouldbetheweightforthefirstgroup(thatis,thegroupcodedwith the lowest value in the data editor). For the superherodata, the group codedwith the lowest valuewas theSupermangroup(whichhadacodeof1)andsoweshouldentertheweightingforthisgroupfirst.Click intheboxlabelledCoefficientswiththemouseandthentype‘2’inthisboxandclickon .Next,weinputtheweightforthesecondgroup,whichwastheSpidermangroup(becausethisgroupwascodedinthedataeditorwiththesecondhighestvalue).ClickintheboxlabelledCoefficientswiththemouseandthentype‘2’inthisboxandclickon .Next,weinputtheweightforHulkgroup(becauseithadthenextlargestcodeinthedataeditor),soclickintheboxlabelledCoefficientswiththemouseandtype‘-2’andclickon .Finally,weinputthecodeforthelastgroup(theonewiththelargestcodeinthedataeditor),whichwastheNinjaTurtlegroup—clickintheboxlabelledCoefficientswiththemouseandtype‘-2’andclickon .TheboxshouldnowlooklikeFigure4(left).
.ThentheHulkgroup:clickintheboxlabelledCoefficients,type‘1’andclickon .Finally,weinputtheweight for theNinjaTurtlegroupbyclicking in thebox labelledCoefficients, typing ‘-1’andclickingon (seeFigure4,right).
Post Hoc Tests in SPSS Normallyifwehavedoneplannedcomparisonsweshouldnotdoposthoctests(becausewehavealreadytestedthehypotheses of interest). Likewise, if we choose to conduct post hoc tests then planned contrasts are unnecessary(becausewehavenohypothesestotest).However,forthesakeofspacewewillconductsomeposthoctestsonthesuperherodata.Clickon inthemaindialogueboxtoaccesstheposthoctestsdialoguebox(Figure5).Thechoiceofcomparisonproceduredependsontheexactsituationyouhaveandwhetheryouwantstrictcontroloverthefamilywiseerrorrateorgreaterstatisticalpower.Ihavedrawnsomegeneralguidelines:
I recommend running theGames-Howell procedure in addition to anyother tests youmight select becauseof theuncertaintyof knowingwhether thepopulation variances are equivalent. For the superherodata there are slightlyunequalsamplesizesandsowewilluseGabriel’stest(seeTipabove).Whenthecompleteddialogueboxlooks likeFigure5clickon toreturntothemaindialoguebox.
. First you can ask for some descriptive statistics, whichwilldisplay a table of the means, standard deviations, standard errors,rangesandconfidenceintervalsforthemeansofeachgroup.Thisisausefuloptiontoselectbecauseitassistsininterpretingthefinalresults.You can also select Homogeneity-of-variance tests. Earlier in themodulewesawthatthereisanassumptionthatthevariancesofthegroupsareequalandselectingthisoptionteststhisassumptionusingLevene’stest(seeyourhandoutonbias).SPSSoffersustwoalternativeversionsoftheF-ratio:theBrown-ForsytheF(1974),andtheWelchF(1951).ThesealternativeFscanbeusedifthehomogeneityofvarianceassumption is broken. If you’re interested in the details of thesecorrectionsthenseeField(2013),butifyou’vegotbetterthingstodowithyourlifethentakemywordforitthatthey’reworthselectingjustin case the assumption is broken. You can also select aMeans plotwhichwillproducealinegraphofthemeans.Again,thisoptioncanbeusefulforfindinggeneraltrendsinthedata.Whenyouhaveselectedtheappropriateoptions,clickon toreturntothemaindialogbox.Clickon inthemaindialogboxtoruntheanalysis.
Figure6:OptionsforOne-WayANOVA
Bootstrapping Alsointhemaindialogboxisthealluring button.Wehaveseeninthemodulethatbootstrappingisagoodwaytoovercomebias,andthisbuttonglistensandtemptsuswiththepromiseofuntoldriches,likeadiamondinabull’srectum.However, ifyouusebootstrappingit’llbeasdisappointingas ifyoureachedforthatdiamondonlytodiscover that it’sapieceofglass.Youmight,notunreasonably, think that ifyouselectbootstrapping it’ddoanicebootstrapof theF-statistic foryou. Itwon’t. Itwillbootstrapconfidence intervalsaroundthemeans (ifyouask fordescriptivestatistics),contrastsanddifferencesbetweenmeans(i.e.,theposthoctests).This,ofcourse,canbeuseful,butthemaintestwon’tbebootstrapped.
û SPSShasscaledthey-axistomakethemeanslookasdifferentashumanlypossible.Thinkbacktoweek1whenwe learnt that itwas verybad to scale your graph tomaximise groupdifferences – SPSShasnot readmyhandoutJ
û Therearenoerrorbars:thegraphjustisn’tveryinformativebecausewearen’tgivenconfidenceintervalsforthemean.
Figure8showsanerrorbarchartoftheinjurydata.Themeansindicatethatsomesuperherocostumesdoresultinmoresevere injuries thanothers.Notably, theNinjaTurtlecostumeseemsto result in lesssevere injuries and theSuperman costume results inmost severe injuries. The error bars (the I shapes) show the95% confidence intervalaroundthemean.
Ifwewere to take 100 samples from the samepopulation, the truemean (themeanof thepopulation)would liesomewherebetweenthetopandbottomofthatbarin95ofthosesamples.Inotherwords,thesearethelimitsbetweenwhichthepopulationvalueforthemeaninjuryseverityineachgroupwill(probably)lie.Ifthesebarsdonotoverlapthenweexpecttogetasignificantdifferencebetweenmeansbecauseitshowsthatthepopulationmeansofthosetwosamplesarelikelytobedifferent(theydon’tfallwithinthesamelimits).So,forexample,wecantellthatNinjaTurtlerelated injuriesare likelytobe lessseverethanthoseofchildrenwearingsupermancostumes(theerrorbarsdon’toverlap)andSpidermancostumes(onlyasmallamountofoverlap).
® IfLevene’stestissignificant(i.e.thevalueofsig.islessthan.05)thenwecanconcludethatthevariancesaresignificantlydifferent.ThiswouldmeanthatwehadviolatedoneoftheassumptionsofANOVAandwewouldhavetotakestepstorectifythismattersby(1)transforming all of the data (see your handout on bias), (2) bootstrapping (notimplementedinSPSSforANOVA,or(3)usingacorrectedtest(seebelow).RememberthathowweinterpretLevene’stestdependsonthesizeofsamplewehave(seethehandoutonbias).
For the main ANOVA, we selected two procedures (Brown-Forsythe and Welch) that should be accurate whenhomogeneityofvarianceisnottrue.So,weshouldperhapsinspecttheseF-valuesinthemainanalysis.Wemightalsochooseamethodofposthoctest thatdoesnotrelyontheassumptionofequalvariances (e.g., theGames-Howellprocedure).
Thetableyou’llseeisdividedintobetweengroupeffects(effectsduetotheexperiment)andwithingroupeffects(thisistheunsystematicvariationinthedata).Thebetween-groupeffect istheoverallexperimentaleffect(theeffectofwearingdifferentcostumesontheseverityofinjuries).Inthisrowwearetoldthesumsofsquaresforthemodel(SSM=4180.62).Thesumofsquaresforthemodelrepresentsthetotalexperimentaleffectwhereasthemeansquaresforthemodelrepresentstheaverageexperimentaleffect.Therowlabelledwithingroupgivesdetailsoftheunsystematicvariationwithinthedata(thevariationduetonaturalindividualdifferencesinphysiqueandtolerancetoinjury).Thetabletellsushowmuchunsystematicvariationexists(theresidualsumofsquares,SSR).Itthengivestheaverageamountof unsystematic variation, the residualmean squares (MSR). The test ofwhether the groupmeans are the same isrepresentedby theF-ratio for thecombinedbetween-groupeffect.Thevalueof this ratio is8.32.The finalcolumnlabelledsig.indicateshowlikelyitisthatanF-ratioofatleastthatsizewouldhaveoccurrediftherewerenodifferencesbetweenmeans.Inthiscase,thereisaprobabilityof0.000(that’slessthana.1%chance!).Wehaveseenthatscientiststendtouseacutofpointof.05astheircriterionforstatisticalsignificance.Hence,becausetheobservedsignificancevalue is less than .05wecansay that therewasasignificanteffectof thecostumewornontheseverityof injuriessustained.However,atthisstagewestilldonotknowexactlywhattheeffectofeachcostumewas(wedon’tknowwhichgroupsdiffered).Also,weknowfrompreviouslecturesthatthinkingaboutsignificanceinthisblackandwhitewayisnotalwayshelpfulandweshouldconsiderotherinformationsuchastheeffectsizescomputedatthebeginningofthishandout.
WetoldSPSStoconductthreeplannedcomparisons:onetotestwhether ‘flying’superherocostumes ledtoworseinjuries than ‘non-flying’ superhero costumes; the second to compare injury severity for the two flying superherocostumes (Superman vs. Spiderman costumes); and the third to compare injury severity for the two non-flyingsuperherocostumes(Hulkvs.NinjaTurtlecostumes).Output5showstheresultsoftheplannedcomparisonsthatwerequested.Thefirsttabledisplaysthecontrastcoefficientsanditiswellworthlookingatthistabletodoublecheckthatthecontrastsarecomparingwhattheyaresupposedto:theyshouldcorrespondtoTable2,whichtheydo.Iftheydon’tthenyou’veenteredtheweightsincorrectly(seeFigure4).
The second table gives the statistics for each contrast. The first thing to notice is that statistics are produced forsituationsinwhichthegroupvariancesareequal,andwhentheyareunequal.Typically,ifLevene’stestwassignificantthenyoushouldreadthepartofthetablelabelledDoesnotassumeequalvariances;ifLevene’stestwasnotsignificantyouusethepartofthetablelabelledAssumeequalvariances.ForthesedataLevene’stestwasnotsignificantimplyingthat we can assume equal variances; however, the variance ratio suggested that actually this assumption ofhomogeneitymight beunreasonable (and that Levene’s testmight havebeennon-significant becauseof the smallsamplesize).Therefore,basedonthevarianceratioweprobablyshouldnotassumeequalvaraiancesandinsteadusethepartofthetablelabelledDoesnotassumeequalvariances.
Thetabletellsusthevalueofthecontrastitself,theassociatedt-testandthetwo-tailedsignificancevalue.Hence,forcontrast1,wecansaythat injuryseveritywassignificantlydifferent inkidswearingcostumesof flyingsuperheroescompared to thosewearingnon-flyingsuperherocostumes, t(15.10)=3.99,p= .001.Contrast2 tellsus that injuryseveritywasnotsignificantlydifferent inthosewearingSupermancostumescomparedtothosewearingSpidermancostumes,t(8.39)=2.21,p=.057.Finally,contrast2tellsusthatinjuryseveritywasnotsignificantlydifferentinthosewearingHulkcostumescomparedtothosewearingNinjaTurtlecostumes,t(11.57)=1.65,p=.126.
Output for Post Hoc Tests Ifwehadnospecifichypothesesabout theeffect thatdifferentsuperherocostumeswouldhaveontheseverityofinjuries,thenwecouldcarryoutposthocteststocompareallgroupsofparticipantswitheachother.Infact,weaskedSPSStodothis(seeearlier)andtheresultsofthisanalysisareshowninOutput6.ThistableshowstheresultsofGabriel’stestandtheGames-Howellprocedure,whichwerespecifiedearlieron.IfwelookatGabriel’stestfirst,eachgroupofchildreniscomparedwithalloftheremaininggroups.Foreachpairofgroupsthedifferencebetweengroupmeansisdisplayed,thestandarderrorofthatdifference,thesignificancelevelofthatdifferenceanda95%confidenceinterval.Firstofall, theSupermangroupiscomparedtotheSpidermangroupandrevealsanonsignificantdifference(Sig. isgreaterthan.05),butwhencomparedtotheHulkgroup(p=.008)andtheTurtlegroup(p<.001)thereisasignificantdifference(Sig.islessthan.05).
ü Theassumptionofhomogeneityofvariancewasviolated;therefore,theBrown-ForsytheF-ratioisreported.Therewasasignificanteffectofthecostumewornontheseverityofinjuriessustained,F(3,16.93)=7.68,p=.005.
ü Theassumptionofhomogeneityofvariancewasviolated;therefore,theWelchF-ratioisreported.Therewasasignificanteffectofthecostumewornontheseverityofinjuriessustained,F(3,13.02)=7.10,p=.002.
Wecanreportcontrastsandtrendsinmuchthesameway:
ü Themeanseverityofinjuriesdecreasedproportionatelyacrossthefoursuperherocostumes,F(1,26)=23.44,p<.001.
ü Plannedcontrastsrevealedthatinjuryseveritywassignificantlydifferentinchildrenwearingcostumesofflyingsuperheroes compared to those wearing non-flying superhero costumes, t(15.10) = 3.99, p = .001. Injuryseverity was not significantly different in those wearing Superman costumes compared to those wearingSpidermancostumes,t(8.39)=2.21,p=.057,norbetweenthosewearingHulkcostumescomparedtothosewearingNinjaTurtlecostumes,t(11.57)=1.65,p=.126.
ü IngeneralhomogeneitycouldnotbeassumedbetweenpairsofgroupsexceptfortheHulkgroupwithbothSupermanandSpiderman.WherehomogeneitycouldnotbeassumedGames-Howellposthoctestswereused,wherelocalhomogeneitycouldbeassumedGabriel’stestwasused.ThesetestsrevealedsignificantdifferencesbetweentheSupermangroupandboththeHulk,p=.008,d=1.62,andNinjaTurtle,p=.016,d=2.60,groupsandtheSpidermanandTurtlegroups,p=.050,d=1.48.TherewerenosignificantdifferencesbetweentheSpidermanandbothSuperman,p=.197,d=1.26,andHulk,p=.907,d=0.49,groups,orbetweentheHulkandNinjaTurtlegroup,p=.392,d=0.82.
Unguided Task 3 I reada story in anewspaper recently claiming that scientistshaddiscovered that the chemical genistein,which isnaturallyoccurringinSoya,waslinkedtoloweredspermcountsinwesternmales.Infact,whenyoureadtheactualstudy,ithadbeenconductedonrats,itfoundnolinktoloweredspermcountsbuttherewasevidenceofabnormalsexual development in male rats (probably because this chemical acts like oestrogen). The journalist naturallyinterpretedthisasaclearlinktoapparentlydecliningspermcountsinwesternmales(bloodyjournalists!).Anyway,asaVegetarianwhoeatslotsofSoyaproductsandprobablywouldliketohavekidsoneday,imagineIwantedtotestthisideainhumansratherthanrats.Itook80malesandsplitthemintofourgroupsthatvariedinthenumberofSoyameals
Complete themultiple choice questions forChapter 11 on the companionwebsite to Field(2013):https://studysites.uk.sagepub.com/field4e/study/mcqs.htm.Ifyougetanywrong,re-readthishandout(orField,2013,Chapter13)anddothemagainuntilyougetthemallcorrect.
Cohen,J.(1988).Statisticalpoweranalysisforthebehaviouralsciences(2ndedition).NewYork:AcademicPress.Cohen,J.(1992).Apowerprimer.PsychologicalBulletin,112(1),155-159.Davies, P., Surridge, J., Hole, L.,&Munro-Davies, L. (2007). Superhero-related injuries in paediatrics: a case series.
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