MODELINGDECISIONSAMONGMANYALTERNATIVESByPeterKvamADISSERTATIONSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofPsychologyŒDoctorofPhilosophy2017ABSTRACTMODELINGDECISIONSAMONGMANYALTERNATIVESByPeterKvamManyoftheactionswetakedependonbeingabletomakeselectionsamongmanyalternativesorevenalongacontinuum.However,ourunderstandingofthedecisionprocessesunderlyingtheseselectionsissparse,largelyduetoatraditionalfocusondevelopingmodelsofbinarydecisions.Re-centforaysintomodelingmulti-alternativedecisionshavebeenforcedtobuildinrelationsbetweenrepresentationsofavailablealternatives.Inthispaper,Iproposeandtestageneralframeworkformodelingdecisionsbetweenarbitrarilylargenumbersofalternativesthatnaturallyincorporatespsychologicalrelationshipsbetweentherepresentationsofavailablealternatives.Inthestudy,Iconstructandevaluatethebasiccomponentsofamodelofthisprocessbyestablishingbench-markempiricalphenomenafordecisionsonacontinuum.Inthesecondstudy,Iexaminehowthenumberofalternativesandtherelationsbetweenthemaffectrepresentationsandthecomponentsofthedecisionprocess.Takentogether,thispaperestablishesbenchmarkempiricalresultsinanewchoicedomain(continuousselection),proposesandtestsanewmodelingframeworkthatac-countsforthesephenomena,andbringstogetherdecisionandrepresentationmodelsinordertodevelopanover-archingtheoryofhowpeoplemakedecisionsamongmanyalternatives.ACKNOWLEDGEMENTSTheauthorwouldliketothankTimothyPleskac,SusanRavizza,AndrewHeathcote,PhilipSmith,TaoshengLiu,ZachHambrick,andDavidJohnsonforhelpfulcommentsonthemanuscriptortheideascontainedherein.TheauthorofthethesiswassupportedbyafellowshipfromtheNationalScienceFoundation(GrantNo.1424871)whilepursuingthiswork.iiiTABLEOFCONTENTSLISTOFTABLES.......................................viLISTOFFIGURES.......................................viiCHAPTER1INTRODUCTION...............................11.1Benchmarkphenomena................................21.2Relativeevidencemodels...............................51.3Absoluteevidencemodels...............................81.4Multiplealternatives..................................111.5Geometricrepresentations...............................131.5.1Relativeevidence...............................131.5.2Absoluteevidence...............................161.6Randomwalks.....................................171.7Psychologicalspaces.................................201.7.1Randomwalksinpsychologicalspace....................211.8Hick'sLaw......................................23CHAPTER2GOALSANDPREDICTIONS........................272.1Study1........................................282.2Study2........................................29CHAPTER3STUDY1-EMPIRICALHURDLESANDMODELELEMENTS.....313.1Methods........................................323.1.1Task......................................333.1.2Participants..................................353.1.3Materials...................................353.1.4Procedure...................................353.2Results.........................................363.2.1Responsetimes................................373.2.2Accuracy/responsedeviation........................393.3Modeling.......................................413.3.1Parameter-freepredictions..........................443.3.1.1Homogeneousresponsetimes...................443.3.1.2VonMisesresponses........................453.3.2Freeeffectsmodel..............................463.3.3cue-inclusivemodel........................483.4Preliminarydiscussion................................51CHAPTER4STUDY2-EXPLORINGRELATIONSBETWEENALTERNATIVES...534.1Methods........................................544.1.1Participants..................................544.1.2Materials...................................54iv4.1.3Decisiontask.................................544.1.3.1Practicetrials............................574.1.4Similarityratingtask.............................584.1.5Procedure...................................594.2Results.........................................604.2.1Numberofalternatives............................604.2.2withdif...........................634.2.3Multidimensionalscaling...........................654.2.4Summaryofimportanteffects........................684.3Modeling.......................................694.3.1Responseboundaries.............................714.3.2Discriminability................................724.3.3Changesinthreshold.............................744.3.4Modelandresults...........................754.3.4.1PosteriorPredictions........................794.4Preliminarydiscussion................................84CHAPTER5GENERALDISCUSSION...........................865.1Limitations......................................875.2Extensions.......................................895.3Conclusions......................................91BIBLIOGRAPHY........................................93vLISTOFTABLESTable3.1Meanestimatesofcoefformaineffectsandinteractionsofstimulusdif,speedmanipulation,cuepresence,andcueorientationonresponsetimes.Therangescontainingthe95%HighestDensityInterval(HDI)arealsoprovided.Intervalsexcludingzeroarestarred....................38Table3.2Meanestimatesofcoefformaineffectsandinteractionsofstimulusdif,speedmanipulation,cuepresence,andcueorientationonaccuracyofresponses.Therangescontainingthe95%HighestDensityInterval(HDI)arealsoprovided.Intervalsexcludingzeroarestarred...............41Table3.3Maximumlikelihoodestimatesfortheparametersoftherestrictedmodel.....49Table4.1Observedresultsandthemodelcomponentsthatcanaccountforthem.Acheckmarkindicatesthatthemodelcomponentcanaccountforaneffect,ablankmeansitcannot,andanXmeansitcanproducetheoppositeeffect.....75Table4.2Bayesianinformationcriterion(afunctionoftheloglikelihoodwithapenaltyforthenumberofparameters;seeSchwarz,1978)foreachmodeltested.ThemodelwiththelowestBICispreferredandshowninbold.............77Table4.3MaximumlikelihoodparameterestimatesfortheBIC-favoredmodelofthedatafromStudy2..................................78viLISTOFFIGURESFigure1.1Diagramofthreebasicelementsofthediffusionmodel:drift,diffusion,andthreshold......................................6Figure1.2Diagramofbasicelementsofanaccumulatormodel:startingpointvariabil-ity,multipleaccumulators(withdrifts),andthreshold.Asshown,driftcanberepresentedasafunctionofsamplearrivalrate.Thisallowstheaccumu-latormodeltoalsobedescribedintermsofsamplingrateandprobabilityofsamplinginfavorofoptionA(oranyotheralternative)..............10Figure1.3Representationofaperson'saccumulatedevidenceandchoicecriteriafor2-,3-,and4-alternativediffusionprocesses(A,B,C)and2-and3-alternativeaccumulatorprocesses(D,E).Analternativeischosenwhenaperson'srep-resentedevidence(yellow/red)crossesthecorrespondingedge(formodelsA,B,andD)orface(formodelsCandE).....................14Figure1.4Latticeandstatetransitionstructurefora3-alternative,relativeevidencemodelofcolor...........................19Figure1.5Predictedresponsetimeproportionaltothenumberofbitsrequiredtodis-criminatebetweenalternatives.Itisunclearwhattheresponsetimepredic-tionshouldbeintermsofbit-basedinformationforthecontinuouscase,asHick'slawmaynotapplytothiscondition.....................24Figure3.1Timecourseofatrialoftheorientationdetectiontask...............34Figure3.2Responsetimes(left)andaccuracyintermsofdegreesdeviationfromthecorrectresponse(right)acrossdiflevelsandspeed-accuracymanipu-lation.Notethathighervaluesindicatelessaccurateresponses.Errorbarsindicatepooledstandarderroracrossparticipants.................39Figure3.3Meanresponsetimes(left)andabsoluteresponsedeviation(right)bytheorientationofthecue.Notethathigherresponsedeviationsindicatelessaccurateresponses.Errorbarsindicatepooledstandarderroracrossparticipants.40Figure3.4Staterepresentation,sampling,threshold,andevidencetrajectoriesfortheorientationmatchingtask..............................42Figure3.5Driftmagnitude(rate),threshold,andnon-decisiontimeestimatesforindi-viduals(colors)andthemeanofthegroup(black)inthefreeeffectsmodel...47Figure3.6Effectsofcue,speed-accuracy,anddifmanipulationsondriftmagni-tude,direction,andthresholdforindividuals(colors)andgroupmeaneffects(black)........................................50viiFigure4.1DiagramofthedecisiontaskinStudy2.Dependingonthecondition,thesewillbecomprisedof2,3,5,8,oracontinuousspanofalternatives.Partici-pantsrespondedbymovingtheirmouseacrossthearccorrespondingtothedesiredresponsealternative.............................56Figure4.2Layoutofthesimilarityratingtask.........................58Figure4.3Relationshipbetweenthenumberofalternativesandmeanresponsetimesinthecoloreddotmixturestask.Meanresponsetimesforthecontinuouscondition(forcomparison)areshownontheright.Eachcolorcorrespondstoadifferentparticipant..............................61Figure4.4Relationshipbetweenthenumberofalternativesandaccuracyinthecoloreddotmixturestask.Accuracyforthecontinuouscondition(forcomparison)isshownontheright.Eachcolorcorrespondstoadifferentparticipant.......62Figure4.5Relationshipbetweenthenumberofalternativesavailableandthediscrim-inabilityofthetargetalternative(hue-baseddistancetothenearestdistractor)..64Figure4.6MeanaccuracyandresponsetimeandaccuracyforParticipant5conditionedonthehueofthetargetcolor.Thecoloredlineisgivenbyakerneldensityestimator(bandwidth=0.1)passedoverthecorrespondinghueaccuracyorRTdata......................................66Figure4.7ResultsoftheMDSanalysisofthesimilaritytaskincludedaspartofStudy2foreachindividualparticipant.Thedistancebetweenanypairofhuesistheapproximatedissimilaritybetweenthehuesaccordingtothatparticipant.....67Figure4.8BasicelementsofthemodelsusedinStudy2.Inadditiontodriftmagnitude,threshold,andnon-decisiontime,decisionboundariesseparatedthedifferentresponsesintocorrectandincorrectresponses.Theseboundswereassumedtobeoptimal,sothattheywerehalfwaybetweentheavailablealternativesinthediscretechoicecases(2and3alternativesshownatthebottomleftandtopright,respectively).Inthecontinuouscase,theyweresetto7degreesfromthemostprominentdotcolor,tothecriterionof7degreestoleranceforresponsesinthecontinuouscondition.....................70Figure4.9Posteriorpredictionsfromthemodelsforeachparticipant,over-laidontheactualresponsetimeandaccuracydata.................80Figure4.10Posteriormodelpredictions(X)andobservedmeanresponsetimes(fadedlines)foreachparticipant(differentiatedbycolor).Errorbarsindicate1unitofstandarderrorinthedata.............................81Figure4.11Posteriormodelpredictions(X)andobservedaccuracy(fadedlines)foreachparticipant.Errorbarsindicate1unitofstandarderrorinthedata........82viiiFigure4.12Posteriormodelpredictions(dottedblackline)oftherelationshipbetweentargethueandresponsetimeoraccuracyforeachparticipant.Bothdata(col-oredcurve)andmodelpredictioncurvesarecomputedbypassingakerneldensityestimatoroverthehueRTorhueaccuracydata(orprediction)fromeverytrial...................................83ixCHAPTER1INTRODUCTIONManyofthetasksweaccomplishinthelaboratoryoroutintheworldrequireustomakeselectionsamongmanyalternativesoralongacontinuum.Whetherwearedecidinghowmuchtimeormoneytoinvest,outwhatdirectiontowalkordrive,reproducingtheorientationofastimulus,comingupwiththenextwordinasentence,orevenproducingamusicalnote,thereisaplethoraofoptionsavailable.However,ourunderstandingofthedecisionprocessesunderlyingthesesortsofselectionsissparse.Thisislargelyduetoatraditionalfocusondevelopingtheoriesandmodelsofbinarydecisions,whereapersononlyhastwoalternativesfromwhichtochoose.Thegoalofthisthesisistoremedythisissuebydevelopingandtestingageneralframeworkthatcanbeusedtomodeldecisionsamonganynumberofalternatives.Inordertobeginbuildingaframeworkformodelingdecisionsamongmanyalternatives,itishelpfultoonwhatapproacheshavebeensuccessfulinexplainingbehavioronbinarychoicetasks.Byfarthemostempiricallysuccessfulaccountsofthisprocessaresequentialsamplingmodels,whichareabletoreproducedecisionmakers'observedchoiceproportionsanddistribu-tionsofresponsetimeswithhigh.Thesectionofthispaperoutlinesthecurrentstateofsequentialsamplingmodels,lookingatbothdiffusionandaccumulatormodelsof2-choicedecision-making.Eachoftheseapproachestomodelingbinarydecisionbehaviorcarrieswithitasetofassumptionsabouthowchoicealternativesrelatetooneanother.Imaketheseassumptionsexplicit,andexaminetheimplicationstheycarryformodelsofmultiple-choicebehavior.Car-ryingthisfurther,Iextendbothdiffusionandaccumulatormodelstodealwithmultiplechoicealternativesinawaythatisconsistentwiththeirapproachtomodelingbinarydecisions.ThesetwoapproachescanberelatedtooneanotherbywayofrepresentingthemgeometricallyŒi.e.asthemovementofanevidencestateinamultidimensionaldecisionspace.Ivisitthedifferentrepresentationsthatallowbothdiffusionandaccumulatormodelstobeanalyzedasrandomwalkprocessesinthesedecisionspaces.Finally,Iprovideageneralframeworkforsequentialsampling1modelsofmulti-alternativeaswellascontinuous-responsedecisions,formedbyarandomwalkprocessunfoldinginamultidimensionalpsychologicalspace(e.g.,afeaturespace).Thisprovidesthebasisforconstructingmodelsofdecisionsamonganynumberofalternatives.Ofcourse,theframeworkbyitselfisperhapsuninterestingwithoutanyevidenceforitspre-dictions.Thesecondsectionofthepaperisdevotedtotheoftwoempiricalstudies,andisaimedatlayingtheempiricalgroundworkforamodelofdecisionsamongmanyalternatives.Itsetsbenchmarkempiricalresultsusingasimpleperceptualtaskwhereparticipantsmustmakeaselectiononacontinuum,testspredictionsofamodelofthisprocess,andappliesthemodeltotheresultingchoiceandresponsetimedata.Indoingso,Iestablishempiricalhurdlesformodelsofthisprocess,exploretheelements(parameters)thatallowthemodeltoaccountforpatternsofbehavioronthetask,andverifyseveralindependentpredictionsgeneratedbythemodel.Thethirdsectionfocusesonansecondstudyaimedatunderstandinghowtheavailablechoicealternativesinteractwiththedecisionprocess.Itexamineshowthenumberofalternativesaswellastheirpsychologicalrepresentationsaffectdecisions,lookingatbothaccuracyandresponsetimesinacolor-basedtask.Itintegratesarepresentationmodelbasedonmultidimensionalscalingwiththedecisionmodelinordertoexplainandpredictbehavioralchangesresultingfromdifferentcombinationsofavailablealternatives.Indoingso,itillustratestheimportanceofconsideringtherelationshipsbetweenalternativesandprovidesaformalmethodfordoingso.Thesectionofthethesisreviewstheandimplicationsofthetwostudies,con-sidersthelimitationsoftheframeworkandpresentstudies,suggestsavenuesforfurtherresearch,andconcludeswithanexaminationofthecontributionthisworkmakestowardunderstandingthecognitiveprocessesunderlyingdecisionsamongmanyalternatives.1.1BenchmarkphenomenaBeforelookingatthestructureofsequentialsamplingmodels,itisworthexaminingthebe-haviorsthattheyseektoexplain.Todoso,ithelpstoconsiderahostofrecurringempirical2phenomenathatserveasbenchmarksformodelperformance.Thebehavioraldatainbinarydeci-siontasksismostfrequentlya)theresponsethatapersonmakestoastimulus,andb)thetimeittakesthemtomakethisresponse.Muchofthemotivationforusingsequentialsamplingmodelsistheirabilitytopredictbothofthesecharacteristicsofdecisionssimultaneously.Inthecaseofinferentialdecisionswhereadecision-makerhastwoalternativesbetweenwhichtochoose,thereisonecorrectandoneincorrectresponse.Themetricforchoiceperformanceinthiscaseisaccuracy,whichissimplytheproportionofresponsesapersonmakesthatarecorrect.Whentheinformationthatdecision-makersreceiveisbetter,onecanexpectthattheiraccuracywillbehigherand/orthattheywillarriveatthecorrectdecisionmorequickly.Conversely,whenthequalityofinformationtheygetislower,decision-makersresponsemoreslowlyorwithloweraccuracy.Werefertothisstraightforwardrelationshipasthedifeffect.Itiswell-supportedbyempiricaldata,andcanoftenbedirectlyestimatedfromtheevidenceadecision-makerisgiven(seeBusemeyer&Townsend,1993;Krajbichetal.,2012;Link&Heath,1975;Palmeretal.,2005;Nosofsky&Palmeri,1997;Ratcliff,2014).Thisprovidesahurdlefordecisionmodels,indicatingthatametricdescribinginformationqualitymaybeimportanttoaccountforchoicebehavior.Whileaccuracyandresponsetimecanoftenbothimprovewitheasiertasks(yieldingfasterresponsetimesandhigheraccuracy),theycantradeoffwithinaleveloftaskdifty.Inthesecases,accuracysufferswhileresponsetimesgetfasterorresponsespeedsufferswhileaccuracyincreases.Thistrade-offarisesbecausehigheraccuracyoftendemandsmoreinformation,whereasfasterresponsetimessetalimitonhowmuchinformationadecisionmakercangatherbeforeresponding.Decision-makerscanimprovetheiraccuracyorresponsespeedbyadjustinghowmuchinformationtheygather,optingtorequiremore(moreaccurateresponses)orless(fasterresponses)informationbeforemakingtheirdecisions.Thisisreferredtoasthespeed-accuracytradeoff,wherepeoplecantheamountofinformationtheygathertorespondmorequicklyorviceversa(Bogaczetal.,2010;Heitz&Schall,2012;Vickers&Packer,1982;Wickelgren,1977).Thisprovidesasecondhurdlefordecisionmodels,andsuggeststhepresenceofaninternalcriterion3indicatingtheamountofinformationapersonrequirestomaketheirdecisions.ThephenomenonIexaminecoversdecisionbiasestorespondinaparticularway.1Theal-ternativetowardwhichapersonisbiasedwillexperiencefasterresponsetimes,andtheoneagainstwhichtheyarebiasedwillexperienceslowerones.Whenapersonisbiasedtowardthealternativewhichhappenstobecorrect(congruentbias),wecanexpectthattheywillbemoreaccurate,andlessaccuratewhenbiasedtowardanalternativewhichisincorrect(incongruentbias).Biasmayariseduetounbalancedrewardsforhitsversuscorrectrejectionsorunbalancedpenaltiesforfalsealarmsversusmisses(Diederich&Busemeyer,2006;Pleskac&Busemeyer,2010),differentbaseratesofstimuli(e.g.moretargetsthandistractorsJ.M.Wolfeetal.,2007),orsignalsoftruedifferencesinpriorplausibilityofvarioushypotheses.ThepatternofresultsarisingfromabiastowardonealternativeoveranotherŒfasterandmoreaccurate(congruent)orslowerandlessaccurate(incongruent)responsestowardthefavoredalternativesŒiswhatIwillrefertoasadecisionbiaseffect.Thisprovidesaempiricalhurdle,suggestingthatdecisionmodelsshouldhavesomemechanismallowingforapersontoholdapre-stimuluspreferenceorbelief.Althoughthesethreephenomenacertainlydonotconstituteanexhaustivelistofchoiceandresponsetimephenomena,theyrepresentthreeofthemostimportantempiricalphenomenathathaveguidedconstructionofbinarychoicemodels(Ratcliff&McKoon,2008).Theyareparticu-larlyinterestingbecauseeacheffectcorrespondstoadifferentcomponentsofthedecision-makingprocess.Difaffectstherateofinformationaccumulationinfavorofthevariousalternatives,thespeed-accuracytrade-offaffectstheinternalcriteriaparticipantsusetomaketheirdecisions,andthedecisionbiaseffectalterstheinformationthatapersonconsidersinfavorofaalternative.Inthenextsection,Ivisitseveralcommonandsuccessfulsequentialsamplingmodels1Inthispaper,biasreferstoatendencytorespondinfavorofonesetofalternativesoveranotherbyvirtueofsomeinformationexternaltothetargetstimulus.Itisnotintendedtorefertopatternsofnon-coherentbehavior,asthetermisusedinthejudgmentanddecision-makingliteratureonheuristicsandbiases(Tversky&Kahneman,1974;Gilovichetal.,2002).Infact,predecisionbiasesinrespondingmayactuallybehighlyadaptiveandcorrespondtoanoptimalpriorforagivendecision(Bogaczetal.,2006).4andexaminehowtheyaccountforeachoftheseeffects.1.2RelativeevidencemodelsSequentialsamplingmodelscanbebroadlybrokendownintotwocategories:relativeevidencemodelsandabsoluteevidencemodels(seeRatcliff&Smith,2004,foradetailedbreakdownofmodels).Thoughtheconclusionsthatonedrawsfromadoptingonetyperatherthantheothertendnottodiffer(Donkinetal.,2011),theassumptionsthattheymakeabouthowalternativesrelatetooneanotherisconceptuallyimportantwhenitcomestoderivingaframeworkformulti-alternativedecisions.Relativeevidencemodelslikethediffusionmodel(Ratcliff,1978;Ratcliff&McKoon,2008)positthatapersonrepresentstheinformationorevidencetheyhaveregardingthedecisionasabalancebetweentwochoicealternatives.Astheygatherpiecesofinformation,thisbalanceshiftstowardonealternativeortheother.Oncethebalanceshiftsfarenoughineitherdirection,ex-ceedingacriterionvaluebyaparticularthreshold,adecisionistriggeredandaresponseentered.Theirevidencestate,whichcanberepresentedasapoint,followsaparticularpathastheaccumulationprocessunfolds.Thisallowsustodisplaythechoiceprocessasanevidencetrajec-tory,whichmovesalongonedimensionbetweenfavoringalternativeAandfavoringalternativeB.TheaccumulationprocessdescribingthismovementisshowninFigure1.1.Oneofthedesirablepropertiesofrelativeevidencemodelsisthatthebalanceofevidencebe-tweenalternativessumstozero.Forinferentialdecisions,thismeansthattheevidencebalancebetweentwoalternatives[hypotheses]islinearlyrelatedtothelogoddsofonehypothesisrelativetoanother.2Thisderivesfromthediffusionmodel'srootsinthesequentialprobabilityratiotest(Edwards,1965;Laming,1968;Wald&Wolfowitz,1949),whichtreatsdecision-makingassequentialtestsofevidenceagainstacriterion.Thecriterioncorrespondstoadesiredinternal2Itshouldbenotedthatmanydiffusionmodelsdepartfromthelogoddsframework(Link&Heath,1975;Ratcliff,1978).Modelscanprovidegooddescriptionsofthedecision-makingprocesswithoutmakingexplicitreferencetoposterioroddsofthevarioushypotheses.5Figure1.1Diagramofthreebasicelementsofthediffusionmodel:drift,diffusion,andthreshold.levelofsetbythethresholdparameter.Foraparticulardesiredlevelofthesample-by-sampleevidenceaccumulationprocessconstitutesarandomwalkinlogoddsspace,de-scribinganoptimalBayesianbeliefupdatingprocedure(Bogaczetal.,2006;Palmeretal.,2005).Aswiththesimplebalanceofevidence,adecisionistriggeredwhenthelogoddsforonehypothe-sisovertheothersexceedsthecriticalthresholdvalueq.Theminimumboundqensuresthatthedecision-makercanbewithatleast(geq):1oddsthatthehypothesistheyhavechooseisthecorrectone,wheregisascalingfactorthatcorrespondstothedifofdiscriminationbetweenthetwohypotheses(ameasureofd'asinsignaldetectiontheory).Averysimplerelativeevidencemodelcanoftenbecharacterizedbyadriftrate,diffusion,initialstate,andthreshold(aswellasstartingpointvariability,non-decisiontime,anddriftratevariability;seeRatcliff&McKoon,2008).Thedriftratecontrolshowquicklytheevidencestatemovestowardoneboundaryortheotheranddiffusioncontrolstherandomnoise,whichmovestheevidencestateindiscriminatelytowardeitheralternative.Thethresholdsetstheamountofinformationthatapersonrequirestomakeadecision,andtheinitialstateindicateswherealongtheA-Bcontinuumapersonbeginsaccumulatingevidence.Thisinitialstateisfrequentlychosen6tosomepre-stimulusbiasinfavorofoptionAoroptionB,asitindicateswhataperson'sstateisbeforeconsideringanyinformation.Inthecasewheretheyareunbiased,itispreciselyinthemiddleofthethresholdscorrespondingtoAandB(inthecaseoflogoddsrepresentations,theinitialstateis0).However,ifapersonreceivessomepredecisioncueorhigherrewardforonealternativeovertheother,theymayfavorthatalternativeinabsenceofanystimulusinformation.Thisfavoritismisdescribedbythestartingpoint/initialstateoftheaccumulationprocess.Thediffusionrateisoftenedat1inordertosetthescaleofthemodelŒdrift,diffusion,initialstate,andthresholdareotherwiseconfusablewhenpermittedtovaryfreelywithinacondition.Therefore,the'practical'numberofparametersinthiscaseisonly3,anditisfrequentlythecasethatdiffusionwillbeignoredasacontributingfactorthedecisionsandresponsetimesbecauseitcannotbedisentangledfromtheothers.EachofthesecomponentsoftherelativeevidenceaccumulationprocessisshowninFigure1.1.Thethreeelementsofdrift,threshold,andinitialstateallowrelativeevidencemodelstoaccountforthethreechoicephenomenaIreviewedearlier.ThedifeffectcorrespondstochangesindriftrateŒlowerdifyieldshigherdriftratestowardthecorrectalternative,givingbothfasterandmoreaccurateresponses.Thespeed-accuracytrade-offcorrespondstoshiftsinthethresholdŒhigherthresholdsyieldslowerbutmoreaccurateresponses,whilelowerthresholdsyieldfasterbutlessaccurateresponses.Finally,thebiaseffectisachievedbymovingtheinitialstatearoundŒinformationorrewardfavoringAwillmoveittowardtheAthreshold,makingresponsesinfavorofAfasterandmorelikely(butdecreasingaccuracyifAisnotthecorrectalternatives).ThereverseistruewheninformationorrewardsfavoroptionB.Anoteregardingthediffusionrateisworthmaking.Ratherthanitto1,itisoftenmoremeaningfulandmoreeffectiveintermsofmodeltoeithersetdiffusionratesdirectlyfromthestimuliorconstrainthemtoveryinaparticularfunctionalwayacrossconditions(Donkinetal.,2009).Insuchacase,itisalsopossibletothinkofadiffusionprocessintermsoftheoverallsamplingrateandtheprobabilitythateachsamplefavorsoptionA(versusoptionB)ŒinKvam&Pleskac(2016),theseareexaminedasindependentfactorscalledweightandstrength,respectively.7Thisre-parametrizationishelpfulfordevelopingadiscreteMarkovrandomwalkformulationofthemodels(seeDiederich&Busemeyer,2003).Additionally,itenablesmorestraightforwardcomparisonswithothermodelsofthedecisionprocess,includingabsoluteevidencemodelsandthemoregeneralcontinuous-responsemodelsIdescribelater.1.3AbsoluteevidencemodelsAbsoluteevidencemodelslikeaccumulatormodels(Vickers&Lee,1998;Smith&VanZandt,2000;Brown&Heathcote,2005,2008),bycontrast,representinformationorevidencefortheal-ternativesasseparatequantities[accumulators].Asadecisionmakergathersorreceivesapieceofinformation,asinglevaluedescribingthealternativethatpiecefavorsisincremented,whilevaluesfortheotheralternativesareunaffected.Thesevaluesarefrequentlyreferredtoas`accumulators,'aseachonestoresthecumulativesumofinformationfavoringthecorrespondingalternative.TheaccumulationprocessiteratesforeachnewpieceofinformationŒasonecomesin,thecorre-spondingaccumulatorisupdated,andadecisionistriggeredonceoneoftheaccumulatorsreachesathresholdvalue.Itiscriticaltonotethateachoftheaccumulatorsisusuallyassumedtobeindependent,suchthatonlyoneaccumulator'svalueisupdatedwitheachnewpieceofinformation.ThisiswhatsetsthemapartfromdiffusionmodelsŒthetotalamountofinformationisretainedandspreadacrossthevariousaccumulators,allowingeachonetostoretheabsoluteamountofevidenceforthealternativeitmatches.Despitethismajordifference,thereareanumberofcommonmechanismsandanalogouspa-rametersthataresharedbetweenabsoluteandrelativeevidencemodels.Likerelativemodels,absoluteevidencemodelsfrequentlyusedriftandthresholdparametersinordertospecifytheev-idenceaccumulationprocess.Eachaccumulatorreceivesitsowndriftrate,thoughtheycanbesetsystematicallyusingfewerthannparametersfornaccumulators.3Thethresholdworksina3Forexample,itistypicaltoassumeonedriftratefortheaccumulatorcorrespondingtothe8similarwaytorelativemodelsŒonceaprocesscrossesit,therelevantalternativeisselectedŒbutincasesofabsolutemodelsthereisfrequentlyonlyonethresholdforallalternativesratherthanoneforeachalternatives.Inthiscase,itiswhichaccumulatorcrossesthelonethresholdratherthanwhichthresholditcrossesthatdetermineschoiceoutcomes.Despitethesimilaritiesintermsofdriftandthreshold,someabsoluteevidencemodelsdepartfurtherfromrelativemodelsintermsofthesourceofresponseandresponsetimevariability.Bal-listicaccumulatorslikethoseofBrown&Heathcote(2008)assumenowithin-trialvariabilityinevidenceaccumulationrateforanindividualaccumulator,sothediffusioncomponentisremovedinfavorofothermechanismsthatgenerateerrorsandresponsetimevariability.Inthesemod-els,psychologicalorstimulusnoisecausesnon-targetaccumulatorstobeincrementedmoreoften,sometimesallowingthemtoreachthresholdbeforethetarget(correct)accumulator.However,thisalonedoesnotnecessarilyleadtoerrors;thedriftrateforthetargetaccumulatorshouldstillusuallybethehighest.Inthesecases,errorscanbemadewhendistractor/incorrectaccumulatorsrandomlystartwithhighervaluesthanthetarget/correctaccumulators.Thisoccursduetovari-abilityintheinitialstateoftheaccumulatorprocesses,whichcausesthevalueofeachaccumulatoratthestartofadecisiontovaryfromtrialtotrial.Returningtothecaseofnon-ballistic,probabilisticaccumulators,thedriftratesoftheseaccu-mulatorscanalsobereplacedbyexponentiallydistributedinter-arrivaltimes(seeFigure1.2).Inthecasewhereeachnewsampleprovidesone`unit'ofevidenceforthecorrespondingaccumu-lator,theinter-arrivaltimessimplydescribetheamountoftimeittakesbetweeneachsequentialpieceofarrivinginformation.InaPoissonprocessŒacommonbasisforaccumulatormodels(Smith&VanZandt,2000;Smith&Vickers,1988;Vickers&Lee,1998)Œtheinter-arrivaltimesforaparticularaccumulatorwillbeexponentiallydistributedwithasingleparameterl.Onav-erage,thesefunctionsimilarlytodriftrates.However,duetothediscretenatureofsamplesandstochasticnatureoftheirarrivaltimes,asingleaccumulatormaybehavesomewhatdifferentlythanintheballisticcase.target/truealternativeandanotherforallofthedistractoraccumulators.9Figure1.2Diagramofbasicelementsofanaccumulatormodel:startingpointvariability,multipleaccumulators(withdrifts),andthreshold.Asshown,driftcanberepresentedasafunctionofsamplearrivalrate.ThisallowstheaccumulatormodeltoalsobedescribedintermsofsamplingrateandprobabilityofsamplinginfavorofoptionA(oranyotheralternative).Aswiththediffusionmodel,theparametersofasetofaccumulatorscanberepresentedintermsofanoverallevidencesamplingrateandtheprobabilitiesofobtainingevidenceinfavorofoptionAandB(thoughthelinearaccumulatorofBrown&Heathcote,2008,hasconstantsamplingratherthanindividualsamples,soitcanonlyberepresentedintermsofrelativerates).SupposethataccumulatorsAandBarePoissonprocessessothataccumulatorAisincrementedwithexponentiallydistributedtimebetweeneachincrement,Exp(l1),andaccumulatorBisalsoincrementedwithexponentiallydistributedinter-arrivaltimesExp(l2).TheycanbecombinedintoasinglePoissonprocesswithexponentiallydistributedarrivaltimesExp(l1+l2),wheretheprobabilityofincrementingaccumulatorAispA=l1l1+l2andtheprobabiltyofincrementingaccumulatorBis1pA(seeRoss,2014,,Chapter5).ThistransformationisalsoshowninFigure1.2.Aswithrelativeevidencemodels,accumulatormodelsfrequentlyaparameterinordertosetthescaleoftheevidenceaccumulationprocess.Themostcommonwayofdoingsoseemstobetothesumoftheaccumulationratesacrossaccumulatorsto1Œinthiscase,thedriftratesforindividualaccumulatorscanbeviewedastheprobabilityofthataccumulatorbeingincremented10next.Conveniently,thisrepresentstheevidenceaccumulationprocessintermsofasamplingrate(1)andprobabilitydistributionoverpossiblesamplingoutcomes(accumulators).Thismakesthesemodelseasilycomparabletothemulti-dimensionalrandomwalksIpresentinthenextsection.Perhapsthegreatestofthesemodelsisthattheirlikelihoodfunctionstendtobeeasilytractableandthereforethemodelsarestraightforwardtotoexperimentaldata.Aswiththediffusionmodel,thedifeffectismodeledasashiftindriftrateforthecorrectrelativetotheincorrectalternative.Correspondingly,thespeed-accuracytrade-offismodeledasachangeinthethresholdparameter:lowervaluesyieldfasterdecisionsandhighervaluesmoreaccurateones.And,decisionbiasforonealternativeismodeledasachangeintheparameterscorrespondingtotheinitialstateorasashiftintherelativevalesofthedriftrates.However,becausetherearemultipleinitialstates(oneforeachaccumulator),thestartingpointsanddriftratesmustbeadjustedseparatelytoaccountforabiasinfavorofonealternativeoveranother.1.4MultiplealternativesMorerecently,thesemodelshavebeenextendedtoaccountfordecisionsbetweenmultiplealterna-tives.Inthecaseofmodelswhereevidenceforalternativesisrepresentedasmultipleindependentaccumulators,thisisstraightforwardŒoneneedonlyaddanadditionalaccumulatorforeachad-ditionalchoicealternative.Adecisionisstilltriggeredonceanyoftheseaccumulatorsreachacriticalthreshold,sothisadditioncanbedoneadoratleastuntilamodelerrunsoutofcomputationalresources.Forrelativeevidencemodels,addingalternativesisslightlymorecomplicated.Ina3-alternativecasewithoptionsA,B,andC,ratherthandecreasingthelogoddsofBby1everytimethelogoddsofAincreaseby1,onewouldhavetodecreasethelogoddsofBandCeachby12whenthelogoddsofAincreaseby1.Asmorealternativesareaddedsothattherearenoutcomes,thisdecrementdecreasessothatincrementingevidenceforasinglealternativeby1alsodecrementsallotheralternativesby1n1.ThesamedecisionruleappliesŒoncethe(logodds)evidenceinfavor11ofonealternativeexceedsacriticalthreshold,adecisionistriggered.Althoughassumingindependenceofaccumulators(absoluteevidence)oratleastauniformdistributionofnegativeevidence(relativeevidence)areconvenientsimplifyingassumptions,theyareperhapsunrealistic.Asubstantialbodyofliteraturehassuggestedthatthereareoftenstrongandunbalancedinteractionsbetweendifferentpairsofitemsinaset.Forexample,contexteffectsarisingfromtheinclusionofathirdalternativeŒsuchasdecoy(Huberetal.,1982),compromise(Simonson,1989),andsimilarityeffects(Tversky,1972)Œsubstantiallyalterchoicesbetweenanoriginalsetoftwo(seee.g.Truebloodetal.,2014).Similarly,inabsolutetasks,adjacentcategories(e.g.50-60and60-70)interactmorestronglythannon-adjacentones(50-60and80-90)(Brownetal.,2008).necessarilyinvolvesresponseswithatleastordinalrelationstooneanother,socategorieshavetobestructuredwithsomeadjacencypropertiesaswell(Pleskac&Busemeyer,2010;Ratcliff&Starns,2009).Modelsofdecisionmakinghavebeeninanumberofwaystoaccountforthesephenomena.Forexample,decisiontheory(Busemeyer&Townsend,1993;Busemeyer&Diederich,2002)introducedanadditionalstepinthedecisionprocesswherepairsofitemsarecontrastedagainstoneanotherbeforecomputingaccumulatorvalues.Theleakycompetingac-cumulatormodel(Usher&McClelland,2001,2004)introducedcompetitionandlossaversiontoasimilareffect,themulti-attributelinearballisticaccumulatormodel(Truebloodetal.,2014)includespairwisecomparisonsaswellassubjectiveattributevalues,andtheselectiveattention,mapping,andballisticaccumulationmodel(Brownetal.,2008)andutilizesadjacencybetweencategoriestotheevidenceaccumulationprocessforseparateaccumulators.Sim-ilarly,modelsof(seePleskac&Busemeyer,2010;Ratcliff&Starns,2009)specifyadjacencyofjudgmentcategoriesusingordinalstatesoraccumulators.Whatallofthesemodelshaveincommonisthattheyspecifypsychologicalrelationshipsbe-tweenalternativesinachoiceset.Indeed,Truebloodetal.(2014)noteexplicitlythatdecisionsaremadebetweenthepsychologicalrepresentationsofalternativesratherthanphysicalones,andthatthisisthekeycomponentallowingthedifferentmodelstoaccountforcontexteffects.These12modelseachaddcomponentsinordertoavoidmakingthesimplifyingassumptionsofindepen-denceanduniformnegativeevidence.Instead,theysuggestthatevidenceforalternativeAmayalsobeevidenceforalternativeBandsimultaneouslybestrongevidenceagainstalternativeC,forexample.Inthenextsection,Iexaminehowthepsychologicalrelationshipsbetweenalternativescanbethoughtofasgeometricrelationsinapsychologicalspace.1.5GeometricrepresentationsInordertointroduceageometricrepresentationofthedecisionprocess,Ireturntothesimplebinarycases.Usingthebinarychoicediffusionmodel,onecanestablisharuletoconstructmodelsrelatingaccumulatoranddiffusionmodels,constructgeometricmodelsofevidenceaccumulationamongmultiplealternatives,andinturnderiveamethodformodelingevidenceaccumulationwhenthenumberofalternativesisverylargeorcontinuous1.5.1RelativeevidenceThebasicuni-dimensionaldiffusionmodelusedinrecognitionmemoryandotherareasofpsychol-ogy(Ratcliff,1978)originallydescribedthebehaviorofphysicalparticlesalongasingledimensioninspace.Itsnaturalgeometricanalogueisarandomwalkonaline(Figure1.3A)(Link&Heath,1975).Ininferencetasks,thissingledimensionmaycorrespondtothelogoddsofonehypothesis(H1)relativetoanother(H2).Putsimply,thecloseraperson'srepresentationofevidence(orpreference)istoaboundarycorrespondingtoachoicealternative,themoretheycurrentlyfavorthatalternative.EvidencethatprovidessupportforanalternativeH1movesaperson'sstatedirectly(atanorthogonalangle)towardtheboundarycorrespondingtoH1indirectionD1andawayfromtheboundaryH2,whichisindirectionD2.Thisgivesrisetoaspatialrelationshipbetweentheamountofevidence(Ev)13Figure1.3Representationofaperson'saccumulatedevidenceandchoicecriteriafor2-,3-,and4-alternativediffusionprocesses(A,B,C)and2-and3-alternativeaccumulatorprocesses(D,E).Analternativeischosenwhenaperson'srepresentedevidence(yellow/red)crossesthecorrespondingedge(formodelsA,B,andD)orface(formodelsCandE).aparticularpieceofevidencefavoringhypothesisH1yieldsandthecorrespondingamountofevidenceitprovidesforhypothesisH2:Ev(H2)(Ev(H1)cos(\D1D2)(1.1)Thecosineofthisangle\D1D2canbequitenaturallyviewedasthesimilaritybetweenthetwochoicealternativesŒthecosinefunctionhasbeenusedasametricforsimilaritybetweenvectorsinafeaturespace,especiallyinlatentsemanticanalysis(Wangetal.,2008;M.B.Wolfe&Goldman,2003)aswellasquantummodelsofsimilarity(Pothosetal.,2013).TheevidencethatinformationfavoringH1providesforH2isthereforedirectlyrelatedtohowsimilarthesealternativesare.Inthetwo-alternativediffusionmodel,thetwochoiceoptionsareviewedasperfectlydissimilarŒtheevidencefavoringalternativeH1mustmovethestateintheoppositedirectionfromevidencefavoringalternativeH2,sothatcos(\D1D2)=1.Inordertoobtainthecaseforthreealternatives,whereevidenceforH1decreasesthelogoddsofH2andH3equally,itmustbethecasethatcos(\D1D2)=cos(\D1D3)=12(andof14courseitwillalsobethecasewithcos(\D2D3)toconservetotallogodds).4Thisresultsinanevidenceaccumulationprocessthatunfoldsinaplane,containedwithinanequilateraltriangle(Figure1.3B).Adecisionistriggeredwhenastatecrossesoneofthesidesofthetriangle,eachofwhichcorrespondstochoosingoneofthealternatives(seeLaming,1968,forasimilarproposal).Extendingthisstrategytomodeldecisionsbetweenalargernumberofalternativesisrelativelystraightforward.Inordertoaccountfordecisionswheretherearenalternatives,onemustcreateasituationwheretherearendirectionsfD1;D2;:::;Dngsatisfyingthepropertycos(\DiDj)=1n1foralli6=j.Inthiscase,evidenceforanyindividualalternativeprovidesevidenceagainstallothersalternativesequally.Inthecaseof4alternatives,theboundariescorrespondingtoH1,H2,H3,andH4wouldeachconsistofaplaneina3-dimensionalspace,togetherformingatetrahedron(Figure1.3C),andevidenceaccumulationwillunfoldina3-dimensionalspace.Inordertoaccommodatenalternatives,thiswouldnaturallybeextendedtopermitevidenceaccumulationin(n1)dimensions.Thestatewouldexistintheinteriorofasimplex(thegeneralversionofatriangleortetrahedronin4+dimensions),withthechoiceboundariescorrespondingtoeachofits(n2)-dimensionalfacets.ItisworthanotethatthecosinerelationinEquation1.1willpreservelogoddsinanyn-dimensionalspacebyvirtueofeveryintegralRR2p0cos(f)df=0.However,caseswheredeci-sionboundsdonotformregularshowthatlogoddsarepreservedacrossthetheoreticallypossiblespaceofalternativesbutnotnecessarilyacrossallavailableones.Ifwesupposethatthelogoddsofdifferenthypotheses(givenstoredevidence)arethebasisofprobabilityandjudgmentsŒanoptimalstrategyandoneadoptedbyseveralcognitiveandneuralmodelsofcon-judgmentproduction(Bogaczetal.,2006;Edwards,1965;Kepecsetal.,2008;Kiani&Shadlen,2009;Meynieletal.,2015;Pleskac&Busemeyer,2010)Œthenthismayleadtosubad-ditivityorsuperadditivityofthesejudgmentsamongsetsofpsychologicallyrelatedalternatives.4Notethatthisapproachoffersanalternativesolutiontotherelative-accumulatorproblemen-counteredbyNosofsky(1997),wheretherewasmorenegativeevidencethanpositiveevidenceaddedacrossaccumulatorsifincrementsanddecrementswererestrictedtovaluesofone.151.5.2AbsoluteevidenceWhileitisoftenpracticallyunnecessarytoenvisionaccumulatormodelsinageometricway,do-ingsoillustratesthepsychologicalassumptionsthatgointothesemodelsandallowsthemtobeanalyzedasarandomwalk.Whenthereare2alternatives,H1andH2,evidenceinfavorofH1pro-videsnoinformationthatchangesaperson'sbeliefsaboutH2.Usingtherelationshipbyequation1.1,thismeansthatthedirectionscorrespondingtoeachalternativemustbeorthogonal,cos(\D1D2)=0.Thechoiceboundariescanthereforeberepresentedastwosidesofarectangle(Figure1.3D),wherethepositionofaperson'sstatealongonedimension(left/right)describesevidenceforonealternativeandthepositionalongtheother(up/down)describesevidencefortheother.However,theevidencestatedoesnotimmediatelyhaveaclearlogoddsinterpretationasitdidinthediffusionmodels.OnecouldpotentiallyaddressthisbyassumingthattherearetwotheoreticalalternativesindirectionsD1andD2(ifD1andD2aregivenbyvectors)andanchorlogoddstobezeroatsomereferencepoint(theinitialstateposition,forexample).Thiswouldallowcomputationofrelativelogoddsofthehypothesesinthecaseapersonwantedtomakearelativejudgmentofthetwoalternatives(e.g.preferenceorHowever,doingsoisnotnecessaryforpredictingchoicesandresponsetimes.Extendingaccumulatormodelstothreeormorealternativesisstraightforward.Oneneedonlyaddadditional,orthogonaldimensionstotheevidenceaccumulationspace,thensetthechoicecriterionandnewdirectionDnforeachnewalternativeasorthogonaltoexistingones.Asinthetwoalternativecase,logoddscouldbepreservedbyalsoallowingatheoreticalalternativedirectionDn.Inthecaseofthreealternatives,theevidenceaccumulationprocesswouldunfoldonaboundedbythreechoicecriteriaconstitutingsidesofarectangularprism(Figure1.3E).Fornalternatives,theorthogonalchoicecriteriawouldcomposeasetofintersectingfacetsofann-orthotopeorinspecialcasesann-cube.161.6RandomwalksAnimportantconferredbythisgeometricwayofconstructingresponsealternativesisthattherelativeandabsoluteevidencemodelscanbeconstructedasdifferenttypesofrandomwalkmodels.Twoimportantdistinctionscanbemadebetweendifferentrandomwalkmodelsofdecision-making.Theisbetweencontinuous-timeanddiscrete-timemodels.Sofar,Ihavecoveredprimarilycontinuous-timemodels,whichexaminehowevidencechangescontinuouslyacrosstime.Theydosobyspecifyingdistributionsthatdescribehowfrequentlyevidenceisup-datedwithnewsamplesortherateatwhichevidencechangesovertime.However,discrete-timemodels,whichbreaktheevidenceaccumulationprocessintodistinctunitsoftime,arealsouse-ful.Forexample,theycanbeusedtodescribehowevidenceisupdatedsamplebysample,aswithdrawingcardsorreceivingdiscretechunksofinformation(Diederich&Busemeyer,2003;Markantetal.,2015).Theprimaryfunctionaldifferencebetweenthetwoapproachesisthatcontinuous-timerandomwalkspredictcontinuousresponsetimes(inminutes,seconds,etc.)asafunctionofdistributionalassumptionsabouthowevidencearrives,whilediscrete-timewalkspredictdiscreteresponsetimesintermsofthenumberofstepsittakestoarriveatadecision.Theseconddistinctionbetweentypesofrandomwalksisdiscretespaceorcontinuousspacerandomwalks.Discrete-spacewalksdescribehowevidencechangesovertimeusingasetofindividualevidencestates.Thesearemostfrequentlyusedwhenthereisameaningfulsetofnon-overlappingcognitivestatesthatapersoncouldbeinwhengatheringevidence.Formally,theyaredescribedbyamixedstatethatgivestheprobabilityofbeingineachparticularstateatagiventime.5Forexample,severalmodelsuseseparateevidencestatesthatcorrespondtodifferentlevelsofallowingforthedistributionofevidenceacrossstatesatanygiventimetogiverisetoadistributionofpossiblejudgmentresponses(Busemeyeretal.,2006;Kvam5Notethattheprobabilitiesinthemixedstatethemodeler'sŒnotthedecision-maker'sŒuncertaintyabouttheirbeliefsorpreferences.Thetruelocationoftheevidenceispresumablyedinthedecision-maker'ssystem(thoughseealsoKvametal.,2015),anditthedecision-maker'slevelofevidence.17etal.,2015;Pleskac&Busemeyer,2010).Continuous-spacerandomwalksaremorefrequentlyusedwhensuchdiscretelevelsofevidencearenotnecessary.Theycanalsobeconceptuallysimpler,asevidencestatesincontinuousspacefrequently(barringtheadditionofunusualstate-ortime-dependentfunctionsintheevidence)canbedescribedusingcomputationallyconvenientdistributionslikeanormaldistribution.Thecontinuousanddiscretespacerepresentationsarecloselyrelated,however;adiscrete-spacemodelconvergestoacontinuous-statemodelasthe(unbounded)statespaceismoreandmorepartitioned,i.e.asthenumberofstatess!¥.ThegeometricframeworkIhaveproposedlendsitselftobothdiscrete-timeandcontinuous-timeaswellasdiscrete-spaceandcontinuous-spacerandomwalkrepresentations,allowingitcon-siderablexibilityintermsofthetypesofresponsesanddecisionprocessesitcanbeusedtodescribe.ForthemodelsIhavedescribedsofar,adiscrete-spacerepresentationŒusefulforgener-atinganalyticdistributionsofresponsesandresponsetimesŒcanbegainedbyimposingalatticestructureuponthem.Forexample,supposeweareinterestedinusinganequalrelativeevidencediffusionmodeltodescribehowapersonsortsacolor-basedstimulusintothreecategories:red,blue,orgreen.ThiscorrespondstothetriangularstructureshowninFigure1.3B.Inordertopro-duceadiscreterandomwalkinthisspace,onecanconstructatriangularlatticeboundedbythethreechoicecriteria.Inthiscase,aperson'sinternalrepresentationofthestimulusintermsofthelogoddsofthethreehypotheses(red,green,blue)correspondstotheirpositiononthelattice.Initially,theymightstartoutinthemiddle,correspondingtoanunbiased0/0/0logoddsdistributionoverthehypotheses,butastheyviewthestimulustheyshouldsamplepiecesofevidencethatfavorgreen,red,orblue,causingthemtostepatanglesp6radians/30degrees,5p6radians/150degrees,or9p6radians/270degreesonthelattice.Theprobabilityoftakingastepineachdirectionisgivenbyp(forgreen),q(forred),and1pq(forblue,tosumtoone).Theedgesofthelatticebychoicecriteriaconsistofabsorbingstates;uponenteringoneofthesestates,thepersonhaltsthetransitionprocessandselectsthecorrespondingalternative.ThisstatetransitionprocesscanberepresentedasaMarkovchainmuchasthe2-alternative18Figure1.4Latticeandstatetransitionstructurefora3-alternative,relativeevidencemodelofcolordiffusionmodelis(seeDiederich&Busemeyer,2003),withthecaveatthateachstatehasthreeratherthantwotransitiondestinations.Itcanbeimplementedasacontinuous-timerandomwalkbyintroducingthestandardexponentiallydistributedtransitiontime(requiringoneadditionalsam-plingrateparameter),allowingpredictionofchoiceprobabilitiesaswellascontinuousresponsetimesforeachofthethreepossiblechoicealternatives.Boththerelativeandabsoluteevidencemodelswithanynumberofalternativescanbede-scribedinasimilarwayŒmanyoftheseareillustratedinFigure1.3.Thelatticeshapewillchangebasedonthetypeofmodelandnumberoftransitiondestinations,andthenumberoftransitiondestinationsateachstepwillgrowalongwiththenumberofalternatives,butthesamplingrateandthresholdsoperateinasimilarway.Notethataperson'sevidencerepresentationwillonlybeabletosteptowardavailablealterna-tives,notdirectlyawayfromthemaswasthecaseinthetwoalternativediffusionmodel.InFigure1.4,thisisindicatedbyunidirectionaltransitions.Similarly,inaccumulatormodelsliketheoneshowninFigures1.3D-E,evidencecanonlyfavoroneofasetoforthogonalalternatives,meaningthattherewillneverbeevidencesampledagainstaparticularone.Atpresent,theaccumulation19processbyabsoluteevidencemodelswillonlyconsistofmovementinhalfoftheavail-abledirections(e.g.rightwardorupwardinthetwoalternativecase,Figure1.3D),thoughthisassumptioncouldbetogenerateinterestingchoiceprocesses.Eachoftheserandomwalksrequiressomeinitialstate.Thiscanbeaedpointoritcanbeasamixedstateofasetofpossibleinitialpoints.Themixedstate,dependingonhowvariableitis,isoftenusedtoproduceerrorresponsesthatareasfastorfasterthancorrectresponses.Biasintheinitialstate,correspondingtoatendencytoselectsomealternativesoverothersinabsenceofanymitigatinginformation,canbeintroducedbymovingtheinitialstateclosertooneboundaryoranother.However,thisisnottheonlywaytointroducebiasintothesystem.Itcaninsteadbeintroducedasasymmetrictransitionprobabilities,indicatingthatmoreinformationisgatheredinfavorofonealternativeovertheother.Thisbiasisincludedinthedriftrateontopoftheinformationprovidedbythestimulus.Supposethatdscorrespondstotheinformationfromthestimulus,anddbcorrespondstothebiasedinformation.Ordinarily,thedriftratewouldbedswhenthestimulusfavoredoptionAanddswhenitfavoredoptionB.However,whendriftbiasfavorsoptionA,itsdriftratewillbeds+dbwhenthestimulusfavorsoptionAbutds+dbwhenthestimulusfavorsoptionB.1.7PsychologicalspacesThusfarIhavefocusedmainlyonthegeometricstructureofmodelswhereevidenceforonealter-nativehasnoneteffectontheevidencebalancebetweenotheralternatives.However,asIcoveredinthesectiononmulti-alternativemodels,thisisoftenanunrealisticassumption.Returningtocolorcategorization,supposethataparticipantmustmatchastimulustooneof4categories:red,yellow,green,orblue.Onemightexpectthatastimulusemittinglightpeakingatawavelength610nm(orange)wouldprovideevidenceinfavorofbothfiredflandfiyellowflresponses,butprovideevidenceagainstafiblueflorafigreenflresponse.20Insuchacase,itmakeslittlesensetotreatred,yellow,blue,andgreencategoriesasindepen-dentorequallyrelatedalternatives.Instead,theymustberelatedtooneanotherbyconstructingapsychologicalspacedescribingthecognitiverepresentationsofthestimulusandchoicealterna-tives.Doingsorequirestwogeneralizationsoftheframeworkdescribedintheprevioussection.First,thedirectionscorrespondingtoalternativesarepermittedtovary.Thereareseveralwaystodoso.Forexample,theycouldbereleasedtovaryasfreeparametersŒin2dimensions,thiscouldsimplybetheanglerelativetoareferencedirection,thoughthiswouldrequiremoreparameterswhenmovingto3ormoredimensions.Alternatively,thedirectionscouldbesetaprioribythemodeler.Thiscouldbedonebyusingthephysicalcharacteristicsofthestimulus,arrangingthemspatiallybytheirfeatures.Thedirectionscouldalsobeconstructedbyusingexistingpsychologicaltheoryorindependentdatalikesimilarityjudgments.Ipursueeachoftheseapproachesinthestudiesthatconstitutetheempiricalcomponentofthispaper.Thesecondoftheframeworkaboveistheassumptionthataperson'srepresen-tationofevidenceistodirectlytheinformationgatheredfromthestimulus.Forexample,ifapersonistryingtoreproducetheorientationofastimulusthatcanvaryanywherefrom0to180degrees,theymustbeabletosampleandrepresentinformationthatfavorsanydirectionbetween0and180degrees.Thiswilloftenmeanthatdiscrete-stateMarkovchainrepre-sentationsoftheevidenceaccumulationprocessarenolongerpossible,exceptasapproximationsorintherarecasethatthechoicealternativesarearrangedinapsychologicalspacesothattheirorientationsallowforaconvenientlatticetobesuperposeduponthem.1.7.1RandomwalksinpsychologicalspaceInsteadoftheusualunidimensionalrandomwalk(orMarkovchain)representation,evidenceaccu-mulationinapsychologicalspaceisenabledbyutilizingamultidimensionalrandomwalk.Inthisframework,aperson'scognitivestaterepresentingevidencetheyhavegatheredfromastimulusisdescribedbyapointinamultidimensional(e.g.,feature-based)space.Asapersonintegratesa21newpieceofinformation,thisstaterepresentationisupdatedbydrawingarandomvariablefthatdescribesadirectioninthespaceandmovingoneunitofdistanceinthatdirection.Thedistributionoffisdeterminedbythestimulusandthepsychologicalspaceinwhichitisrepresented.Thearrivaltimeofeachpieceofevidenceisagaindescribedbyanexponentialdistribution,Exp(l).Asbefore,oncethestaterepresentationcrossesoneoftheboundariescorre-spondingtoachoicealternative,thatalternativeischosen,yieldingachoiceandresponsetime.Muchasthediscreterandomwalkconvergestoacontinuousdiffusionprocessasthestepsizebecomessmall,thesamplingdistributionapproachesadiffusionprocessastheunitofdistanceforeachstepofthemultidimensionalrandomwalkbecomessmall.Thisdiffusionprocesscanbedescribedusingadriftdirection,whichwhatinformationisbeingaccumulatedmostoften,anddriftmagnitude,whichhowrapidlythisinformationisaccumulated(signaltonoiseratio).ThisdiffusionprocessisdescribedbySmith(2016),andithasanumberofinterestingproperties.Ifwesupposethatapersonisusingcirculardecisionboundsandthatdecision-makingstartsinthecenterofthiscircle,thetimeittakesthemtoreachadecision(hittingtheedgeofthecircle)isindependentofwheretheyhittheboundary.Thisisthecontinuousanalogueofsymmetriccorrectandincorrectresponsetimesthatisfoundindiffusionmodelswithnostartingpointorbetween-trialdriftratevariability.WhiletheevidenceatanygivenpointintimecanbegivenbyitsrectangularorCartesiancoordinates(x,y,z,etc.),itisoftenmoreconvenienttorepresentitintermsofpolarorsphericalcoordinates,whereonecoordinatedescribesitsdistancefromtheoriginandtheothersdescribeitsangle.Inthecaseofthetwodimensionalmodel,thisgivesaradiusrandsingleangleq(notethatthisshouldnotbeconfusedwiththethresholdparameterdiscussedearlier,whichdescribestheradiusatwhichadecisionistriggered).Animportantcharacteristicofthetwo-dimensionalmodelisthatthedistributionofevidenceatanygivenpointintimeshouldavonMisesdistributionoveritsqcoordinate.Thisistrueduringevidenceaccumulationandatthetimeofchoice,meaningthatthedistributionofresponsesonacircleshouldalsobedescribedbyavonMisesdistribution.Bothoftheseconstituteaprioripredictionsofthecurrent2-dimensionaldiffusionmodel:re-22sponsetimesshouldbeuncorrelatedwithresponseerror(accuracy)withinacondition,andre-sponsesshouldbevonMisesdistributedwhenentered.Asanimportantgoalofthispaperistodevelopanempiricallyaccuratemodelofdecisionsonacontinuum,Itestbothofthesepredictionsofthemodelontopofthethreeempiricalphenomenadescribeearlier(difeffect,speed-accuracytrade-off,decisionbiaseffect)inacontinuousresponsetask.Study1investigatesthese5predictions.1.8Hick'sLawOneprominentphenomenonthatrelatescloselytothemodelsIhavedescribedhereiscalledHick'sLaw(Hick,1952),whichstatesthatthetimerequiredtoreachadecisionincreasesasafunctionofthelogofthenumberofalternatives.Thisrelationshipwasoriginallyexplainedintermsofthenumberofbits(Shannoninformation,Shannon&Weaver,1949)itwouldrequireinordertodistinguishbetweennalternatives.Intheory,itshouldrequire1discriminatingbittodistinguishbetween2alternatives,2bitstodiscriminatebetween4,3bitsbetween8,andsoon,yieldingalogbase2relationbetweenthenumberofalternativesandthediscriminationcomponentofresponsetime.ThisrelationshipisshowninFigure1.5.Alternatively,Usheretal.(2002)showedthattherelationshipmightalsobeexplainedintermsofaccuracyandthenumberofaccumulatorsrequiredtorepresentevidenceforeachofthemŒthethresholdparameterinanaccumulatormodelwouldhavetobeincreasedincrementallywiththenumberofindependentaccumulatorsaddedinordertomaintainthesamedesiredlevelofaccuracy.Therefore,ifparticipantssettheirinternalchoicecriteriasothattheycouldmaintainaconstantlevelofaccuracy,thiswouldproducethelog-linearrelationshipthatHickandothersobservedbetweenthenumberofalternativesandresponsetime.Study2investigatesHick'slawandhowitrelatestobothdiscretesetsandacontinuousspanofalternatives.Ofcourse,thenumberofalternativesinthecontinuouscaseleadstothebizarrepredictionthatresponsetimesshouldbeessentially(log2(¥)=¥).However,peo-23Figure1.5Predictedresponsetimeproportionaltothenumberofbitsrequiredtodiscriminatebetweenalternatives.Itisunclearwhattheresponsetimepredictionshouldbeintermsofbit-basedinformationforthecontinuouscase,asHick'slawmaynotapplytothiscondition.plecouldbeusingasmaller`practical'numberofalternatives.Ifso,itisunclearwhatnumbertheymaybeusingtoapproximateacontinuousspanofalternatives.Assuch,anyinthecontinuousconditionisinformative,whetherittakesaparticularlylongtimeorsomehowfallswithintherangeofdiscretechoiceconditions.ThemodelingcomponentofStudy2alsoinvestigatespossibleexplanationsforwhyresponsetimeschangesasafunctionofthenumberofalternatives.AsUsheretal.(2002)suggested,Iinvestigatewhethershiftsinthresholdsmayberesponsible.However,Ialsoproposeandtestanotherpotentialexplanation.Itislikelythatalargernumberofalternativescan'clutter'thedecisionspace,suchthatalternativestendtobemoresimilarasmoreareintroducedwithinthesamerange(e.g.2randomvaluesdrawnbetween0and1willonaveragebefurtherapartthan3,4,or5randomvalues).Assuch,pairsofalternativeswithinalargersetonthesamerangewillonaveragebemorediftodistinguishfromoneanother.Inpracticalterms,thismayhavetwoeffects.Theisthattheportionoftheresponsespacedevotedtoeachalternativeissmaller,andthereforeeachoneislesslikelytobeselected24whenselectiondependsonarandomvariable.Forexample,considerasituationwheretherearetwoalternativesŒfigreaterthan50flandfilessthan50flŒaspredictionsforarandom0-100draw.Comparethistothecasewheretherearethreealternatives,filessthan33florfi33-66florfigreaterthan66.flInallcases,theprobabilityoftheandlastoptionsinthe3-alternativecasewillbelessthanorequaltotheandsecondinthe2-alternativeone.Asimilarconceptmaybeatplaninthecaseofmultiplealternativesinothersituations.However,thisonlyexplainsdifferencesinaccuracy,whichcanonlybecomedifferencesinresponsetimeifparticipantsmakesometrade-offinordertomaintainaccuracyattheexpenseofresponsespeed.Asecondeffectofresponsecrowdingisthattheremaybeashiftinthediscriminabilityofalternativesasafunctionofsetsize.Thiscorrespondstochangesintheratioofsamplingcorrecttoincorrectinformation,achangeindriftrateacrossconditionsratherthanthresholds.Themajordifferencebetweentheseexplanationsisthatadecreaseindriftwithmorealternativeswillresultinbothloweraccuracyandslowerresponsetimes,ratherthanmaintainingaccuracyattheexpenseofslowerresponsetimes.Additionally,theprecisenatureoftherelationshipbetweennumberofalternativesandresponsetimeswillnotnecessarilybelog-linear.Ifthealternativesthatareintroducedaremoresimilartothosealreadyinthechoiceset,theywillhaveadifferenteffectonresponsetimesandaccuracythanalternativesthataredissimilartothosealreadyintheset.Forexample,addingorangetoasetofcolorresponsealternativesfred;bluegislikelytohaveamuchmoredistractingeffectwhenthetargetcolorisred.Thediscriminabilitybetweenorangeandredismuchsmallerthanbetweenorangeandblue,sotherateoftrueinformationforredshouldbesmallerwhenthereisaredstimuluswithorangedistractorthantherateforbluewhenthereisabluestimuluswithorangedistractor.Interestingly,thisisperhapsamoreaccuratedescriptionoftherelationshipbetweenthenumberofalternativesandresponsetimesthanthestrictlylog-linearone.Longstrethetal.(1985)havesuggestedthat,whiletherelationbetweenthenumberofalternativesandresponsetimeappearstobepositive,itisnotnecessarilyoftheformdescribedbyHick(1952)andinsteadmayvarydependingontheassortmentofavailablealternatives.Byimplementingtheproposedelements25intoacomputationalmodelinStudy2,Ishowaricherrelationshipbetweensetsizeandresponsetimethatfactorsinthediscriminabilitybetweenalternativesandtheirinteractionwithinternalchoicecriterialikethresholds.26CHAPTER2GOALSANDPREDICTIONSSofar,Ihaveoutlinedageneralframeworkformodelingdecisionsbetweenanynumberofal-ternatives.Whenamodelerisinterestedinmakingassumptionsofindependenceorequalin-terdependencebetweenanumberofalternatives,theycanusethesimplexorhypercuberepresentationsofalternativesshowninFigure1.3.Doingsoissimilartousingamulti-alternativediffusionoraccumulatormodel,butaddstheadditionaloptionofusingrandomwalkvariationswhenstepsareamongdiscretestatesinalattice.Itisthereforeusefulfordealingwithdiscreteandcontinuousmodelsofdecisionsamonguniformly-relatedorunrelatedalternatives.However,psy-chologicalrelationsbetweenalternativesmustbeintroducedinordertoaccountforacontinuousspanofalternatives(otherwise,wewouldhavetoworkinanspace)aswellastoaccountfordecisionsbetweenalternativesthatshareunequalsimilarityrelationsamongthem.Theprimarygoalofthisthesisistobuildandtestmodelsthataccountfordecisionsamongalternativesthatarerelatedwithoneanotherinvaryingways,andparticularlytoexplorehowpeoplemakeresponsesonacontinuum.Tothisend,Ipresenttwomainstudies.Thefocusesonwhatcomponentsareimportanttoincludeinsuchmodelsbyexaminingcontinuousanaloguesofcommonaccuracyandresponsetimedecisionphenomena.Itinvestigateswhetherthemodelparameterschangeinsensibleways,andexaminestwoadditionalaprioripredictionsthatthemodelmakesinordertogaugeitsappropriatenessformodelingdecisionsonacontinuum.Thesecondstudyexamineshowresponsetimesandaccuracychangeacrossvaryingnumbersandarrangementsofalternatives.Itfocusesonthecaseofcolor,wherephysicallyrela-tions(hue)betweenalternativesmaynotmapdirectlyontopsychologicalrelationsbetweenthem.Themodelingcomponentofthisstudyillustrateshowtheserelationscanbebuiltintoourmodel-ingapproachbycombiningamultidimensionalscalingrepresentationofcolorswiththedecisionmodel.272.1Study1Study1coversmuchofthegroundworkinestablishingwhetherthemodelframeworkisappro-priatetoexplainandpredictdecisionsonacontinuum.BecausethestructureofthemodelIuseforthecontinuouscaseisanalogoustothe1-dimensionaldiffusionmodel,weshouldsimi-laraccuracyandresponsetimephenomenabetweenthe2-alternativeandthecontinuouscaseaswell.Ithereforetestcontinuousanaloguesofthedifeffect,speed-accuracytrade-off,andpredecisioncuebiaseffect.Thepredictionsofthestudyarerelativelystraightforward.Iexpecteachofthemanipula-tionstohaveparalleleffectstothosethatappearinthe2alternativecases.Manipulationsofthedifofthetaskshouldresultinslowerresponsesandlargererrorsforharderstimuli.Intermsofmodelmanipulatingdifshouldaffecttheestimatesofthedriftmagnitudeparameter(jmj)inthemodel.Responsetimesandaccuracythespeed-accuracytrade-offshouldchangeinthesamewayasinbinarydecisionstudiesaswell.InthespeedconditionŒrelativetotheaccuracyconditionŒIexpecttofasterresponsetimesandlargererrors.Thisshouldbeinthemodelviachangesinthethresholdparameter(q)estimates,whichshouldbehigherintheaccuracyconditionandlowerinthespeedcondition.Inthecasewherethereisapredecisioncuedesignedtobiasresponses,responsesshouldbefasterandmoreaccuratewhenthecueiscongruentwiththestimulusandslowerandlessaccuratewhenthecueislesscongruentwiththestimulus.Ordinarily,thiswouldproducechangesintheinitialstateofevidence.However,duetocomputationallimitations,ithasnotbeenpossibletoallowtheinitialstatetovary.Instead,thebiascanbemodeledasifparticipantsareintegratingtwosourcesofevidenceŒthecueandthestimulus.Thisshouldbeasshiftsintwoparameters:thedriftdirection(f;thecueshouldpullevidencesamplingtowarditandawayfromthestimulusmeanwhenincongruent)andthedriftmagnitude(thecueshoulddiluteevidencewhenitisincongruentwiththestimulus,andenhanceitwhenitiscongruent).Asthediscrepancy28betweenthecueandthestimulusgetslarger,Iexpectthatthedriftanglewilldeviatefurtherfromthestimulusmeanandthedriftmagnitudewillgetsmaller,i.e.thesamplingdistributionwillbemorediffuse.Non-decisiontimeestimatesshouldnotchangetoosubstantiallyacrossthevariousmanipula-tions,asresponsesareenteredinthesamewaywithineachtask.Participantsshouldbeabletorespondatrelativelyconsistentspeedsacrossconditions.Inadditiontothemanipulationsderivedfrombinarychoice,the2-dimensionaldiffusionmodelmakesadditionalpredictionsregardingthedistributionofresponsesandresponsetimes.Itsuggeststhatresponsetimedistributionsshouldbeunaffectedbyresponsedeviation(withinacondition),andthatresponsesshouldbevonMisesdistributed.Study1testseachoftheseclaimsaswell.Ultimately,Study1shouldestablishempiricalphenomenathatcanbefoundincontinuousresponsetasks,andestablishwhatcomponentsshouldbeincludedinamodelofbehavioronthesetasks.2.2Study2Theoutcomesofthesecondstudyaresomewhatlesscertain.IfHick'slawholds,responsetimesshouldshiftasafunctionofthelog2ofthenumberofalternativeswhentheyarediscrete,butitisunclearwhatwillhappenwhenthereisacontinuousspanofalternatives.Itislikelythataccuracywilldecreasewiththenumberofalternatives(thoughthisisnotpredictedbyHick'slaw),particularlybecausestimulusdiscriminabilitywillchangeasmorealternativesareintroducedintothesamespace.Thethreshold-basedexplanationofHick'slaw(Usheretal.,2002)suggeststhatthresholdestimatesinthemodelshouldincreaseasthenumberofalternativesincreases,andthatthisshouldberesponsiblefortheresponsetime(andaccuracy)differencesbetweenconditions.Participant'ssimilarityratingsbetweenpairsofalternativesshouldhoweasythoseal-ternativesaretodiscriminate.Therefore,weshouldthatalternativestheyrateaslesssimilar29ŒaboveandbeyondthephysicaldifferencesŒshouldbefasterandmoreaccuratetodiscriminate.Thosethattheyrateasmoresimilarshouldbeslowerandlessaccuratetodiscriminate.Thereareanumberofcomponentsofthemodel,whicharedescribedinmoredetailinthesectiondetailingStudy2.Iexpectthatmulti-dimensionalscalingtoobtaindriftrates,shiftsincategory/choiceboundariesbasedonthepositionsofalternatives,shiftsinthresholdforthenumberofalternatives,andthresholdsthatshiftbasedonthe(multi-dimensionalscaling-based)discriminabilityofalternativesareallpossibleelementsofthemodel.Themodelingcomponentofthisstudytestswhichcomponentscanbeomittedwithoutdamagingthemodel'sperformance.Ultimately,Study2seekstoexaminehowdecisionsamongcontinuousanddiscretenumbersofalternativesarerelated,andexplorehowmodelscancapturethepsychologicalrelationsbetweenalternatives(beyondphysicalrelations)byintegratingsimilarityanddecisioncomponentstogetherinasinglemodel.30CHAPTER3STUDY1-EMPIRICALHURDLESANDMODELELEMENTSThepurposeofStudy1wastoexploreempiricalphenomenarelatingtodecisionsonacontinuumandtheirimplicationsformodelingthesedecisions.Partofthereasonthatmodelsofbinaryde-cisionhavebeensowell-developedisthepresenceofestablishedempiricalphenomenathatrelatetheaccuracyofresponsestothespeedatwhichtheyareentered.Thesephenomenaimposestrongconstraintsonformaltheories,indicatingwhatparametersandstructuresmightberequiredinor-dertoaccommodatetypicalchoicedata.Ratcliff&McKoon(2008)focusedonthreeparticularthathaveshapedtheorydevelopment:thespeed-accuracytrade-off,whereemphasisonresponseaccuracyslowsdownresponseswhileemphasisonresponsespeedmakesresponseslessaccurate(Bogaczetal.,2010;Heitz&Schall,2012;Wickelgren,1977);stimulusdif,whichcandecreaseaccuracyaswellasincreaseresponsetimes(Link&Heath,1975);andpredecisionbiasestomakeresponses,whichimproveresponsespeedandaccuracyforfavoredresponsesandhurtthemfordisfavoredones(Diederich&Busemeyer,2006;J.M.Wolfeetal.,2007).Accuracyandresponsetimearethebasicmetricsbywhichweevaluatedecisionmodels(Brown&Heathcote,2008;Vickers&Lee,1998),andyetnoestablishedaccuracyandresponsetimephenomenaexistforcontinuous-responsedecisions;somepreviousworkhaslookedateitheraccuracyorresponsetimesincontinuousresponsetasks,butneverboth.Forexample,theaccuracyoforientationreproductionshasbeenusedasameasuretodistinguishbetweenmodelsofvisualshort-termmemory(VanDenBergetal.,2012).However,theinterplaybetweenaccuracyandresponsetimesisnontrivialandmayhaveseriousconsequencesforstudiesthatonlyconsideroneortheother.Forexample,responsedistributionsthatappeartobeamixtureofadiscretesetorcontinuumofvonMisesdistributions(whichmightbeusedtomotivateadiscreteorcontinuous-resourcemodelofvisualshorttermmemory,respectively)couldactuallyresultfromcontinuousordiscretecombinationsofdif,speedoraccuracyemphasis,ordecisionbiases.Study1seekstorectifythelackofempiricalresultsincontinuouschoicebyexploringcontin-31uousanaloguesofthreeclassicbinarychoicephenomena.Itexaminestheeffectofspeedversusaccuracyincentives,theeffectofstimuluscoherence(difonaccuracyandresponsetimes,andtheeffectofpredecisionresponsecuesonthesesameoutcomes.Ifthedecisionprocessissimilarbetweenbinaryandcontinuouschoice,weshouldthepresenceofaspeed-accuracytrade-off,higherresponsetimesandloweraccuracyformoredifstimuli,fasterandmoreaccurateresponsesforaccuratepredecisioncues,andslowerandlessaccurateresponsesforin-accuratepredecisioncues.IalsoexaminetwoadditionalpredictionsmadebytheparticularmodelIusehere:vonMisesdistributedresponsesandresponsetimesthatareunrelatedtoerrormagnitudewithinacondition.Inthefollowingsections,Itestthesepredictionsusingasimpleperceptualtaskrequiringresponsesonacontinuum.3.1MethodsInthecontinuouscase,accuracyisnotbinary.Rather,allresponsescanbeseenasvaryingdegreesofincorrect,astheycanbecloserorfurtherfromatruepointanswer.Therefore,'accuracy'inthesestudiesreferstohowclosearesponseistothetrueanswer.Frequently,Iwillrefertothedeviationofaresponsefromthetrueansweroracue.BecausethestimuliinStudy1allfallon0to180degreeorientations,anyresponsedeviationsareboundedbetween0and90degrees.1Hence,perfectaccuracywillresultinaverageresponsedeviationsof0,guessingresponseswillresultinaverageresponsedeviationsof45degrees,andperfectinaccuracy(consistentlygivingthemostwrongresponse)willresultinaverageresponsedeviationsof90degrees.Asinstudiesofbinarychoice,aspeed-accuracytrade-offwasimplementedbychangingtheincentivestructureofthetasktorewardeitherresponsespeedorresponseaccuracy.Difwasmanipulatedbychangingthecoherenceofthestimulus,andpredecisionbiaswasmanipulatedby1Notethatbeing120degreesclockwiseisthesameasbeing60degreescounterclockwisefromthetruemeanorientation.Becausethestimulifallonlyonahalf-circle,responsesat180degreesfromoneanotherareequivalent.Thesmallestdistancetothetarget/trueresponseistakenastheactualresponsedeviation.32providinginformativecuespriortoshowingthestimulus.Eachofthesewasimplementedinanorientation-basedtask,describedbelow.3.1.1TaskThetaskisshowninFigure3.1.Thiswasprimarilyanorientationdetectiontask,wherepartic-ipantssawanoisilyjitteringGaborpatchstimulusandhadtoselectitsmeanorientation.ThejitteringGaborstimuluswasgeneratedbyintiallychoosingarandommeanorientationfromauni-formdistributionon[0,180)degrees.Oneachframe,anewGaborpatchwouldbedrawnfromawrappednormaldistributioncenteredonthismeanorientation.Thestandarddeviationofthiswrappednormaldependedonthedifmanipulation.Thestandarddeviationswere15,30,or45degrees,correspondingtoeasy,medium,andharddif(largerstandarddeviationsresultinnoisierstimulusinformation).Thedifwasdrawnrandomlybetweentheselevelsoneachtrial.Participants'taskwastoproducethemeanorientationofthestimulusbymovingtheirmouseacrosstheedgeofaresponsecircle(outerwhitecircle).TheirresponsecouldbemadeonthelowerorupperhalfofthecircleŒthosemadeat30and210degreeswereequivalent.Participantsreceivedpointsinthetaskforboththespeedandaccuracyoftheirresponses,contingentonwhetherablockoftrialswasaspeed-emphasisblockoraccuracy-emphasisone.Inthespeedcondition,theyreceived100pointsforrespondingwithin800millisecondsofthestimulusonsetandupto100pointsbasedonhowclosetheyweretothetruemeanorientationofthestimulus:eachdegreeawaydecreasedtheirrewardby1.1points.Intheaccuracycondition,theyreceivedupto200pointsbasedsolelyonhowclosetheyweretothetruemeanorientationŒinthiscase,eachdegreeawaydecreasedtheirrewardby2.2points.Participantswhoaccumulatedasufnumberofpointsduringthetaskreceivedadditionalresearchcreditfortheirparticipation.Tobeginatrial,participantsclickedonasmallwhitecircleinthemiddleofthescreen.Theircursorwasthencenteredonthescreen,whichinvolvedonlyminoradjustmentsasthewhitecirclewasalreadyinthecenterofthescreen.Inthecuedcondition(topleftofFigure3.1),constituting33Figure3.1Timecourseofatrialoftheorientationdetectiontask.halfthetrials,theythensawagreenlineonthescreen.Thislinecorrespondedtothetruemeanorientationofthestimulusin50%ofcuedtrials,butwasrandomlysetto20,50,or70degreesawayfromthetrueorientationontheremaining50%ofcuedtrials.Thisallowedthecuetobeinformativebutnotoverwhelmstimulusinformationintermsofusefulness.Foruncuedtrials,theysimplysawanuninformativegreencircleappear(bottomleftofFigure3.1).Eitherthegreenlinecueorcircledisappearedafter1second,andthenthejitteringGaborstim-uluswaspresented.Participantscouldrespondatanytimebymovingtheirmouseacrosstheedgeofthewhiteresponsecircleattheirdesiredorientation.Theywereencouragedtohaveballisticmousemovementbypenalizingthembasedontheamountoftimethemousepointerspentbe-tweenthecenter(circlethattheyclickedtostartthetrial)andtheedgeoftheresponsecircle.Thiswasdonebothtocontrolthenon-decisiontime(timerequiredfornon-decisionresponseprocesseslikemovingthemouse)acrosstrialsaswellasavoidhavingparticipantsexternalizetheirbeliefsbeforeresponding.ThelattercancreateissuesintermsofmultiplesequentialmeasurementsŒexpressingone'sbeliefsinsuchawaymayaffectthedistributionofsubsequentresponses.343.1.2ParticipantsAtotalof12MichiganStateUniversitystudents(8female,4male)eachcompleted960trialsoftheexperimentforclasscredit.Participantswereprimarily18-26yearsold.Oneadditionalparticipantcompletedthetask,butwasremovedfromfurtheranalysesforhavingmedianresponsetimeswelloutsidethenormalrangeforparticipants(>5000milliseconds[ms],comparedto500-1500msforotherparticipants).Thefulltask,includingintroductionandpracticetrials,tookapproximately1to1.5hourstocomplete.Thesamplesizeof12participantswasselectedsothatthegroup-levelposteriordistributionsofparameterestimateswerebothsufconstrainedandsuitablywell-sampled.BecauseeachparticipantprovidedalargenumberofdatapointsandallparameterswereestimatedinahierarchicalBayesianway,thissamplewassuftoobtainpreciseestimatesonbothindividualandgroup-levelparameters.3.1.3MaterialsAllstimuliweregeneratedandresponsesrecordedusingMATLABandPsychtoolbox3(Brainard,1997;Kleineretal.,2007).Responseswererecordedonthemouse.3.1.4ProcedureUponenteringthelaboratory,participantswerebriefedontheroughcontentofthetaskandcom-pletedinformedconsent.Theywerethenseatedinadark,windowlessoftocompletethetask.Theycompletedapproximately60practicetrialsofthetask(moretrialsweregeneratedwhenre-sponsesweretooinaccurateorslow),withimmediatefeedbackonhowquickandhowaccuratetheirresponseswereoneachpracticetrial.Uponthepracticetrials,participantscompleted960trialsofthemaintaskasde-scribedabove.Thesewereorganizedinto12blocksof80trials.Theblockswereorganizedsothatparticipantswouldsee3blocksofaccuracywithapredecisionorientationcue,3blocksof35accuracytrialswithnoorientationcue,3blocksofspeedwiththeorientationcue,and3blocksofspeedtrialswithnopredecisionorientationcue.Difmanipulationstookplacewithinblocks.Theorderoftheblocksoftrialsandtheorderofdifoftrialswithinablockwerebothrandomized.Beforeeachblock,participantsreceivedinstructionsfortheupcomingblock'scondition,in-dicatingbothhowtheywouldearnpoints(forthespeed/accuracymanipulation)aswellasthepresenceofthecue.Theywerealsoremindedbeforeeachtrialwhethertheywereinaspeedorac-curacyblockwithasmallpieceoftextabovethewhitepre-trialcentercircle,whichread`SPEED'or`ACCURACY'.Uponcompletingtheexperiment,participantsweredebriefedonthepurposeofthestudyandtoldhowmuchcredittheywouldreceivebasedonthenumberofpointstheyaccumulatedduringthetask.3.2ResultsTheanalysesfocusedontwomainoutcomes.ThewassimpleresponsetimesŒthenumberofsecondsittookforapersontoentertheirresponse.Thesecondwastheirresponsedeviationsfromthetrueorientation.Forthismetric,ItookthemeanorientationoftheGaborstimuliandcomparedittoparticipants'responsesonthecircle.Theirresponsedeviationwasthenumberofdegreesclockwisethattheirresponsefellrelativetothetrueorientation.Forexample,iftheirresponsewas2degreescounterclockwisefromtheactualmeanorientation,theirresponsedeviationwouldbe2degrees(allresponsedeviationsthereforefellbetween90and+90degrees).Onaverage,responsedeviationswereapproximatelyzero,indicatingnoclockwiseorcounterclockwisebias.Inordertogaugeaccuracy,Iusedtheabsoluteresponsedeviation,whichwassimplytheabsolutevalue/magnitudeofthedeviationwithoutregardtoitsdirection.Responsetimesandresponsedeviationsdependedonfourmainfactors.Thetwasstimulusdif,operationallyasthestandarddeviationoftheorientationofthejitteringGabor36patches(15/30/45degrees).Thesecondwasthespeedoraccuracymanipulation,codedinthemodelsas0foraccuracyand1forspeed.Thirdwasthepresenceofthecue;thedefaultcuewasassumedtobetheorientationthatmatchedthetruestimulusorientation,sothisfactordescribestheconferredbyaninformativecue.Thefactorwastheorientationofthecue,whichcorrespondedtothenumberofdegreesthatthecuedeviatedfromthetruestimulusorientation(i.e.0/20/50/70degrees).Theinclusionofthisfactordependedonthethirdfactor,sothatcueorientationwasnotconsideredwhentherewasnocue.Eachofthesefactorswasstandardizedbeforeusingthemtopredictresponsesorresponsetimes.AllanalysesusedahierarchicalBayesianlinearmodel,whereindividual-levelparameters(coefforthesizeofeachstandardizedeffect)wereconstrainedbyagroup-leveldistribution.Here,Ireportonlythegroup-levelparameterestimatesforeacheffectforthesakeofbrevity.UnlessotherwisetheBayesianmodelsusediffusepriorssuchthatthedatawillhavemaximaloverparameterestimates.Inreporting,Iprovidethemeanposteriorparameterestimatesaswellastherangethatincludesthe95%mostcrediblevaluesfromthisposteriordis-tributionofpossibleparametervalues.Thisisreferredtoasthe95%highestdensityinterval[95%HDI](seeKruschke,2014).ThedataaswellasthecodeforeachoftheJAGSmodelsisavailableontheOpenScienceFrameworkatosf.io/ufe96.3.2.1ResponsetimesToexaminetheeffectsofthesemanipulationsonresponsetimes,IusedahierarchicalBayesianmodelwithanintercept,alinearmaineffectofthedifofthestimulus,abinarymaineffectofwhethereachtrialwasaspeed-emphasistrial,abinarymaineffectofwhetheratrialincludedacue,alinearmaineffectfortheorientationofthecue(whichwaspairedwiththebinaryindicatorforcuedtrialssoitdidnotcomeintoplayonnon-cuedtrials)andall2-,3-,and4-wayinteractionsbetweenthesefactors.However,notethatcueorientationisnotpermittedtointeractdirectlywithdiforspeedfactorsasitiscontingentuponthepresenceofthecue.Itcanthereforeonly37appearasahigherorderinteraction.Beforeanalysis,responsetimeswerelogtransformedinordertomakethemapproximatelynormallydistributed.Allpredictorsandoutcomeswerethenstandardized.Thecoefforeachofthemaineffectsandinteractionsweresethierarchicallybyparticipant,andthegrouplevelestimatesareshownhere.Table3.1Meanestimatesofcoefformaineffectsandinteractionsofstimulusdif,speedmanipulation,cuepresence,andcueorientationonresponsetimes.Therangescontainingthe95%HighestDensityInterval(HDI)arealsoprovided.Intervalsexcludingzeroarestarred.MeanEstimate95%HDIDif0:06[0:04;0:08]*Speed0:13[0:19;0:06]*Cue0:09[0:14;0:03]*Cueorientation0:04[0:02;0:05]*Difspeed0:02[0:03;0:00]Difcue0:02[0:04;0:00]Difspeedcue0:00[0:02;0:02]Difcueorientation0:00[0:02;0:01]Speedcue0:02[0:06;0:02]Speedcueorientation0:01[0:01;0:00]Difspeedcueorientation0:01[0:01;0:02]Themeanestimatesand95%HighestDensityIntervalsforeachcoefareshowninTable3.1.Asshown,theonlyeffectswithsubstantialcontributionstodifferencesinresponsetimeswerethemaineffects,includingthemaineffectofcueorientationthatwascontingentonthecuebeingpresent(andishencepresentedasaninteraction).Responsetimesincreasedwithdifaswellaswithlessaccuratecueorientations:thefurtherthecuewasfromthetruemeanstimulusorientation,thelongerparticipantstooktorespond.Bycontrast,responsetimesdecreasedwithspeedinstructionsaswellaswiththepresenceofaninformativecue.Eachoftheinteractionsinthemodelincludedzeroasacrediblevalue,suggestingthatthevariousmanipulationsdidnotinteractstronglyenoughtoproducesubstantialchangesinmeanresponsetimes.Thismodelthatdif,speed-accuracy,andcue/cueorientationmanipulationshadtheiranticipatedeffects.Therawdatafordifandspeed-accuracyconditionscanbeseen38Figure3.2Responsetimes(left)andaccuracyintermsofdegreesdeviationfromthecorrectresponse(right)acrossdiflevelsandspeed-accuracymanipulation.Notethathighervaluesindicatelessaccurateresponses.Errorbarsindicatepooledstandarderroracrossparticipants.inFigure3.2(leftpanel),anddataforcueorientationconditionscanbeseeninFigure3.3(leftpanel).3.2.2Accuracy/responsedeviationTopredictthelocationsofparticipants'responsesrelativetothetruemeanstimulusorientation,Iusedamodelthatishighlysimilartotheoneusedforresponsetimes.However,ratherthanthemeanlocationofresponsesŒwhichisessentiallyzero,astheyaregenerallycenteredonthecorrectresponseŒeachfactorpredictedthevarianceofthedistributionofresponsedeviations.2Thisgivesusanestimateofhowfarawayfromthetrueanswerwecanexpectresponsestobe,whetherthatdeviationisintheclockwiseorcounterclockwisedirection.Asbefore,theseoutcomeswereallowedtochangeasafunctionofmaineffectsofdif2Formally,theypredictedthelogofthevarianceofthisdistributioninordertoallowthesumofallfactorstotakeanyvalueandtomakethepredictedvarianceslog-normallydistributed.Asimilarmodelingapproachpredictingthevarianceofadrift-diffusionprocesscanbefoundinKvam&Pleskac(2016).39Figure3.3Meanresponsetimes(left)andabsoluteresponsedeviation(right)bytheorientationofthecue.Notethathigherresponsedeviationsindicatelessaccurateresponses.Errorbarsindicatepooledstandarderroracrossparticipants.(linear),speedmanipulation(binary),cuepresence(binary),andcueorientation(conditionallin-ear),aswellastheinteractionsbetweenallcombinationsofthesefactors.Forsimplicity,Idiscusstheresultsintermsofaccuracy.Thismeansthatlowercoefforeffectsonresponsedeviationcorrespondtohigheraccuracyandhighercoefcorrespondtoloweraccuracy.EstimatesforallcoefareshowninTable3.2.Themaineffectsofmanipulationswereasexpected,withdif,speedemphasis,andlessaccuratecues[orientation]alldecreasingoverallaccuracyandthepresenceofaccuratecues[cue]increasingaccuracy.However,unliketheresponsetimeresults,theaccuracyresultsweresomewhatcomplicatedbyinteractionsbetweenmanipulations.The2-wayinteractionbetweendifandspeedemphasisaswellasthe3-wayinteractionbetweendif/cue/orientationand4-wayinteractionbetweendif/speed/cue/orientation,eachofwhichincreasedaccuracy,canbeatleastpartiallyattributedtoaeffect.ThesemanipulationsputtogetherwouldresultinthelowestaccuracyconditionsŒnotethemaineffectsofdif,speed,andcueorientationonaccuracyŒbutaverageresponsedeviationshaveamaximumvalue.Guesseswillyieldresponsesthatareonaverage4540Table3.2Meanestimatesofcoefformaineffectsandinteractionsofstimulusdif,speedmanipulation,cuepresence,andcueorientationonaccuracyofresponses.Therangescontainingthe95%HighestDensityInterval(HDI)arealsoprovided.Intervalsexcludingzeroarestarred.MeanEstimate95%HDIDif0:79[0:70;0:89]*Speed0:16[0:07;0:26]*Cue0:31[0:49;0:14]*Cueorientation0:52[0:40;0:65]*Difspeed0:09[0:15;0:04]*Difcue0:24[0:46;0:02]*Difspeedcue0:03[0:06;0:13]Difcueorientation0:09[0:15;0:03]*Speedcue0:05[0:21;0:09]Speedcueorientation0:06[0:00;0:12]Difspeedcueorientation0:05[0:09;0:00]degreesawayfromthetrueorientation,someanresponsedeviationswillgenerallynotexceedthesevaluesinanyconditions.Theinteractionbetweendifandcuepresencewasthestrongesteffectoftheseandtheonlyonetorivalanyofthemaineffects,butthisinteractionissimpletoexplain.Whenparticipantsreceivedanaccuratecue,itcouldbeusedtomakeaccurateinferencesregardlessofwhatwasshowninthestimulus.Hence,theilleffectsofdifwerecurtailedwhenacuewaspresent.3.3ModelingBecausethesestimuliaresymmetricacrosshorizontalandverticalaxes,meaningtheyhavenotopandbottom,theorientationsofthesestimuliandthepossibleresponsestothemvaryfrom0to180degrees.Forsimplicity,assumethatstimuliatorthogonalrotations(i.e.45/135degreesorhorizontal/vertical)provideevidenceagainstoneanother.Therefore,theycanbearrangedinacircleasshowninFigure3.4.Themodelassumesthatparticipantscompletethetaskbysamplingorientationinformationpiecebypiece,andeachpieceofinformationtheygatherisassumedtobepulledfromavon41Figure3.4Staterepresentation,sampling,threshold,andevidencetrajectoriesfortheorientationmatchingtask.Misesdistribution.3Theysampleinformationuntilacriterionlevelofcertaintyismet,givenbythecircularthreshold,andthepointatwhichthewalkcrossesthethresholdgivestheorientationresponse.Withtheseassumptions,themodelisfunctionallyequivalenttothe2-dimensionalrandomwalkonadiscproposedbySmith(2016).ThisallowsustoapplytheanalyticlikelihoodfunctionsthatSmithderivedtothepresenttask.Theparticularfunctionalformofthemodelusedherehas5mainparts:thedriftmagnitudejmj(howquicklyinformationinfavorofthecorrectdirectionissampled),thedriftdirectionr(themeanorientationgatheredfromthestimulusorstimulus+cue),thethresholdq(howmuchinformationisneededtomakeadecisions),non-decisiontimendt(thelengthofresponsetimescomposedofprocessesotherthanthedecision,suchasperceivingthe3Thisissimilartoanormaldistributiononacircle.Eachpieceofevidencecouldbesam-pledfrommomentaryactivationacrossorientationcolumnsinthevisualcortex.ThisopensupthequestionofwhetheractivationacrossthecolumnsmimicsavonMisesdistribution,whichisinterestingbutbeyondthescopeofthethesis.42stimulusandmovingthemouse),andtheinitialpositionoftheevidence.Unfortunately,duetothecomplexityofitslikelihoodfunction,theversionofthemodelincludingvariationintheinitialstateisintractable,sotheinitialstateissettothecenteroftheresponsecircle[0,0]andotherparametersmustcompensateforitsomission.Theresultinglikelihoodfunction,coveringthejointdistributionofresponsetimesandresponsesPr(x;t),isgivenbySmith(2016)andshownbelowinEquation3.1.Pr(x;t)=expqjmj(cos(x)cos(r)+sin(x)sin(r))jmj2(tndt)2s2q2s2nåk=1j0;kJ1(j0;k)expj20;ks2(tndt)2q2(3.1)Becausetheparametersarewhenthenoiseparameters2isallowedtovary,itwassetto1.ThefunctionJ1()isaBesselfunctionofthekind,andtheelementsj0;karethezeroesofthezero-orderBesselfunctionofthekind.Theseriesånk=1computesandevaluatesthefunctionforthenzeroes.Thetruelikelihoodisgivenbyanseries,n=¥.However,forthepracticalpurposesofthisstudy,Ifoundthatn=151yieldssufpreciseapproximations.Theremaining4parametersusedinthelikelihoodŒdriftdirection,magnitude,threshold,andnon-decisiontimeŒvaryfromconditiontoconditionbasedontheexperimentalmanipulations.Theymustthereforebeandestimatedeitherfreelyorinsomefunctionalwayacrosstheseconditions.Allowingeachoftheseparameterstovaryfreelyacrossconditionswouldyield4(parameterspercondition)2(speed/accuracy)3(diflevels)8(nocue+7cuede-70,50,20,0,20,50,and70degrees)=192freeparameters.Insteadofattemptingtoestimatethemallsimultaneously,Iexplorethemanipulationsintwostagesinordertosettheparametersprogressively.First,Ilookatjustthedifandspeed-accuracymanipulations,tingeachconditionindependently.Thisallowsmetoexplorewhatrelationshipseachofthesetwomanipulationshaswithdrift,threshold,andnon-decisiontime.Inturn,theserelationscanbeusedtosimplifyparametersettingwhenweincludecuemanipulationsinthemodelŒforexample,drift43magnitudeappearstobeclosetoalinearfunctionofthestandarddeviationofthestimulus,soitcanbesetacrossconditionsusingjustaninterceptandslopeparameter.3.3.1Parameter-freepredictionsThemodelusedheremakestwopredictionsthatcanbetestedwithoutactuallyhavingtodoanyparameterestimationinthecognitivemodel.Theoftheseisthepredictionthatresponsetimeswillnotvarywithinaconditiondependentonthemagnitudeofresponsedeviations.Thisisanal-ogoustothesymmetriccorrectanderrorresponsetimedistributionspredictedby1-dimensionalrandomwalks.Introducingdriftratevariabilityintothemodelwillre-introducearelationshipbetweenresponsetimeanddeviation,sothistestisdesignedtotestwhethersuchacomponentisnecessaryforthe2-dimensionalmodel.Irefertotheindependenceofresponsetimesfromresponsedeviationsasfiresponsetimehomogeneityflbelow.Thesecondpredictionthatcanbetestedwithoutestimatingthecognitivemodel'sparametersisthatresponsesshouldbevonMisesdistributedwhentheyarrive.Thispredictioncouldpotentiallybebyintroducingvariabilityintheinitialstateordifferentsamplingdistributionstotheevidence,butthismaynotbenecessary.IrefertothispropertysimplyasfivonMisesresponsesflbelow.3.3.1.1HomogeneousresponsetimesItestedthisassumptionbylookingatthecorrelationbetweenresponsetimeandabsolutere-sponsedeviation.Thegroup-levelmeanestimate(fromahierarchicalBayesianmodel)forthestandardizedcorrelationbetweenresponsetimeanddeviation,acrossallconditionsandpartici-pants,wascenteredalmostexactlyatzero,M(b1)=0:01(95%HDI=[0:04;0:06]).Withinparticipants,therewereoccasionalinstanceswhereresponsetimewascorrelatedpositivelywithaccuracywithinthespeedcondition,andtheHDIindicatedthattheycouldbegreaterthanzero.Thesecasesarelikelyduetotheseparticularparticipantsguessinginordertokeeptheirresponse44timesunder800msthresholdrequiredforspeed-basedrewards,andcouldbemodeledasacon-taminantdistributionifoneisparticularlyinterestedintherelationshipbetweenRTandabsoluteresponsedeviationineachcondition.3.3.1.2VonMisesresponsesThesecondassumptionmadebythemodel,ifthereisavonMisessamplingdistribution(orBrow-nianmotionwithdrift)andnochangesintheinitialstateoftheprocess,isthatallresponsesshouldbevonMisesdistributedwhentheyhittheresponsethreshold.Althoughtestingthisassumptiondoesn'trequirethe2-dimensionaldiffusionmodel,itdoesrequireadescriptivemodeloftheresponsesforeachparticipantandcondition.Totestthisassumption,IestimatedthemaximumlikelihoodparametervaluesofavonMisesdistributionforeachconditionandparticipant.Thisyieldsacentraltendencyormeanparametermandaconcentrationorvarianceparameterk.Ithenusedthesemaximumlikelihoodparameterstogeneratearandomsampleof10,000datapoints.Inturn,Icomparedtherandomlygenerateddatatotheactualdataforeachofthe(12participants3diflevels2speed/accuracyconditions8cueconditions[7cueorientations+1non-cue]=)576sets.ThecomparisonwasdoneusingaKuipertest(Louter&Koerts,1970),thecircularanalogueofaKolmogorov-Smirnovtest.Itteststhehypothesisthattwoindependentsamplesweregeneratedfromthesameunderlyingperiodicdistribution.Inonly12ofthe576sets(2%)wasthehypothesisthatthesamplescamefromthesameunderlyingdistributionrejectedata=:05orlower.4Thisisareasonableratetoexpectsimplybychance,indicatingthatmostsetsofresponsesinthedatacanreasonablybeconsideredvonMisesdistributed.4AtpresentthereappearstobenoBayesiananaloguetothistest,soIaskthereadertoforgiveamixofclassicalandBayesianinferentialstatisticsforthepresentcircumstances.453.3.2FreeeffectsmodelThecognitivemodelexaminesonlynon-cuedconditionsinordertoestablishrelationshipsbetweenthespeed-accuracyanddifmanipulationsandthedrift,threshold,andnon-decisiontimeparameters.Therewasnopre-stimulusinformationandnobiastorespondclockwiseorcounterclockwiseintheuncuedconditions,sowecansafelyassumethatparticipantswereonaveragesamplingorientationinformationbasedonwhatthestimulusindicated.Therefore,thedriftdirectionwasedforeachtrialatthetruemeanorientationofthestimulus.Driftmagnitude(therateofsamplingtrueinformationrelativetorandomnoise),threshold(theamountofrequiredinformation),andnon-decisiontimewerepermittedtovaryfreelyacrossthe6speed/accuracyanddiflevelconditions.Eachoftheseparameterswasestimatedforeachparticipantandcondition(asidefromcuedones)usingmaximumlikelihoodestimation.Theresultingestimatesforeachparticipant,aswellasthemeanacrosstheseestimates,isshowninFigure3.5Asexpected,driftmagnitudeestimatesdecreasedwithincreasingdif,indicatingthatlesscoherentstimuliwereprovidingnoisierinformation.Driftmagnitudeestimatesweresimilarforspeedandaccuracyconditionsaswell,suggestingthatparticipantsweregatheringinformationatroughlythesamerateinbothconditions.Importantly,driftmagnitudedecreasesapproximatelylinearlywithstimulusdifŒasaresult,Iassumealinearfunctionalrelationshipbetweenthetwointhemodelpredictingbehaviorthecuedcondition.Thresholdestimateswereconsistentlyhigherintheaccuracyrelativetothespeedcondition,suggestingthatparticipantswereapplyingstrictercriteriafortheirdecisionstotheevidenceintheaccuracyconditionrelativetothespeedcondition.ThisshiftexplainsthelongerresponsetimesandhigheraccuracyintheaccuracyconditionasshowninFigure3.2.However,thresholdestimatesseemedtoalsodecreasewithincreasingdif.Whilethiswassomewhatunexpected,similarresultswerefoundbyKvam&Pleskac(2016).Theauthorssuggestedthatsuchaneffectmaybetheresultofincreasingthresholdswhenhigh-qualityinformationisavailable,suggesting46Figure3.5Driftmagnitude(rate),threshold,andnon-decisiontimeestimatesforindividuals(colors)andthemeanofthegroup(black)inthefreeeffectsmodel.thatparticipantsareadjustingtheirchoicecriteriaon-line.Alternatively,itcouldbetheresultofdecisionboundariesthatcollapseovertime,indicatingthatparticipantsaretradingoffaccuracyfortheamountoftimeittakestoatrial(Bowmanetal.,2012;Drugowitschetal.,2012;Ratcliff&Frank,2012).Fortunately,ed-boundaryversuscollapsing-boundarymodelspredicthighlysimilardistributionsofresponsetimes(Voskuilenetal.,2016),suggestingthatcollapsingboundariesareunlikelytobeanecessarycomponentofthemodelatpresent.Ithereforethe47thresholdofthemodelacrossdiflevelsinsubsequentsections.Finally,non-decisiontimeappearsnottochangesubstantiallybetweenspeedandaccuracyconditionsoracrossdiflevelsŒanyshiftsarewithinapproximately40milliseconds.Ithereforenon-decisiontimeacrossconditionswhenmodelingtheeffectofthecuemanipulation.3.3.3cue-inclusivemodelGiventheresultsofthefree-effectsmodelandtheuncueddata,wecansimplifytheparameterspaceofthemodel.Insteadofallowingthemtobefreeacrossconditions,driftmagnitudecansetasalinearfunctionofstimulusdifandthresholdcanvarybetweenspeedandaccuracymanipulations.Non-decisiontimeisedacrossconditions.Despitesimplifyingthemodel,eventheseassumptionsdon'tbuyenoughsimplicityunlesswemakesomeassumptionsabouthowtheparametersmightchangewiththecuemanipulationsaswell.Ordinarily,onemightassumethatinitialstatewouldbetheparametertochangeacrosscuemanipulations,butthelikelihoodfunctionincludingnonzeroinitialstatesistoocomplextoefestimatetheparametersofthemodel.Instead,thedriftmagnitudeanddriftdirectionareallowedtovaryaslinearfunctionsofthecueangle.Thisshiftinparameterizationisakintoclaimingthatthecueisintegratedasinformationalongwiththestimulusinformationaspartofthedecisionprocessratherthanasaedpre-stimulusbias.Itseffectsarenotexactlythesameasallowinginitialstatetovary,butthismodelistractableandstillinterestingtoexamine.Becausetherewere7levelsofcueorientations(-70/-50/-20/0/20/50/70),itseffectondriftmagnitudeanddriftdirectionwerealsolinearfunctions.Intotal,thisyielded7freeparametersinthemodel.Driftmagnitudehadanintercept(m0)and2slopeparametersthatsetitasalinearfunctionofthedifmanipulation(m1)andthecueorientation(m2).Driftdirectionhadanintercept(d0)andalinearslopethatdependedonthecueorientation(d1),thethresholdshiftedbetweenspeed(qs)andaccuracy(qa)conditions,andnon-decisiontime(ndt)wasedtoasinglevalueacrossconditions.Aswemightexpectbasedontheresultsofthefreeeffectsmodel,estimatesofthethreshold48Table3.3Maximumlikelihoodestimatesfortheparametersoftherestrictedmodel.Participant#m0m1d0d1ndtqsqa10.001-0.0570.1390.0040.3600.9471.30520.001-0.0510.0990.0050.3111.0051.17230.000-0.036-0.0150.0110.2041.1241.33240.001-0.0380.0240.0050.3440.8741.42550.010-0.0460.1150.0060.1930.8411.01160.001-0.0480.0110.0040.2041.1151.20370.001-0.0470.0600.0040.2750.8410.92180.002-0.0400.1980.0020.1270.9611.20390.001-0.0390.1190.0000.1281.0931.623100.001-0.0440.0990.0100.2400.8481.657110.000-0.0390.195-0.0040.1240.9631.198120.001-0.0430.0060.0090.1391.0541.383parameterwerehigherfortheaccuracyrelativetothespeedconditionforeveryparticipant(seeTable3.3andFigure3.6,bottomright).Driftmagnitudealsostilldecreasedwithhigherdif(Figure3.6,topright),inthenegativeslopeofthelinearfunctionmappingdifontodriftmagnitude.Thecuemanipulationimpactedbothdriftmagnitudeanddirection.Inthecaseofdriftdi-rection,nearlyallparticipantssawtheaverageinformationtheysampledshifttowardthecueddirection,indicatedbyapositiveslope(Figure3.6,topleft).Giventhatthecuewasofteninfor-mative,thisispreciselywhatweshouldexpectŒitprovidedsomeinformationthatwasintegratedwithstimulusinformation,movingthedriftawayfromcenterwhenthecuewasnotcongruentwiththestimulus.Perhapsmoreinterestingistheeffectofthecueondriftmagnitude,shownatthebottomleftofFigure3.6.Avalueofzeroindicatesthatthedriftwasthesamebetweenthecuedandnon-cuedconditions.Inthecaseswherethecueindicatedthecorrectanswer,thedriftratewashigher,thedualsourcesoftrueinformation.However,thedriftmagnitudewasstillhigherinthecasewherethecuewas20degreesoffofthetruestimulusorientationthanintheuncuedcondition,indicatingthatthecuewashelpfulevenwhenitwasnotpreciselycorrect.Thedriftmagnitudefelloffasthecuebecamelessinformative,with50and70degreecueorientationshurtingoverall49Figure3.6Effectsofcue,speed-accuracy,anddifmanipulationsondriftmagnitude,direction,andthresholdforindividuals(colors)andgroupmeaneffects(black).performance.Oneinterestingcharacteristicofthisparticularrelationshipbetweencueorientationanddriftmagnitudeisthatitshouldberelatedtothetuningfunctionsoforientationcolumnsinthebrain.Verticalcolumnswillbemosthighlyactivatedbyverticalorientationsofacue,buttheymayalsobeactivatedsomewhatbyorientationsof5,10,15,orperhaps20degreesawayfromvertical(lessandlessasthisdiscrepancyincreases,ofcourse).Itmaybethecasethatactivityofthetruestimulusdirectionmaybefacilitatedbyoff-meanorientationssimplybecausethetuningfunctionoforientationcolumnsoverlapswithnearby(butnotexactlymatching)cueorientations.503.4PreliminarydiscussionTheresultsprovidestrongsupportforthepresenceforeachofthethreeeffects:speedoraccuracyincentivesproducedatrade-offbetweenresponsetimeandresponseaccuracy,stimulusdifproducedslowerandlessaccurateresponses,andpredecisioncuesyieldedfasterandmoreaccurateresponseswhentheyindicatedthetrueorientationofthestimulusbutslowerandlessaccurateresponsesforcuesthatwerelessaccurate.Giventheprevalenceofanalogousphenomenainbinarychoice,wemightexpectthatamodelwithsimilarstructuretodiffusionoraccumulatormodelswouldbewell-suitedtodescribingbe-haviorincontinuous-responsetasksaswell.Thisisinhowtheparametersofthemodelchangewithmanipulations.Asinbinarychoice,speed-accuracymanipulationsyieldshiftsinthresholdsanddifmanipulationsyieldchangesindrift.Thoughitwasunfortunatelynotpossibletoexaminehowinitialstateschangewithpredecisioncues,thedriftmagnitudeanddirec-tiondid,indicatingthatcuesmayhaveasimilareffectaswell.Inaddition,twoaprioripredictionsofthe2-dimensionaldiffusionmodelwerevalidatedaswell.Acrossthevariousconditionsandparticipants,errormagnitudeswerelargelyunrelatedtoresponsetimes,indicatingthatthereisnoneed(yet)forlikedriftrateorstartingpointvariabilityinthemodel.Furthermore,responsesonthetaskappearedtobeapproximatelyvonMisesdistributed,asthesamplingdistributionsuggests.Overall,thissuggeststhattherecentproposalsofSmith(2016)andKvam(2016)provideasolidfoundationformodelinghowresponsetimesandresponsesonacontinuumaredistributed.Driftmagnitude,direction,threshold,andnon-decisiontimeseemtobeinformativeparametersfordescribingthedecisionprocessinsuchcases.However,thisdoesnotquitegivethefullpicture.Animportantfeatureofthegeometricframe-workisthattherelationshipsbetweenresponsealternativesarebuiltintothedecisionspace.TherelationsbetweenresponseoptionsinthisstudyarerelativelytrivialŒorientationcanbeintermsofdegreesandeasilyarrangedonacircle.Butwhataboutmorecomplexstimulithatdo51nothaveaneasyphysicaltopsychologicalmapping?Iexaminethisquestioninasecondstudy,presentedinthenextchapter.52CHAPTER4STUDY2-EXPLORINGRELATIONSBETWEENALTERNATIVESThesecondstudywasaimedatshowingthepotentialofusingthegeometricframeworkformodelingdecision-makingamonginterrelatedalternatives.ItfocusedonhowtheavailablealternativesŒratherthanmanipulationsofthestimulus,cues,orrewardsŒaffectedthedecisionprocess.Toinvestigatetheircontribution,Imanipulatedboththenumberofalternativesavailabletoparticipantsaswellasthealternatives'(similarity)relationstooneanother.Thisstudyalsousedperceptualstimuli,butinvestigatedthepotentialsourcesandcaveatsofHick'sLawbymanipulatingthenumberofalternativesandtheirsimilaritytooneanother.Inaddi-tiontolookingadiscretenumbersofalternatives,thisstudyexaminedhowtheseconditionsrelatedtoonewhereparticipantshadacontinuum(atheoreticallybutnotpracticallyofalterna-tivesfromwhichtochoose.RecallthatHick'slawdoesnoteasilyextrapolatetoacontinuous-responseconditionbecausesuchanalogscalesdon'tlendthemselveseasilytobit-basedpredic-tions;amajorgoalofthisstudywastouncoverhowitrelatestodiscrete-choiceconditionsandhow(if)Hick'slawholds.Beyondjustresponsetimes,thisstudyalsolookedathowaccuracychangedwiththenumberandassortmentofalternatives.Thisgave4majoroutcomestoexamine:2manipulations(numberofalternatives,similaritybetweenalternatives)2outcomes(responsetimes,accuracy).Eachofthe4patternsofresultsconstitutedahurdlethatamodelofdecisionsamongmanyalternativesshouldbeabletoclear.ThemodelingsectionfocusesonwhatcomponentsoftheframeworkŒandwhatadditionsandtoitŒallowittoaccountforvariationinbehavioronall4counts.ThissectionconcludeswithareviewofthemajorempiricalandmodelcomponentssuggestedbyStudy2.Ifocusinparticularonhowdecisionmodelsandrepresentationmodelscanbeusedtoinformoneanother,providingthegroundworkforamethodofconstructingadecisionspacethatrelatestheavailablechoicealternatives.534.1MethodsStudy2examinedhowparticipants'psychologicalrepresentationsofcolorsrelatedtotheirphysi-callyhue.Itconsistedoftworelatedtasks,whichwererunwithinparticipantsinordertomodeleachperson'sbehaviorindividually.Assuch,therewerefewparticipantsbutaverylargeamountofdatagatheredfromeachone.4.1.1ParticipantsAtotalof6MichiganStateUniversitygraduatestudents(4female,2male)eachcompleted5ses-sionsofadecisiontaskand1sessionofasimilarityratingtask,describedbelow.Allparticipantswere22-30yearsold.Theycompletedapproximately500-1000practicetrialsofthedecisiontask,1400-2300fulltrialsofthedecisiontask,and435trialsofthesimilarityratingtask.Eachsessiontookapproximately1hourtocomplete.Participantswerepaid$10persessionforparticipating,andinformedoftheiraverageaccuracyattheendofeachsession.4.1.2MaterialsAllstudystimuliweregeneratedandpresentedinMATLABusingPsychtoolbox(Brainard,1997;Kleineretal.,2007).Analysesusedthemachinelearningandcircularstatisticstoolboxes(Berens,2009).Allresponseswererecordedfromthemouse.4.1.3DecisiontaskThedecisiontaskusedinStudy2isshowninFigure4.1.Inthistask,participantsviewedadisplayof78dotsscatteredaroundadiscofapproximately10visualdegreesindiameter.Thedotsvariedinhuesuchthatnotwocolorshadahuethatwaswithin0.04unitsofoneanother(withhuerangingfrom0.0to1.0,wrappingaroundsuchthat0.0=1.0).Thesaturationofthedotcolorswassetto1.0andthevaluewassetto0.8.Thedisplayhadasingledominantdothue,ofwhichtherewere54alwaysexactly18dots.Thedominantdotcolorwasalwaysahuethatexactlymatchedahueintheavailablechoicealternatives.Ontopofthedominantdotcolor,therewere15otherdothuespresentinthedisplay,with4dotsineachofthesehues.Ineachblockoftrials,therewere8huesthatwouldappearoneverytrialwhethertheywerethedominantcolorornot.Thesehueswerepartiallydeterminedbytheavailablealternatives.Forexample,whentherewere8choicealternatives,eachoftheir8hueswouldappearoneverytrialoftheblock.Iftherewere3alternatives,those3huesplus5moreedhueswouldappearoneverytrial,andsoon.Withinablock,thecorrectalternativewasalwaysoneofthese8edhues,eveninthecontinuouscondition(wherethe8edhuesweregeneratedrandomlyatthestartoftheblock).Theremaining8edhuespresentinthedotdisplayweredrawnrandomlyoneverytrial,againwiththerestrictionthatnopairofhuesinthedisplaybecloserthan0.04hueunitsapart.Theedhueswereneverthetargetcolor,sotheyservedstrictlyasdistractorsornoiseinthestimulus.Intotal,thisyieldedthe78(18target+74non-targeted+84non-targetrandomhues)dotsinthedisplay.Eachblockconsistedof30trialsofthedecisiontask,pluspractice(describedbelow).Withinablock,thesetofalternativeswased.Iftherewere3alternatives,thesame3wouldbepresentonall30trials.Arandomalternativeoutofthoseavailablewouldbethedominantcoloroneachtrial.Itispossiblethatthismayhavemadedecisionseasierintheconditionswheretherewerefeweralternatives,asthisnarroweddownthecolorsthataparticipantneededtoattend,butitensuredthattherewasalwaysamatchbetweenthetrueanswerandtheavailableones.Thealternativesavailabletoaparticipantwereplacedaroundtheedgesofacircleatapprox-imately20visualdegreesfromthecenter,asshowninFigure4.1.Eachalternativetookupa14-degreearcalongtheedgeofthiscircleinthediscretecondition(top/leftpanels).Inthecontin-uouscondition(bottomrightpanel),ahuecirclewasshownwhereeverydegreeofthecirclewasadifferenthue,approximatingacontinuousgradientofhues.AsinStudy1,aparticipantbeganeachtrialbyclickingwithinasmallwhitecircleinthemiddleofthescreen,atwhichpointtheirmousecursorwascentered.Thealternativesforablockwere55Figure4.1DiagramofthedecisiontaskinStudy2.Dependingonthecondition,thesewillbecomprisedof2,3,5,8,oracontinuousspanofalternatives.Participantsrespondedbymovingtheirmouseacrossthearccorrespondingtothedesiredresponsealternative.alwaysonscreen,butthestimulusdidnotappearuntiltheyclickedwithinthewhitecircle.Onceatrialhadstarted,aparticipantenteredtheirresponsebymovingthemouseacrosstheedgeofthecircleonwhichthealternativesweredrawn.Assoonasthemousecursorcrossedthisboundary,theirresponsetimewasrecordedandtheirresponsewasgradedascorrectorincorrect.Inordertomatchtheaccuracycriterionacrossallconditions,responseswereconsideredcorrectiftheywerewithin7degreesofthecenterofthelocationofthetruedominantdotcolor.Inthediscretecase,thismeantthatresponseswerecorrectiftheycrossedthearccoloredinthetruedominantdothue.Inthecontinuouscondition,itsimplymeantthatresponsesthatwerewithin7degreesofthetruedominanthuewereconsideredcorrect.Participantswereinformedofthisgradingcriterionprior56tobeginningthestudy.Withinasession,thenumberofresponsealternativesinablockwasevenlysplitbetween2,3,5,8,andacontinuumofalternatives.Thehuecolorwheelandthelocationsofalternativeswererotatedrandomlyforeachsession,butkeptconstantwithinasession.Thiswasdonetoensurethatresponsetimeswerenotunevenlydistributedacrosshuesduetotheirspatiallocationsonthescreen.Attheendofeachdecisiontrial,participantswouldseefeedbackonwhetherornottheirchoicewascorrectbyreceiving100(correct)or0(incorrect)pointsforthattrial.SimilartoStudy1,ballisticmousemovementwasencouragedbypenalizingparticipantsforstrayingbetweenthedotdisplayandavailablealternatives.Thispenaltywas1pointperevery20msafterthemousecursorwasinbetweenthedotsandresponsealternativesfor300ms.4.1.3.1PracticetrialsInordertoensurethatresponsetimeswereaffectedaslittleaspossiblebypracticeeffects,thephysicallocationsofalternatives,andthetimeittooktomakeaballisticmovementtotheedgeofthecircle,therewerealargenumberofpracticetrialsthatprecededeveryblockofdecisiontrials.Eachpracticetrialwassimilartothedecisiontrials,exceptthatasinglelarge,coloreddotwasshownratherthananoisymulticoloreddotdisplay.Therefore,insteadofpickingthedominantdotoutofthedisplay,participantssimplyhadtomatchthecolorshowntothealternativesavailable.Asinthedecisiontask,theiraccuracyandresponsespeedweregatheredandstored.Thealternativesshownduringthepracticetrialswerethesameasthoseinthesucceedingdecisiontrials.Oneofthegoalsofthepracticewastomakesureparticipantsknewexactlywhereeachofthealternativeswas.Therefore,foreachchoiceoptionpresentinthedisplayofalternatives(2,3,5,or8forthediscreteconditions),aparticipantsawatleast3instancesofthecorrespondingcolorappearinthecenterforthemtomatch.Inthecontinuouscondition,theysaw10ormorerandomhuesappearinsequenceinthecenter.Anytimeaparticipantmadeanincorrectassignmentduringthepracticetrials,anadditionalpracticetrialwasadded.57Aftereachofthepracticetrials,participantswouldreceiveimmediatefeedbackontheaccuracyoftheirselections,includingthehuetheychose,itslocationonthescreen(intermsofdegreesaroundthecircle),thecorrecthue,thecorrecthue'slocationonthescreen(itscenter),andhowfarawayindegreestheirresponsewasfromthecenterofthecorrecthue.4.1.4SimilarityratingtaskPriortothesessionswiththedecisiontask,participantscompletedasimilarityratingsessionwheretheywouldseepairsofcolorsinthemiddleofthescreenandascalearoundtheedges(seeFigure4.2).TheirtaskwassimpleŒtheysimplyhadtocomparethetwocolorsonthescreenandassignavaluefrom0(opposite)to100(identical)indicatinghowsimilartooneanothertheythoughtthecolorswere.Figure4.2Layoutofthesimilarityratingtask.Thecolorsthemselveswere30huesequallyspacedalongthecolorwheel.Participantswerepresentedwitheachpossiblecombinationof2non-identicalcolorsexactlyonce,givingaratingforeachpairwisecomparison.Thesepairwiseratingswereusedtopopulatetheupperdiagonalofasimilaritymatrix,whichwasusedtogenerateamultidimensionalscaling[MDS]solutionthat58arrangedthecolorsin2dimensions.Thisprocedureisdescribedindetailinthecorrespondingmodelingsection.4.1.5ProcedureAftercompletinginformedconsent,participantswereplacedinadark,windowlessofandcom-pletedthesimilarityratingtaskforthedurationofthesession.Onlaterdates,theycompleted5sessionsofthedecisiontask.Thesimilarityratingtaskwasself-paced,sothatparticipantscouldtakeaslongastheywantedtomakeexactsimilarityjudgmentsandtakebreaksastheyneeded.Theywereencouragedtotakeaconsistentamountoftimeoneachtrialandtomakesurethattheirjudgmentswereconsistent(i.e.aratingof30shouldindicatethatapairofcolorsismoresimilarthan25,regardlessofscale).Participantsreceived$10forcompletingthesimilarityratingtaskandforeachsessionofthedecisiontask.Duringthesessionofthedecisiontask,theexperimenterwoulddemonstratehowtoper-formthetaskinbothdiscreteandcontinuousconditions,emphasizingthatmousemovementsshouldbeconsistentandballisticŒi.e.,thatparticipantsshouldnotmovethemouseuntiltheywerereadytorespond,butmoveitdirectlytothealternativewhentheywere.Inadditiontotheirinitialanddemonstration,participantscompletedanextra30practicetrialsattheoutsetofthesessionthatcoveredallnumbersofalternativestheymightseeduringthetask.Inthesession,theywouldthencomplete10or15blocks(dependentontime)ofthedecisiontask,in-cludingpracticetrials.Insubsequentsessions,theywouldcomplete15or20blocksofthedecisiontask,includingpracticetrials.Onceall6sessionswerecompleted,participantsweredebriefedonthepurposeofthestudyandpermittedtoseesomeoftheresultsoftheirperformance,ifdesired.594.2ResultsAsinthepreviousstudy,theresultsarebrokenintodescriptiveandprocess/cognitivemodelingsections.Bothtypesofmodelsexamineaccuracyaswellastheamountoftimeittookparticipantstomakearesponse.Recallthatinthedotdisplaytask,acorrectanswerinvolvesmovingthemouseacrossthearccorrespondingtothemostprominentdotcolorinthediscreteconditionsorwithin7degreesofthemostprominentdotcolorinthecontinuouscondition.Anyresponsenotincludedintheserangeswerecountedasincorrect,includingonesthatdidnotcorrespondtoanyavailablealternativeinthediscretecondition(i.e.,iftheirmousecrossedthecircleonwhichalternativeswereplaced,butnotinalocationthathadacolorthere).Responsetimewassimplythelengthoftimeinsecondsbetweenwhenthestimulusappearedonthescreenandwhenaparticipant'scursorcrossedtheresponsearc.Theinitialdescriptiveandinferentialstatisticsexaminehowdifferentmanipulationsofthenumberofresponsealternatives,thecombinationsofalternatives,andthediscriminabilityoftheseaffectedparticipants'responsetimesandaccuracyonthetask.TheyutilizehierarchicalBayesianmethodstoestimategroup-levelrelationshipsbetweenthesemanipulationsandoutcomes.4.2.1NumberofalternativesAstraightforwardquestionregardingtheresultsissimplyhowresponsetimesandaccuracyrelatetomanipulationsofthenumberofresponsealternativesavailable.TherelationshipsbetweenthemareshowninFigures4.3and4.4.Clearly,meanresponsetimesincreasewiththenumberofalternativesandmeanaccuracyde-creaseswiththenumberofalternativesifwelooksimplyatthediscreteconditions.However,thepreciserelationshipbetweenthenumberofalternativesandmeanresponsetimedoesnotnecessar-ilyfollowalog2relationshipasHick'slawpredicts.Infact,amodelwithalinearfunctionalrela-tionshipbetweenthenumberofalternativesandresponsetimesbetterthanonewhichpredictedresponsetimesasthelog2ofthenumberofalternatives,DIC(linear)=22176<DIC(log2)=60Figure4.3Relationshipbetweenthenumberofalternativesandmeanresponsetimesinthecoloreddotmixturestask.Meanresponsetimesforthecontinuouscondition(forcomparison)areshownontheright.Eachcolorcorrespondstoadifferentparticipant.24290.Ofcourse,someparticipants(seeFigure4.3)stillseemtohavenegativecurvatureinhowtheirresponsetimesincreasewiththenumberofalternatives.PerhapsmoreproblematicforHick'slawisthepatternofresponsetimesthatappearsinthecontinuouscolorwheelcondition.Forhalfoftheparticipants,meanresponsetimesinthecontinu-ousconditionarecontainedwithintherangeofthoseproducedbydiscretenumbersofalternatives(usuallybetween5to8).Inboththelinearandlog-linearcases,onewouldpredictmuchlongerresponsetimesinthecontinuousthaninanyofthediscrete-alternativeconditionssimplybecausetherearemanymoreoptionsavailable.InordertotheresultsofthecontinuouscasewithintheboundsofHick'slaw,wewouldbeforcedtomaketheclaimthattherewassomefipracticalflnumberofalternativesthatparticipantswereusinginthecontinuouscase,andthatthispracticalnumberwasfrequentlyfewerthan8(oftenabout7alternatives).61Figure4.4Relationshipbetweenthenumberofalternativesandaccuracyinthecoloreddotmixturestask.Accuracyforthecontinuouscondition(forcomparison)isshownontheright.Eachcolorcorrespondstoadifferentparticipant.Thesuperiorperformanceachievedbysomeparticipantsinthecontinuouscaserelativetothe8-alternativeoneisparticularlycurious.Itsuggeststhatthestrategyparticipantsimplementinthecontinuousconditionisactuallymoreeffectivethantheonetheyusefordiscretenumbersofalter-natives.Indeed,ignoringtheavailablealternativesandinsteadapplyingthecontinuous-responsestrategyforthe8-alternativeconditionwouldyieldperformancethatisatleastasgoodasinthecontinuouscondition.Itmaybethecasethatthereisamappingcomponent(fromevidenceontoavailablealternatives)thatisintroducedinthe8-alternativeconditionthatslowsdownparticipants'responses.Alternatively,participantscouldbesettingdifferentthresholdsforevidenceacrossdifferentconditions.Thiswouldsuggestthatparticipantstradespeedforaccuracybetweenthe8-alternativeconditionandthecontinuousone,meaningthatthoseparticipantswithshorterresponsetimesinthecontinuousconditionwouldhaveloweraccuracy,andthosewithlongerresponsetimeswouldhave62higheraccuracy(asinStudy1).Thereissomesuggestionthatthismaybeaplausibleexplanation,asthoseparticipantswhoseresponsetimesinthecontinuousconditionwereshorterthaninthe8-alternativeconditionshowedloweraccuracyintheirresponsesinthecontinuousconditionthanthe8-alternativeone(seeyellowandcyanparticipantsinFigures4.3and4.4).IinvestigatethispossibilityfurtherintheModelingsection.4.2.2withOneissuewithexaminingdifferencesbetweenthenumberofalternativesacrossconditionsisthatthenumberofalternativesinthese(andmostother)tasksiswithhoweasyitistodistinguishbetweenthealternatives.AnytimethereisaedrangefromwhichalternativescanbedrawnŒsuchaswithinthehuewheel,audiblesounds,visiblecolors,thesizeofthescreen,orwithanyboundsonstimulidimensionsŒagreaternumberofalternativeswillresultinmore`crowding'inlocalareasofthespaceofalternatives.Toillustratethispoint,considerthecasewherewedrawuniformrandomvariableson[0,1].Whenwedraw2randomvariables,eachonewillbeabout0.33unitsawayfromoneanother.With3randomdrawstheywouldbeonaverageabout0.25unitsawayfromoneanother,with4drawswewouldexpectthemtobe0.20unitsaway,andsoon.Asthenumberofrandomdraws(alternativesadded)increases,thespaceisdividedintosmallerandsmallerparts,witheachdrawbeingclosertoitsneighborsonaverage.Theimplicationforanalysesbasedonthenumberofalternativesisquiteimportant.Ifthereisagreaternumberofalternatives,wecanexpectthatanygiventargetalternativewillhavemoredistractoralternativesincloseproximitythanwhentherearefeweralternatives.Itisthereforemorediftodiscriminatetargetsfromdistractorswhentherearemorealternatives,meaningthataccuracyshouldbelowerandresponsetimeshigherwhenagreaternumberofalternativesareavailable(asIshowedwiththedifeffectinStudy1).ThedifferenceindiscriminabilitybetweenconditionsforthisstudyisshowninFigure4.5.Aswemightexpect,thedistancebetweenthetargethueanditsnearestdistractordecreaseswiththenumberofalternatives.Notethatbecauseeachavailablehuealternativeinthediscretecondition63spanned7.2degreesineitherdirection,nopairofalternativeswascloserthan0.04unitsofcolorhuedistance(14.4degrees).ThisyieldsthevalueforthecontinuousconditionaswellŒthenearestdistractorwastakentobethenearesthuethatwouldbecountedas`incorrect.'Figure4.5Relationshipbetweenthenumberofalternativesavailableandthediscriminabilityofthetargetalternative(hue-baseddistancetothenearestdistractor).Ultimately,thereisastrongcasetobemadethatthediscriminabilityofalternativesislargelyresponsiblefortherelationshipbetweenthenumberofalternativesanddecisionmakers'accu-racyandresponsetimes.Thisstudywasnotdesignedtoinvestigatethisissue,butitcertainlyhintsthatthisisalineofinvestigationworthpursuing.Ifincreasingthenumberofalter-nativeswhilemaintainingtheirdiscriminabilityresultsinconsistentresponsetimesandaccuracy,thenthenumberofalternativesisnotactuallyanimportantindependentfactorinalldecisions.In-stead,Hick'slawmayarisefrom`downstream'factorslikethedifoftasksfeaturingdifferentnumbersofalternatives.64Despitethis,therestillseemstobeageneralnegativetrendinaccuracyandpositivetrendinresponsetimesasafunctionofthenumberofresponsealternativesavailable.AsIsuggestedbefore,thiscouldpotentiallybeduetovaryingdecisionthresholdsthatpeoplesetacrossdifferentconditionsŒapossibilitywhichIinvestigateinthemodelingsection.4.2.3MultidimensionalscalingThehueofcolorsasshownonthecolorwheelisacasewherethephysicaldistances(inhueunits)donotmapdirectlyontothepsychologicalrepresentationsofthestimuli.Forexample,thedistancebetweentwoshadesofgreenmaybethesameasthedistancebetweenagreenandayellow,butthelatterwillalmostcertainlybemoredistinguishable.Thissuggeststhatthereisanadditionalcomponentthatoughttobeaddedtothemodelinordertomapthephysicalrelationsbetweenalternativesontopsychologicalones.Butitisworthvisitingtheeffectsthatphysical-psychologicaldifferencesintroduceintothechoicedata.ThemeanaccuracyandresponsetimesforParticipant5asafunctionofthemostprominentdothueinthedisplay(target)areshowninFigure4.6.Notethatbothshiftmassivelydependingonwhatthemostprevalentdotcolorwasinthedisplay,andinreversetooneanotherŒwhenaccuracydecreases,responsetimeincreasesandviceversa.Thissuggeststhatdecisionsforthisparticipanttendedtobemoredifwhenthemostprominentdotcolorwas(e.g.)green,andeasierwhenitwas(e.g.)purple.Thedifferenceindifbetweentheseconditionssuggeststhatsomehueswereeasiertodiscriminatefromtheirneighborsthanotherswere.Greencolorswerehardtotellapartfromnearbyhues,andpurpleoneswereeasytotellapartfromnearbyhues.Notably,thetroughsinaccuracy(andcorrespondingpeaksinresponsetimes)tendtocorrespondtothelocationsoftwoofthethreemaincolorsofconereceptorsintheretina:greenandblue.Inotherparticipants,similartroughstendedtoappeararoundredaswell(seeFigure4.12).Thisislikelybecausecolorsarehardtodiscriminateatthepeakrangesofthesereceptors,asthedifferenceinactivation(andthereforeeaseofdiscrimination)betweentwocolorsdependsontheslopeoftheactivationfunctionforred,65Figure4.6MeanaccuracyandresponsetimeandaccuracyforParticipant5conditionedonthehueofthetargetcolor.Thecoloredlineisgivenbyakerneldensityestimator(bandwidth=0.1)passedoverthecorrespondinghueaccuracyorRTdata.green,andbluedetectors.Theslopeissmallestatthepeakactivationlevels,andthereforethedistancebetweentwohueswillyieldthesmallestdifferenceinactivationnearthecenterofthesethreemainreceptors.Thesedifferencesindiscriminabilityfromthedecisiontaskareintheresultsofthesimilarityratingtaskaswell,givingindependentof(andcapacitytoestimate)differ-encesindifacrosstargethues.Foreachparticipant,thesimilaritydatawasusedtoconstructtheupperdiagonalofa3030similaritymatrix.Thesevalueswerescaledtobetween0and1(from0to100)andthenfedintoastress-based2-dimensionalmultidimensionalscaling[MDS]process.11Formally,thisminimizedKruskal'snormalizedstresscriterion,aprocesswhichisimple-mentedinMATLABusingmdscale.66TheresultsoftheMDSprocedureareshowninFigure4.7.Thedistancebetweenanypairofpointsinthesediagramscorrespondsapproximatelytotherateddissimilaritybetweenthem.Recallthatthehueswereoriginallysampleduniformlyacrossthecolorwheel,soclusteringofhuesindicatesthatpeopleratedclusteredhuesasmoresimilarthannon-clusteredones.Figure4.7ResultsoftheMDSanalysisofthesimilaritytaskincludedaspartofStudy2foreachindividualparticipant.Thedistancebetweenanypairofhuesistheapproximatedissimilaritybetweenthehuesaccordingtothatparticipant.Themostnotableclustersofhuesoccurtowardthemiddleofthegreen,red,andblueranges.Thisindicatesthatparticipantsfeltthatthesehuesweremoresimilartooneanotherthantootheradjacenthues,andtheirinabilitytodistinguishbetweenthesehuesrelativeto'boundary'colorsbetweenthethreemajorhues.Thesedifferencesindiscriminabilitymayberelatedtolan-guageaswellasthemajorconetuningfrequencies(Heider,1972;Witthoftetal.,2003),buttheofbothofthesecontributionsshouldbeinboththemultidimensionalscalingoutputaswellasdecisiondata.67TheMDSresultswereusedtoinformthedecisionmodelbyusingittoestimatethedifofeachdotsetofresponseoptions.ThemetricdistancebetweenapairofcolorsintheMDSspacewastakenasanindependentsourcetopredictthediscriminabilityofapairofcolors.Thelocationsofhues,includingbothtargetcolorsanddistractors,wereimputedusingcubicsplineinterpolation(DeBoor,1978)basedonthe30examplehuesthatwereincludedinthesimilarityratingtask.2Thisyieldedalocationforeachhueusedinthestudy,andallowedustocomputethedistancebetweenthetargethueanddistractorhuesintheMDS-generatedspace.Thisdistancewastakenastheinversediscriminabilityofatargetfromitsdistractors.Inturn,thediscriminabilitywasusedasapredictorofthedriftmagnitudeandthresholdinthecognitivemodelbelow.4.2.4SummaryofimportanteffectsAcrossthesecondstudy,therewereafewmainresultsthatstoodoutasparticularlyimportantforamodelofthedecisionprocesstocapture.Theofthesewasthepositiverelationshipbetweenthenumberofresponsealternativesandmeanresponsetimes(Figure4.3).Forallpartic-ipants,regardlessoftheparticularfunctionalform,thereseemedtobeamonotonicallyincreasingrelationshipbetweennumberofalternativesinthediscreteconditionsandmeanresponsetimes.Whetherthiscanbeattributedtothedifofthetaskisanopenquestion,butthepatternofresultsisclear.Thesecondthingtonoteisthatthereisadecreasingrelationshipbetweenaccuracyandthenumberofresponsealternatives(Figure4.4).Likemeanresponsetimes,thiscouldbeatleastpartiallyattributedtoofthenumberofalternativeswithtaskdif,butamodelshouldbeabletoaccountforthispatternaswell.Third,theheterogeneityofresponsetimesasafunctionofthetargethue(Figure4.6,bottompanel)isaparticularlystrikingfeatureofthedata.Thismaybeduetoshiftsindriftrate(because2NotethatthisincludedallcolorsusedinthestudyŒsincehuevalueswerecontinuous,thecol-orsshowninFigure4.7wereneverperfectlyreplicatedintheexperiment.ThesplineinterpolationmethodisexecutedwithinMATLAB'ssplinefunction.68ofthediscriminabilityofalternatives)orthreshold(alsobasedondiscriminability)inthemodel.Fourth,theheterogeneityofaccuracyasafunctionofthetargethue(Figure4.6,toppanel)shouldalsobeexplainablebythemodel.However,itcannotbeexplainedsimultaneouslywiththeheterogeneityinresponsetimesbyhavingthresholdschangeacrosstargethues(ordiscrim-inability)Œanincreaseinthresholdwouldseeaccuracyincreaseandresponsetimesincrease,butaccuracyappearstoincreasewhenresponsetimesdecreases.Thissuggeststhatdriftratesarelikelytobeinvolvedinexplainingbotheffectsratherthanthreshold,butshiftsinthresholdcouldbeatplayaswell.Oneexplanationfortheaccuracy-relatedeffectscouldbethatparticipantsareusingdif-ferentdecisionboundariesacrossdecisions.Thisisdiscussedintheresponseboundariessectionbelow.4.3ModelingThebasicelementsofthe2-dimensionaldiffusionmodelusedinStudy1werealsousedinthemodelsofStudy2.Itsinterpretationisessentiallythesame:ateachpointintime,aparticipantgathersinformationfromthestimulusdisplaythatindicatesthepresenceofaparticulardotcolor.Astheyviewthestimulus,theygathermanyofthesesamplesandintegratethemovertime.Oncetheparticipanthasenoughinformation(thedistancefromtheevidencestatetothecenter/unbiasedinitialstateislargeenough),adecisionistriggeredinfavorofthedotcolortheparticipantfavorsatthatpointintime.Thisgivesrisetoaresponsewhichdependsonthematchbetweentheavailablealternativesandtheparticipant'scurrentbeliefsaboutthemostprominentcolorinthedisplay.Finally,thetimeatwhichthedecisionprocesscrossestheboundarybythethreshold,inadditiontothetimetakenfornon-decisioncomponentsofmakingaresponse,givesrisetoaparticipant'soverallresponsetimeforthetrial.Themodelagainpredictsbothresponsesandresponsetimes.AsinStudy1,thedecisionprocesscanbebythedriftmagnitude,initialstate,threshold,andnon-decisiontime.69Figure4.8BasicelementsofthemodelsusedinStudy2.Inadditiontodriftmagnitude,threshold,andnon-decisiontime,decisionboundariesseparatedthedifferentresponsesintocorrectandincorrectresponses.Theseboundswereassumedtobeoptimal,sothattheywerehalfwaybetweentheavailablealternativesinthediscretechoicecases(2and3alternativesshownatthebottomleftandtopright,respectively).Inthecontinuouscase,theyweresetto7degreesfromthemostprominentdotcolor,tothecriterionof7degreestoleranceforresponsesinthecontinuouscondition.Asbefore,theinitialstatewassetto[0,0].Thiswaspartlydoneinordertoensurecomputationaltractability,butsincetherewerenopre-stimuluscuesinthisstudytherewasalsonoreasontoassumeabiasedinitialstate.Forthisstudy,thedriftdirectionwasassumedtobecenteredonthetruedominantdotcolorpresentinthestimulusdisplay.3Adiagramofthismodelisshownin3Thisassumptioncouldbemadebecausetherewasnopre-stimuluscueastothemostpromi-nentcolor,thoughafairarcumentcouldbemadeforallowingittovaryacrossconditions.Notably,itcouldbeadjustedinthediscrete-choicecasetothefactthattheavailablealternativesac-tuallygavesomeindicationofwhatthemostprominentdotcolorcouldbe.70Figure4.8.4.3.1ResponseboundariesBecauseStudy2responsesaseithercorrectorincorrect,themodelalsoincludesamech-anismfortransformingthelocationatwhichtheevidencetrajectorycrossesthethresholdintoacorrect/incorrectdiagnosis.Forexample,whenaparticipanthastochoosebetweengreenandyellowresponses(bottomleftofFigure4.8),theymayhitthethresholdwithevidencethatfavorsan`orange'response.Thismustbemappedontooneoftheavailablealternatives,sothepartici-pantmustapplysomecriterioninordertomapitontoeithergreenoryellow.Todoso,themodelassumesthatresponsesaregivenbasedonthealternativethatismostsimilartotherepresentedevidenceatthetimeitcrossesthedecisionthreshold.Inessence,decisionboundariesareplacedevenlybetweentheavailableresponsealternativesonalldiscrete-choicetrials.Forcontinuoustrials,theseresponseboundariesarenotnecessary,butresponsescanbeascorrectorincorrectbasedonwhetherornottheyarewithin7degreesofthetruedominantdotcolor.There-fore,decisionboundariescanbeplacedat7degreesfromthetruedominantcolorinordertoseparateresponsesintocorrectandincorrect.Thissetupallowsforoptimaldiscriminationbasedontheinformationavailabletoapartic-ipant;however,itisanassumptionthatcouldbeamended.Forexample,ifonecategorywererewardedmorethananother,thechoicecriteriamaybeadjustedasymmetrically.Ifgreenwererewardedmorethanblue,forexample,blue-greenevidencecouldbeasgreenratherthanblue.Suchchangestotheresponseboundariescouldbemodeledbysettingtheboundariesasfreeparametersoradjustingthemassomefunctionoftheexperimentalmanipulationsappliedtomovethemaround.However,forthepurposesofthepresentstudythisisnotnecessaryandsotheboundariesareassumedtobeunbiased.Notethattheresponseboundariesthemselvesactuallymayaccountfordifferencesinaccuracybetweendifferentnumbersandsetsofresponsealternatives.ComparethetoprightagainstthebottomleftpanelsofFigure4.8Œintheformercase,thereare3relativelydistinctresponsealter-71natives(blue,green,red)thatareeasytotellapartandsoall3responsesshowninthediagramareasgreen/correct.However,inthecasewherethealternativesaremorediftytotellapart(bottomleft,greenversusyellow),oneoftheseresponsesisasyellowsimplybecausetheresponseboundariesmustbeplacedmorecloselyinordertodistinguishbetweenthem.Thissamepropertymaybeatleastpartiallyresponsibleforthedifferencesinaccuracybe-tweennumbersofresponsealternativesshowninFigure4.4.Becausethealternativesarenaturallygroupedmorecloselytogetherwhenthereisalargernumberofthem,wecanexpectthatthespanofthedecisionboundarieswillonaveragebenarrowerinthesecases(comparetheboundariesonthetoprightagainstthebottomrightinFigure4.8,forexample).Thiswillresultinfewerevidencerepresentationsbeingcorrectly,andcorrespondinglyloweraccuracy.Therefore,theincorporationofchoiceboundariesmayactuallybeabletoaccountforbothimportantaccuracyeffectsobservedintheempiricaldata.Decisionboundarieswillbenarrowerwhenalternativesaregroupedcloselytogether(lowerdiscriminability),resultinginloweraccu-racywhentherearemorealternativesormoresimilaralternativesavailable.However,theydonotaccountforthevariationinresponsetimes,neitherasafunctionofdiscriminabilitynorasafunctionofthenumberofalternatives.Theremustthereforebeanothercomponentofthedecisionprocessthatleadstovariationinresponsetimesinthetask.4.3.2DiscriminabilityThemajoradvantageconferredbythemultidimensionalscalingsolutionisthatwecanestimatethediscriminabilityofanypairorsetofchoicealternativespresentedinthestudy.ThemoststraightforwarduseofthisinformationistofeeditintothedriftrateofthedecisionprocessŒwhenthesetofalternativesiseasiertodiscriminate,thedriftmagnitudeshouldbehigher.Becausethemultidimensionalscalingsolutionwascomputedindependently,thiscanbedonebyaddingjustoneadditionalfreeparameterthatsetsthedriftrateasa(linear)functionofthedistancebetweenthetargethueandthenearestnon-target/distractorresponsepresentinthedisplay.Thereasonforusingthedistancetothenearestdistractorandnotthediscriminabilityofall72availablealternativesistwofold:itisnecessaryinordertoconsiderthecontinuouscondition.Becausethereisannumberofalternativesinthecontinuouscondition,itmakeslittlesensetotrytoconsiderthesimilarityofallofthemtothetargethue.Instead,thenearestdistractorcanbeusedtoestimatethedifofthedecision,whichwillbesmallerwhenaparticularhueisdiftodiscriminatefromthehuesarounditandlargerwhenitiseasier.Thisallowsforaconsistentmetricacrossallconditionstobeusedtosetthedriftmagnitude.Second,thedriftmagnitudeonlyreallyneedsoneadditionalreferencepointtobeset.Ifwethinkaboutthemagnitudeastheinverseofthewidthofthesamplingdistribution(itsvariance),itcanbeassessedbyexamininghowquicklythedistributionfallsofffromitscenter.Thiscanbedonebytakingtwopointswithaedrelativedensity(e.g.3:1)andidentifyingwheretheyareonthecircleŒforexample,comparingthecenterofthesamplingdistributiontothelocationofanotherpointwith13thedensity.Thenearestdistractorcanbeseenasapointwith1ntimesthedensityofthemean,andthereforeitslocationcanbeusedasanindexthatgivestheapproximatevarianceofthesamplingdistribution.Thelocationofthenearestdistractoristhereforesuftoprovideanapproximateestimateofthedriftmagnitude.Asimplerwaytoputthisisthatthesamplingdistributionrequires2freeparameters,ameanandavariance(concentration).Themeanissetatthetarget,sowecansetthesamplingdistributionbyreferringtothelocationofthedistractortosetthevariance/concentrationasthesecondparameter.Thegreatestadvantageofincorporatingmultidimensionalscaledistanceintothedriftmagni-tudeinthemodelisthatitcantheoreticallyaccountforallfourmajorphenomenaobservedintheempiricaldata.Itaccountsforvariationinresponsetimeandaccuracyasafunctionoftargethuebecausethenearestdistractortothesehueswilltendtobefurther(higherdiscriminability,faster&moreaccurateresponses)orcloser(lowerdiscriminability,slower&lessaccurateresponses)dependingonthetarget.Inaddition,becausehuestendtonaturallybegroupedclosertogetherwhentherearemorealternatives,itpredictsfasterandmoreaccuratedecisionswhentherearefeweralternativesfromwhichtochoose.734.3.3ChangesinthresholdAlthoughchangesindriftmagnitude(asafunctionofmultidimensionalscalingdistance)areinprinciplesuftoaccountforvariationinaccuracyandresponsetimesacrosstargetsandcon-ditions,itmaywellbethecasethatparticipantssystematicallyadjusttheirowninternalchoicecriteriaacrosstrialsaswell.Thiswouldadesiretogathermoreorlessinformationdepen-dentonsomecharacteristicsofaparticulardecision.Perhapsthemostlikelyinstancewhereparticipantscouldchangetheirthresholdsiswhenthenumberofalternativeschanges.Usheretal.(2002)suggestthatthisisthemechanismthatactuallyproducesthechangesinresponsetimesacrossnumbersofalternatives.Giventheevidencetheyprovide,itseemsreasonabletoatleasttestthisassumptionbyallowingthresholdstochangeasafunctionofthenumberofalternativesavailabletoaparticipantonatrial.Therefore,oneofthepotentialmodelcationsthatItestedwastoallowthethresholdtovaryasalinearfunctionofthenumberofalternatives.Becauseitisunclearhowthecontinuousconditionrelatesintermsofthediscretenumberofalternativesitcorrespondsto,thethresholdwaspermittedtovaryindepen-dentlyinthecontinuouscondition.Thisresultedinthreeadditionalfreeparameters:oneinterceptparameterforthethresholdinthediscreteconditions,oneparameterthatsettheslopeasafunctionofthenumberofalternatives,andonethatsetthethresholdforthecontinuouscondition.However,thisisnottheonlymanipulationthatmayhavechangedthethresholdsthatpartici-pantsapplied.Thediscriminabilityofpairsofalternativeswithinthechoicesetcouldalsocausethemtoapplystricterormorelenientchoicecriteria.Thethresholdwasthereforepermittedtovaryasalinearfunctionofthediscriminabilityofthetargethueanditsnearestdistractor,justasthedriftmagnitudevaried.Thisintroducedonefreeparametertothemodel,aslopethatsetthethresholdasafunctionofdiscriminability.Eitherofthecationstothresholdscouldaccountforsomechangesinresponsetimeandaccuracyacrossconditions.However,theycannotaccountforbothsimultaneously.BecauseresponsetimeincreasedwhileaccuracydecreasedŒbothasthenumberofalternativesvariedas74wellasthemultidimensionalscalingdistancebetweenthemŒchangesinthresholdcannotaccountforall4effects.Thisisbecauseashiftinthresholdcancreateeitherincreasebothresponsetimeandaccuracyordecreasebothresponsetimeandaccuracy.Alone,itdoesnotpredictdiscordantchangesinthetwooutcomes.4.3.4ModelandresultsInall,therewerefourpossibleadditionstothebasicmodelwhichincludedabasethreshold,non-decisiontime,anddriftmagnitude.TheeffectsofeachoftheseisshowninTable4.1.Theofthese,responseboundaries,isnecessarytoincludebecauseitseparatescorrectfromincorrectresponses.Thereisnostraightforwardalternativemodelwhichdoesnotincorporateasimilarmechanism.However,addingthiselementtothebasicframeworkmaybyitselfaccountforvariationinaccuracyacrosstheconditionsofthetask.Becauseitwasrequiredinordertogenerateaccuracypredictionsandbecauseitaddednofreeparameterstothedecisionmodel,nomodelwhichexcludedthiswastested.Table4.1Observedresultsandthemodelcomponentsthatcanaccountforthem.Acheckmarkindicatesthatthemodelcomponentcanaccountforaneffect,ablankmeansitcannot,andanXmeansitcanproducetheoppositeeffect.ModelcomponentEmpiricaleffectResponseboundariesMDSdistance7!driftMDSdistance7!threshold#alternatives7!thresholdAccuracy#alternatives(Figure4.4)XXXXRT#alternatives(Figure4.3)XXXAccuracytargethue(Figure4.6,top)XXXRTtargethue(Figure4.6,bottom)XXThesecondadditionwastosetthedriftmagnitudeasafunctionoftheMDSdistancebetweenthetargethueanditsnearestdistractor.Thisadditioncanintheoryaccountforallfourimportant75effects,anditistheonlyconsideredthatcancreatedifferencesinresponsetimesasafunctionoftargethue(withoutadverselycreatingtheoppositeoftheobservedeffectinaccuracy).Thetwoadditionsarenotabsolutelynecessaryinordertocreatetheobservedaccuracyandresponsetimephenomena.Thesearethatallowedthresholdstovaryacrossconditions,eitherasafunctionofthenumberofalternativesorasafunctionofthediscriminabilityofthetarget.Inordertogaugewhethertheinclusionofeachoftheseelementsoutweighedtheassociatedincreaseincomplexity,Icompared3modelsexcludingeachindividualelementagainstthefullmodel.Thealternativemodelallowedonlyasingledriftmagnitudeacrossconditions,ignor-ingthediscriminabilitybetweenalternativesasafactorimpactingdrift.Thesecondalternativemodelallowedthresholdstovaryonlyasafunctionofthenumberofalternatives,andthethirdalternativemodelallowedthresholdstovaryonlyasafunctionofthediscriminabilityofalterna-tives(generatedfromtheMDSsolution).Eachmodelincludedallotherelementsnotomittedfromthefullmodel,sothateachalternativetestedthehypothesisthatasinglecouldbeomittedwithoutreducingmodelperformance.Eachofthefourmodelswasusingmaximumlikelihoodestimation.Thelikelihoodfunc-tionincorporatedboththeaccuracyandresponsetimeofeachdecisionparticipantsmade.Thepredictedaccuracyofaresponsewasgivenbyintegratingtheprojecteddistributionofresponsesovertherangeofvaluesthatwouldbeasthetruedominantcolorgivenoptimallyplaceddecisionbounds(seeFigure4.8).NotethatpredictedresponseswillbevonMisesdistributedasafunctionofthedriftmagnitude(jmj)andthreshold(q),andthelocationofthecounterclockwise/lowerdecisionbound(L)andclockwise/upperdecisionbound(U)willvarybytrial.There-fore,thelikelihoodofacorrectresponsePr(C)willvaryasafunctionofthecondition,choicealternatives,andtargetcolor.Pr(C)=ZULexpqjmjs2cos(f)2pI0(qjmjs2)df(4.1)76Inthiscase,fistheangleatwhicharesponsecanhitthethreshold,Lisanegativenumberindicatinghowfarawaythenearestcounterclockwisedecisionboundaryis,andUisapositivenumberindicatinghowfarawaythenearestclockwisedecisionboundaryis.ThefunctionI0()isaBesselfunctionofthekind,oforder0.Asthedistributionofresponsesisassumedtobecenteredonthetruedominantdotcolor,thecenterofthevonMisesdistributionwillbe0,givingtheaboveequation.Computingthejointdistributionofresponsesandresponsetimesrequiresthatwemultiplytheconditionalresponsetimedistributionsbytheprobabilityofbeingcorrect.Theconditionaldistri-butionofresponsetimesisgivenbyconditioningEquation3.1ontheresponseanglex.Notably,thepredictedresponsetimedistributionsaresimplyscaledcopiesofoneanother,astheirrelativefrequencydoesnotdependontheresponseangle.Thesewereusedtoderiveloglikelihoodsforeverymodelandparticipant,whichwereinturnusedtocomputeaBayesianinformationcriterion[BIC](Kass&Raftery,1995;Schwarz,1978).TheBICincludesapenaltyforthenumberofparametersinthemodel,favoringthemoreparsimoniousmodelwithfewerfreeparameterswhentheirlikelihoodsaresimilar.Therefore,ifoneofthemodelelements(freeparameters)conferredinsuftheBICshouldfavorthemodelwhichomitsthatfeature.TheresultsofthemodelareshowninTable4.2.Table4.2Bayesianinformationcriterion(afunctionoftheloglikelihoodwithapenaltyforthenumberofparameters;seeSchwarz,1978)foreachmodeltested.ThemodelwiththelowestBICispreferredandshowninbold.ModelParticipant#FullFixeddriftmagnitudeThresholdinvariantto#alternativesThresholdinvarianttodiscriminability1497.39509.48737.483153.522845.92866.53502.53072.538430.68553.19769.19449.141136411477140371159152885131502344823481667449.98288.57728.57524.5Foreachparticipant,theBICfavoredthefullmodelincludingdriftthatvariedbyMDSdis-77tance,thresholdsthatvariedbyMDSdistance,andthresholdsthatvariedbythenumberofalterna-tives.Themodelingresultssuggestthatthediscriminabilityandnumberofalternativesbothplayimportantrolesinthedecisionprocess.However,giventhatbothofthemarepartsofthebest-model,theeffectofthenumberofalternativescannotbeisolatedtojustshiftsinthresholdorjustchangesinthediscriminabilityofalternatives.Instead,theresultspaintasomewhatricherpicturewherethenumberofalternativesaffectsmultiplefacetsofthedecisionprocess,includingboththeinformationparticipantsconsideraswellasthechoicecriteriatheyapply.Sincethefullmodelwasfavoredforallparticipants,thesameparametersarepresentineachofthemodels.ThemaximumlikelihoodparameterestimatesaregiveninTable4.3,fordriftmagnitudeinterceptm0,driftslopeasafunctionofMDSdistancems,non-decisiontimendt,thresholdinthecontinuousconditionqc,thresholdinterceptforthediscretealternativeconditionqd,thresholdslopeasafunctionofthenumberofalternatives(forthediscreteconditions)qsa,andthethresholdslopeasafunctionoftheMDSdistancebetweentargetandnearestdistractorqsb.Table4.3MaximumlikelihoodparameterestimatesfortheBIC-favoredmodelofthedatafromStudy2.Participantm0msndtqcqdqsaqsb13.198.740.163.152.970.320.3321.369.910.392.600.990.120.3531.266.420.373.873.350.332.2642.272:300.4812.880.240.910:2151.3516.550.467.793.680.441:3561.952.060.442.561.580.381:31Fornearlyallparticipants,driftmagnitudeincreasedwithalargerMDSdistancebetweenthetargethueanditsnearestdistractor(i.e.,discriminability;ms>0).Thresholdsalsoconsistentlyincreasedwiththenumberofalternatives(qsa>0),assuggestedbyUsheretal.(2002).Curiously,therewassubstantialvariationintheeffectofdiscriminabilityonthresholds(qsb).Non-decisiontimes(ndt)werelargelywithinnormalrangesexpectedforaratingtaskofthissort.Interestingly,thethresholdestimatesforthecontinuouscondition(qc)fellallalongthespectrumofthresholdsforthediscreteconditions(compareqctoqd+qsa#alternatives).Thissuggeststhattheremay78besubstantialdisagreementbetweenparticipantsastohowmuchinformationtheyshouldgatherinthecontinuousconditionrelativetothediscreteonesŒthethresholdsinthecontinuousconditionwereequivalenttothethresholdsforanywherefrom1to14alternativesinthediscreteconditions.Theparameterestimatesthemselvesaresomewhatinteresting,buthowdoestheresultingmodelperform?Intheory,themodelelementsincludedshouldallowittopickupthevariationinbehavioracrossnumbersofalternatives,thecontinuouscondition,anddifferentdiscriminabilitiesofalternatives.4.3.4.1PosteriorPredictionsForeachparticipant,Itookthemodel(thefullmodel)andderiveditspredictedresponsetimeandaccuracydata.Becausetheparametersvariedbasedonwhattheavailablealternativeswere,howmanyofthemwereavailable,andwhichonewasthedominantcolor,Igeneratedaresponsetimedistributionandpredictedaccuracyforeverytrial.ThesewereusedtogenerateaggregatedistributionsofresponsetimesandaccuracyasshowninFigure4.9.WiththeslightexceptionofParticipant#4,theresultingmodelsprovidedagoodaccountoftheaggregateresponsetimeandaccuracydata.Becausethemodelgeneratesajointpredictionforeachtrial,wecanalsoexamineitspredic-tionsforresponsetimesandaccuracyacrosstargethuesaswellasacrossnumbersofalternatives.Predicted(Xs)andactual(fadedlineswitherrorbars)meanresponsetimesbythenumberofavailablealternativesareshowninFigure4.10.Themodelsrecoverthesemeanresponsetimesquitewell,buttendtoslightlyunderestimatetheamountoftimeittakestorespondinthecon-tinuouscondition.ThemostlikelyexplanationforthiserroristhatparticipantsaretakinglongertoactuallyentertheirresponsesinthecontinuousconditionŒthedifferencesaretedmainlyinameanshiftinresponsetimesbutnotinaccuracy.Thissuggeststhattheremaybeanadditionalcomponentofnon-decisiontimeinthecontinuousconditionthatisnotpresentinthediscrete-alternativeconditions.Ifthisisthecase,thenitislikelythatdifferentexperimentaldesignswillproducevaryingdifferencesbetweencontinuousanddiscretechoiceconditions.79Figure4.9Posteriorpredictionsfromthemodelsforeachparticipant,overlaidontheactualresponsetimeandaccuracydata.Notethatthemodelsdonotnecessarilypredictexactlylinearnorexactlylog-linearrelation-shipsbetweenthenumberofalternativesandmeanresponsetimes.Asthedataandmodelresponsetimeisalsoheavilybythediscriminabilityofalternatives.Thisfactorsubstantiallychangesthedriftmagnitude,andtoalesserextentthethresholdaswell(seeTable80Figure4.10Posteriormodelpredictions(X)andobservedmeanresponsetimes(fadedlines)foreachparticipant(differentiatedbycolor).Errorbarsindicate1unitofstandarderrorinthedata.4.3).Asaresult,thepredictedandobservedrelationshipsbetweenthenumberofalternativesandmeanresponsetimesaremuchricherthanthatwhichissuggestedbyHick'slaw.Similarmodelpredictionsforaccuracyasafunctionofthenumberofalternativescanalsobegenerated.Aswithresponsetime,themodelprediction(Xs)andobservedaccuracy(fadedlineswitherrorbars)areshown.Aswiththeresponsetimedistributions,thereissubstantialforthecontinuousconditiondataofParticipant#4(blue).Itseemslikelythatthisparticipanthadsubstantialtroublewiththecontinuouscondition,tothepointwhereeachresponsewasbothveryslowandveryinaccurate.Despitethis,themodelpredictswelltheaccuracydataforthediscrete-alternativeconditions.Althoughthresholdsdidseemtochangeacrossthenumberofalternatives(indicatedbythemodelwithandwithoutthisfactor),thiswasapparentlyinsuftomaintainconstantac-curacyacrossthediscrete-choiceconditions.Asaresult,bothmeanresponsetimeandaccuracyshiftedasafunctionofthemanipulationsofthenumberofalternatives.Itispossiblethatifpartic-81Figure4.11Posteriormodelpredictions(X)andobservedaccuracy(fadedlines)foreachparticipant.Errorbarsindicate1unitofstandarderrorinthedata.ipantswereabletomaintainaccuracy,theirresponsetimeswouldbelog-linearlydistributed,butthisishighlyunlikely.TheextraprocessingtimeneededtomaintainaccuracyinconditionswithalargenumberofalternativeswouldlengthenresponsetimesintheseconditionsŒi.e.,wecouldexpectthatstrivingtomaintainaccuracywouldonlyincreasethemarginalresponsetimeaddedbyeachalternative.Asaresult,meanresponsetimewouldprobablybeaconvexfunctionofthenumberofalternativesŒnotaconcaveoneasHick'slawpredicts.Finally,thevariationinresponsetimesandaccuracyacrossthediscriminabilityofalternativesisshowninFigure4.12.Whiletheposteriormodelpredictionsdonotexplainthisvariationasneatly,theydidpickuponsomeimportantcharacteristicsofthedata.Participant#5providesagoodexampleŒtheslowerandlessaccurateresponseswhengreenorbluewasthetargetarepredictedbyvirtueofthesecolorsbeingclosertotheirneighborsintheMDSspace(Figure4.7)andthereforehardertodiscriminate.Aswithsomeoftheotherposteriorpredictions,Participant#4wasabitofanoutlier;whilethemodelpredictionsfollowedthegeneraltrendofaccuracyandresponsetimesforotherparticipants,itpredictedalmostthereverseoftheempiricaltrendfor82Figure4.12Posteriormodelpredictions(dottedblackline)oftherelationshipbetweentargethueandresponsetimeoraccuracyforeachparticipant.Bothdata(coloredcurve)andmodelpredictioncurvesarecomputedbypassingakerneldensityestimatoroverthehueRTorhueaccuracydata(orprediction)fromeverytrial.Participant#4.Overall,thefullmodelseemstoaccountquitewellforthepatternsofmeanresponsetimesandaccuracyacrossdifferentnumbersofalternatives,andcapturessomeoftheimportanttrendsinhowaccuracyandRTrespondtodifferencesintargethue.834.4PreliminarydiscussionStudy2providedanexaminationofhowthenumberandassortmentofalternativesaffectsthedecisionprocess.Ifoundthataccuracydecreasesandmeanresponsetimesincreaseasthenumberofalternativesapersonhastochoosebetweengetslarger.Thefunctionalrelationship,especiallytakingthecontinuousconditionintoconsideration,doesnotseemtostrictlyfollowHick'slaw.Instead,thepatternsofresponsetimeandaccuracythatresultfrommanipulationsofthenumberofalternativesarisefromseveralcontributingfactors.Thediscriminabilityofthetargetfromdistractorsinthestimulusplayedalargerole,asdidshiftsinchoicecriteriaresultingdirectlyfromthenumberofalternativesavailable.Thelocationsofchoiceboundariesthatdividedresponsesintotheavailablealternativesalsocontributedtoshiftsinaccuracyacrossvaryingnumbersofalternatives.Althoughthepatternsofresponsetimesandaccuracyarefairlystraightforwardfordiscretechoiceconditions,thecontinuous-responseconditiondidnotneatlyinwiththeseresults.Someparticipantswereabletomaintainaccuracyandresponsetimesintheseconditionsasiftheyweredealingwithonly2-8alternatives;othershadgreatdifandsufferedbothintermsofaccuracyandresponsespeed.Itseemslikelythatdifferentparticipantsapplydifferentstrategiesinordertocopewithacontinuumofresponses,meetingwithvariablelevelsofsuccess.Despitethis,responsesinthecontinuouscasedidseemtobeimpactedbythediscriminiabilityofthetargethuejustastheywereinthediscretealternativecases.ThissuggeststhatthecharacteristicsoftheinformationparticipantsuseseemtobethesameŒwhichwemightexpectgiventhatthestimuliwerenearlyidenticalŒbutthatdifferentdecisionrulesareapplieddependingonwhetherthetaskhasdiscreteoptionsoracontinuum.Therearehintsthatthewayparticipantsentertheirresponses(i.e.non-decisioncomponentsofthetask)maydifferinthecontinuousconditionaswell.Theadditionofamultidimensionalscalingcomponentbasedonsimilarityratingdatapartic-ularlyhelpedtounderstandthefactorsgivingrisetovariationinbehavior.Byusingthisdatatoconstructamultidimensionalscalingrepresentationofthecolorsforeachparticipant,Iwasable84topredictthediscriminabilityofapairofcolorsindependentofthedecisiondata.Andasthemodelsuggest,incorporatingthismultidimensionalscalingrepresentationintothedecisionmodelhelpedtoaccountforbothresponsetimeandaccuracyoutcomes.Thisstudythereforepro-videsandfavorablytestsamethodofintegratingrepresentationanddecisionmodelsintothesameframework,andemphasizestheimportanceofconsideringtherelationshipsbetweenalternativeswhenaccountingforbehavior.85CHAPTER5GENERALDISCUSSIONTheframeworkproposedintheintroductionservesasthefocalpointofthispaper,buttheempiricalinvestigationsarerelevantsansmodelaswell.Thespeed-accuracytrade-off,difeffect,andpredecisioncuesallhaveanaloguesinacontinuous-responsetask,providingsomeofthephe-nomenathatrelateerrormagnitudestoresponsetimesonacontinuum.ThevonMisesdistributedresponsesand(non-)relationshipbetweenRTanderrormagnitudeprovideadditionaldimensionstotheempiricalresults.Furthermore,theeffectsofthenumberandsimilarityofalternativesinbothdiscreteandcontinuouschoiceconditionsprovidehurdlesforanytheoryofhowpeoplemakeselectionsonthesetasks.Theresponsesandresponsetimesalonetellusthatstimuluscharacteristics,cues,rewards,andassortmentofchoiceoptionsallhavedramaticimpactsondecisionprocesses.Theaddedvalueofthemodelistodescribepreciselyandconcretelyhowtheoutcomeschangeasafunctionofthesemanipulations.Inturn,ithasmanypotentialuses.Oneoftheseisthatitcanserveasanembodimentofourunderstandingofdecisionprocessesinchoicesamongmanyalternatives.Wecanunderstandtheeffectsofourmanipulationsintermsofshiftsinparametervalues,allowingtheframeworktoservesasameasurementmodel.Theparametersarethemselvespsychologicallymeaningful:driftmagnitudetellsushowquicklyapersoncangathervalidinformationinforma-tion,thresholddescribesthestrictnesswithwhichtheymakedecisions,driftdirectiongivesusalensontobiasesandinaccuracyininformationtheyuse,andnon-decisiontimeletsusdiscriminatethecomponentsofactionandperceptionbeyonddecisionprocesses.Estimatesoftherelationshipsbetweenalternativesarealsoparticularlyimportant:wecanuncoverhowpeoplearerepresentingtheiravailablechoiceoptions,andinturnprojectwhaterrorsarelikelyandwhatdecisionswillbemostdifInturn,wecandeviseinterventionsbasedontheseparameterestimates,focusingonspeedingupdecisions,alleviatingbias,improvingdiscriminationbetweensetsofalternatives,orencouragingpeopletosetbettercriteriaformakingtheirdecisions.86Theintegrationofthedecisionmodelwithrepresentationcomponentsisparticularlynotableinthatitbringstogethermultipleindependentprocessesinordertomakebetterpredictions.Eventu-ally,ourmodelofthewholebrainanditscomputationalprocesseswillhavetoincorporatemulti-plelayersofcognitivemodelsŒfromperceptiontorepresentation,beliefandpreferenceupdating,decision,action,feedback,andstrategyrevision.Usingmultidimensionalscalingtoconstructde-cisionspacesputsatleasttwopiecesofthispuzzletogether,allowingustoexplainphenomenainbehaviorthatresultsfromtheinteractionofmultiplecognitiveprocesses.Myhopeisthattheframeworkanddataproposedherehaveopenedupnewquestionsaswellasprovidedhintsastohowwerepresentandprocessinformation.Atworst,itatleastprovidesapassatattemptingtomodelhowpeoplemakeselectionsamongmanyalternativesŒataskwhichdecisionmodelinghasstalledonforperhapstoolong(Townsend,2008).5.1LimitationsOfcourse,thepresentstudiesexploreonlyasmallportionofpotentialtasksandmodelsthatcouldbeusedinresearchoncontinuousormulti-choicedecisions.Perhapsmostnotably,theexper-imentspresentedherearebothperceptual,inferentialchoicetaskswherethereisatruecorrectanswer.Giventhedifferencesbetweenperceptualandpreferentialchoiceevenwiththesamestim-uli(Zeigenfuseetal.,2014;Dutilh&Rieskamp,2015),itseemslikelythatthedecisionprocessesinselectionsamongmanyalternativesarelikelytochangebasedonthegoalsoftheparticipantaswell.Fortunately,thesechangesmaybepossibletoexplainthroughchangesinparametersratherthancompleteshiftsinmodelstructure.Morecomplexmulti-attributedecisionsarealsolikelytorequiresometothestructureofthemodelspresentedhere.Fortunately,thearchitecturepresentedintheintroductioniswell-suitedtosuchtasks,allowingthespaceofavailablealternativestoamulti-dimensionalfeaturespaceinwhichtheymightexist.Inthesecases,thealternativesmayhavetobeconstructedbasedontheirrawphysicalcharacteristics,thoughmultidimensionalscalingmayhelptoconstruct87thedecisionspaceinsuchcasesaswell.Asthenumberoffeaturedimensionsincreases,however,sodoesthecomplexityofthedecisionspaceandcorrespondingdecisionmodel.Whenthealternativescanbearrangedsothattheycreateahypersphere(forexample,onealternativetakingupeachoctantofasphere),thereareanalyticalsolutionstothelikelihoodfunctionthatcanbederivedbysubstitutingthevonMisesdistributionofevidencewithavonMises-Fischerdistribution.Giventhetractabilityofeventhe2-dimensionalcase,thesearelikelytoquitecomplexanddiftousepractically.Inmanyothercaseswherethealternativesdonotformaneatclosedhypersphere,responsetimeandresponsedistributionsmayhavetobederivedbyrepeatedlysimulatingthedecisionprocess.Theparticularfunctionsattachedtothesamplingdistribution(drift),threshold,andnon-decisiontimesmayalsonotbethebestcandidatesinallsituations.Whenapersonswitchesslowlybetweentwoattributesofastimulus,forexample,thesamplingdistributionmayneedtobetime-dependent,asinthebinarydecisionmodelsofDiederich&Busemeyer(2006).Similarly,asStudy2suggests,non-decisioncomponentsmayvarybetweentasks(evendiscreteversuscontinuousselectiontasks)intheirmeanorfunctionalform,ratherthanbeingsetasedoruniformdistributions(Verdonck&Tuerlinckx,2016).Giventheapparentshiftsinthresholdsasafunctionofstimulusmanipula-tionsinStudy1(Figure3.5),thresholdsmayneedtovaryacrosstime.Theseassumptionsleadtodifferentdistributionsofresponsetimesanddecisions,andfurtherworkshouldexploreexactlyhowsensitivemodelpredictionsaretotheseparticulardistributionalassumptions.Finally,recentworkonthesequentialeffectsbetweenandwithintrialshavesuggestedthatdecisionsandjudgmentsmadeinsequenceŒwhetheraboutdifferentstimuliorthesameoneŒaffectoneanother(Brownetal.,2008;Kvametal.,2015).Trial-to-trialeffectsmayhaveadifferentunderlyingsourcethanwithin-trialeffects,butresponsetimeandaccuracydistributions(aswellasforsequentialjudgmentshavebeenwell-describedbydecisionmodelsconstructedusingaquantumprobabilityframework(Fuss&Navarro,2013).Itispossiblethataquantumrandomwalkinmultipledimensionsmayprovideamorecompleteofmulti-choiceorcontinuous-responsedecisionsaswell.885.2ExtensionsMostofthelimitationsIhavediscussedsofaractuallyofferopportunitiesforfutureworkaswell.Notably,sequentialchoiceeffects,collapsingresponseboundaries,quantumrandomwalks,andpreferentialdecisionsallofferdirectionsforfurtherinvestigation.However,thereareanumberofotherapplicationsofthemodelthatdomorethanexplorebinarychoicephenomenainmulti-choiceorcontinuous-responsecases.Oneparticularlyinterestingextensionistospatialtasks,wheredifferentlocationscorrespondtodifferentrewardsorrisks.Decidingwheretolookforrewardsdrawsstrongparallelstoforag-ing,whereagentshavetomakespatialselectionswhengatheringfood.Thesedifferentlocationsalmostcertainlycorrespondtovariationsinpredationrisks,environmentalhazards,andpotentialpayoffs(food),whereforagershavetomakecompromisesinordertosurviveandreproduce.Ourunderstandingofriskydecisionprocessesinhumansandotheranimalscanbewell-informedbyexaminingwhatconditionsaffecttheutilityofdifferentoutcomes,produceriskavoidanceorriskseekingbehavior,impactexplorationorexploitationorientation,oryieldvaryingrepresentationsofrareevents.Itseemslikelythatexperienceinaspatialdomainproducesdifferentrepresentationsandforaging(decision)behaviorthandescriptionsormapsoftheterritory,andthesecontinuousrepresentationsshouldbericherthanthosethatgiverisetothedescription-experiencegapinbinarygambles(Hertwig&Erev,2009).Anotherimmediateextensionofthemodelingframeworkistobeginpredictingorlikelihoodratingsthatpeoplegive.Therearetwodirectionstopursueinthiscase.Inthesimplecasewheretherearetwoalternativesbutacontinuumofjudgmentstobemadebetweenthem,themodelframeworkcanofferamethodofpredictingthedistributionsofjudgmentsaswellasthespeedwithwhichthesejudgmentsaremade.Itislikelythatthemodelframeworkcanbeusedtoreproducemanyoftheimportanteffectsobservedinjudgmentdistributionsandjudgmentresponsetimes(Pleskac&Busemeyer,2010;Moranetal.,2015),butasofyetthisisanuntestedclaim.89Anotherdirectionthatcouldbepursuedistoexaminehowjudgmentsaremadebetweenmanyalternatives,orevenoncontinuousspans(e.g.fiHowlikelyisitthatthepopulationofcityXfallsbetween170,000and200,000?fl).Inprinciple,thelocationofthemultidimensionalrandomwalkprocess(whenitunfoldsonahypersphere)correspondstothelogoddsacrossallavailablehypothesesoralternativesinthatspace.Thereisthereforean'true'/optimalprobabilityorcon-ratingthatoneshouldgivebasedontheavailableinformationorevidence.Moreover,theoptimalcaseimposesstrongconstraintsontheadditivityofjudgmentsŒbecausethelogoddsalwayssumtozero,probabilityjudgmentsmadeacrossallalternativesshouldsumtoone.Itispossiblethatevidenceisre-mappedontoratings(Ratcliff&Starns,2009;Pleskac&Busemeyer,2010),inwhichcasethisrelationshipcouldbebroken.Ineithercase,themodelframeworkprovidesabasisfromwhichwecanstudyjudgmentsofmanyalternativesontopofjustchoicesamongmanyalternatives.Understandinghowthesejudgmentsareapportionedmayalsohelpusunderstandhowpeoplestoreandreviseevidence,allowingforappropriaterevisionsofthemodelstructureintermsofhowinformationisrepresented.Behavioraldataisnottheonlyprovincefordecisionmodelsliketheoneproposedhere,either.Morerecently,sequentialsamplingmodelshavebeenlinkedtoneuraldata(Hanes&Schall,1996;Schurgeretal.,2012;Shadlen&Newsome,2001;Ratcliff&McKoon,2008)andmodel-basedanalysesofjointneuralandbehavioraldataarebecomingeasierandmoreeffective(Turneretal.,2013).Themultidimensionalwalkproposedherecouldcertainlybeusedforthesepurposesaswell.Forexample,activationoforientationcolumnscouldbemappedontothesamplingdistribu-tionfromthemodelinStudy1.TheparameterestimatesinFigure3.6(bottomleft)givesanideaofhowvaryingcueorientationsmightfacilitateorinhibitdecisionsinfavorofrelatedorientationsŒthisisalmostcertainlyrelatedtothetuningcurvesoforientationdetectorsinthebrain.Eachofthemodelparametersshoulddescribesomefacetsofbothbehaviorandneuralactivity;theymotivateexplorationofwhatbehaviorsandregionsarerelatedaswellaswhatparametersaremostappropriatefordescribingboththeactivityandproductsofthebrain.Inprinciple,themodelingframeworkcanbeextendedtoexplainandpredictanyselection90amongalternativesthatarepsychologicallyrelated.Inmostcases,theprimaryhurdleisconstruct-ingthedecisionspacethatrelatesthealternativestooneanother.ResearchinsomedomainshasalreadymappedthisstructureŒforexample,pitchesandtonesonecanselectareoftenmodeledastoroidalorhelicalshapes(Lerdahl,2004;Shepard,1982).Wemightreasonablyexpectthatadecisionspacewheresomeonehastoselect(orproduce)apitchortoneshouldthesepsy-chologicalrepresentations,bothintermsofwhaterrorstheymakeandhowquicklytheyareabletoreachtheirdecisions.ThisisonlythetipoftheicebergŒanydomaininwhichthepsychologi-calrelationsbetweenalternativescanbeempiricallyestimatedisadomaininwhichthemodelingframeworkcanbeappliedtopredictdecisionsamongthem.5.3ConclusionsThisthesishascoveredabroadrangeofconsiderationsthatgointodecisionsamongmanyalterna-tives,butthemostimportantpointsarerelativelysimple.First,thedecisionprocessthatgoesintomultiplechoiceandcontinuousresponsedecisionsasequentialsamplingevidenceaccu-mulationprocess.Aswesawinthestudy,theempiricalphenomenaandmodelalladecisionarchitecturethatresembles(butisageneralcaseof)thatusedtopredictbinarydecisionsandresponsetimes.Second,thecombinationsofavailableoptionshavesubstantialeffectsonthedecisionprocess.Aswesawinthesecondstudy,thenumberandthesimilaritybetweenalternativescanslowdownchoicesandmakethemmoreerror-prone.However,thearchitectureofthedecisionmodelcanbeinformedbyindependentdataontherelationshipsbetweenthepsychologicalrepresentationsoftheavailablealternatives.Thisintegrationallowsittoaccountformuchofthevariationindecisionpatternsacrosscombinationsofalternatives.Finally,themodelframeworkproposedhereopensupawiderangeofnewbehaviorstopre-dictandnewquestionstoaddress.Itletsusmovetowardpredictingandexplainingbehaviorontaskslikeinvestment,timemanagement,andlikelihoodjudgments,pricesetting,spatial91navigation,andeventhingslikepitchproductionorforaging.Empiricalstudiesofeachofthesedomainscaninformourunderstandingofdecisionprocesses,andtheframeworkIhaveestab-lishedinthisthesisprovidestheformalquantitativetheorytoembodyandextendourknowledgeofbehaviorondecisionsamongmanyalternatives.92BIBLIOGRAPHY93BIBLIOGRAPHYBerens,P.(2009).Circstat:AMatlabtoolboxforcircularstatistics.JournalofStatisticalSoftware,31(10),1Œ21.Bogacz,R.,Brown,E.,Moehlis,J.,Holmes,P.,&Cohen,J.D.(2006).Thephysicsofoptimaldecisionmaking:Aformalanalysisofmodelsofperformanceintwo-alternativeforced-choicetasks.PsychologicalReview,113(4),700-765.doi:10.1037/0033-295X.113.4.700Bogacz,R.,Wagenmakers,E.-J.,Forstmann,B.U.,&Nieuwenhuis,S.(2010).Theneu-ralbasisofthespeedŒaccuracytradeoff.Trendsinneurosciences,33(1),10Œ16.doi:10.1016/j.tins.2009.09.002Bowman,N.E.,Kording,K.P.,&Gottfried,J.A.(2012).Temporalintegrationofolfactoryperceptualevidenceinhumanorbitofrontalcortex.Neuron,75(5),916Œ927.Brainard,D.H.(1997).Thepsychophysicstoolbox.SpatialVision,10,433Œ436.Brown,S.D.,&Heathcote,A.(2005).Aballisticmodelofchoiceresponsetime.Psychologicalreview,112(1),117Œ128.doi:10.1037/0033-295X.112.1.117Brown,S.D.,&Heathcote,A.(2008).Thesimplestcompletemodelofchoicere-sponsetime:Linearballisticaccumulation.Cognitivepsychology,57(3),153Œ178.doi:10.1016/j.cogpsych.2007.12.002Brown,S.D.,Marley,A.,Donkin,C.,&Heathcote,A.(2008).AnintegratedmodelofchoicesandresponsetimesinabsoluteidentPsychologicalReview,115(2),396Œ425.doi:10.1037/0033-295X.115.2.396Busemeyer,J.R.,&Diederich,A.(2002).Surveyofdecisiontheory.MathematicalSocialSciences,43(3),345Œ370.Busemeyer,J.R.,&Townsend,J.T.(1993).Decisiontheory:adynamic-cognitiveapproachtodecisionmakinginanuncertainenvironment.Psychologicalreview,100(3),432Œ459.doi:10.1037/0033-295X.100.3.432Busemeyer,J.R.,Wang,Z.,&Townsend,J.T.(2006).Quantumdynamicsofhumandecision-making.JournalofMathematicalPsychology,50(3),220-241.doi:10.1016/j.jmp.2006.01.003DeBoor,C.(1978).Apracticalguidetosplines(Vol.27).Springer-VerlagNewYork.Diederich,A.,&Busemeyer,J.R.(2003).Simplematrixmethodsforanalyzingdiffusionmodelsofchoiceprobability,choiceresponsetime,andsimpleresponsetime.JournalofMathematicalPsychology,47(3),304Œ322.doi:10.1016/S0022-2496(03)00003-8Diederich,A.,&Busemeyer,J.R.(2006).Modelingtheeffectsofpayoffonresponsebiasinaperceptualdiscriminationtask:Bound-change,drift-rate-change,ortwo-stage-processinghypothesis.Perception&Psychophysics,68(2),194Œ207.doi:10.3758/BF0319366994Donkin,C.,Brown,S.,Heathcote,A.,&Wagenmakers,E.-J.(2011).Diffusionversuslinearbal-listicaccumulation:differentmodelsbutthesameconclusionsaboutpsychologicalprocesses?Psychonomicbulletin&review,18(1),61Œ69.doi:10.3758/s13423-010-0022-4Donkin,C.,Brown,S.D.,&Heathcote,A.(2009).Theoverconstraintofresponsetimemodels:Rethinkingthescalingproblem.PsychonomicBulletin&Review,16(6),1129Œ1135.Drugowitsch,J.,Moreno-Bote,R.,Churchland,A.K.,Shadlen,M.N.,&Pouget,A.(2012).Thecostofaccumulatingevidenceinperceptualdecisionmaking.TheJournalofNeuroscience,32(11),3612Œ3628.Dutilh,G.,&Rieskamp,J.(2015).Comparingperceptualandpreferentialdecisionmaking.Psychonomicbulletin&review,23,1Œ15.Edwards,W.(1965).Optimalstrategiesforseekinginformation:Modelsforstatistics,choicereactiontimes,andhumaninformationprocessing.JournalofMathematicalPsychology,2(2),312Œ329.Fuss,I.G.,&Navarro,D.J.(2013).Openparallelcooperativeandcompetitivedecisionprocesses:Apotentialprovenanceforquantump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