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©Prof.AndyField,2016 www.discoveringstatistics.com Page1
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.
©Prof.AndyField,2016 www.discoveringstatistics.com Page2
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).
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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
©Prof.AndyField,2016 www.discoveringstatistics.com Page4
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
©Prof.AndyField,2016 www.discoveringstatistics.com Page5
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.
©Prof.AndyField,2016 www.discoveringstatistics.com Page6
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
©Prof.AndyField,2016 www.discoveringstatistics.com Page7
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.
©Prof.AndyField,2016 www.discoveringstatistics.com Page8
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:
©Prof.AndyField,2016 www.discoveringstatistics.com Page9
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.
©Prof.AndyField,2016 www.discoveringstatistics.com Page10
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.
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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.
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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
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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.
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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
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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).
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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:
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Basedontheerrorbarchart,whichgroups(ifany)doyouthinkwillbesignificantlydifferentandwhy?
YourAnswer:
Whatistheassumptionofhomogeneityofvariance?Hasthisassumptionbeenmet(quoterelevantstatisticsinAPAformat)?Iftheassumptionhasnotbeenmetwhatactionshouldbetaken?
YourAnswer:
Doesthetypeofstatisticsclasssignificantlyaffectaggression(quoterelevantstatisticsinAPAformat)?WhatdoestheF-valuerepresent?
YourAnswer:
Whatdoesthepvaluereportedabovemean?
YourAnswer:
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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.
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® 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
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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
©Prof.AndyField,2016 www.discoveringstatistics.com Page21
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.
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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.