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Issues with Repeated Measures Designs Repeatedmeasuresisatermusedwhenthesameentitiestakepartinallconditionsofanexperiment.So,forexample,youmightwanttotesttheeffectsofalcoholonenjoymentofaparty.Inthistypeofexperimentitisimportanttocontrolfor individual differences in tolerance to alcohol: somepeople can drink a lot of alcoholwithout really feeling theconsequences,whereasothers,likeme,onlyhavetosniffapintoflagerandtheyfalltothefloorandpretendtobeafish.Tocontrolfortheseindividualdifferenceswecantestthesamepeopleinallconditionsoftheexperiment:sowewouldtesteachsubjectaftertheyhadconsumedonepint,twopints,threepintsandfourpintsoflager.Aftereachdrinktheparticipantmightbegivenaquestionnaireassessingtheirenjoymentoftheparty.Therefore,everyparticipantprovidesascorerepresentingtheirenjoymentbeforethestudy(noalcoholconsumed),afteronepint,aftertwopints,andsoon.Thisdesignissaidtouserepeatedmeasures.
What is Sphericity? Wehaveseenthatparametrictestsbasedonthenormaldistributionassumethatdatapointsareindependent.Thisisnotthecaseinarepeatedmeasuresdesignbecausedatafordifferentconditionshavecomefromthesameentities.Thismeans that data from different experimental conditionswill be related; because of thiswe have tomake anadditional assumption to those of the independent ANOVAs you have so far studied. Put simply (and not entirelyaccurately), we assume that the relationship between pairs of experimental conditions is similar (i.e. the level ofdependencebetweenpairsofgroupsisroughlyequal).Thisassumptionisknownastheassumptionofsphericity.Themoreaccuratebutcomplexexplanationisasfollows.Table1showsdatafromanexperimentwiththreeconditions.Imaginewecalculatedthedifferencesbetweenpairsofscoresinallcombinationsofthetreatmentlevels.Havingdonethis,wecalculatedthevarianceofthesedifferences.Sphericityismetwhenthesevariancesareroughlyequal.InthesedatathereissomedeviationfromsphericitybecausethevarianceofthedifferencesbetweenconditionsAandB(15.7)isgreaterthanthevarianceofthedifferencesbetweenAandC(10.3)andbetweenBandC(10.7).However,thesedatahavelocalcircularity(orlocalsphericity)becausetwoofthevariancesofdifferencesareverysimilar.
What is the Effect of Violating the Assumption of Sphericity? Theeffectofviolatingsphericityisalossofpower(i.e.anincreasedprobabilityofaTypeIIerror)andateststatistic(F-ratio)thatsimplycannotbecomparedtotabulatedvaluesoftheF-distribution(formoredetailsseeField,2009;2013).
Assessing the Severity of Departures from Sphericity Departuresfromsphericitycanbemeasuredinthreeways:
® If Mauchly’s test statistic is significant (i.e. has a probability value less than .05) weconcludethattherearesignificantdifferencesbetweenthevarianceofdifferences:theconditionofsphericityhasnotbeenmet.
o In small samples large deviations from sphericity might be deemed non-significant.
o Inlargesamples,smalldeviationsfromsphericitymightbedeemedsignificant.
Correcting for Violations of Sphericity Fortunately, if data violate the sphericity assumption we simply adjust the defrees of freedom for the effect bymultiplyingitbyoneoftheaforementionedsphericityestimates.Thiswillmakethedegreesoffreedomsmaller;byreducing the degrees of freedom we make the F-ratio more conservative (i.e. it has to be bigger to be deemedsignificant).SPSSappliestheseadjustmentsautomatically.
One-Way Repeated Measures ANOVA using SPSS “I’macelebrity,getmeoutofhere”isaTVshowinwhichcelebrities(well,Imean,they’renotreallyarethey…I’mstrugglingtoknowwhoanyoneisintheseriesthesedays)inapitifulattempttosalvagetheircareers(orjusthavecareersinthefirstplace)goandliveinthejungleandsubjectthemselvestoritual humiliation and/or creepy crawlies in places where creepy crawliesshouldn’tgo.It’scruel,voyeuristic,gratuitous,carcrashTV,andI loveit.Aparticular favourite bit is the Bushtucker trials in which the celebritieswillinglyeatthingslikestickinsects,Witchettygrubs,fisheyes,andkangarootesticles/penises,nomnomnoms….
Entering the Data Theindependentvariablewastheanimalthatwasbeingeaten(stick,insect,kangarootesticle,fisheyeandwitchettygrub)andthedependentvariablewasthetimeittooktoretch,inseconds.
Themaindialogbox(Figure3)hasaspacelabelledwithinsubjectsvariablelistthatcontainsalistof4questionmarksproceeded by a number. These questionmarks are for the variables representing the 4 levels of the independentvariable.Thevariablescorrespondingtotheselevelsshouldbeselectedandplacedintheappropriatespace.Wehaveonly4variablesinthedataeditor,soitispossibletoselectallfourvariablesatonce(byclickingonthevariableatthetop,holdingthemousebuttondownanddraggingdownovertheothervariables).Theselectedvariablescanthenbetransferredbydraggingthemorclickingon .
Post Hoc Tests ThereisnoproperfacilityforproducingposthoctestsforrepeatedmeasuresvariablesinSPSS(youwillfindthatifyouaccesstheposthoctestdialogboxitwillnotlistanyrepeated-measuredfactors).However,youcangetabasicsetofposthoctestsclicking inthemaindialogbox.Tospecifyposthoctests,selecttherepeatedmeasuresvariable
(inthiscaseAnimal)fromtheboxlabelledEstimatedMarginalMeans:Factor(s)andFactorInteractionsandtransferittotheboxlabelledDisplayMeansforbyclickingon (Figure4).Onceavariablehasbeentransferred,theboxlabelledComparemaineffects( )becomesactiveandyoushouldselectthisoption.Ifthisoptionisselected,theboxlabelledConfidenceintervaladjustmentbecomesactiveandyoucanclickon toseeachoiceofthreeadjustmentlevels.ThedefaultistohavenoadjustmentandsimplyperformaTukeyLSDposthoctest(thisisnot recommended). The secondoption is a Bonferroni correction (whichwe’ve encounteredbefore), and the finaloptionisaSidakcorrection,whichshouldbeselectedifyouareconcernedaboutthelossofpowerassociatedwithBonferronicorrectedvalues.Whenyouhaveselectedtheoptionsof interest,clickon to return to themaindialogbox,andthenclickon toruntheanalysis.
Figure4:Optionsdialogbox
Output for Repeated Measures ANOVA Descriptive statistics and other Diagnostics
The Main ANOVA Output3showstheresultsoftheANOVAforthewithin-subjectsvariable.Thetableyouseewilllookslightlydifferent(itwilllooklikeOutput4infact),butforthetimebeingI’vesimplifieditabit.Bearwithmefornow.ThistablecanbereadmuchthesameasforOne-wayindependentANOVA(seeyourhandout).ThesignificanceofFis.026,whichissignificantbecauseitislessthanthecriterionvalueof.05.Wecan,therefore,concludethattherewasasignificantdifferenceinthetimetakentoretchaftereatingdifferentanimals.However,thismaintestdoesnottelluswhichanimalsresultedinthequickestretchingtimes.
Although this result seems very plausible,we saw earlier that the assumption ofsphericity had been violated. I also mentioned that a violation of the sphericityassumptionmakes theF-test inaccurate. So,whatdowedo?Well, Imentionedearlieron thatwe can correct thedegreesoffreedominsuchawaythatitisaccuratewhensphericityisviolated.ThisiswhatSPSSdoes.Output4(whichistheoutputyouwillseeinyourownSPSSanalysis)showsthemainANOVA.Asyoucanseeinthisoutput,thevalueofF does not change, only the degrees of freedom. But the effect of changing the degrees of freedom is that thesignificanceofthevalueofFchanges:theeffectofthetypeofanimalislesssignificantaftercorrectingforsphericity.
Thenextissueiswhichofthethreecorrectionstouse.EarlierIgaveyousometipsandtheywerethatwhenε > .75then use the Huynh-Feldt correction, andwhen ε < .75, or nothing is known about sphericity at all, then use theGreenhouse-Geissercorrection;εistheestimateofsphericityfromOutput2andthesevaluesare.533and.666(thecorrectionofthebeast….);becausethesevaluesarelessthan.75weshouldusetheGreenhouse-Geissercorrectedvalues.Usingthiscorrection,Fisnotsignificantbecauseitspvalueis.063,whichismorethanthenormalcriterionof.05.
® Inthisexampletheresultsarequiteweirdbecauseuncorrectedtheyaresignificant,andapplying the Huynh-Feldt correction they are also significant. However, with theGreenhouse-Geissercorrectionappliedtheyarenot.
Post Hoc Tests Giventhemaineffectwasnotsignificant,weshouldnotfollowthiseffectupwithposthoctests,butinsteadconcludethatthetypeofanimaldidnothaveasignificanteffectonhowquicklycontestantsretched(perhapsweshouldhaveusedbeansontoastasabaselineagainstwhichtocompare…).
Reporting One-Way Repeated Measures ANOVA WecanreportrepeatedmeasuresANOVAinthesamewayasanindependentANOVA(seeyourhandout).Theonlyadditional thingwe should concern ourselveswith is reporting the corrected degrees of freedom if sphericitywasviolated.Personally,I’malsokeenonreportingtheresultsofsphericitytestsaswell.Therefore,wecouldreportthemainfindingas:
Two-Way Repeated Measures ANOVA Using SPSS Aswehaveseenbefore,thenameofanyANOVAcanbebrokendowntotellusthetypeofdesignthatwasused.The‘two-way’partofthenamesimplymeansthattwoindependentvariableshavebeenmanipulatedintheexperiment.The‘repeatedmeasures’partofthenametellsusthatthesameparticipantshavebeenusedinallconditions.Therefore,thisanalysisisappropriatewhenyouhavetworepeated-measuresindependentvariables:eachparticipantdoesalloftheconditionsintheexperiment,andprovidesascoreforeachpermutationofthetwovariables.
A Speed-Dating Example Itseemsthatlotsofmagazinesgoonallthetimeabouthowmenandwomenwantdifferentthingsfromrelationships(orperhapsit’sjustmywife’scopiesofMarieClare’s,whichobviouslyIdon’tread,honestly).Thebigquestiontowhichweallwanttoknowtheanswerisarelooksorpersonalitymoreimportant.Imagineyouwantedtoputthistothetest.Youdevisedacunningplanwherebyyou’dsetupaspeed-datingnight.Littledidthepeoplewhocamealongknowthatyou’dgotsomeofyourfriendstoactasthedates.Specificallyyoufound9mentoactasthedate.Ineachofthesegroupsthreepeoplewereextremelyattractivepeoplebutdifferedintheirpersonality:onehadtonnesofcharisma,one had some charisma, and the third personwas as dull as this handout. Another three peoplewere of averageattractiveness,andagaindifferedintheirpersonality:onewashighlycharismatic,onehadsomecharismaandthethirdwasadullard.Thefinalthreewere,notwishingtobeunkindinanyway,butt-uglyandagainonewascharismatic,onehadsomecharismaandthefinalpoorsoulwasmind-numbinglytedious.Theparticipantswereheterosexualwomenwhocametothespeeddatingnight,andoverthecourseoftheeveningtheyspeed-datedall9men.Aftertheir5minutedate,theyratedhowmuchthey’dliketohaveaproperdatewiththepersonasapercentage(100%=‘I’dpaylargesumsofmoneyforyourphonenumber’,0%=‘I’dpayalargesumofmoneyforaplanetickettogetmeasfarawayaspossiblefromyou’).Assuch,eachwomanrated9differentpeoplewhovariedintheirattractivenessandpersonality.So,therearetworepeatedmeasuresvariables:looks(withthreelevelsbecausethepersoncouldbeattractive,averageorugly)andpersonality(againwiththreelevelsbecausethepersoncouldhavelotsofcharisma,havesomecharisma,orbeadullard).
Data Entry ToenterthesedataintoSPSSweusethesameprocedureastheone-wayrepeatedmeasuresANOVAthatwecameacrossinthepreviousexample.
ThedataareinthefileFemaleLooksOrPersonality.savfromthecoursewebsite.Firstwehavetodefineourrepeatedmeasuresvariables,soaccessthedefinefactorsdialogboxselect .As with one-way repeatedmeasures ANOVA (see the previous example) we need to give names to our repeatedmeasuresvariablesand specifyhowmany levels theyhave. In this case thereare twowithin-subject factors: looks(attractive,averageorugly)andcharisma(highcharisma,somecharismaanddullard).Inthedefinefactorsdialogboxreplacethewordfactor1withtheword looks.Whenyouhavegiventhisrepeatedmeasuresfactoraname,tellthecomputerthatthisvariablehas3levelsbytypingthenumber3intotheboxlabelledNumberofLevels(Figure5).Clickon toaddthisvariabletothelistofrepeatedmeasuresvariables.Thisvariablewillnowappearinthewhiteboxatthebottomofthedialogboxandappearsaslooks(3).
ThemaindialogboxisshowninFigure6.AtthetopoftheWithin-SubjectsVariablesbox,SPSSstatesthattherearetwofactors:looksandcharisma.Intheboxbelowthereisaseriesofquestionmarksfollowedbybracketednumbers.Thenumbers in brackets represent the levels of the factors (independent variables). In this example, there are twoindependentvariablesandsotherearetwonumbersinthebrackets.Thefirstnumberreferstolevelsofthefirstfactorlistedabovethebox(inthiscaselooks).Thesecondnumberinthebracketreferstolevelsofthesecondfactorlistedabovethebox(inthiscasecharisma).Wehavetoreplacethequestionmarkswithvariablesfromthelistontheleft-handsideofthedialogbox.Withbetween-groupdesigns,inwhichcodingvariablesareused,thelevelsofaparticularfactorarespecifiedbythecodesassignedtotheminthedataeditor.However,inrepeatedmeasuresdesigns,nosuchcodingscheme isusedandsowedeterminewhichconditiontoassigntoa levelat thisstage.Thevariablescanbeenteredasfollows:
Other Options Theadditionofanextravariablemakesitnecessarytochooseadifferentgraphtotheoneintheprevioushandout.Clickon toaccessthedialogboxinFigure7.PlacelooksintheslotlabelledHorizontalAxis:andcharismainslot labelled Separate Line. When both variables have been specified, don’t forget to click on to add thiscombinationtothelistofplots.ByaskingSPSStoplotthelooks´charismainteraction,weshouldgettheinteractiongraph for looks and charisma. You could also think about plotting graphs for the twomain effects (e.g. looks andcharisma).Asfarasotheroptionsareconcerned,youshouldselectthesameonesthatwerechosenforthepreviousexample.Itisworthselectingestimatedmarginalmeansforalleffects(becausethesevalueswillhelpyoutounderstandanysignificanteffects).
Descriptives and Main Analysis Output6showstheinitialoutputfromthisANOVA.Thefirsttablemerelyliststhevariablesthathavebeenincludedfromthedataeditorandthelevelofeachindependentvariablethattheyrepresent.Thistableismoreimportantthanitmightseem,becauseitenablesyoutoverifythatthevariablesintheSPSSdataeditorrepresentthecorrectlevelsoftheindependentvariables.Thesecondtableisatableofdescriptivesandprovidesthemeanandstandarddeviationforeachofthenineconditions.ThenamesinthistablearethenamesIgavethevariablesinthedataeditor(therefore,ifyoudidn’tgivethesevariablesfullnames,thistablewilllookslightlydifferent).Thevaluesinthistablewillhelpuslatertointerpretthemaineffectsoftheanalysis.
Attractive and Highly CharismaticAttractive and Some CharismaAttractive and a DullardAverage and Highly CharismaticAverage and Some CharismaAverage and a DullardUgly and Highly CharismaticUgly and Some CharismaUgly and a Dullard
Mean Std. Deviation N
Mauchly's Test of Sphericityb
Measure: MEASURE_1
.904 .810 2 .667 .912 1.000 .500
.851 1.292 2 .524 .870 1.000 .500
.046 22.761 9 .008 .579 .791 .250
Within Subjects EffectLooksCharismaLooks * Charisma
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 Within Subjects Design: Looks+Charisma+Looks*Charisma
Output8showstheresultsoftheANOVA(withcorrectedFvalues).Theoutputissplitintosectionsthatrefertoeachoftheeffectsinthemodelandtheerrortermsassociatedwiththeseeffects.Theinterestingpartisthesignificancevaluesof theF-ratios. If these values are less than .05 thenwe can say that aneffect is significant. Lookingat thesignificancevaluesinthetableitisclearthatthereisasignificantmaineffectofhowattractivethedatewas(Looks),asignificantmaineffectofhowcharismaticthedatewas(Charisma),andasignificant interactionbetweenthesetwovariables.Iwillexamineeachoftheseeffectsinturn.
Output8
The Main Effect of Looks WecameacrossthemaineffectoflooksinOutput8.
® Thiseffect tellsus that ifwe ignoreallothervariables, ratingsweredifferent forattractive,averageandunattractivedates.
IfyourequestedthatSPSSdisplaymeansforthelookseffect(I’llassumeyoudidfromnowon)youwillfindthetableina section headed EstimatedMarginalMeans. Output 9 is a table of means for themain effect of looks with theassociatedstandarderrors.Thelevelsoflooksarelabelledsimply1,2and3,andit’sdowntoyoutorememberhowyouenteredthevariables(oryoucanlookatthesummarytablethatSPSSproducesatthebeginningoftheoutput—seeOutput6).IfyoufollowedwhatIdidthenlevel1isattractive,level2isaverageandlevel3isugly.Tomakethingseasier,thisinformationisplottedinFigure8:asattractivenessfalls,themeanratingfallstoo.Thismaineffectseemstoreflectthatthewomenweremorelikelytoexpressagreaterinterestingoingoutwithattractivementhanaverageoruglymen.However,wereallyneedto lookatsomecontraststofindoutexactlywhat’sgoingon(seeField,2013 ifyou’reinterested).
Guided Example A clinical psychologistwas interested in the effects of antidepressants and cognitive behaviour therapy on suicidalthoughts.Fourpeoplediagnosedwithdepressiontookpartinfourconditions:placebotabletwithnotherapyforonemonth,placebotabletwithcognitivebehaviourtherapy(CBT)foronemonth,antidepressantwithnotherapyforonemonth,andantidepressantwithcognitivebehaviourtherapy(CBT)foronemonth.Theorderofconditionswasfullycounterbalancedacrossthe4participants.Participantsrecordedthenumberofsuicidalthoughtstheyhadduringthefinalweekofeachmonth.
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 aredisplayed in the Tests of Within-Subjects Effects table.
a.
Design: Intercept Within Subjects Design: DRUG+THERAPY+DRUG*THERAPY
® Conduct the appropriate analysis to seewhether the number of suicidal thoughtspatientshadwassignificantlyaffectedbythetypeofdrugtheyhad,thetherapytheyreceivedortheinteractionofthetwo..
In your own time … Task 1 Thereisalotofconcernamongstudentsastotheconsistencyofmarkingbetweenlecturers.Itisprettycommonthatlecturersobtainreputationsforbeing‘hardmarkers’or ‘lightmarkers’butthere isoften littletosubstantiatethesereputations. So, a groupof students investigated the consistencyofmarkingby submitting the sameessay to fourdifferentlecturers.Themarkgivenbyeachlecturerwasrecordedforeachofthe8essays.Itwasimportantthatthesameessayswereusedforalllecturersbecausethiseliminatedanyindividualdifferencesinthestandardofworkthateachlecturerwasmarking.Thedataarebelow.
Task 2 In a previous handout we came across the beer-goggles effect: a severe perceptual distortion after imbibing vastquantitiesofalcohol. Imaginewewantedtofollowthisfindingupto lookatwhatfactorsmediatethebeergoggleseffect.Specifically,wethoughtthatthebeergoggleseffectmightbemadeworsebythefactthatitusuallyoccursinclubs,whichhavedimlighting.Wetookasampleof26men(becausetheeffect isstronger inmen)andgavethemvarious doses of alcohol over four different weeks (0 pints, 2 pints, 4 pints and 6 pints of lager). This is our firstindependentvariable,whichwe’llcallalcoholconsumption,andithasfourlevels.Eachweek(and,therefore,ineachstateofdrunkenness)participantswereaskedtoselectamateinanormalclub(thathaddimlighting)andthenselecta secondmate in a specially designed club that had bright lighting. As such, the second independent variablewaswhethertheclubhaddimorbrightlighting.Theoutcomemeasurewastheattractivenessofeachmateasassessedbyapanelofindependentjudges.Torecap,allparticipantstookpartinalllevelsofthealcoholconsumptionvariable,andselectedmatesinbothbrightly-anddimly-litclubs.ThisistheexampleIpresentedinmyhandoutandlectureinwritinguplaboratoryreports.
Task 3 ImagineIwantedtolookattheeffectalcoholhasonthe‘rovingeye’(apparentlyIamratherobsessedwithexperimentsinvolvingalcoholanddating…).The‘rovingeye’effectisthepropensityofpeopleinrelationshipsto‘eye-up’membersof theopposite sex. I took 20men and fitted themwith incredibly sophisticated glasses that could track their eyemovements and recordboth themovement and theobject beingobserved (this is thepoint atwhich it shouldbeapparentthatI’mmakingitupasIgoalong).Over4differentnightsIpliedthesepoorsoulswitheither1,2,3or4pintsofstronglagerinapub.EachnightImeasuredhowmanydifferentwomentheyeyed-up(awomenwascategorizedashavingbeeneyedupiftheman’seyemovedfromherheadtotoeandbackupagain).Tovalidatethismeasurewealsocollectedtheamountofdribbleontheman’schinwhilelookingatawoman.
Task 4 Westernpeoplecanbecomeobsessedwithbodyweightanddiets,andbecausethemediaareinsistentonrammingridiculousimagesofstick-thincelebritiesdownintooureyesandbrainwashingusintobelievingthattheseemaciatedcorpsesareactuallyattractive,weallendupterriblydepressedthatwe’renotperfect.Thisgivesevilcorporatetypestheopportunitytojumponourvulnerabilitybymakingloadsofmoneyondietsthatwillapparentlyhelpusattainthebodybeautiful!Well,notwishingtomissoutonthisgreatopportunitytoexploitpeople’sinsecuritiesIcameupwithmy own diet called the ‘Andikins diet’1. The basic principle is that you eat likeme: you eat nomeat, drink lots ofDarjeelingtea,eatshed-loadsofsmellyEuropeancheesewithlotsoffreshcrustybread,pasta,andeatchocolateateveryavailableopportunity,andenjoyafewbeersattheweekend.Totesttheefficacyofmywonderfulnewdiet,Itook10peoplewhoconsideredthemselvestobeinneedoflosingweight(thiswasforethicalreasons–youcan’tforcepeopletodiet!)andputthemonthisdietfortwomonths.TheirweightwasmeasuredinKilogramsatthestartofthedietandthenafter1monthand2months.
Complete themultiple choice questions forChapter 14 on the companionwebsite to Field(2013):https://studysites.uk.sagepub.com/field4e/study/mcqs.htm.Ifyougetanywrong,re-readthishandout(orField,2013,Chapter14)anddothemagainuntilyougetthemallcorrect.
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