Analysis of Covariance (ANCOVA) - Discovering … of Covariance (ANCOVA) Some background ANOVA can be extended to include one or more continuous variables that predict the outcome

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

Analysis of Covariance (ANCOVA)

Some background ANOVAcanbeextendedtoincludeoneormorecontinuousvariablesthatpredicttheoutcome(ordependentvariable).Continuousvariablessuchasthese,thatarenotpartofthemainexperimentalmanipulationbuthaveaninfluenceonthedependentvariable,areknownascovariatesandtheycanbeincludedinanANOVAanalysis.Forexample,intheViagraexamplefromField(2013),wemightexpecttheretobeotherthingsthatinfluenceaperson’slibidootherthanViagra.Somepossibleinfluencesonlibidomightbethelibidooftheparticipant’ssexualpartner(afterall‘ittakestwoto tango’), other medication that suppresses libido (such as antidepressants), and fatigue. If these variables aremeasured,thenitispossibletocontrolfortheinfluencetheyhaveonthedependentvariablebyincludingtheminthemodel.What,ineffect,happensisthatwecarryoutahierarchicalregressioninwhichourdependentvariableistheoutcome,andthecovariateisenteredinthefirstblock.Inasecondblock,ourexperimentalmanipulationsareentered(intheformofwhatarecalledDummyvariables).So,weendupseeingwhateffectanindependentvariablehasaftertheeffectofthecovariate.Field(2013)explainsthesimilaritybetweenANOVAandregressionandthisisusefulreadingtounderstandhowANCOVAworks.

ThepurposeofincludingcovariatesinANOVAistwo-fold:

1. To reducewithin-grouperror variance: InANOVAweassess theeffectof anexperimentby comparing theamountofvariabilityinthedatathattheexperimentcanexplain,againstthevariabilitythatitcannotexplain.Ifwecanexplainsomeofthis‘unexplained’variance(SSR) intermsofcovariates,thenwereducetheerrorvariance,allowingustomoreaccuratelyassesstheeffectoftheexperimentalmanipulation(SSM).

2. EliminationofConfounds:Inanyexperiment,theremaybeunmeasuredvariablesthatconfoundtheresults(i.e.avariablethatvariessystematicallywiththeexperimentalmanipulation).Ifanyvariablesareknowntoinfluencethedependentvariablebeingmeasured,thenANCOVAisideallysuitedtoremovethebiasofthesevariables.Onceapossibleconfoundingvariablehasbeenidentified,itcanbemeasuredandenteredintotheanalysisasacovariate.

The Example ImaginethattheresearcherwhoconductedtheViagrastudy inField(2013)suddenlyrealizedthatthe libidooftheparticipants’sexualpartnerswouldeffectthatparticipant’sownlibido(especiallybecausethemeasureoflibidowasbehavioural).Therefore,theresearcherrepeatedthestudyonadifferentsetofparticipants,buttookameasureofthepartner’slibido.Thepartner’slibidowasmeasuredintermsofhowoftentheytriedtoinitiatesexualcontact.

Assumptions in ANCOVA ANCOVAhasthesameassumptionsasanylinearmodel(seeyourhandoutonbias)exceptthattherearetwoimportantadditionalconsiderations:(1)independenceofthecovariateandtreatmenteffect,and(2)homogeneityofregressionslopes.Thefirstonebasicallymeansthatthecovariateshouldnotbedifferentacrossthegroupsintheanalysis(inotherwords,ifyoudidanANOVAort-testusingthegroupsastheindependentvariableandthecovariateastheoutcome,thisanalysisshouldbenon-significant).ThisassumptionisquiteinvolvedsoallI’llsayisreadmybookchapterformoreinformation,orreadMillerandChapman(2001).

WhenanANCOVAisconductedwelookattheoverallrelationshipbetweentheoutcome(dependentvariable)andthecovariate:wefitaregressionlinetotheentiredataset,ignoringtowhichgroupapersonbelongs.Infittingthisoverallmodelwe,therefore,assumethatthisoverallrelationshipistrueforallgroupsofparticipants.Forexample,ifthere’sapositive relationship between the covariate and the outcome in one group, we assume that there is a positiverelationshipinalloftheothergroupstoo.If,however,therelationshipbetweentheoutcome(dependentvariable)andcovariatediffers across thegroups then theoverall regressionmodel is inaccurate (it doesnot representall of thegroups).Thisassumptionisveryimportantandiscalledtheassumptionofhomogeneityofregressionslopes.Thebestwaytothinkofthisassumptionistoimagineplottingascatterplotforeachexperimentalconditionwiththecovariate


ononeaxis and theoutcomeon theother. If you then calculated, anddrew, the regression line for eachof thesescatterplotsyoushouldfindthattheregressionlineslookmoreorlessthesame(i.e.thevaluesofbineachgroupshouldbeequal).

Figure 1 shows scatterplots that display the relationship betweenpartner’s libido (the covariate) and theoutcome(participant’s libido) for each of the three experimental conditions (different colours and symbols). Each symbolrepresentsthedatafromaparticularparticipant,andthetypeofsymboltellsusthegroup(circles=placebo,triangles= lowdose, squares=highdose). The lines are the regression slopes for theparticular group, they summarise therelationshipbetweenlibidoandpartner’slibidoshownbythedots(blue=placebogroup,green=low-dosegroup,red=high-dosegroup).Itshouldbeclearthatthereisapositiverelationship(theregressionlineslopesupwardsfromlefttoright)betweenpartner’slibidoandparticipant’slibidoinboththeplaceboandlow-doseconditions.Infact,theslopesofthelinesforthesetwogroups(blueandgreen)areverysimilar,showingthattherelationshipbetweenlibidoandpartner’slibidoisverysimilarinthesetwogroups.Thissituationisanexampleofhomogeneityofregressionslopes(theregressionslopesinthetwogroupsaresimilar).However,inthehigh-doseconditionthereappearstobenorelationshipatallbetweenparticipant’slibidoandthatoftheirpartner(thesquaresarefairlyrandomlyscatteredandtheregressionlineisveryflatandshowsaslightlynegativerelationship).Theslopeofthislineisverydifferenttotheothertwo,andthisdifferencegivesuscausetodoubtwhetherthereishomogeneityofregressionslopes(becausetherelationshipbetweenparticipant’slibidoandthatoftheirpartnerisdifferentinthehigh-dosegrouptotheothertwogroups).We’llhavealookhowtotestthisassumptionlater.

Figure1:ScatterplotofLibidoagainstPartner’slibidoforeachoftheexperimentalconditions

ANCOVA on SPSS Entering Data ThedataforthisexampleareinTable1,whichshowstheparticipant’slibidoandtheirpartner’slibido.Themeanlibido(andSDinbrackets)oftheparticipants’libidoscoresareinTable2.Inessence,thedatashouldbelaidoutintheDataEditorastheyareTable1.Withoutthecovariate,thedesignissimplyaone-wayindependentdesign,sowewouldenterthesedatausinga coding variable for the independent variable, and scoreson thedependent variablewill go in adifferentcolumn.Allthatchangesisthatwehaveanextracolumnforthecovariatescores.

® CovariatesareenteredintotheSPSSdataeditorinanewcolumn(eachcovariateshouldhaveitsowncolumn).

® CovariatescanbeaddedtoanyofthedifferentANOVAswehavecoveredonthiscourse!

o Whenacovariateisaddedtheanalysisiscalledanalysisofcovariance(so,forexample,youcouldhaveatwo-wayrepeatedmeasuresAnalysisofCovariance,orathreewaymixedANCOVA).


Table1:DatafromViagraCov.sav

Dose Participant’sLibido

Partner’sLibido

Placebo 3 42 15 52 12 22 27 72 44 5

LowDose 7 55 33 14 24 27 65 44 2

HighDose 9 12 36 53 44 34 34 26 04 16 32 08 15 0

So,createacodingvariablecalleddoseandusetheLabelsoptiontodefinevaluelabels(e.g.1=placebo,2=lowdose,3=highdose).Therewerenineparticipantsintheplacebocondition,soyouneedtoenter9valuesof1intothiscolumn(sothatthefirst9rowscontainthevalue1),followedbyeightvaluesof2torepresentthepeopleinthelowdosegroup,andfollowedbythirteenvaluesof3torepresentthepeopleinthehighdosegroup.Atthispoint,youshouldhaveonecolumn with 30 rows of data entered. Next, create a second variable called libido and enter the 30 scores thatcorrespondtotheparticipant’slibido.Finally,createathirdvariablecalledpartner,usetheLabelsoptiontogivethisvariableamoredescriptivetitleof‘partner’slibido’.Then,enterthe30scoresthatcorrespondtothepartner’slibido.

Table2:Means(andstandarddeviations)fromViagraCovariate.sav

Dose Participant’sLibido

Partner’sLibido

Placebo 3.22(1.79) 3.44(2.07)LowDose 4.88(1.46) 3.12(1.73)HighDose 4.85(2.12) 2.00(1.63)

Main Analysis MostoftheGeneralLinearModel(GLM)proceduresinSPSScontainthefacilitytoincludeoneormorecovariates.Fordesignsthatdon’tinvolverepeatedmeasuresitiseasiesttoconductANCOVAviatheGLMUnivariateprocedure.To


access themaindialogboxselect (seeFigure2).Themaindialogbox issimilartothatforone-wayANOVA,exceptthatthereisaspacetospecifycovariates.SelectLibidoanddragthisvariabletotheboxlabelledDependentVariableorclickon .SelectDoseanddragittotheboxlabelledFixedFactor(s)andthenselectPartner_LibidoanddragittotheboxlabelledCovariate(s).

Figure2:MaindialogboxforGLMunivariate

Contrasts and Other Options Thereare variousdialogboxes that canbeaccessed from themaindialogbox. The first thing tonotice is that if acovariateisselected,theposthoctestsaredisabled(youcannotaccessthisdialogbox).Posthoctestsarenotdesignedforsituationsinwhichacovariateisspecified,however,somecomparisonscanstillbedoneusingcontrasts.

Figure3:OptionsforstandardcontrastsinGLMunivariate

Clickon toaccessthecontrastsdialogbox.ThisdialogboxisdifferenttotheonewemetforANOVAinthatyoucannotentercodestospecifyparticularcontrasts.Instead,youcanspecifyoneofseveralstandardcontrasts.Thesestandardcontrastswerelistedinmybook.Inthisexample,therewasaplacebocontrolcondition(codedasthefirstgroup),soasensiblesetofcontrastswouldbesimplecontrastscomparingeachexperimentalgroupwiththecontrol.Toselectatypeofcontrastclickon toaccessadrop-downlistofpossiblecontrasts.Selectatypeofcontrast(inthiscaseSimple)fromthislistandthelistwillautomaticallydisappear.Forsimplecontrastsyouhavetheoptionofspecifyingareferencecategory(whichisthecategoryagainstwhichallothergroupsarecompared).Bydefaultthereferencecategoryisthelastcategory:becauseinthiscasethecontrolgroupwasthefirstcategory(assumingthatyoucodedplaceboas1)weneedtochangethisoptionbyselecting .Whenyouhaveselectedanewcontrast


option,youmustclickon toregisterthischange.ThefinaldialogboxshouldlooklikeFigure3.Clickon toreturntothemaindialogbox.

Figure4:OptionsdialogboxforGLMunivariate

Anotherwaytogetposthoctestsisbyclickingon toaccesstheoptionsdialogbox(seeFigure4).Tospecifyposthoctests,selecttheindependentvariable(inthiscaseDose)fromtheboxlabelledEstimatedMarginalMeans:Factor(s)andFactorInteractionsanddragittotheboxlabelledDisplayMeansfororclickon .Onceavariablehasbeen transferred, the box labelled Compare main effects becomes active and you should select this option (

).Ifthisoptionisselected,theboxlabelledConfidenceintervaladjustmentbecomesactiveandyoucanclickon toseeachoiceofthreeadjustmentlevels.Thedefaultistohavenoadjustmentandsimply performa Tukey LSDpost hoc test (this option is not recommended); the second is to ask for a Bonferronicorrection (recommended); the final option is to have a Sidak correction. The Sidak correction is similar to theBonferronicorrectionbutislessconservativeandsoshouldbeselectedifyouareconcernedaboutthelossofpowerassociatedwithBonferronicorrectedvalues.ForthisexampleusetheSidakcorrection(wewill use Bonferroni later in the book). As well as producing post hoc tests for theDosevariable,placingdoseintheDisplayMeansforboxwillcreateatableofestimatedmarginalmeansforthisvariable.Thesemeansprovideanestimateoftheadjustedgroupmeans(i.e.themeansadjustedfortheeffectofthecovariate).Whenyouhaveselectedtheoptionsrequired,clickon toreturntothemaindialogbox.

Aswithone-wayANOVA,themaindialogboxhasa button.Selectingthisoptionwillbootstrapconfidenceintervalsaroundtheestimatedmarginalmeans,parameterestimatesandposthoctests,butnotthemainFtest.ThiscanbeusefulsoselecttheoptionsinFigure5.Clickon inthemaindialogboxtoruntheanalysis.


Figure5:Bootstrapdialogbox

Output from ANCOVA Main Analysis Output1shows(forillustrativepurposes)theANOVAtableforthesedatawhenthecovariateisnotincluded.Itisclearfromthesignificancevaluethattherearenodifferencesinlibidobetweenthethreegroups,thereforeViagraseemstohavenosignificanteffectonlibido.

Output1

Output2showstheresultsofLevene’stestwhenpartner’slibidoisincludedinthemodelasacovariate.Levene’stestissignificant,indicatingthatthegroupvariancesarenotequal(hencetheassumptionofhomogeneityofvarianceislikleybeenviolated).However,Levene’stestisnotnecessarilythebestwaytojudgewhethervariancesareunequalenoughtocauseproblems(seeyourhandoutfromweek2orField,2013chapter5).Wesawinweek2thatagooddoublecheckistolookatthevarianceratio1.Thevarianceratioforthesedatais4.49/2.13=2.11.Thisvalueisgreaterthan2indicatingthatourvariancesareprobablyheterogeneous!Wesawlasttermthatwecouldtrytotransformourdatatocorrectthisproblem(haveagoifyou’refeelingkeen),butforthetimebeingdon’tworrytoomuchaboutthedifferencesinvariances.

1Reminder1:thevarianceratioisthelargestvariancedividedbythesmallestandshouldbelessthanabout2.YoucangetthesevariancesbysquaringtheSDsinTable1.


Output2

Output3showstheANOVAtablewiththecovariateincluded.Comparethistothesummarytablewhenthecovariatewasnotincluded.TheformatoftheANOVAtableislargelythesameaswithoutthecovariate,exceptthatthereisanadditionalrowofinformationaboutthecovariate(partner).Lookingfirstatthesignificancevalues,itisclearthatthecovariatesignificantlypredictsthedependentvariable,becausethesignificancevalueislessthan.05.Therefore,theperson’slibidoisinfluencedbytheirpartner’slibido.What’smoreinterestingisthatwhentheeffectofpartner’slibidoisremoved,theeffectofViagrabecomessignificant(pis.027whichislessthan.05).Theamountofvariationaccountedforbythemodel(SSM)hasincreasedto31.92units(correctedmodel)ofwhichViagraaccountsfor25.19units.Mostimportant,thelargeamountofvariationinlibidothatisaccountedforbythecovariatehasmeantthattheunexplainedvariance(SSR)hasbeenreducedto79.05units.NoticethatSSThasnotchanged;allthathaschangedishowthattotalvariationisexplained.

This example illustrates how ANCOVA can help us to exert stricter experimental control by taking account ofconfoundingvariablestogiveusa‘purer’measureofeffectoftheexperimentalmanipulation.Withouttakingaccountofthelibidooftheparticipants’partnerswewouldhaveconcludedthatViagrahadnoeffectonlibido,yetclearlyitdoes.Lookingbackat thegroupmeans fromTable1:e1 it seemsprettyclear that thesignificantANOVAreflectsadifferencebetweentheplacebogroupandthetwoexperimentalgroups(becausethelowandhighdosegrouphaveverysimilarmeanswhereastheplacebogrouphavealowermean).However,weneedtocheckthecontraststoverifythisconclusion.

Output3

WecanreportthemaineffectofDoseinAPAformatas:

ü Therewas a significant effect ofViagraon levels of libido after controlling for theeffect ofpartner’slibido,F(2,26)=4.14,p=.027.

Contrasts Output4showstheresultofthecontrastanalysisspecifiedinFigure3andcompareslevel2(lowdose)againstlevel1(placebo)asafirstcomparison,andlevel3(highdose)againstlevel1(placebo)asasecondcomparison.Thesecontrastsareconsistentwithwhatwasspecified:allgroupsarecomparedtothefirstgroup.Thegroupdifferencesaredisplayed:adifferencevalue,standarderror,significancevalueand95%confidenceinterval.Theseresultsshowthatboththelow-dosegroup(contrast1,p=.045)andhigh-dosegroup(contrast2,p=.010)hadsignificantlydifferentlibidosthantheplacebogroup.

These contrasts tell us that therewere groupdifferences, but to interpret themweneed to know themeans.WeproducedthemeansinTable2sosurelywecanjustlookatthesevalues?Actuallywecan’tbecausethesegroupmeanshavenotbeenadjustedfortheeffectofthecovariate.TheseoriginalmeanstellusnothingaboutthegroupdifferencesreflectedbythesignificantANCOVA.Output5givestheadjustedvaluesofthegroupmeansanditisthesevaluesthat


should be used for interpretation (this is themain reason for selecting theDisplayMeans foroption). From theseadjustedmeansyoucanseethatlibidoincreasedacrossthethreedoses.

Output4

Output5

WecanreportthesecontrastsinAPAformatas:

ü Plannedcontrastsrevealedthathavingahigh,p=.010,95%CI[0.58,3.88],andlow,p=.045,95%CI[0.04,3.53],doseofViagrasignificantlyincreasedlibidocomparedtohavingaplacebo.

Post Hoc Tests Output6showstheresultsoftheSidakcorrectedposthoccomparisonsthatwererequestedaspartoftheoptionsdialogbox.Thebottomtableshowsthebootstrappedsignificanceandconfidenceintervalsforthesetestsandbecausethesewillberobustwe’llinterpretthistable(again,remember,yourvalueswilldifferbecauseofhowbootstrappingworks).Thereisasignificantdifferencebetweentheplacebogroupandboththelow(p=.003)andhigh(p=.021)dosegroups.Thehighandlow-dosegroupsdidnotsignificantlydiffer(p=.56).Itisinterestingthatthesignificantdifferencebetweenthelow-doseandplacebogroupswhenbootstrapped(p=.003)isnotpresentforthenormalposthoctests(p=.130).Thiscouldreflectpropertiesofthedatathathavebiasedthenon-robustversionoftheposthoctest.

Interpreting the Covariate Onewaytodiscovertheeffectofthecovariateissimplytodrawascatterplotofthecovariateagainsttheoutcome.Theresultingscatterplotforthesedatashowsthattheeffectofcovariateisthataspartner’slibidoincreases,sodoestheparticipant’slibido(asshownbytheslopeoftheregressionline).

WecanreporttheeffectofthecovariateinAPAformatas:

ü Thecovariate,partner’s libido,wassignificantly relatedtotheparticipant’s libido,F(1,26)=4.96,p=.035.


Output6

Figure6:Scatterplotofparticipants’libidoscoresagainstthoseoftheirpartner

Testing the Assumption of Homogeneity of Regression Slopes

To test the assumption of homogeneity of regression slopes we need to rerun the ANCOVA but this time use acustomizedmodel.Accessthemaindialogboxasbeforeandplacethevariablesinthesameboxesasbefore(sothefinishedboxshouldlooklikeFigure2).Tocustomizethemodelweneedtoaccessthemodeldialogboxbyclickingon

.Tocustomizeyourmodel,select toactivatethedialogbox(Figure7).Thevariablesspecifiedinthemaindialogboxarelistedontheleft-handside.Totesttheassumptionofhomogeneityofregressionslopes,weneed


tospecifyamodelthatincludestheinteractionbetweenthecovariateandindependentvariable.Hence,tobeginwithyoushouldselectDoseandPartner_Libido(youcanselectbothofthematthesametimebyholdingdownCtrl).Then,clickonthedrop-downmenuandchangeitto .Havingselectedthis,clickon tomovethemaineffectsofDoseandPartner_LibidototheboxlabelledModel.Nextweneedtospecifytheinteractionterm.Todothis,selectDoseandPartner_Libidosimultaneously(byholdingdowntheCtrlkeywhileyouclickonthetwovariables),thenselect

inthedrop-downlistandclickon .ThisactionmovestheinteractionofDoseandPartner_LibidototheboxlabelledModel.ThefinisheddialogboxshouldlooklikeFigure7.Havingspecifiedourtwomaineffectsandtheinteractionterm,clickon toreturntothemaindialogboxandthenclickon toruntheanalysis.

Figure7:GLMunivariatemodeldialogbox

SPSSOutput6showsthemainsummarytablefortheANCOVAusingonlytheinteractionterm.Lookatthesignificancevalueofthecovariatebydependentvariableinteraction(dose*partner),ifthiseffectissignificantthentheassumptionofhomogeneityofregressionslopeshasbeenbroken.Theeffecthereissignificant(p=.028);thereforetheassumptionisnottenable.AlthoughthisfindingisnotsurprisinggiventhepatternofrelationshipsshowninFigure1itdoesraiseconcernaboutthemainanalysis.Thisexampleillustrateswhyitisimportanttotestassumptionsandnottojustblindlyaccepttheresultsofananalysis.

Output7

Guided Example AfewyearsbackIwasstalked.You’dthinktheycouldhavefoundsomeoneabitmoreinterestingtostalk,butapparentlytimeswerehard.Itcouldhavebeenalotworsethanitwas,butitwasn’tparticularlypleasant.Iimaginedaworldin


which a psychologist tried two different therapies on different groups of stalkers (25 stalkers in each group—thisvariableiscalledGroup).Tothefirstgroupofstalkershegavewhathetermedcruel-to-be-kindtherapy(everytimethestalkers followedhimaround,or senthima letter, thepsychologist attacked themwitha cattleprod). The secondtherapywaspsychodyshamictherapy,inwhichstalkerswerehypnotisedandregressedintotheirchildhoodtodiscusstheirpenis(orlackofpenis),theirfather’spenis,theirdog’spenis,theseventhpenisofaseventhpenisandanyotherpenisthatsprangtomind.Thepsychologistmeasuredthenumberofhoursintheweekthatthestalkerspentstalkingtheirpreybothbefore(stalk1)andafter(stalk2)treatment.Analysetheeffectoftherapyonstalkingbehaviouraftertherapy,covaryingfortheamountofstalkingbehaviourbeforetherapy.

CrueltobeKindTherapy PsychodyshamicTherapy

InitialStalking StalkingAfterTherapy InitialStalking StalkingAfterTherapy

47 11 52 47

50 18 53 47

51 34 54 50

52 40 57 55

53 50 58 56

57 54 60 56

57 55 61 61

60 58 61 61

63 59 62 61

66 60 65 61

68 61 66 62

72 61 66 62

72 62 66 62

73 63 71 64

75 64 71 64

77 65 72 64

79 65 75 70

85 78 77 74

62 55 80 78

71 63 87 78

53 52 75 62

64 80 57 71

79 35 59 55

75 70 46 46

60 61 89 79

® EnterthedataintoSPSS.(Hint:Thedatashouldnotbeenteredastheyareinthetableabove).

® Savethedataontoadiskinafilecalledstalker.sav.


® Conducttheappropriateanalysistoseewhetherthetwotherapieshadasignificanteffectonstalkingbehaviourwhencontrolling for theperson’sgeneral tendency tostalk.

Whatis/aretheindependentvariable(s)andhowmanylevelsdotheyhave?

YourAnswer:

Whatisthedependentvariable?

YourAnswer:

Whatisthecovariate?

YourAnswer:

Whatanalysishaveyouperformed?

YourAnswer:

Hastheassumptionofhomogeneityofvariancebeenmet?(QuoterelevantstatisticsinAPAformat).

YourAnswer:

Reporttheeffectof‘therapy’inAPAformat.Isthiseffectsignificantandhowwouldyouinterpretit?

YourAnswer:


Reporttheeffectof‘initialstalking’inAPAformat.Isthiseffectsignificantandhowwouldyouinterpretit?

YourAnswer:

Someanswerstothisquestioncanbefoundonthecompanionwebsiteofmybook.

Unguided Example Amarketingmanagerwasinterestedinthetherapeuticbenefitofcertainsoftdrinksforcuringhangovers.Hetook15peopleoutonthetownonenightandgotthemdrunk.Thenextmorningastheyawoke,dehydratedandfeelingasthoughthey’dlickedacamel’ssandyfeetcleanwiththeirtongue,hegavefiveofthemwatertodrink,fiveofthemLucozade(averyniceglucose-basedUKdrink)andtheremainingfivea leadingbrandofcola (thisvariable iscalleddrink).Hemeasuredhowwelltheyfelt(onascalefrom0=Ifeellikedeathto10=Ifeelreallyfullofbeansandhealthy)twohourslater(thisvariableiscalledwell).Hemeasuredhowdrunkthepersongotthenightbeforeonascaleof0=assoberasanunto10=flappingaboutlikeahaddockoutofwateronthefloorinapuddleoftheirownvomit.

Water Lucozade Cola

Well Drunk Well Drunk Well Drunk

5 5 5 6 5 2

5 3 4 6 6 3

6 2 6 4 6 2

6 1 8 2 6 3

3 7 6 3 6 2

® EnterthedataintoSPSS.(Hint:Thedatashouldnotbeenteredastheyareinthetableabove).

® SavethedataontoadiskinafilecalledHangoverCure.sav.


® Conducttheappropriateanalysistoseewhetherthedrinksdiffer intheirabilitytocontrolhangoverswhencontrollingforhowmuchwasdrunkthenightbefore.

Someanswerstothisquestioncanbefoundonthecompanionwebsiteofmybook.

Multiple Choice Questions

Complete themultiple choice questions forChapter 12 on the companionwebsite to Field(2013):https://studysites.uk.sagepub.com/field4e/study/mcqs.htm.Ifyougetanywrong,re-readthishandout(orField,2013,Chapter12)anddothemagainuntilyougetthemallcorrect.

References Field,A.P.(2013).DiscoveringstatisticsusingIBMSPSSStatistics:Andsexanddrugsandrock'n'roll(4thed.).London:

Sage.

Miller,G.A.,&Chapman,J.P.(2001).Misunderstandinganalysisofcovariance.JournalofAbnormalPsychology,110(1),40-48.doi:Doi10.1037//0021-843x.110.1.40

Terms of Use Thishandoutcontainsmaterialfrom:

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

ThismaterialiscopyrightAndyField(2000-2016).

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

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

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Analysis of Covariance (ANCOVA) - Discovering … of Covariance (ANCOVA) Some background ANOVA can be extended to include one or more continuous variables that predict the outcome