Mixed Factorial ANOVA

©Prof.AndyField,2016 www.discoveringstatistics.com Page1

Mixed Factorial ANOVA

Introduction ThefinalANOVAdesignthatweneedtolookatisoneinwhichyouhaveamixtureofbetween-groupandrepeatedmeasuresvariables.Itshouldbeobviousthatyouneedatleasttwoindependentvariablesforthistypeofdesigntobepossible,but youcanhavemore complex scenarios too (e.g. twobetween-groupandone repeatedmeasures,onebetween-groupandtworepeatedmeasures,oreventwoofeach).SPSSallowsyoutotestalmostanydesignyoumightwanttoofvirtuallyanydegreeofcomplexity.However,interactiontermsaredifficultenoughtointerpretwithonlytwovariablessoimaginehowdifficulttheyareifyouinclude,forexample,four!

Two-Way Mixed ANOVA using SPSS Aswehaveseenbefore,thenameofanyANOVAcanbebrokendowntotellusthetypeofdesignthatwasused.The‘two-way’partofthenamesimplymeansthattwoindependentvariableshavebeenmanipulatedintheexperiment.The ‘mixed’ part of the name tells us that the same participants have been used tomanipulate one independentvariable,butdifferentparticipantshavebeenusedwhenmanipulatingtheother.Therefore,thisanalysisisappropriatewhenyouhaveonerepeated-measuresindependentvariables,andonebetween-groupindependentvariables.

An Example: The Real Santa AllisnotwellinLapland.Theorganisation‘StatisticiansHateInterestingThings’haveexecutedtheirlongplannedCampaignAgainstChristmasbyabductingSanta.Spearheadedby theirevil leaderProfessorN.O.Life,achubbybeardedmanwithapenchantforredjumperswhogetsreallyenviousofotherchubbybeardedmenthatpeopleactuallylike,andhiscrackS.W.A.T.team(StatisticiansWithAutisticTendencies),theyhavetakenSantafromhishomeandhavebrickedhimupbehindcopiesofaratherheavyandimmoveableStatstextbookwrittenbysomebrainlessgibboncalled‘Field’.

It’sChristmasEveandtheelvesareworried.Theelfleader,twallybliddle(don’tblameme,Ididn’tnamehim…)hasralliedhiselftroopsandusingRudolph’sincrediblypowerfulnose,theytrackeddownthebaseofS.H.I.T.andplannedtobreakdownthedoor.Theythenrealisedtheydidn’tknowwhatadoorwasandwentdownthechimney instead,much to thesurpriseofa roomfullofsweaty men with abacuses. Armed with a proof of Reimann’s Hypothesis1 they overcame thebemusedhuddleofstatisticiansandlocatedSanta.Theyslowlypeeledawaythetowerofbooks.Onebyone,thebarriercamedownuntiltheycouldseehetipofaredhat,theycouldhearaheartychuckle.

ImaginetheirsurpriseasthelastbookwasremovedrevealingthreeidenticalSantas…Allthreewerestaticasifhypnotised.TheStatisticianscackledwithjoy,andProfessorLifeshouted“haha,Santa is transfixed,held inacatatonicstatebythepowerofstatisticalequations.ItisstatisticallyimprobablethatyouwillidentifytherealSantabecauseyouhavebuta1in3chanceofguessingcorrectly.Theoddsarestackedagainstyou,youpointyeared,funloving,non-significantelves;soconfidentamIthatyouwon’tidentifytherealSanta,thatifyoucanIwillreleasehimandrenouncemylifeofnumberstojointheelfinclan”

Somehow,theyhadtoidentifythe‘real’SantaandreturnhimtoLaplandintimetodeliverthe Christmas presents, what should they do? They decided that each elf in turn wouldapproachallthreeoftheSantas(incounterbalancedorderobviously…)andwouldstrokehis

1Thereiscurrentlya$1millionprizeonoffertoproveReimann’shypothesis,whichforreasonsIwon’tboreyouwithisaveryimportanthypothesis—theproofofwhichhaseludedthegreatestmathematicians(includingReimann)ofthelast150yearsorso.Anelfpoppingoutofachimneywithproofof this theorywoulddefinitelysendaroomfullofmathematiciansintoapoplexy.


beard,sniffhimandticklehimunderthearms.HavingdonethistheelfwouldgivetheSantaaratingfrom0(definitelynottherealSanta)to10(definitelytherealSanta).Justtomakedoublysure,theydecidedtoenlistthehelpofRudolphandsomeofhisfellowreindeerswho’dbeenenjoyinganicebucketofwateroutside.TheseReindeershaveparticularlysensitivetastebudsandeachonelickedthethreeSantas(whichtheElvescouldn’tdobecauseofElf&Safetyregulations…)andliketheelvesgaveeachSantaaratingfrom0to10basedonhistaste.

Table1:DatafortheSantaexample

Rater RatingsofSanta1 RatingsofSanta2 RatingsofSanta3

Elves 1 3 1

2 5 3

4 6 6

5 7 4

5 9 1

6 9 3

Reindeer 1 10 2

4 8 1

5 7 3

4 9 2

2 10 4

5 10 2

Entering the Data

TheindependentvariablesweretheSantathatwasbeingassessed(Santa12or3)andwhethertheratingwasmadebyanelforareindeer.Thedependentvariablewastheelf’sratingoutof10.

ToenterthesedataintoSPSSweusethesameprocedureastherepeatedmeasuresANOVAthatwecameacrosslastweek,exceptthatwealsoneedavariable(column)thatcodeswhetherthehelperwasanelforareindeer.

® LevelsofrepeatedmeasuresvariablesgoindifferentcolumnsoftheSPSSdataeditor.

® Datafromdifferentpeoplegoindifferentrowsofthedataeditor,therefore,levelsofbetween-groupvariablesgoinasinglecolumn(acodingvariable).

Therefore,separatecolumnsshouldrepresenteachlevelofarepeatedmeasuresvariableandafourthcolumnshouldbemadewithnumbersrepresentingwhethertheraterwasanelforareindeer.So,createacolumnandcallitrater.Usethevalue1torepresentelves,and2torepresentreindeer(andremembertochangethe‘values’propertysothatweknowwhatthesenumbersrepresent,andtochangethe‘measure’propertyto‘nominal’sothatSPSSknowsthatraterisacategoricalvariable.

ü SavethesedatainafilecalledTheRealSanta.sav

The Main Analysis

ToconductanANOVAusingarepeatedmeasuresdesign,selectthedefinefactorsdialogboxbyfollowingthemenupath


Figure1:DefineFactorsdialogboxforrepeatedmeasuresANOVA

IntheDefineFactorsdialogbox,youareaskedtosupplyanameforthewithin-subject(repeated-measures)variable.InthiscasetherepeatedmeasuresvariablewastheSantathattheElves/Reindeertested,soreplacethewordfactor1withthewordSanta.Thenameyougivetotherepeatedmeasuresvariablecannothavespaces.Whenyouhavegiventherepeatedmeasuresfactoraname,youhavetotellthecomputerhowmanylevelsthereweretothatvariable(i.e.howmanyexperimentalconditionstherewere).Inthiscase,therewere3differentSantasthattheElves/Reindeershadtorate,sowehavetoenterthenumber3intotheboxlabelledNumberofLevels.Clickon toaddthisvariabletothelistofrepeatedmeasuresvariables.ThisvariablewillnowappearinthewhiteboxatthebottomofthedialogboxandappearsasSanta(3).Ifyourdesignhasseveralrepeatedmeasuresvariablesthenyoucanaddmorefactorstothelist.Whenyouhaveenteredalloftherepeatedmeasuresfactorsthatweremeasuredclickon togototheMainDialogBox.

TheMaindialogboxhasaspacelabelledwithinsubjectsvariablelistthatcontainsalistof3questionmarksproceededbyanumber.Thesequestionmarksareforthevariablesrepresentingthe3 levelsofthe independentvariable.Thevariablescorrespondingtotheselevelsshouldbeselectedandplacedintheappropriatespace.Wehaveonly3variablesinthedataeditor,soitispossibletoselectallthreevariablesatonce(byclickingonthevariableatthetop,holdingthemousebuttondownanddraggingdownovertheothervariables).Theselectedvariablescanthenbetransferredbydraggingthemorclickingon .

Whenallthreevariableshavebeentransferred,youcanselectvariousoptionsfortheanalysis.Thereareseveraloptionsthatcanbeaccessedwiththebuttonsatthebottomofthemaindialogbox.Theseoptionsaresimilartotheoneswehavealreadyencountered.

Sofartheprocedurehasbeensimilartoaone-wayrepeatedmeasuresdesign(lastweek).However,wehaveamixeddesignhere,andsowealsoneedtospecifyourbetween-group factoraswell.Wedo thisbyselecting rater in thevariableslistanddraggingittotheboxlabelledBetween-SubjectsFactors(orclickon ).ThecompleteddialogboxshouldlookexactlylikeFigure3.I’vealreadydiscussedtheoptionsforthebuttonsatthebottomofthisdialogbox,soI’lltalkonlyabouttheonesofparticularinterestforthisexample.


Figure2:MaindialogboxforrepeatedmeasuresANOVA

Figure3

Post Hoc Tests

ThereisnoproperfacilityforproducingposthoctestsforrepeatedmeasuresvariablesinSPSS!However,consultyourhandoutfromlastweektotellyouaboutusingBonferronicorrectedt-tests.

Graphing Interactions

Whentherearetwoormorefactors,theplotsdialogboxisaconvenientwaytoplotthemeansforeachlevelofthefactors.Thisplotwillbeusefulforinterpretingthemeaningoftheinteractioneffects.Toaccessthisdialogboxclickon

.Selectsanta fromthevariables listontheleft-handsideofthedialogboxanddragittothespacelabelledHorizontalAxis (or clickon ). In the space labelledSeparate Linesweneed toplace the remaining independentvariable:Rater. It isdowntoyourdiscretionwhichwayroundthegraph isplotted.Whenyouhavemovedthetwoindependentvariablestotheappropriatebox,clickon andthisinteractiongraphwillbeaddedtothelistatthe


bottomofthebox(seeFigure4).Whenyouhavefinishedspecifyinggraphs,clickon toreturntothemaindialogbox.

Figure4

Additional Options

Thefinaloptions,thathaven’tpreviouslybeendescribed,canbeaccessedbyclicking inthemaindialogbox.Theoptionsdialogbox(Figure5)hasvarioususefuloptions.Youcanaskfordescriptivestatistics,whichwillprovidethemeans, standard deviations and number of participants for each level of the independent variable. The option forhomogeneityofvariancetestsisactivebecausethereisabetweengroupfactorandweshouldselectthistogetLevene’stest(seeyourhandoutonBiasfromweek1).

Figure5:Optionsdialogbox

Perhapsthemostusefulfeatureisthatyoucangetsomeposthoctestsviathisdialogbox.Tospecifyposthoctests,selecttherepeatedmeasuresvariable(inthiscaseSanta)fromtheboxlabelledEstimatedMarginalMeans:Factor(s)


andFactorInteractionsanddragittotheboxlabelledDisplayMeansfor (orclickon ).Onceavariablehasbeentransferred, the box labelledComparemain effects ( )becomes active and you should select thisoption.Ifthisoptionisselected,theboxlabelledConfidenceintervaladjustmentbecomesactiveandyoucanclickon

toseeachoiceofthreeadjustmentlevels.ThedefaultistohavenoadjustmentandsimplyperformaTukeyLSDposthoctest(thisisnotrecommended).ThesecondoptionisaBonferronicorrection(recommendedforthereasonsmentionedabove),andthefinaloptionisaSidakcorrection,whichshouldbeselectedifyouareconcernedaboutthelossofpowerassociatedwithBonferronicorrectedvalues.

Whenyouhaveselectedtheoptionsofinterest,clickon toreturntothemaindialogbox,andthenclickon toruntheanalysis.

Descriptive statistics and other Diagnostics

SPSSOutput1showstheinitialdiagnosticsstatistics.First,wearetoldthevariablesthatrepresenteachleveloftheindependent variable. This box is usefulmainly to check that the variableswere entered in the correct order. Thefollowingtableprovidesbasicdescriptivestatisticsforthefourlevelsoftheindependentvariable.Fromthistablewecanseethat,onaverage,Santa2wasratedhighest(i.e.mostlikelytobetherealSanta)bybothelvesandreindeer,notethatthereindeersgaveparticularlyhighratingsofSanta2.

SPSSOutput1

Assessing Sphericity

Lastweekyouwere told thatSPSSproducesa test that looksatwhether thedatahaveviolatedtheassumptionofsphericity.Thenextpartoftheoutputcontainsinformationaboutthistest.

® Mauchly’stestshouldbenonsignificantifwearetoassumethattheconditionofsphericityhasbeenmet.

® If it is significantwemustuseGreenhouse-GeisserorHuyn-Feldt correcteddegreesoffreedomtoassesthesignificanceofthecorrespondingF.

SPSSOutput2showsMauchly’stestfortheSantadata,andtheimportantcolumnistheonecontainingthesignificancevale.Thesignificancevalueis.788,whichismorethan.05,sowecanrejectthehypothesisthatthevariancesofthedifferencesbetweenlevelsweresignificantlydifferent.Inotherwordstheassumptionofsphericityhasbeenmet.

SPSSOutput3showstheresultsofthe‘repeatedmeasures’partoftheANOVA(withcorrectedFvalues).Theoutputissplitintosectionsthatrefertoeachoftheeffectsinthemodelandtheerrortermsassociatedwiththeseeffects(abitlikethegeneraltableearlieroninthishandout).TheinterestingpartisthesignificancevaluesoftheF-ratios.Ifthesevaluesarelessthan.05thenwecansaythataneffectissignificant.Lookingatthesignificancevaluesinthetableitisclearthatbothoftheeffectsaresignificant.Theeffectofthebetweengroupvariable(Rater) isfoundinadifferenttable,whichwe’lllookatnextbecauseIwillexamineeachoftheeffectsthatweneedtoanalyseinturn.

Within-Subjects Factors

Measure: MEASURE_1

santa1santa2santa3

Santa123

DependentVariable

Descriptive Statistics

3.83 1.941 63.50 1.643 63.67 1.723 126.50 2.345 69.00 1.265 67.75 2.221 123.00 1.897 62.33 1.033 62.67 1.497 12

Type of RaterElfReindeerTotalElfReindeerTotalElfReindeerTotal

Rating of Santa # 1

Rating of Santa # 2

Rating of Santa # 3

Mean Std. Deviation N


SPSSOutput2

SPSSOutput3

The Effect of Rater

ThemaineffectofRaterislistedseparatelyfromtherepeatedmeasureeffectsinatablelabelledTestsof Between-Subjects Effects. Before looking at this table it is important to check the assumption ofhomogeneityofvarianceusingLevene’stest(seehandoutonexploringdata).SPSSproducesatablelistingLevene’stestforeachoftherepeatedmeasuresvariablesinthedataeditor,andweneedtolookforanyvariablethathasasignificantvalue.SPSSOutput4showsbothtables.Thetableshowing

Levene’stestindicatesthatvariancesarehomogeneous(i.e.moreorlessthesameforelvesandReindeers)foralllevelsoftherepeatedmeasuresvariables(becauseallsignificancevaluesaregreaterthan.05).HadanyofthevaluesbeensignificantthenitwouldhavecompromisedtheaccuracyoftheF-testforRaterandyoucouldconsidertransformingallofthedatatostabilize

thevariancesbetweengroups(seeField,2009,Chapter5oryourhandout).Luckilyforus,thiswasn’tthecaseforthesedata.ThesecondtableshowstheANOVAsummarytableforthemaineffectofRater,andthisrevealsanon-significanteffect(becausethesignificanceof.491ismorethanthestandard

cut-offpointof.05).

IfyourequestedthatSPSSdisplaymeansfortheRatereffectyoushouldscanthroughyouroutputandfindthetableinasectionheadedEstimatedMarginalMeans.SPSSOutput5isatableofmeansforthemaineffectofRaterwiththeassociatedstandarderrors.ThisinformationisplottedinFigure6.Itisclearfromthisgraphthatelvesandreindeer’sratingsweregenerallythesame(althoughrememberthatthisdoesnottakeintoaccountwhichSantawasbeingrated,itjusttakestheaverageratingacrossallSantas).

Mauchly's Test of Sphericityb

Measure: MEASURE_1

.948 .477 2 .788 .951 1.000 .500Within Subjects EffectSanta

Mauchly's WApprox.

Chi-Square df Sig.Greenhouse-Geisser Huynh-Feldt Lower-bound

Epsilona

Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables isproportional to an identity matrix.

May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed inthe Tests of Within-Subjects Effects table.

a.

Design: Intercept+Rater Within Subjects Design: Santa

b.

Tests of Within-Subjects Effects

Measure: MEASURE_1

174.056 2 87.028 36.946 .000174.056 1.902 91.519 36.946 .000174.056 2.000 87.028 36.946 .000174.056 1.000 174.056 36.946 .00018.167 2 9.083 3.856 .03818.167 1.902 9.552 3.856 .04118.167 2.000 9.083 3.856 .03818.167 1.000 18.167 3.856 .07847.111 20 2.35647.111 19.019 2.47747.111 20.000 2.35647.111 10.000 4.711

Sphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-bound

SourceSanta

Santa * Rater

Error(Santa)

Type III Sumof Squares df Mean Square F Sig.


® TheFratiorepresentstheratiooftheexperimentaleffectcomparedto‘error’.Therefore,whenitislessthat1itmeanstherewasmoreerrorthanvariancecreatedbytheexperiment.Assuch,anF<1isalwaysnon-significantandsometimespeoplereportthesestatisticsassimplyF<1.However,itismoreinformativetoreporttheexactvalue.

® Wecanreportthat‘therewasanon-significantmaineffectofrater,F(1,10)=0.51,p=.491.

® This effect tellsus that ifwe ignore the santabeing rated, elves and reindeers gave similarratings’.

SPSSOutput4

Elf Reindeer

Mea

n R

atin

g of

San

ta (0

-10)

0

2

4

6

8

10

SPSSOutput5 Figure6

The Effect of Santa

ThefirstpartofSPSSOutput3tellsustheeffectoftheSantathatwasevaluatedbytheelves/reindeers(i.e.didthethreeSanta’sreceivedifferentaverageratings?).Forthiseffectsphericitywasn’tanissueand

sowecanlookattheuncorrectedF-ratio,whichwassignificant.

YoucanrequestthatSPSSproducemeansofthemaineffects(seeField,2009)andifyoudothis,you’llfindthetableinSPSSOutput6inasectionheadedEstimatedMarginalMeans.SPSSOutput

6isatableofmeansforthemaineffectofSantawiththeassociatedstandarderrors.Thelevelsofthisvariablearelabelled1,2and3andwemustthinkbacktohowweenteredthevariabletosee

whichrowofthetablerelatestowhichcondition.WeenteredthisvariableintheorderofSantasrated(Santa#1,Santa#2andSanta#3).Figure7usesthisinformationtodisplaythemeansforeachSanta.Itis

clearfromthisgraphthatthemeanRatingwashighestforSanta#2,M=7.75(i.e.theelvesandReindeersweremostconfidentthatthisSantawastherealSanta).TheotherSanta’swereratedfairlyequally,Santa#1M=3.67andSanta#3M=2.67.Therefore,elvesandReindeersincombinationweremostconfidentthatSanta#2wastherealSanta.If

Levene's Test of Equality of Error Variancesa

.207 1 10 .6592.232 1 10 .1661.025 1 10 .335

Rating of Santa # 1Rating of Santa # 2Rating of Santa # 3

F df1 df2 Sig.

Tests the null hypothesis that the error variance of the dependentvariable is equal across groups.

Design: Intercept+Rater Within Subjects Design: Santa

a.

Tests of Between-Subjects Effects

Measure: MEASURE_1Transformed Variable: Average

793.361 1 793.361 180.082 .0002.250 1 2.250 .511 .491

44.056 10 4.406

SourceInterceptRaterError


Estimates

Measure: MEASURE_1

4.444 .495 3.342 5.5474.944 .495 3.842 6.047

Type of RaterElfReindeer

Mean Std. Error Lower Bound Upper Bound95% Confidence Interval


youasked forPostHoc tests (SPSSOutput7)youcouldgoon to say that ratingsofSanta#1andSanta#3didnotsignificantlydiffer,butratingsofSanta#2weresignificantlyhigherthanbothSanta#1andSanta#3.

Santa 1 Santa 2 Santa 3

Mea

n R

atin

g of

San

ta (0

-10)

0

2

4

6

8

10

SPSSOutput6 Figure7

SPSSOutput7

Thiseffectshouldbereportedas:

® TherewasasignificantmaineffectoftheSantabeingrated,F(2,20)=36.95,p<.001

® Thiseffecttellsusthatifweignorewhethertheratingcamefromanelforreindeer,theratingsofthethreeSantassignificantlydiffered.

® BonferronicorrectedposthoctestsshowedthatratingsofSanta#1andSanta#3didnotsignificantlydiffer (p= .45),but ratingsofSanta#2weresignificantlyhigher thanbothSanta#1andSanta#3(bothps<.001).

The Santa ´ Rater Interaction Effect

SPSSOutput3indicatedthattheorganismdoingtherating(elforreindeer)interactedwiththeSantathatwasbeingrated.Inotherwords,theratingsforthethreeSantasdifferedinsomewayinelvesandreindeers.ThemeansforallconditionscanbeseeninSPSSOutput8.


® TherewasasignificantSanter×Raterinteraction,F(2,20)=3.86,p=.038.ThiseffecttellsusthattheratingsofthethreeSantassignificantlydifferedinelvesandreindeers.

Estimates

Measure: MEASURE_1

3.667 .519 2.510 4.8237.750 .544 6.538 8.9622.667 .441 1.684 3.649

Santa123


Pairwise Comparisons

Measure: MEASURE_1

-4.083* .554 .000 -5.673 -2.4931.000 .643 .453 -.846 2.8464.083* .554 .000 2.493 5.6735.083* .676 .000 3.143 7.023

-1.000 .643 .453 -2.846 .846-5.083* .676 .000 -7.023 -3.143

(J) Santa231312

(I) Santa1

2

3

MeanDifference

(I-J) Std. Error Sig.a Lower Bound Upper Bound

95% Confidence Interval forDifferencea

Based on estimated marginal meansThe mean difference is significant at the .05 level.*.

Adjustment for multiple comparisons: Bonferroni.a.


WecanusethemeansinSPSSOutput8toplotaninteractiongraph,whichisessentialforinterpretingtheinteraction.Figure8showsthattheratingsofelvesandreindeerswerefairlysimilarforSanta#2andSanta#3(thebarsaresimilarheights).However,forSanta#2reindeerratingsseemtobequitealothigherthanelves.So,althoughbothelvesandreindeersseemedmostconfidentthatSanta#2wastherealSanta,reindeersappearedtobemoreconfidentaboutthisthanelves(theirratingswerehigher).Toverifytheinterpretationoftheinteractioneffect,wewouldneedtolookatsomecontrasts(seeField,2009,chapter13).

Number of Treats Consumed

Santa 1 Santa 2 Santa 3

Mea

n R

atin

g of

San

ta (0

-10)

0

1

2

3

4

5

6

7

8

9

10

Elf Reindeer

SPSSOutput8 Figure8

A Happy Ending?

Well, you’re probably wondering what happened? Both the elves and the reindeers were mostconfidentthatSanta#2wastherealSantaandthat’stheSantatheychose.Onhearingtheirchoice,ProfessorN.O. Life let out a blood-curdling cry, “Damn you, you pesky elves and reindeers” hewailed,“youhavecorrectlyidentifiedtherealSanta”.TruetohiswordhereleasedtherealSanta,andSantas#1and3removedtheirveryrealisticdisguisestorevealtwomorestatisticians.Santaemergedfromhisspell,“Thankyou,myfriends”hesaid“myheadwasfilledwithnumbers,Icouldn’tmove,Ifearedformymental‘elf”.“That’sOk”saidTwallybliddle,“Thekidsneedyoutodeliverthepresents,andwecan’tletthemdown”.“Buthowonearthdidyouidentifyme”said Santa? “Well”, said Twallybliddle “my elves tickled you andwe have big pointy super-sensitive ears that allowed us to distinguish your laugh from the impostors”; at which pointRudolphbuttedinandsaid“andmyreindeerfriendsandIhavebigtongues,andwetastedyou,andaseveryoneknowsthe Real Santa tastes of Christmas pudding2”, he added “the other two Tasted of B.O., sowe knew theymust bemathematiciansindisguise”.Santaroaredabigbellylaugh“HoHoHo”hesaid,“comeon,there’sworktobedone”.“Abitlesswork”addedTwallybliddle,“becausewehaveanextraelfhelper”,andashepassedapairoflittlegreenshortsandapointygreenhattoProfessorN.O.Lifeeveryonebegantolaugh,evenProfessorN.O.LifewhohadalreadybegancalculatinganequationtominimiseSanta’sflightpathonChristmasnight.TheyallgotbacktoLaplandintime,andalltheChildrengottheirpresentsandtheworldwasahappyplace.Hooray!

Three-Way Mixed ANOVA Aswehaveseenbefore,thenameofanyANOVAcanbebrokendowntotellusthetypeofdesignthatwasused.The‘three-way’partofthenamesimplymeansthatthreeindependentvariableshavebeenmanipulatedintheexperiment.The‘mixed’partofthenametellsusthatthesameparticipantshavebeenusedtomanipulateoneormoreindependentvariables,butdifferentparticipantshavebeenusedwhenmanipulatingoneormoreindependentvariables.Therefore,

2Youcanputittothetestinacoupleofweeksshouldyouwantto…

3. Type of Rater * Santa

Measure: MEASURE_1

3.833 .734 2.198 5.4696.500 .769 4.786 8.2143.000 .624 1.611 4.3893.500 .734 1.864 5.1369.000 .769 7.286 10.7142.333 .624 .944 3.723

Santa123123

Type of RaterElf

Reindeer



thisanalysisisappropriatewhenyouhaveoneormorerepeated-measuresindependentvariables,andoneormorebetween-groupindependentvariables.

An Example: Are Fairies or Elves More Susceptible to Christmas Treats? AtChristmaswenormallyleavetreatsforSantaClausandhishelpers(mincepies,aglassofsherryandabucketofwaterforRudolph).SantaClausnoticedthathewasstrugglingtodeliverallthepresentson Christmas Eve andwonderedwhether these treatsmight be slowing down his Elves3. He alsowantedtoseewhetheradifferenttypeofhelpermightbelesssusceptibletothesetreats.So,Santadida littleexperiment.He randomly selected9Elves fromhisworkforceandalso tookon9new

helpersfromtheFairykingdomandtimedhowlongittookeachofthemtodeliverthepresentsto5houses. About half of the elves/fairies were told that they could eat anymince pies or Christmas

puddingbutthattheymustnothaveanysherry,whiletheotherhalfweretoldtodrinksherrybutnottoeatanyfoodthatwasleftforthem.ThefollowingyearSantatookthesame9elvesandthesame9fairiesandagaintimedhowlongittookthemtodeliverpresentstothesame5housesasthepreviousyear.Thistime,however,theoneswhohaddrunksherrythepreviousyearwerebannedfromdrinkingitandtoldinsteadtoeatanymincepiesorChristmaspudding.Conversely,theoneswhohadeatentreatstheyearbeforeweretoldthisyearonlytodrinksherryandnottoeatanytreats.Assuch,overthetwoyearseachofthe9elvesand9fairieswastimedfortheirspeedofpresentdeliveryafter1,2,3,4and5dosesofsherry,andalsoafter1,2,3,4,&5dosesofmincepies.(Fairyimageisfromwww.drawingbusiness.com/)

® WhydoyouthinkSantagothalfoftheelves/fairiestodrinksherrythefirstyearandate treats the secondyear,while theotherhalfate treats the first yearanddranksherrythesecondyear?

Thinkaboutthedesignofthisstudyforamoment.Wehavethefollowingvariables:

® Treat:IndependentVariable1isthetreatthatwasconsumedbytheelves/fairiesandithas2levels:SherryorMincePies.

® Dose:IndependentVariable2isthedoseofthetreat(remembereachelf/fairyhadatreatatthefivehousestowhichtheydeliveredandsothetotalquantityconsumedincreasedacrossthehouses).Thisvariablehas5levels:house1,house2,house3,house4&house5.

® Helper:Independentvariable3waswhetherthehelperwasanelforafairy.

® Thedependentvariablewasthetimetakentodeliverthepresentstoagivenhouse(innanoseconds:elvesdeliververyquickly!)

Thesedatacould,therefore,beanalysedwitha2´5×2three-waymixedANOVA.AswithotherANOVAdesigns,thereisnolimittothenumberofconditionsforeachoftheindependentvariablesintheexperiment;however,inpractice,you’llfindthatyourparticipantsgetveryboredandinattentiveifthereweretoomanyconditions(althoughifyou’refeedingthemcakeandsherrythenperhapsnot…)!

Treat: Sherry MincePies/ChristmasPudding

Dose: 1 2 3 4 5 1 2 3 4 5Elf Trampy 11 11 30 15 42 8 15 17 28 18

Hughy 15 16 19 28 43 13 8 27 21 31Lardy 15 16 15 22 45 11 15 31 12 26Alchi 12 9 31 33 25 11 16 20 38 41Goody 7 14 23 29 39 14 7 9 13 42Pongo 8 14 21 34 36 10 12 17 24 29

3Hewasstartingtowonderifthealcoholandcakeswerebadfortheir’elf…(boy,amIgoingtogetsomemileageoutofthatpitifulattemptatajoke).


Bringitup 16 14 15 21 50 16 8 15 14 33Dio 12 15 24 22 35 12 16 12 30 43

Chunder 6 15 32 34 39 13 11 15 14 3Fairy Duncan 12 13 35 35 53 7 12 20 7 32

Ozzy 6 16 35 40 59 13 16 20 28 28Skinny 6 18 29 52 72 7 15 25 32 27Speedy 8 10 23 50 65 6 18 18 19 37Floaty 10 12 25 30 82 7 13 8 20 31Smiley 17 16 22 19 36 10 13 20 16 25Retch 6 8 12 50 46 12 9 24 18 40Wibble 9 10 35 38 36 12 14 23 23 33Condrick 9 15 22 37 72 12 14 22 14 19

Table2:Dataforexampletwo

Entering the Data

ToenterthesedataintoSPSSweusethesameprocedureasthepreviousexample,rememberingthegoldenrulesofthedataeditor:

® LevelsofrepeatedmeasuresvariablesgoindifferentcolumnsoftheSPSSdataeditor.

® Datafromdifferentpeoplegoindifferentrowsofthedataeditor,therefore,levelsofbetween-groupvariablesgoinasinglecolumn(acodingvariable).

Ifapersonparticipatesinallexperimentalconditions(inthiscaseallelvesandfairiesexperiencebothSherryandMincePiesinthedifferentdoses)theneachexperimentalconditionmustberepresentedbyacolumninthedataeditor.Inthisexperimenttherearetenexperimentalconditionsandsothedataneedtobeenteredintencolumns(so,theformatisidenticaltotheoriginaltableinwhichIputthedata).Youshouldcreatethefollowingtenvariablesinthedataeditor(variableview)withthenamesasgiven.Foreachone,youshouldalsoenterafullvariablenameforclarityintheoutput.

Sherry1 1DoseofSherrySherry2 2DosesofSherrySherry3 3DosesofSherrySherry4 4DosesofSherrySherry5 5DosesofSherryPie1 1MincePiePie2 2MincePiesPie3 3MincePiesPie4 4MincePiesPie5 5MincePies

Inaddition,weneedavariable(column)thatcodeswhetherthehelperwasanelforafairy.So,createacolumnandcall itHelper.Use thevalue1 to representelves, and2 to represent fairies (and remember to change the ‘values’propertysothatweknowwhatthesenumbersrepresent,andtochangethe‘measure’propertyto‘nominal’sothatSPSSknowsthatHelperisacategoricalvariable.

Oncethesevariableshavebeencreated,enterthedataasinTable2(above)andsavethefileontoadiskwiththenamesanta.sav.

Running the analysis

TheanalysisisruninthesamewayasforrepeatedmeasuresANOVA:accessthedefinefactorsdialogboxusethemenupath . In thedefine factors dialog box you are asked to supply anameforthewithin-subject(repeatedmeasures)variable.Inthiscasetherearetwowithin-subjectfactors:treat(Sherry


orMincePie)anddose(1,2,34or5doses).Replacethewordfactor1withthewordTreat.Whenyouhavegiventhisrepeatedmeasuresfactoraname,youhavetotellthecomputerhowmanylevelsthereweretothatvariable.Inthiscase,thereweretwotypesoftreat,sowehavetoenterthenumber2intotheboxlabelledNumberofLevels.Clickon

toaddthisvariabletothelistofrepeatedmeasuresvariables.ThisvariablewillnowappearinthewhiteboxatthebottomofthedialogboxandappearsasTreat(2).Wenowhavetorepeatthisprocessforthesecondindependentvariable.EnterthewordDoseintothespacelabelledWithin-SubjectFactorNameandthen,becausetherewerefivelevelsof this variable, enter thenumber5 into the space labelledNumberof Levels. Clickon to include thisvariable inthelistoffactors; itwillappearasDose(5).ThefinisheddialogboxisshowninFigure9.Whenyouhaveenteredbothofthewithin-subjectfactorsclickon togotothemaindialogbox.

Figure9:Define factorsdialogbox for factorialrepeatedmeasuresANOVA

Figure10

Themaindialogbox isessentiallythesameaswhenthere isonlyone independentvariable (seeprevioushandout)exceptthattherearenowtenquestionmarks(Figure10).AtthetopoftheWithin-SubjectsVariablesbox,SPSSstatesthattherearetwofactors:treatanddose.Intheboxbelowthereisaseriesofquestionmarksfollowedbybracketednumbers.Thenumbersinbracketsrepresentthelevelsofthefactors(independentvariables).

_?_(1,1) variablerepresenting1stleveloftreatand1stlevelofdose

_?_(1,2) variablerepresenting1stleveloftreatand2ndlevelofdose

_?_(1,3) variablerepresenting1stleveloftreatand3rdlevelofdose

_?_(1,4) variablerepresenting1stleveloftreatand4thlevelofdose

_?_(1,5) variablerepresenting1stleveloftreatand5thlevelofdose

_?_(2,1) variablerepresenting2ndleveloftreatand1stlevelofdose

_?_(2,2) variablerepresenting2ndleveloftreatand2ndlevelofdose

_?_(2,3) variablerepresenting2ndleveloftreatand3rdlevelofdose

_?_(2,4) variablerepresenting2ndleveloftreatand4thlevelofdose

_?_(2,5) variablerepresenting2ndleveloftreatand5thlevelofdose


Inthisexample,therearetwoindependentvariablesandsotherearetwonumbersinthebrackets.Thefirstnumberreferstolevelsofthefirstfactorlistedabovethebox(inthiscasetreat).Thesecondnumberinthebracketreferstolevelsofthesecondfactorlistedabovethebox(inthiscasedose).Aswithone-way repeatedmeasuresANOVA,youare required to replace thesequestionmarkswithvariablesfromthelistontheleft-handsideofthedialogbox.Withbetween-groupdesigns,inwhichcodingvariablesareused,thelevelsofaparticularfactorarespecifiedbythecodesassignedtotheminthedataeditor.However,inrepeatedmeasuresdesigns,nosuchcodingschemeisusedandsowedeterminewhichconditiontoassigntoalevelatthisstage.Forexample,ifweenteredsherry1into

the list first, thenSPSSwill treatsherryas the first leveloftreat,anddose1as the first levelof thedose variable.However,ifweenteredpie5intothelistfirst,SPSSwouldconsidermincepiesasthefirstleveloftreat,anddose5asthefirstlevelofdose

Itshouldbereasonablyobviousthatitdoesn’treallymatterwhichwayroundwespecifythetreats,butisveryimportantthatwespecifythedosesinthecorrectorder.Therefore,thevariablescouldbeenteredasfollows:

Sherry1 _?_(1,1)Sherry2 _?_(1,2)Sherry3 _?_(1,3)Sherry4 _?_(1,4)Sherry5 _?_(1,5)Pie1 _?_(2,1)Pie2 _?_(2,2)Pie3 _?_(2,3)Pie4 _?_(2,4)Pie5 _?_(2,5)

Whenthesevariableshavebeentransferred,thedialogboxshouldlookexactlylikeFigure11.ThebuttonsatthebottomofthescreenhavealreadybeendescribedfortheoneindependentvariablecaseandsoIwilldescribeonlythemostrelevant.

Figure11

Sofartheprocedurehasbeensimilartootherfactorialrepeatedmeasuresdesigns.However,wehaveamixeddesignhere,andsowealsoneedtospecifyourbetween-groupfactoraswell.Wedothisbyselectinghelperinthevariableslistanddraggingit(orclickingon )totransferittotheboxlabelledBetween-SubjectsFactors.ThecompleteddialogboxshouldlookexactlylikeFigure12.I’vealreadydiscussedtheoptionsforthebuttonsatthebottomofthisdialogbox,soI’lltalkonlyabouttheonesofparticularinterestforthisexample.


Figure12

Graphing Interactions

Theadditionofanextravariablemakes itnecessarytochooseadifferentgraphtotheone inthepreviousexample.Clickon toaccessthedialogboxinFigure13.PlacedoseintheslotlabelledHorizontalAxis:andtreatintheslotlabelledSeparateLine:,finally,placehelperintheslotlabelledSeparatePlots.Whenallthreevariableshavebeenspecified,don’tforgettoclickon toaddthiscombinationtothelistofplots.ByaskingSPSStoplotthedose´treat´helperinteraction,weshould get the interaction graph for dose and treat, but a separate version of this graphwill beproducedforelvesandfairies.Youcouldalsothinkaboutplottinggraphsforthetwowayinteractions(e.g.dose´treat,dose´helper,andtreat´helper).Asbefore,itisdowntoyourdiscretionwhich

wayroundthegraphisplotted,butitactuallymakessensethistimetohavedoseonthehorizontalaxisbecausethisvariableisordered(2dosesarebiggerthan1andsoon),andbyplacingthisvariableonthehorizontalaxisitwillenableustoeasilytracktheeffectondeliverytimesasthedoseincreases.Also,becausedosehaslotsoflevels(5),ifweaskedforthisvariabletobeplottedasseparatelinesthenwe’dhavealotoflines(veryconfusing),orworsestill,ifweaskedforthisvariabletobeplottedondifferentgraphswe’dendupwith5differentgraphs(theroadtomadness).However,ultimatelyit’suptoyou,whatIhavesuggestedissimplywhatIfindeasiest.Whenyouhavefinishedspecifyinggraphs,clickon toreturntothemaindialogbox.

Other Options

Asfarasotheroptionsareconcerned,youshouldselectthesameonesthatwerechosenforthepreviousexample.Itisworth selecting estimatedmarginalmeans for all effects (because these valueswill help you to understand anysignificanteffects).Whenalloftheappropriateoptionshavebeenselected,runtheanalysis.


Figure13

Interpreting the Output from Three-Way Mixed ANOVA

DescriptivesandMainAnalysisSPSSOutput9showstheinitialoutputfromthisANOVA.Thefirsttablemerelyliststhevariablesthathavebeenincludedfromthedataeditorandthelevelofeachindependentvariablethattheyrepresent.Thistableismoreimportantthanitmightseem,becauseitenablesyoutoverifythatthevariablesintheSPSSdataeditorrepresentthecorrect levels of the independent variables. The second table is a tableof descriptives and

providesthemeanandstandarddeviationforeachofthetenconditions.ThenamesinthistablearethenamesIgavethe variables in thedata editor (therefore, if youdidn’t give these variables full names, this tablewill look slightlydifferent).

SPSSOutput9

Within-Subjects Factors

Measure: MEASURE_1

sherry1sherry2sherry3sherry4sherry5pie1pie2pie3pie4pie5

Dose1234512345

Treat1

2

DependentVariable

Descriptive Statistics

11.33 3.674 99.22 3.563 9

10.28 3.675 1813.78 2.333 913.11 3.371 913.44 2.833 1823.33 6.538 926.44 7.812 924.89 7.169 1826.44 6.766 939.00 10.689 932.72 10.818 1839.33 7.089 957.89 16.412 948.61 15.542 1812.00 2.345 9

9.56 2.789 910.78 2.798 1812.00 3.674 913.78 2.539 912.89 3.197 1818.11 6.990 920.00 5.025 919.06 5.985 1821.56 9.139 919.67 7.433 920.61 8.140 1829.56 12.905 930.22 6.340 929.89 9.869 18

Type of HelperElfFairyTotalElfFairyTotalElfFairyTotalElfFairyTotalElfFairyTotalElfFairyTotalElfFairyTotalElfFairyTotalElfFairyTotalElfFairyTotal

Time Taken to Deliver Presents After 1 Sherry

Time Taken to Deliver Presents After 2 Sherries




Time Taken to Deliver Presents After 1 Mince Pie

Time Taken to Deliver Presents After 2 Mince Pies




Mean Std. Deviation N


Thedescriptivesareinterestinginthattheytellusthatthevariabilityamongscoreswasgreatestafter5Sherriesandwasgenerallyhigherwhensherrywasconsumed(comparethestandarddeviationsofthelevelsofthesherryvariablecomparedtothoseofthemincepievariable).Thestandarddeviationsalsolookbiggerforfairiescomparedtoelves.Thevaluesinthistablewillhelpuslatertointerpretthemaineffectsoftheanalysis.

SPSSOutput10showstheresultsofMauchly’ssphericitytestforeachofthethreerepeatedmeasureseffectsinthemodel(twomaineffectsandoneinteraction).Thesignificancevaluesofthesetests indicatethatthemaineffectofdoseandthedose×treatinteractionhasviolatedthisassumptionandsotheF-valuesforanyeffectinvolvingdoseorthedose×treatinteractiontermshouldbecorrected(seeearlierandchapter13ofField,2009).

® WhyistheSig.columnofMauchley’stestsometimesempty?

® The assumption of sphericity is all about the variance of the differences of differentconditionsbeingequal.Whenyouhaveonly two levelsof a variable (aswehavewithtreat),thenthereisonlyonesetofdifferences(sherrycomparedtomincepies),sothereisn’tanythingtocomparethevarianceofthesedifferencesagainst.

® Therefore,when youhaveonly two levels of a repeatedmeasures variable, Sphericitysimplyisn’tanissue.

® That’swhySPSSleavestheSigcolumnblank,becausesphericitycan’tbetested.

® ThemaineffectoftreathastwolevelssotheassumptionofsphericityisnotanissueandweneednotcorrectitsF-ratio.

SPSSOutput10

SPSSOutput11showstheresultsoftheANOVA(withcorrectedFvalues).Theoutputissplitintosectionsthatrefertoeachoftheeffectsinthemodelandtheerrortermsassociatedwiththeseeffects(abitlikethegeneraltableearlieroninthishandout).TheinterestingpartisthesignificancevaluesoftheF-ratios.Ifthesevaluesarelessthan.05thenwecansaythataneffectissignificant.Lookingatthesignificancevaluesinthetableitisclearthatalloftheeffectsaresignificant.Iwillexamineeachoftheseeffectsinturn.

The Effect of Helper

ThemaineffectofHelperislistedseparatelyfromtherepeatedmeasureeffectsinatablelabelledTestsofBetween-SubjectsEffects.BeforelookingatthistableitisimportanttochecktheassumptionofhomogeneityofvarianceusingLevene’s test (seehandoutonexploringdata). SPSSproducesa table listing Levene’s test foreachof the repeatedmeasuresvariablesinthedataeditor,andweneedtolookforanyvariablethathasasignificantvalue.SPSSOutput12showsbothtables.ThetableshowingLevene’stest indicatesthatvariancesarehomogeneous(i.e.moreor lessthesameforelvesandfairies)foralllevelsoftherepeatedmeasuresvariables(becauseallsignificancevaluesaregreaterthan.05)exceptfor5dosesofsherry.ThissignificantvaluecompromisestheaccuracyoftheF-testforHelper;however,becauseit’sonlyoneofthe10combinationsoflevelsthatissignificantwecanprobablynotlosesleepoverthisresult.You could consider transforming all of thedata though to stabilize the variancesbetween groups (see Field, 2009,

Mauchly's Test of Sphericityb

Measure: MEASURE_1

1.000 .000 0 . 1.000 1.000 1.000.099 33.321 9 .000 .583 .732 .250.198 23.332 9 .006 .648 .833 .250

Within Subjects EffectTreatDoseTreat * Dose

Mauchly's WApprox.

Chi-Square df Sig.Greenhouse-Geisser Huynh-Feldt Lower-bound

Epsilona

Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables isproportional to an identity matrix.

May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed inthe Tests of Within-Subjects Effects table.

a.

Design: Intercept+Helper Within Subjects Design: Treat+Dose+Treat*Dose

b.


Chapter5oryourhandout).ThesecondtableshowstheANOVAsummarytableforthemaineffectofHelper,andthisrevealsasignificanteffect(becausethesignificanceof.006islessthanthestandardcut-offpointof.05).

® Wecanreportthat‘therewasasignificantmaineffectofHelper,F(1,16)=9.87,p=.006’.

® Thiseffecttellsusthatifweignorethetypeoftreatthatwasgivenandhowmanyofthosetreatswereconsumed,elvesandfairiesdifferedintheirdeliveryspeeds’.

SPSSOutput11

Tests of Within-Subjects Effects

Measure: MEASURE_1

2427.339 1 2427.339 42.352 .0002427.339 1.000 2427.339 42.352 .0002427.339 1.000 2427.339 42.352 .0002427.339 1.000 2427.339 42.352 .000444.939 1 444.939 7.763 .013444.939 1.000 444.939 7.763 .013444.939 1.000 444.939 7.763 .013444.939 1.000 444.939 7.763 .013917.022 16 57.314917.022 16.000 57.314917.022 16.000 57.314917.022 16.000 57.314

19025.256 4 4756.314 92.188 .00019025.256 2.334 8152.053 92.188 .00019025.256 2.928 6498.711 92.188 .00019025.256 1.000 19025.256 92.188 .000

748.144 4 187.036 3.625 .010748.144 2.334 320.569 3.625 .030748.144 2.928 255.554 3.625 .020748.144 1.000 748.144 3.625 .075

3302.000 64 51.5943302.000 37.341 88.4293302.000 46.841 70.4943302.000 16.000 206.3752358.744 4 589.686 10.217 .0002358.744 2.594 909.419 10.217 .0002358.744 3.333 707.646 10.217 .0002358.744 1.000 2358.744 10.217 .006761.589 4 190.397 3.299 .016761.589 2.594 293.632 3.299 .035761.589 3.333 228.484 3.299 .023761.589 1.000 761.589 3.299 .088

3693.867 64 57.7173693.867 41.499 89.0113693.867 53.332 69.2623693.867 16.000 230.867

Sphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-bound

SourceTreat

Treat * Helper

Error(Treat)

Dose

Dose * Helper

Error(Dose)

Treat * Dose

Treat * Dose * Helper

Error(Treat*Dose)



SPSSOutput12

Elf Fairy

Spee

d of

Del

iver

y (m

s)

0

5

10

15

20

25

30

SPSSOutput13 Figure14

IfyourequestedthatSPSSdisplaymeansfortheHelpereffectyoushouldscanthroughyouroutputandfindthetableinasectionheadedEstimatedMarginalMeans.SPSSOutput13isatableofmeansforthemaineffectofHelperwiththeassociatedstandarderrors.ThisinformationisplottedinFigure14.Itisclearfromthisgraphthatdeliverytimesweredifferent,withelvesdeliveringquickerthanfairies.

The Effect of Treat

ThefirstpartofSPSSOutput11tellsustheeffectofthetypeoftreatconsumedbytheelves/fairies.Forthiseffectsphericitywasn’tanissue,sowelookattheuncorrectedF-ratios.YoucanrequestthatSPSSproducemeansofthemaineffects(seeField,2009;2013)andifyoudothis,you’llfindthetableinSPSSOutput14inasectionheadedEstimatedMarginalMeans.SPSSOutput14isatableofmeansforthemaineffectoftreatwiththeassociatedstandarderrors.Thelevelsof

thisvariablearelabelled1and2andsowemustthinkbacktohowweenteredthevariabletoseewhich

Levene's Test of Equality of Error Variancesa

.194 1 16 .6662.279 1 16 .151

.325 1 16 .577

.909 1 16 .3557.305 1 16 .0161.693 1 16 .2122.854 1 16 .1111.294 1 16 .272

.933 1 16 .3482.267 1 16 .152

Time Taken to Deliver Presents After 1 SherryTime Taken to Deliver Presents After 2 SherriesTime Taken to Deliver Presents After 3 SherriesTime Taken to Deliver Presents After 4 SherriesTime Taken to Deliver Presents After 5 SherriesTime Taken to Deliver Presents After 1 Mince PieTime Taken to Deliver Presents After 2 Mince PiesTime Taken to Deliver Presents After 3 Mince PiesTime Taken to Deliver Presents After 4 Mince PiesTime Taken to Deliver Presents After 5 Mince Pies

F df1 df2 Sig.

Tests the null hypothesis that the error variance of the dependent variable is equal across groups.

Design: Intercept+Helper Within Subjects Design: Treat+Dose+Treat*Dose

a.

Tests of Between-Subjects Effects

Measure: MEASURE_1Transformed Variable: Average

8964.605 1 8964.605 1989.065 .00044.494 1 44.494 9.872 .00672.111 16 4.507

SourceInterceptHelperError


Estimates

Measure: MEASURE_1

20.744 .708 19.244 22.24523.889 .708 22.389 25.389

Type of HelperElfFairy



rowofthetablerelatestowhichcondition.Weenteredthisvariablewiththesherryconditionfirstandthemincepieconditionlast.Figure15usesthisinformationtodisplaythemeansforeachcondition.Itisclearfromthisgraphthatmeandeliverytimeswerehigheraftersherry(M=25.99)thanaftermincepies(M=18.64).Therefore,sherrysloweddownpresentdeliverysignificantlycomparedtomincepies.


® Therewasasignificantmaineffectofthetypeoftreat,F(1,16)=42.35,p<.001

® This effect tells us that ifwe ignore the number of treats consumed andwhether thehelperwasanelforfairy,deliverytimesforpresentswereslowerafteronetypeoftreatthanaftertheothertype.

Sherry Mince Pies

Spee

d of

Del

iver

y (m

s)

0

5

10

15

20

25

30


The Effect of Dose

SPSSOutput11alsoreportstheeffectofthenumberoftreatsconsumed(dose)bytheelvesandfairies.ThiseffectviolatedtheassumptionofsphericityandsowelookatthecorrectedF-ratios.AllofthecorrectedvaluesaresignificantandweshouldreporttheGreenhouse-Geissercorrectedvaluesbecausetheyarethemostconservative.Youshouldreportthesphericitydata(Mauchley’stestetc.)asexplainedinthefirstexample.Theeffectitselfcouldbereportedas:

Thiseffectviolatedsphericitysowereportthat(notethedf):

® Therewasasignificantmaineffectofthenumberoftreatsconsumed,F(2.33,37.34)=92.19,p<.001.

® Thiseffecttellsusthatifweignorethetypeoftreatthatwasconsumed,andwhetherthehelperwasanelforafairydeliverytimesofpresentswereslowerafterconsumingcertainamountsoftreats.

Note thedegreesof freedomrepresent theGreenhouse-Geisser correctedvalues.Wedon’t know from thiseffect,whichamountsoftreats(doses)inparticularslowedtheelvesdown,butwecouldlookatthiswithposthoctests–seeexample1.

Ifwerequestedmeansofthemaineffects(seeField,2009,Chapter13)thenyou’llseethetableinSPSSOutput15,whichisatableofmeansforthemaineffectofdosewiththeassociatedstandarderrors.Thelevelsofthisvariablearelabelled1,2,3,4,&5andsowemustthinkbacktohowweenteredthevariablestoseewhichrowofthetablerelatestowhichcondition.Figure16usesthisinformationtodisplaythemeansforeachcondition.Itisclearfromthisgraphthatmeandeliverytimesgotprogressivelyhigherasmoretreatswereconsumed(infactthetrendlookslinear).

Estimates

Measure: MEASURE_1

25.989 .735 24.431 27.54718.644 .773 17.006 20.283

Treat12



Dose 1 Dose 2 Dose 3 Dose 4 Dose 5

Spee

d of

Del

iver

y (m

s)

0

10

20

30

40

50


The Treat × Helper Interaction

SPSSOutput11indicatedthatthetypeofhelperinteractedinsomewaywiththetypeoftreatconsumed.

® Wecanreportthat‘therewasasignificantinteractionbetweenthetypeoftreatconsumedandwhetherthehelperwasanelforafairy,F(1,16)=7.76,p=.013’.

® This effect tells us that the effect of different treats on thedelivery speedof presentswasdifferentinelvescomparedtofairies.

Wecanusetheestimatedmarginalmeanstointerpretthisinteraction(orwecouldhaveaskedSPSSforaplotofTreat´HelperusingthedialogboxinFigure13).Themeansandinteractiongraph(Figure17andSPSSOutput16)showthemeaningof this result. The graph shows theaveragedelivery times for elves and fairies for the two typesof treat(regardlessofhowmanywereeaten).Thegraphclearlyshowsthatformincepiesdeliverytimeswerevirtuallyidenticalforelvesandfairies(thebarsarethesameheight).However,foraftersherry,fairiestookmuchlongerthanelvestodeliverpresents.Ingeneralthisinteractionseemstosuggestthanalthoughelvesandfairieswereequallyaffectedbymincepies,fairieswereaffectedmorebysherrythanelveswere.

Sherry Mince Pie

Spee

d of

Del

iver

y (m

s)

0

5

10

15

20

25

30

35

Elves Fairies


Estimates

Measure: MEASURE_1

10.528 .522 9.422 11.63413.167 .495 12.117 14.21621.972 .994 19.865 24.08026.667 1.606 23.262 30.07239.250 1.703 35.640 42.860

Dose12345


4. Type of Helper * Treat

Measure: MEASURE_1

22.844 1.039 20.641 25.04818.644 1.093 16.327 20.96229.133 1.039 26.930 31.33718.644 1.093 16.327 20.962

Treat1212

Type of HelperElf

Fairy



The Dose × Helper Interaction

SPSSOutput11indicatedthatthetypeofhelperinteractedinsomewaywiththenumberoftreatsconsumed.

TheinteractionviolatedsphericityandsowereportfromtheANOVAtablethat:

® We can report that ‘therewas a significant interaction between the Dose of the treat andwhetherthehelperwasanelforafairy,F(2.33,37.34)=3.63,p=.030.

® Thiseffecttellsusthatthedeliverytimesofelvesandfairieswasaffecteddifferentlybythenumberoftreatsthattheyconsumed.

Wecanusetheestimatedmarginalmeanstointerpretthisinteraction(orwecouldhaveaskedSPSSforaplotofDose´HelperusingthedialogboxinFigure13).Themeansandinteractiongraph(Figure18andSPSSOutput17)showthemeaningofthisresult.Thegraphshowstheaveragedeliverytimesforelvesandfairiesateachofthedoses(ignoringwhichtreatwasconsumed).Thegraphshowsthatdeliverytimesareverysimilarforelvesandfairiesatdoses1to3,butatdose4and5fairiesstarttobecomequitealotslowerthanelves.Ingeneralthisinteractionseemstosuggestthanalthoughelvesandfairieswereequallyaffectedatlowerdosesoftreat,fairieswereaffectedmorethanelvesbyhighdoses(4or5doses).



Spee

d of

Del

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y (m

s)

0

10

20

30

40

50

60

Elf Fairy


The Treat ´ Dose Interaction Effect

SPSSOutput11indicatedthatthenumberoftreatsconsumedinteractedinsomewaywiththetypeoftreat.Inotherwords,theeffectthatthenumberoftreats(dose)hadonthespeedofdeliverywasdifferentformincepiesandsherry.ThemeansforallconditionscanbeseeninSPSSOutput18(andthesevaluesarethesameasinthetableofdescriptives).

Theinteractionviolatedsphericityandsowereportthat:

® Therewasasignificantinteractionbetweenthetypeoftreatconsumedandthenumberoftreatsconsumed,F(2.59,41.50)=10.22,p<.001.

® Thiseffecttellsusthattheeffectofconsumingmoretreatswasstrongerforoneofthetreatsthanfortheother.

WecanusethemeansinSPSSOutput18toplotaninteractiongraph,whichisessentialforinterpretingtheinteraction.Figure19showsthatthepatternofrespondingforthetwotreatsisverysimilarforsmalldoses(thebarsarealmostidenticalheightsfor1and2doses).However,asmoretreatsareconsumed,theeffectofdrinkingsherrybecomesmorepronounced(deliverytimesarehigher)thanwhenmincepiesareeaten.Thisisshownbytheincreasinglylargedifferencesbetweenthepairsofbarsforlargenumbersoftreats.Toverifytheinterpretationoftheinteractioneffect,wewouldneedtolookatsomecontrasts(seeField,

5. Type of Helper * Dose

Measure: MEASURE_1

11.667 .738 10.103 13.23112.889 .700 11.404 14.37320.722 1.406 17.742 23.70324.000 2.272 19.185 28.81534.444 2.408 29.340 39.5499.389 .738 7.825 10.953

13.444 .700 11.960 14.92923.222 1.406 20.242 26.20329.333 2.272 24.518 34.14944.056 2.408 38.951 49.160

Dose1234512345

Type of HelperElf

Fairy



2009,chapter13).However,ingeneralterms,SantaClausshouldconcludethatthenumberoftreatsconsumedhadamuchgreatereffectinslowingdownhishelperswhenthetreatwassherry(presumablybecausetheyallgetshit-facedandstartstaggeringaroundbeingstupid),butmuchlessofaneffectwhenthetreatsweremincepiesalthougheventhepiesdidslowthemdowntosomeextent).However,atthisstagewedon’tknowwhetherthiseffectisfoundinallhelpersorwhetheritdiffersinelvesandfairies.


Dose 1 Dose 2 Dose 3 Dose 4 Dose 5Sp

eed

of D

eliv

ery

(ms)

0

10

20

30

40

50

60

SherryMince Pies


The Treat × Dose × Helper Interaction

SPSSOutput11tellsusthatthereisasignificantthree-wayTreat´Dose´Helperinteraction:

® Wecanreportthat‘therewasasignificantinteractionbetweenthetypeoftreateaten,howmanytreatswereeatenandwhetherthehelperwasanelforafairy,F(2.59,41.50)=3.30,p=.035’.

® Thethree-wayinteractiontellsuswhetherthedose´treatinteractiondescribedaboveisthesame for elves and fairies (i.e. whether the combined effect of the type of treat and theirnumberoftreatsconsumedwasthesameforelvesasforfairies).

ThenatureofthisinteractionisshownupinFigure20,whichshowsthehelperbydoseinteractionforthetwotreatsseparately(sherryontheleftandmincepiesontheright).ThemeansinthisgraphweretakenfromSPSSOutput19.FirstlookattheMincepiesgraph(righthandside).Hopefullyit’sclearthatelvesandfairiesdeliverytimesarerelativelysimilaratalldoses(thebarsaremoreorlessthesameheight).Inotherwordstheeffectthatdosehasondeliverytimesisprettysimilarinelvesandfairiesatalldoses.Now,let’slookattheSherrygraph(lefthandside).Thepictureisverydifferenthere:for1,2oreven3dosesofsherrytheelvesandfairiesareaffectedtoasimilardegree (thebarsarethesameheightapproximately).However,at the fourthdoseof sherry fairies slowdownconsiderably compared toelves,and thiseffectisevenmorepronouncedat5doses.Assuch,thisinteractionseemstosuggestthatelvesand

fairiesarefairlycomparableatalldosesofmincepiesandsmalldoesofsherry(upto3doses),butatlargedosesofsherry,fairiesslowdowndeliveringpresentssubstantiallymorethanelvesdo.

6. Treat * Dose

Measure: MEASURE_1

10.278 .853 8.469 12.08613.444 .683 11.996 14.89324.889 1.698 21.290 28.48832.722 2.108 28.253 37.19248.611 2.980 42.295 54.92810.778 .607 9.490 12.06512.889 .744 11.311 14.46719.056 1.435 16.014 22.09720.611 1.963 16.449 24.77329.889 2.396 24.809 34.969

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Figure20

Conclusions

WhatshouldbeclearfromthishandoutisthatwhenmorethantwoindependentvariablesareusedinanANOVA,ityieldscomplexinteractioneffectsthatrequireagreatdealofconcentrationtointerpret(imagineinterpretingafour-way interaction!). Therefore, it is essential to takea systematic approach to interpretationandplottinggraphs is aparticularlyusefulwaytoproceed.

Guided Example Thereisevidencethatattitudestowardsstimulicanbechangedusingpositiveandnegativeimagery(e.g.Stuart,ShimpandEngle,1987,butseeFieldandDavey,1999).Someresearcherswereinterestedinansweringtwoquestions.Onthe

7. Type of Helper * Treat * Dose

Measure: MEASURE_1

11.333 1.206 8.776 13.89113.778 .966 11.729 15.82623.333 2.401 18.243 28.42426.444 2.982 20.124 32.76539.333 4.214 30.400 48.26612.000 .859 10.179 13.82112.000 1.053 9.769 14.23118.111 2.029 13.810 22.41321.556 2.777 15.669 27.44229.556 3.389 22.371 36.7409.222 1.206 6.665 11.780

13.111 .966 11.063 15.15926.444 2.401 21.354 31.53539.000 2.982 32.679 45.32157.889 4.214 48.956 66.8229.556 .859 7.735 11.376

13.778 1.053 11.546 16.00920.000 2.029 15.698 24.30219.667 2.777 13.780 25.55330.222 3.389 23.038 37.406

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onehand,thegovernmenthadfundedthemtolookatwhethernegativeimageryinadvertisingcouldbeusedtochangeattitudestowardsalcohol.Conversely,analcoholcompanyhadprovidedfundingtoseewhetherpositiveimagerycouldbeusedtoimproveattitudestowardsalcohol.Thescientistsdesignedtwostudiestoaddressbothissues.

In the first study,participantsvieweda totalof3mockadvertsover threesessions. Inonesession, theysawthreeadvertscontainingthreedifferentproductswithanegativeimage(adeadbodywiththeslogan‘drinkingthisproductmakesyourliverexplode’:Theproductswere:(1)abrandofbeer(BrainDeath));(2)abrandofwine(Dangleberry);and(3)abrandofwater(Puritan).Thesexoftheparticipantswasnoted.Table1containsthedata(eachrowrepresentsasinglesubject).Aftereachadvertsubjectswereaskedtoratethedrinksonascalerangingfrom-100(dislikeverymuch)through0(neutral)to100(likeverymuch).Theorderofadvertswasrandomised.Therearetwoindependentvariablesineachexperiment:thetypeofdrink(beer,wineorwater)andthesexoftheparticipant(maleorfemale).

Table2:DataforExperiment1

Experiment Experiment1:NegativeImageryDrink Beer Wine WaterMale 6 -5 -14

30 -12 -1015 -15 -1630 -4 -1012 -2 517 -6 -621 -2 -2023 -7 -1220 -10 -927 -15 -6

Female -19 -13 -2-18 -16 -17-8 -23 -19-6 -22 -11-6 -9 -10-9 -18 -17-17 -17 -4-12 -15 -4-11 -14 -1-6 -15 -1

® EnterthedataintoSPSS.

® Savethedataontoadiskinafilecalleddrinkimagery.sav.

® Conduct the appropriate analysis to seewhethermale’s and female’s attitudes todifferentdrinksaredifferentiallyaffectedbynegativeimagery.

Whataretheindependentvariablesandhowmanylevelsdotheyhave?

YourAnswer:


Whatisthedependentvariable?

YourAnswer:

Whatanalysishaveyouperformed?(i.e.andxbyywhattypeofANOVA).

YourAnswer:

Hastheassumptionofsphericitybeenmet?(QuoterelevantstatisticsinAPAformat).

YourAnswer:

ReportthemaineffectoftypeofdrinkinAPAformat.Isthiseffectsignificantandhowwouldyouinterpretit?

YourAnswer:

ReportthemaineffectofsexinAPAformat.Isthiseffectsignificantandhowwouldyouinterpretit?(Youshouldalsocommentontheassumptionofhomogeneityofvariance)


YourAnswer:

ReporttheinteractioneffectbetweensexandtypeofdrinkinAPAformat.Isthiseffectsignificantandhowwouldyouinterpretit?

YourAnswer:

Answerstothisguidedquestioncanbefoundonthemodulewebsiteinthefileanswertomixedanovaguidedexample.pdf

Unguided Example 1: Inasecondexperiment(aweeklater),theparticipantssawthesamethreebrands,butthistimepresentedwithpositiveimages(asexynakedmanorwomen—dependingontheparticipant’ssex—andtheslogan‘drinkingthisproductmakesyouahornystud-muffin’).Aftereachadvertparticipantswereaskedtoratethedrinksonthesamescale.

® Analyse thedata tosee if imageryaffectspreferences for thedifferentdrinks,andwhethertheseeffectsaredifferentinmenandwomen.

® ReporttheresultsinAPAformat.

® Isthepredictionthattheeffectofpositiveimageryonpreferencesfordrinkswilldifferinmenandwomensupported?


Experiment Experiment2:PositiveImageryDrink Beer Wine WaterMale 1 38 10

43 20 915 20 610 28 208 11 2717 17 930 15 1934 27 1234 24 1226 23 21

Female 1 28 337 26 2322 34 2130 32 1740 24 1515 29 1320 30 169 24 1714 34 1915 23 29

Unguided Example 2 Thedatafromthetwopreviousexamplescouldbecombinedandanalysedinathree-waymixedANOVA.Imaginethataswellaspositiveandnegativeimagery,neutralimageryhadbeenused.Participantsviewedninemockadvertsoverthreesessions.Intheseadvertstherewerethreeproducts(abrandofbeer,BrainDeath,abrandofwine,Dangleberry,andabrandofwater,Puritan).Thesecouldeitherbepresentedalongsidepositive,negativeorneutralimagery.Overthethreesessions,andnineadverts,eachtypeofproductwaspairedwitheachtypeof imagery.Aftereachadvertparticipantsratedthedrinksonthesamescaleastheprevioustwoexamples.Thedesign,thusfar,hastwoindependentvariables:thetypeofdrink(beer,wineorwater)andthetypeofimageryused(positive,negativeorneutral).Thesetwovariablescompletelycrossover,producingnineexperimentalconditions.Nowimaginethat Ialsotooknoteofeachperson’ssex.ThedatacanbefoundinthefileMixedAttitude.sav.

® Analysethedatatoseeifdifferenttypesofimageryhavedifferenteffectsonpreferencesforthedifferentdrinks,andwhethertheseeffectsaredifferentinmenandwomen.

® Whatanalysishaveyoudone?


® Doestheeffectofimageryondrinkpreferencesdifferinmenandwomen?Answersareonthecompanionwebsiteofmybook.

Unguided Example 3 LastweekwesawaSpeeddatingexampledoneonfemales(seelastweek’shandout).Imaginethissameexperimentwasalsodoneonmaleparticipants(ratingspeeddatestheyhadwithwomen).Theexamplefromlastweekwouldnowbecome aMixed ANOVA (with sex as an additional variable). These data can be found in LooksOrPersonality.sav.Analysethesedatawithathree-waymixedANOVA.Answerscanbefoundinchapter14ofmybook(thisisthedatasetIuseinthebooktoexplainmixedANOVA).


Unguided Example 4 TextmessagingisverypopularbutthereareconcernsthatchildrenwillusethisformofcommunicationsomuchthattheywillnotlearncorrectwrittenEnglish.Oneresearcherconductedanexperimentinwhichonegroupofchildrenwasencouragedtosendtextmessagesontheirmobilephonesoverasix-monthperiod.Asecondgroupwasforbiddenfromsendingtextmessagesforthesameperiod.Toensurethatkidsinthislatergroupdidn’tusetheirphones,thisgroupweregivenarmbandsthatadministeredpainfulshocksinthepresenceofmicrowaves(likethoseemittedfromphones).Therewere50differentparticipants:25wereencouragedtosendtextmessages,and25wereforbidden.Theoutcomewasascoreonagrammaticaltest(asapercentage)thatwasmeasuredbothbeforeandaftertheexperiment.Thefirstindependentvariablewas,therefore,textmessageuse(textmessagersversuscontrols)andthesecondindependentvariablewasthetimeatwhichgrammaticalabilitywasassessed(beforeoraftertheexperiment).ThedataareinthefileTextMessages.sav.

® Analysethedatatoseeiftextmessagingaffectsgrammaticalability.


® Doestextmessagingreducegrammaticalability?AnswersareonthecompanionwebsiteofmybookandsomemoredetailedcommentsaboutcanbefoundinField&Hole(2003).

Multiple Choice Questions

Go to https://studysites.uk.sagepub.com/field4e/study/mcqs.htm and test yourself on themultiplechoicequestionsforChapter15.Ifyougetanywrong,re-readthishandout(orField,2013,Chapter15)anddothemagainuntilyougetthemallcorrect.

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/

Mixed Factorial ANOVA

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