one-way independent anova - discovering statistics€¦ · · 2017-01-07one-way independent anova...

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One-Way Independent ANOVA

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:

• Costumesof‘flying’superheroes(i.e.Thatis,theonesthattravelthroughtheair:SupermanandSpiderman)willleadtomoresevereinjuriesthannon-flyingones(theHulkandNinjaTurtles).

• Therewillbeadiminishingtrendininjuriesdependingonthecostume:Superman(mostinjuriesbecauseheflies),Spiderman(nexthighestinjuriesbecausealthoughtechnicallyhedoesn’tfly,hedoesclimbbuildingsandthrowshimselfabouthighupintheair),Hulk(doesn’ttendtoflyaboutintheairmuchbutdoessmashbuildingsandpunchhardobjectsthatwoulddamageachildiftheyhitthem)1,andNinjaTurtles(let’sfaceit,theyengageinfairlytweeNinjaroutines).

Table1:Datashowingtheseverityofinjurysustainedby30childrenwearingsuperherocostumes

Costume Injury

Superman

69

32

85

66

58

52

Spiderman

51

31

58

20

47

37

49

40

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.


Hulk

26

43

10

45

30

35

53

41

NinjaTurtle

18

18

30

30

30

41

18

25

Generating Contrasts Basedonwhatyoulearntinthelecture,rememberthatweneedtofollowsomerulestogenerateappropriatecontrasts:

• Rule1:Choosesensiblecomparisons.Rememberthatyouwanttocompareonlytwochunksofvariationandthatifagroupissingledoutinonecomparison,thatgroupshouldbeexcludedfromanysubsequentcontrasts.

• Rule2:Groupscodedwithpositiveweightswillbecomparedagainstgroupscodedwithnegativeweights.So,assignonechunkofvariationpositiveweightsandtheoppositechunknegativeweights.

• Rule3:Thesumofweightsforacomparisonshouldbezero.Ifyouadduptheweightsforagivencontrasttheresultshouldbezero.

• Rule4:Ifagroupisnotinvolvedinacomparison,automaticallyassignitaweightof0.Ifwegiveagroupaweightof0thenthiseliminatesthatgroupfromallcalculations

• Rule5:Foragivencontrast,theweightsassignedtothegroup(s)inonechunkofvariationshouldbeequaltothenumberofgroupsintheoppositechunkofvariation.

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.

Togettheweights(Table2),weapplyrules2to5.Contrast1comparesflying(Superman,Spiderman)tonon-flying(Hulk,Turtle)superheroes.Eachchunkcontainstwogroups,sotheweightsfortheoppositechunksareboth2.Weassignonechunkpositiveweightsandtheothernegativeweights(inTable2I’vechosentheflyingsuperheroestohavepositiveweights,butyoucoulddoittheotherwayaround).Contrasttwothencomparesthetwoflyingsuperheroestoeachother.Firstweassignbothnon-flyingsuperheroesa0weighttoremovethemfromthecontrast.We’releftwithtwochunks:onecontainingtheSupermangroupandtheothercontainingtheSpidermangroup.Eachchunkcontainsonegroup,sotheweights fortheoppositechunksareboth1.Weassignonechunkpositiveweightsandtheothernegativeweights(inTable2I’vechosentogiveSupermanthepositiveweight,butyoucoulddoittheotherwayaround).


Figure1:ContrastsfortheSupermandata

Table2:WeightsforthecontrastsinFigure1

Contrast Superman Spiderman Hulk NinjaTurtle

Contrast1 2 2 -2 -2

Contrast2 1 -1 0 0

Contrast3 0 0 1 -1

Finally,Contrastthreecomparesthetwonon-flyingsuperheroestoeachother.Firstweassignbothflyingsuperheroesa0weighttoremovethemfromthecontrast.We’releftwithtwochunks:onecontainingtheHulkgroupandtheothercontainingtheTurtlegroup.Eachchunkcontainsonegroup,sotheweightsfortheoppositechunksareboth1.Weassignonechunkpositiveweightsandtheothernegativeweights(inTable2I’vechosentogivetheHulkthepositiveweight,butyoucoulddoittheotherwayaround).

Notethatifweaddtheweightsweget0ineachcase(rule3):2+2+(-2)+(-2)=0(Contrast1);1+(-1)+0+0=0(Contrast2);and0+0+1+(-1)=0(Contrast3).

Effect Sizes: Cohen’s d Wediscussedearlierinthemodulethatitcanbeusefulnotjusttorelyonsignificancetestingbutalsotoquantifytheeffectsinwhichwe’reinterested.Whenlookingatdifferencesbetweenmeans,ausefulmeasureofeffectsizeisCohen’sd.Thisstatisticisveryeasytounderstandbecauseitisthedifferencebetweentwomeansdividedbysomeestimateofthestandarddeviationofthosemeans:

𝑑𝑑 =𝑋𝑋$ − 𝑋𝑋&𝑠𝑠

Ihaveputahatonthedtoremindusthatwe’rereallyinterestedintheeffectsizeinthepopulation,butbecausewecan’tmeasurethatdirectly,weestimateitfromthesamples(Thehatmeans‘estimateof’).Bydividingbythestandarddeviationweareexpressingthedifference inmeans instandarddeviationunits(abit likeaz–score).Thestandarddeviationisameasureof‘error’or‘noise’inthedata,sodiseffectivelyasignal-to-noiseratio.However,ifwe’reusing

SSM Variance in injury severity explained by

different costumes

'Flying' Superheroes Superman and

Spiderman

Non-'Flying' Superheroes

Hulk and TurtleContrast 1

Superman Spiderman Contrast 2

Hulk Ninja Turtle Contrast 3


twomeans,thentherewillbeastandarddeviationassociatedwitheachofthemsowhichoneshouldweuse?Therearethreechoices:

1. Ifoneofthegroupisacontrolgroupitmakessensetousethatgroupsstandarddeviationtocomputed(theargumentbeingthattheexperimentalmanipulationmightaffectthestandarddeviationoftheexperimentalgroup,sothecontrolgroupSDisa‘purer’measureofnaturalvariationinscores)

2. Sometimesweassumethatgroupvariances(andthereforestandarddeviations)areequal(homogeneityofvariance)andiftheyarewecanpickastandarddeviationfromeitherofthegroupsbecauseitwon’tmatter.

3. Weusewhat’sknownasa‘pooledestimate’,whichistheweightedaverageofthetwogroupvariances.Thisisgivenbythefollowingequation:

𝑠𝑠( =𝑁𝑁$ − 1 𝑠𝑠$& + 𝑁𝑁& − 1 𝑠𝑠&&

𝑁𝑁$ + 𝑁𝑁& − 2

Let’slookatanexample.SaywewantedtoestimatedfortheeffectofSupermancostumescomparedtoNinjaTurtlecostumes.Output1showsusthemeans,samplesizeandstandarddeviationforthesetwogroups:

• Superman:M=60.33,N=6,s=17.85,s2=318.62• NinjaTurtle:M=26.25,N=8,s=8.16,s2=66.50

Neithergroupisanaturalcontrol(youwouldneeda‘nocostume’conditionreally),butifwedecidedthatNinjaTurtle(forsomereason)wasacontrol(perhapsbecauseTurtlesdon’tflybutsupermendo)thendissimply:

𝑑𝑑 =𝑋𝑋-.(/012/3456 − 𝑋𝑋7834086

𝑠𝑠7834086=60.33 − 26.25

8.16= 4.18

Inotherwords,themeaninjuryseverityforpeoplewearingsupermancostumesis4standarddeviationsgreaterthanforthosewearingNinjaTurtlecostumes.Thisisaprettyhugeeffect.

Cohen(1988,1992)hasmadesomewidelyusedsuggestionsaboutwhatconstitutesa largeorsmalleffect:d=0.2(small),0.5(medium)and0.8(large).Becarefulwiththesebenchmarksbecausetheyencouragethekindoflazythinkingthatweweretryingtoavoidandignorethecontextoftheeffectsuchasthemeasurementinstrumentsandgeneralnormsinaparticularresearcharea.

Let’shavealookatusingthepooledestimate.

𝑠𝑠( =6 − 1 17.85& + 8 − 1 8.16&

6 + 8 − 2=

1593.11 + 466.1012

= 171.60 = 13.10

Whenthegroupstandarddeviationsaredifferent,thispooledestimatecanbeuseful;however,itchangesthemeaningofdbecausewe’renowcomparingthedifferencebetweenmeansagainstallofthebackground‘noise’inthemeasure,notjustthenoisethatyouwouldexpecttofindinnormalcircumstances.Usingthisestimateofthestandarddeviationweget:

𝑑𝑑 =𝑋𝑋-.(/012/3456 − 𝑋𝑋7834086

𝑠𝑠7834086=60.33 − 26.25

13.10= 2.60

Noticethatdissmallernow;theinjuryseverityforSupermancostumesisabout2standarddeviationsgreaterthanforNinjaTurtleCostumes(whichisstillprettybig)

SELF-TEST:ComputeCohen’sdfortheeffectofSupermancostumesoninjuryseveritycomparedtoHulkandSpidermancostumes.Tryusingboththestandarddeviationofthecontrol(thenon-Supermancostume)andalsothepooledestimate.(Answersattheendofthehandout)

Running One-Way Independent ANOVA on SPSS Let’sconductanANOVAonthe injurydata.Weneedtoenterthedata intothedataeditorusingacodingvariablespecifyingtowhichofthefourgroupseachscorebelongs.Weneedtodothisbecausewehaveusedabetween-groupdesign(i.e.differentpeople ineachcostume).So,thedatamustbeenteredintwocolumns(onecalledherowhich


specifiesthecostumewornandonecalledinjurywhichindicatestheseverityofinjuryeachchildsustained).YoucancodethevariableheroanywayyouwishbutIrecommendsomethingsimplesuchas1=Superman,2=Spiderman,3=TheHulk,and4=NinjaTurtle.

® SavethesedatainafilecalledSuperhero.sav.

® IndependentVariablesaresometimesreferredtoasFactors.

To conduct one-way ANOVA we have to first access the main dialogue box by selecting(Figure1).Thisdialogueboxhasaspacewhereyoucanlistoneormore

dependentvariablesandasecondspacetospecifyagroupingvariable,orfactor.Factorisanothertermforindependentvariable.FortheinjurydataweneedselectonlyinjuryfromthevariablelistandtransferittotheboxlabelledDependentListbyclickingon (ordraggingitthere).ThenselectthegroupingvariableheroandtransferittotheboxlabelledFactorbyclickingon (ordraggingit).Ifyouclickon youaccessthedialogboxthatallowsyoutoconductplannedcomparisons,andbyclickingon youaccesstheposthoctestsdialogbox.Thesetwooptionswillbeexplainedduringthenextpracticalclass.

Figure2:Maindialogueboxforone-wayANOVA

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.

Toconductplannedcomparisons,thefirststepistodecidewhichcomparisonsyouwanttodoandthenwhatweightsmustbeassignedtoeachgroupforeachofthecontrasts(seeField,2013,Chapter11).Wesawearlierinthishandoutwhatsensiblecontrastswouldbe,andwhatweightstogivethem(seeFigure1andTable2).ToentertheweightsinTable2weusethelowerpartofthedialogueboxinFigure3.


Figure3:Dialogboxforconductingplannedcomparisons

Entering Contrast 1

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).

Figure4:ContrastsdialogboxcompletedforthethreecontrastsoftheSuperherodata

Onceyouhaveinputtheweightingsyoucanchangeorremoveanyoneofthembyusingthemousetoselecttheweightthatyouwanttochange.Theweightwill thenappear inthebox labelledCoefficientswhereyoucantype inanewweightandthenclickon .Alternatively,youcanclickonanyoftheweightsandremoveitcompletelybyclicking

.Underneath theweights SPSS calculates the coefficient total, should equal zero (If you’ve used the correct


weights).Ifthecoefficientnumberisanythingotherthanzeroyoushouldgobackandcheckthatthecontrastsyouhaveplannedmakesenseandthatyouhavefollowedtheappropriaterulesforassigningweights.

Entering Contrast 2

Onceyouhavespecifiedthefirstcontrast,clickon .Theweightingsthatyouhavejustenteredwilldisappearandthedialogueboxwillnowreadcontrast2of2.Theweightsforcontrast2shouldbe:1(Supermangroup),-1(Spidermangroup),0(Hulkgroup)and0(NinjaTurtlegroup).Wecanspecifythiscontrastasbefore.RememberingthatthefirstweightweenterwillbefortheSupermangroup,wemustenterthevalue1asthefirstweight.ClickintheboxlabelledCoefficientswiththemouseandthentype‘1’andclickon .Next,weneedtoinputtheweightfortheSpidermangroupbyclickingintheboxlabelledCoefficientsandthentyping‘-1’andclickingon .ThentheHulkgroup:clickintheboxlabelledCoefficients,type‘0’andclickon .Finally,weneedtoinputtheweightfortheNinjaTurtlegroupbyclickingintheboxlabelledCoefficientsandthentyping‘0’andclickingon (seeFigure4,middle).

Entering Contrast 3

Clickon ,andyoucanentertheweightsforthefinalcontrast.Thedialogueboxwillnowreadcontrast3of3.Theweightsforcontrast3shouldbe:0(Supermangroup),0(Spidermangroup),1(Hulkgroup)and-1(NinjaTurtlegroup).Wecanspecifythiscontrastasbefore.RememberingthatthefirstweightweenterwillbefortheSupermangroup,wemustenterthevalue0asthefirstweight.ClickintheboxlabelledCoefficients,type‘0’andclickon .Next,weinputtheweightfortheSpidermangroupbyclickingintheboxlabelledCoefficientsandthentyping‘0’andclickingon

.ThentheHulkgroup:clickintheboxlabelledCoefficients,type‘1’andclickon .Finally,weinputtheweight for theNinjaTurtlegroupbyclicking in thebox labelledCoefficients, typing ‘-1’andclickingon (seeFigure4,right).

Whenalloftheplannedcontrastshavebeenspecifiedclickon toreturntothemaindialoguebox.

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:

Field(2013)recommends:

® WhenyouhaveequalsamplesizesandyouareconfidentthatyourpopulationvariancesaresimilarthenuseR-E-G-W-QorTukeybecausebothhavegoodpowerandtightcontrolovertheTypeIerrorrate.

® If sample sizes are slightlydifferent thenuseGabriel’s procedurebecause it has greaterpower,butifsamplesizesareverydifferentuseHochberg’sGT2.

® IfthereisanydoubtthatthepopulationvariancesareequalthenusetheGames-Howellprocedurebecausethisseemstogenerallyofferthebestperformance.

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.


Figure5:Dialogueboxforspecifyingposthoctests

Options Theadditionaloptionsforone-wayANOVAarefairlystraightforward.Thedialogboxtoaccesstheseoptionscanbeobtainedbyclickingon

. 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.

Output from One-Way ANOVA

Descriptive Statistics

Figure7showsthe‘meansplot’thatweaskedSPSSfor.Afewimportantthingstonoteare:


û Itlookshorrible.

û SPSShasscaledthey-axistomakethemeanslookasdifferentashumanlypossible.Thinkbacktoweek1whenwe learnt that itwas verybad to scale your graph tomaximise groupdifferences – SPSShasnot readmyhandoutJ

û Therearenoerrorbars:thegraphjustisn’tveryinformativebecausewearen’tgivenconfidenceintervalsforthemean.

Pastesomethinglikethisintooneofyourlabreportsandwatchyourtutorrecoilinhorrorandyourmarkplummet!Themoralis,neverletSPSSdoyourgraphsforyouJ

® Usingwhatyoulearntinweekonedrawanerrorbarchartofthedata.(YourchartshouldideallylooklikeFigure8).

Figure7:CrapgraphautomaticallyproducedbySPSS Figure8:Nicelyeditederrorbarchartoftheinjurydata

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.

® Thinkbacktothestartofthemodule,whatdoesaconfidenceintervalrepresent?

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).

Thetableofdescriptivestatisticsverifieswhatthegraphshows:thattheinjuriesforSupermancostumesweremostsevere,andforninjaturtlecostumeswereleastsevere.Thistablealsoprovidestheconfidenceintervalsuponwhichtheerrorbarswerebased.


Output1

Levene's test

Levene’stest(thinkbacktoyourlectureandhandoutonbias)teststhenullhypothesisthatthevariancesofthegroupsarethesame.InthiscaseLevene’stestistestingwhetherthevariancesofthefourgroupsaresignificantlydifferent.

® 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).

Output2

Forthesedatathevariancesarerelativelysimilar(hencethehighprobabilityvalue).Typicallypeoplewouldinterpretthisresultasmeaningthatwecanassumehomogeneityofvariance(becausetheobservedp-valueof.459isgreaterthan.05).However,oursamplesizeisfairlysmall(somegroupshadonly6participants).Thesmallsample(pergroup)willlimitthepowerofLevene’stesttodetectdifferencesbetweenthevariances.Wecouldalsolookatthevarianceratio.ThesmallestvariancewasfortheNinjaTurtlecostume(8.162=66.59)andthelargestwasforSupermancostumes(17.852=318.62).Theratioofthesevaluesis318.62/66.59=4.78.Inotherwords,thelargestvarianceisalmostfivetimelargerthanthesmallestvariance.Thisdifferenceisquitesubstantial.Therefore,wemightreasonablyassumethatvariancesarenothomogenous.

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).

The Main ANOVA

Output3showsthemainANOVAsummarytable.TheoutputyouwillseeisthetableatthebottomofOutput3,thisisamorecomplicatedtablethanasimpleANOVAtablebecauseweaskedforatrendanalysisofthemeans(byselectingtheselectthepolynomialoption inFigure3).ThetopofOutput3showswhatyouwouldseeifyouhadn’tdonethetrendanalysisjustnotethattheBetweenGroups,WithinGroupsandTotalrowsinbothtablesarethesame—it’sjustthatthebottomtabledecomposestheBetweenGroupseffectintolinearandquadratictrends.


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.

Output3

Trend Analysis

Thetrendanalysisbreaksdowntheexperimentaleffectintothatwhichcanbeexplainedbybothalinearandquadraticrelationship.It’sconfusingthatforbothtrendsyougetthreerows(labelledUnweighted,WeightedandDeviation)2butfocusontherowlabelledWeighted3.First,let’slookatthelinearcomponent.Thiscomparisontestswhetherthemeanschangeacrossgroupsinalinearway.Thesumofsquaresandmeansquaresaregiven,butthemostimportantthingstonotearethevalueoftheF-ratioandthecorrespondingsignificancevalue.ForthelineartrendtheF-ratiois23.44andthisvalueissignificantatap<.001levelofsignificance.LookingatFigure8wecaninterpretthistrendasthemean2Ifyouhaveequalsamplesizesyoujustgettwoversions:onelabeledContrastandtheotherlabeledDeviation.Withunequalsamplesizes,SPSSproducesanunweightedandweightedversionofthecontrast.Theweightedversionfactorsinthedifferentsamplessizes.3IfyouhaveequalsamplesizesthenfocusontherowlabelledContrast.


severityofinjuriesdecreasedproportionatelyacrossthefoursuperherocostumes.Movingontothequadratictrend,thiscomparisonistestingwhetherthepatternofmeansiscurvilinear(i.e.isrepresentedbyacurvewithonebendin).Figure8doesnotparticularlysuggestthatthemeanscanberepresentedbyacurveandtheresultsforthequadratictrendbearthisout.TheF-ratioforthequadratictrendisnon-significant,F(1,26)=0.96,p=.336.

Robust tests

Wesawearlieronthattheassumptionofhomogeneityofvariancewasquestionable(atleastintermsofthevarianceratio).Therefore,weshouldinspectOutput4,whichhastheBrown-ForsytheandWelchversionsoftheF-ratio.Ifyoulookatthistableyoushouldnoticethatbothteststatisticsarestillhighlysignificant(thevalueofSig.inthetableislessthan.05).Therefore,wecansaythattherewasasignificanteffectofthecostumewornontheseverityofinjuries.

Output4

Tofindoutwherethedifferencesbetweengroupslie,youneedtocarryoutfurthercomparisons.Therearetwochoicesofcomparison:thefirstisaplannedcomparison,inwhichpredictionsaboutwhichgroupswilldifferweremadepriortotheexperiment,andposthoctests forwhichallgroupsarecomparedbecausenopriorhypothesesaboutgroupdifferencesweremade.Let’slookattheseinturn.

Output for Planned Comparisons

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.

® Contrast2:Bearinmindwhatwe’vediscussedbeforeonthismoduleaboutsamplesizeandsignificance.Thiseffectisquiteclosetosignificance(p=.057)andisbasedonasmallsample.Notealso,thatifwehadassumedequalvariancesthep-valueisbelowthe.05threshold.Itwouldbeparticularlyusefulheretolookattheeffectsize–infactifyoudid


theearlierselftestyou’llhavefoundthatd=1.53and1.26(usingthepooledvarianceestimate),whichisaverylargeeffect.ThereisclearlysomethinggoingonbetweentheSupermanandSpidermanconditionsthatisnotreflectedinthesignificancevaluebecauseoftheverysmallsample(rememberthatpisaffectedbythesamplesize).

Output5

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).

Next,theSpidermangroupiscomparedtoallothergroups.Thefirstcomparison(toSuperman)isidenticaltotheonethatwehavealreadylookedat.TheonlynewinformationisthecomparisonoftheSpidermangrouptotheHulk(p=.907,notsignificant)andTurtle(notsignificant,p=.136)groups.

Finally,theHulkgroupiscomparedtoallothergroups.Again,thefirsttwocomparisonsreplicateeffectsthatwehavealreadyseeninthetable,theonlynewinformationisthecomparisonoftheHulkgroupwiththeNinjaTurtlegroup(p=.650,notsignificant).

TherestofthetabledescribestheGames-HowelltestsandaquickinspectionrevealstwodifferencestotheconclusionsfoundwiththeGabrieltest:(1)theSupermanandHulkgroupsnolongerdiffersignificantly(p=.073insteadof.008);(2)theSpidermanandTurtlegroupsjustaboutdiffersignificantly(p=.050insteadof.136).Inthissituation,whatyouconcludedependsuponwhetheryouthinkit’sreasonabletoassumethatpopulationvariancesdiffer.Wecanusethesamplesasaguide.Table3showsthevariancesineachgroup,andalsothevarianceratiosforallpairsofgroups(i.e.thelargerofthetwovariancesdividedbythesmaller).Notethatallbutonevarianceratiosareclosetoorabove2(indicatingheterogeneity).Mostimportantlet’slookatthetwocomparisonswheretheGames-HowelltestdifferstotheGabrieltest:

1. Supermanvs.Hulk:thevarianceratioisbelow2(althoughclose)sowemightchoosetoreportGabriel’stestandacceptasignificantdifference.

2. Spidermanvs.Turtle:thevarianceratioisabove2sowemightchoosetoreporttheGames-Howelltestandacceptasignificantdifference.


Themainpoint to takehome is thatdataanalysiscan leadyou intocomplexsituations inwhichyouhave tomakeinformeddecisionsabouthowtointerpretthedata.

Output6

Table3:VariancesandvarianceratiosforallgroupsintheSuperherodata

Variance VarianceRatiotoSpiderman

VarianceRatiotoHulk

VarianceRatiotoNinjaTurtle

Superman 318.66 2.14 1.78 4.79

Spiderman 149.13 0.83 2.24

Hulk 179.13 2.69

NinjaTurtle 66.50


Reporting Results from One-Way Independent ANOVA WhenwereportanANOVA,wegivedetailsoftheF-ratioandthedegreesoffreedomfromwhichitwascalculated.FortheexperimentaleffectinthesedatatheF-ratiowasderivedfromdividingthemeansquaresfortheeffectbythemeansquaresfortheresidual.Therefore,thedegreesoffreedomusedtoassesstheF-ratioarethedegreesoffreedomfortheeffectofthemodel(dfM=3)andthedegreesoffreedomfortheresidualsofthemodel(dfR=26):

ü Therewasasignificanteffectofthecostumewornontheseverityofinjuriessustained,F(3,26)=8.32,p<.001.

NoticethatthevalueoftheF-ratioisprecededbythevaluesofthedegreesoffreedomforthateffect.However,inthisexample,wehadsomeevidencethathomogeneityofvariancewasviolated,sowemightreportthealternativestatistics(whichcanbedoneinthesameway).NoticethatthedegreesoffreedomhavechangedtoreflecthowtheF-ratiowascalculated,andthatthevalueofFitselfisdifferent.Notealsothatunlessp<.001itisgoodpracticetoreporttheexactp-value;thisisbecauseit ismoreinformativetoknowtheexactvalueofpthantoknownonlythatitwasbiggerorsmallerthan.05.TheAPArecommendsreportingexactp-values.

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.

SELF-TEST:ComputeCohen’sdfortheeffectofSpidermancostumesoninjuryseveritycomparedtoHulkandNinjaTurtlecostumes,andbetweentheHulkandNinjaTurtleconditions.Tryusingboththestandarddeviationofthecontrol(thenon-Supermancostume)andalsothepooledestimate.(Answersattheendofthehandout)

Posthoctestsareusuallyreportedjustwithp-valuesandeffectsizes:

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.

Guided Task TheUniversitywasinterestedontheeffectsofdifferentstatisticsclassesonaggressioninundergraduates.Followingoneofthreetypesofstatisticclass(workshops,lecturesandanexam)6studentswereplacedinarelaxationroominwhichtherewasadartboardwiththefaceoftheirlecturerpinnedtoit.Thenumberofdartsthateachstudentthrewatthedartboardwasmeasured(Table4).


Table4:Datafortheaggressionexample

Class Darts

Workshop 10

8

15

26

28

12

Lecture 12

6

30

24

18

13

Exam 18

40

35

29

30

25

® EnterthedataintoSPSS.

® SavethedataontoadiskinafilecalledStatsClass.sav

® Drawanerror-barchartofthedata.

® Carryoutone-wayANOVAtofindoutwhetherthetypeofstatisticsclassaffectsaggression.

® Extraactivityifyouhavetime:Usewhatyoulearntinweek1toscreenthedata.

Arethedatanormallydistributed?(ReportsomerelevantstatisticsinyouranswerinAPAformat)

YourAnswer:


Basedontheerrorbarchart,whichgroups(ifany)doyouthinkwillbesignificantlydifferentandwhy?

YourAnswer:

Whatistheassumptionofhomogeneityofvariance?Hasthisassumptionbeenmet(quoterelevantstatisticsinAPAformat)?Iftheassumptionhasnotbeenmetwhatactionshouldbetaken?

YourAnswer:

Doesthetypeofstatisticsclasssignificantlyaffectaggression(quoterelevantstatisticsinAPAformat)?WhatdoestheF-valuerepresent?

YourAnswer:

Whatdoesthepvaluereportedabovemean?

YourAnswer:


Unguided Task 1 Inthe lectures(andmybook)weusedanexampleaboutthedrugViagra.SupposewetestedthebeliefthatViagraincreaseslibidobytakingthreegroupsofpeopleandadministeringonegroupwithaplacebo(suchasasugarpill),onegroupwithalowdoseofViagraandonewithahighdose.Thedependentvariablewasanobjectivemeasureoflibido.Inthelectureweestablishedtwousefulplannedcomparisonsthatwecoulddototestthesehypotheses:

1. HavingadoseofViagrawillincreaselibidocomparedtonothavingany.2. HavingahighdoseofViagrawillincreaselibidomorethanalowdose.

® EnterthedataintoSPSScreatingtwovariables:onecalleddosewhichspecifieshowmuchViagrathepersonwasgivenandonecalledlibidowhichindicatestheperson’slibidooverthefollowingweek).YoucancodethevariabledoseanywayyouwishbutIrecommendsomethingsimplesuchas1=placebo,2=lowdoseand3=highdose.

® SavethedataontoadiskinafilecalledViagra.sav.

® Usingwhatwelearntinthelectures,doaone-wayANOVAtotestwhetherViagrahadasignificanteffectonlibidoandalsodefinecontraststotestthetwohypotheses.

® AcompleteanswercanbefoundinChapter11ofmytextbook(Field,2013)

Table5:DatashowinghowlibidodiffersafterdifferentdosesofViagra

Dose LibidoPlacebo

32114

LowDoseViagra

52423

HighDoseViagra 74536

Unguided Task 2 TheorganisersoftheRugbyworldcupwereinterestedwhethercertainteamsweremoreaggressivethanothers.Overthecourseofthecompetition,theynotedtheinjurypatternsofplayersintheEnglandsquadaftercertaingames.Thedependentvariablewasthenumberofinjuriessustainedbyeachplayerinamatch,andtheindependentvariablewastheteamtheyhadplayed.Differentplayerswereusedinthedifferentmatches(toavoidinjuriesfrompreviousmatchescarryingoverintonewmatches).

® EnterthedataintoSPSSandsavethefileasrugby.sav.

® Conductone-wayANOVAtoseewhetherthenumberofinjuriesinflicteddifferedacrosstherugbyteams.


® Conductplannedcomparisonstotestthesehypotheses:(1)Tongacausemoreinjuriesthanalloftheotherteams;(2)JapancausefewerinjuriesthanWalesandNewZealand;(3)WalesandNewZealandinflictsimilarnumbersofinjuries.

Table6:Rugbydata

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

Team Injuries

Japan

03531292074316

Wales

3040272540153046

NewZealand

1633253220545719

Tonga

5557555756535955


theyateperweekoverayearlongperiod.ThefirstgroupwasacontrolgroupandtheyhadnoSoyamealsatallperweek(i.e.noneinthewholeyear);thesecondgrouphad1Soyamealperweek(that’s52overtheyear);thethirdgrouphad4Soyamealsperweek(that’s208overtheyear)andthefinalgrouphad7Soyamealsaweek(that’s364overtheyear).Attheendoftheyear,alloftheparticipantsweresentawaytoproducesomespermthatIcouldcount(whenIsay‘I’,Imeansomeoneinalaboratoryasfarawayfrommeashumanlypossible).Dataarebelow(althoughinadifferentformattohowitshouldbeenteredintoSPSS).

® EnterthedataintoSPSS(bearinmindthattheyarenotenteredinthesamewayasthetablebelow).

® SavethedataontoadiskinafilecalledSperm.sav

® Arethedatanormallydistributed?

® Istheassumptionofhomogeneityofvariancemet?

® Carryoutone-wayANOVAtofindoutwhetherGenisteinaffectsspermcounts.

® Testforalineartrendanddoposthoctests.Whichgroupsdifferfromwhich?

® Answerscanbefoundonthecompanionwebsiteformytextbook(SmartAlexanswers)

Table7:Soyadata

NoSoyaSperm(Millions)

1SoyaMealSperm(Millions)

4SoyaMealsSperm(Millions)

7SoyaMealsSperm(Millions)

0.35 0.33 0.40 0.310.58 0.36 0.60 0.320.88 0.63 0.96 0.560.92 0.64 1.20 0.571.22 0.77 1.31 0.711.51 1.53 1.35 0.811.52 1.62 1.68 0.871.57 1.71 1.83 1.182.43 1.94 2.10 1.252.79 2.48 2.93 1.333.40 2.71 2.96 1.344.52 4.12 3.00 1.494.72 5.65 3.09 1.506.90 6.76 3.36 2.097.58 7.08 4.34 2.707.78 7.26 5.81 2.759.62 7.92 5.94 2.8310.05 8.04 10.16 3.0710.32 12.10 10.98 3.2821.08 18.47 18.21 4.11

Unguided Task 4 Peoplelovetheirmobilephones,whichisratherworryinggivensomerecentcontroversyaboutlinksbetweenmobilephoneuseandbraintumours.Thebasicideaisthatmobilephonesemitmicrowaves,andsoholdingonenexttoyourbrainforlargepartsofthedayisabitlikestickingyourbraininamicrowaveovenandselectingthe‘cookuntilwelldone’button.Ifwewantedtotestthisexperimentally,wecouldget6groupsofpeopleandstrapamobilephoneontheirheads(thattheycan’tremove).Then,byremotecontrol,weturnthephonesonforacertainamountoftimeeach


day.After6months,wemeasurethesizeofanytumour(inmm3)closetothesiteofthephoneantennae(justbehindtheear).Thesixgroupsexperienced0,1,2,3,4or5hoursperdayofphonemicrowavesfor6months.Dataarebelow(althoughinadifferentformattohowitshouldbeenteredintoSPSS).(ThisexampleisfromField&Hole,2003,sothereisaverydetailedanswerinthereifyou’reinterested).

® EnterthedataintoSPSS(bearinmindthattheyarenotenteredinthesamewayasthetablebelow).

® SavethedataontoadiskinafilecalledTumour.sav

® Arethedatanormallydistributed?

® Istheassumptionofhomogeneityofvariancemet?

® Carryoutone-wayANOVAtofindoutwhethermobilephoneusecausesbraintumours.Testforalineartrend.

® Answerscanbefoundonthecompanionwebsiteformytextbook(SmartAlexanswers)

Table8:Mobilephonedata

0Hoursperday

1Hoursperday

2Hoursperday

3Hoursperday

4Hoursperday

5Hoursperday

0.02 0.77 1.29 4.31 4.65 5.170.00 0.74 1.08 2.47 5.16 5.030.01 0.22 1.07 2.04 4.06 6.140.01 0.94 0.48 3.32 4.61 4.900.04 0.62 1.26 3.18 5.32 4.650.04 0.33 0.52 2.24 4.84 3.880.01 0.47 0.64 3.80 5.20 5.250.03 0.78 1.69 2.86 5.40 2.700.00 0.76 1.85 2.58 3.04 5.310.01 0.03 1.22 4.09 4.73 5.360.01 0.93 1.69 3.51 5.18 5.430.02 0.39 2.34 3.02 5.09 4.430.03 0.62 1.54 3.63 5.65 4.830.02 0.48 1.87 4.14 3.83 4.130.01 0.15 0.83 2.82 4.88 4.050.04 0.72 0.98 1.77 5.21 3.670.03 0.38 0.98 2.56 5.30 4.430.02 0.27 1.39 3.59 6.05 4.620.01 0.00 1.63 2.64 4.13 5.110.01 0.69 0.87 1.84 5.42 5.55

Multiple Choice Test

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.


Effect Size Answers Table9:Cohen’sdforthesuperherodatagroups

Mean1 Mean2 d(control) d(pooled)

SupermanvSpiderman 60.33 41.63 1.53 1.26

SupermanvHulk 60.33 35.38 1.86 1.62

SupermanvNinja 60.33 26.25 4.18 2.60

SpidermanvHulk 41.63 35.38 0.47 0.49

SpidermanvNinja 41.63 26.25 1.88 1.48

HulkvNinja 35.38 26.25 1.12 0.82

References

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.

ArchivesofDiseaseinChildhood,92(3),242-243.doi:10.1136/adc.2006.109793Field,A.P.(2013).DiscoveringstatisticsusingIBMSPSSStatistics:Andsexanddrugsandrock'n'roll(4thed.).London:

Sage.

Terms of Use Thishandoutcontainsmaterialfrom:

Field,A.P.(2013).DiscoveringstatisticsusingSPSS:andsexanddrugsandrock‘n’roll(4thEdition).London:Sage.

ThismaterialiscopyrightAndyField(2000-2016).

This document is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 InternationalLicense,basicallyyoucanuseitforteachingandnon-profitactivitiesbutnotmeddlewithitwithoutpermissionfromtheauthor.

http://creativecommons.org/licenses/by-nc-nd/4.0/

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