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Reasoning with uncertain categories
Name: Reasoning with uncertain categories
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Year: 2012
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neighbour'sdoor.Ifyou areaskedtotakecareof thedogfor aday, youcan predictwhatisgoing toberequir edofyou. Withoutanyspecific informationaboutthispet, knowinga boutdogsin generalallows youto makepredicti onsandinterpretinformationaboutit . Researchoncategory basedinduc tiontakesplace attheintersectionof thetopicsof conceptsand reasoning.Category basedinductionisa reasoningprocessthattakesc onceptualrepresentati onsasitsinpu ts.The waythatpe opleretrieve,c ombine,andevaluatetheinfor mationtheyhave storedaboutcategor iesiscomplex,an dresearchershavedevelopeda numberofmodelsofth eindu ctionprocess(e.g.,O shers on,Smith, Wilkie, Lo pez,&Sha fir,1990; Sloman,1993).Workinour laboratoryhasfocused onaspecific reasoningproblem thatcanarise intheinductionprocess, namelyhowpeopledeal withuncertain categories. Intheexampl eabove, whenyourneighbourtellsyou shehas boughta dogyoucan becertain thatitis infacta dog.Ifyoursister saysshehas boughtanewNissan,you probablyfeel sureitis aNissan. However,in manysituatio nsyoumaynothaveenoughinformati ontoclas sifyanobj ect withcertainty. Acargoingbyseemsto beaNis sanbutis possiblyone of thosenewHyundai s;your illnessisprobablya coldbutpossiblytheflu;a paintingcouldbelateimpres sionistorpossi blyfau vistinst yle.Insuchcases youcansti llusecate goryinformation ,butyoumu sttakeintoaccountyour uncertainty.Normatively,thebestpred ictionintegratesanswersacross categoriesor,aswe shallreferto it,bet hedging.IfthecarmaybeaNissan, withprobabil ityp,or aHyund ai, withprobabilityq,thenyourprediction aboutitsmiles per gallon(mpg) shouldbe theNissanmpgtimespplusthe Hjundaimpgtimes q;thatis,weightingeachcategorybythelikelihoodthat thecaris init.For predictingdiscr eteproperties, onecannottake the averageinquite thisway,but onecancalcul ateapro bability(Anderson, 1991).Ifyou wanttoesti matethepro babilitythatthe carisa diesel,andif youknowthe proportion ofdieselcars eachmanufacturermakes,youcould estimatetheprobabilityasp6proportionofNissandieselsplus q6pro portionofHyundaidies els.Ag ain,thisweightseachc ategory'sinductionby itslikelihood. Anderson(1991)proposedt hissolutioninwhathe describedasa Bayesiananalysisofinduct ion 1 aspart ofhislargerth eoryof category formation.Hisproposalcanbe seenasan applicationof thelawof total probability(Mood,Graybill, &Boes,1974): P A P AjB P B P Aj B P B ; 1 Inallour experimentsthecategories arenoveland equallyprobable,so weignorethe prior probabilitycompone ntofBayesianreasoning.We continuetouse thetermBayesianbecauseof thecommon featureofBayesian modelsofinductionthatp redic tionsareintegratedacros s multiplecategories, weightedbytheirlikelihood. 82MURPHY,CHEN,ROSS Downloaded by at 06:49 08 March 2012


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whereAreferstothe predictionandB and B totherelev antcategor ies (Nissanandnot N issan, i.e.,Hyundai).Althoughpeopleapproximately followtherulewhen theyare askedtogive estimatesof thecomponentsin somecontexts (seeZhao&Osherson,2010, andtheGenera lDiscussion), theymaynot alwaysdoso incategory based induction. MurphyandRoss( 1994;see alsoMalt,Ross,&Murphy,1995)investigated whetherpeopleuseds uchanapproachwhenma kinginducti onswithuncert ain categories,an dtheyfoundthatp eoplegenerallydidnot. Inp articular,varying theproperties ofthelesslikely category(e.g.,the Hyu ndai)didnotinflu ence people'sprediction s(seealsoHayes&Chen,2008).In contrast,varyingthe propertiesofthe mostlikely(target)categorydidaffectpeople'spredictions (Murphy&Ross,2005;R oss &Murphy,1996).MurphyandRossconcluded thatpeop legenerallyfocusonthemostl ikelycategoryand deriveaprediction basedonthatsing lec ategory,inc ontrasttothebet he dgingideal.Murphyand Ross(2010b)proposedthatthisfocusingonasinglecategorywasrelatedtoother reasoningshortfalls inwhichpeople focusonasinglepo ssibleoutcomeeven whenit wouldbe relativ elyeasytot akeintoaccounttwoormore possibilities (Stanovich,2009).Forexample,Evan s(2007)proposedthesingularity principle asag eneralpr opertyofhypotheticalreason ing,thatpeopleconsideronlyone situationatatime,unlesssomethingpromptsthemtoexpandtheirthinkingsuch asah int orfailuretod erivea nanswer.Researchondecisi onmakin gsuggests thatpeop ledislikeuncertaintyandwillpayt oreduce it,evenwhenthis has noe ff ecton theirultimatedecision s(Shafir,Simonson,&Tversky,1993). Thegoalofthepresentarticleistobetterunderstandwhenpeopledoanddo notusemultiple categories incategory based induction.Forminga profileof circumstancesthatleadtosingleversusmultiplecategoryusewillbecriticalfor atheoryofwhenandwhypeopleengageinBayesianreasoningofthissort. Fromapractical standpointit isimportantto discoverwhatsituations might increasepeople's accuracyinthis task.Focusingon asinglecategory when anothercategoryisals ofairlylikelyleadstosubo ptimal inductions. Techniquesthatimprove people'sperformancein ourtaskmay alsoworkin majorreal lifepredictions madeinmedical orcareerdecisions. Wefirst brieflyreviewa paradigmwehave usedinprevious work andthen explainwhato urnewe xperimentswilladdtowh atwehave learnedfro mthat work.Althoughfocusingonasingleparadigmhaslimitations,italsoallowsusto accrueknowledgeacrossrelatedexperimentsinordertoarriveatamorecomplete conclusion. PASTSTUDIESOF CATEGORY BASEDINDUCTION UNDER UNCERTAINTY Figure1shows adisplaybased ononewe usedinMurphy andRoss (2010b).Inthisexperiment participantsv iewedcolouredfig uresthatwere REASONINGWITHUNCERT AINCATEGORIES 83 Downloaded by at 06:49 08 March 2012


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purportedlypicturesthatdi fferentchildrendrewus ingacomputerdraw ing program,inwhichtheycouldc hoo seshapesandcol ours.(Ourfigu res representcoloursbydi ff erentpatterns, sowewillrefertopatterns insteadof colourthroughout thisarticle.)Itwaspoin tedoutthat di ff erentchildren preferreddi ff erentshapesand patterns.Then participants readaboutanew shapethathad beenfound, forexample anew square.Theywereasked whichcategorytheythought thesquarewas in(i.e.,which childdrewit),the probabilitythatitwas inthiscategor y,and thepatternthey thoughtthis squarewould have theinduction. InFigu re1itcanbeseenthatF ed eri cowasmost likely tohavemade asquare,butCyrusalsomadeafewsquares.Peoplegenerallyrecognised thatFederico wasnotguaranteedto havemade thesquare,as theyrated theprobab ilityofthiscategorisationaroun d65%.I ftheynoneth eless usedonlyFede rico's categorytomaketheinduction,sinceit isthemost likely,theywoul dpredict thatthesquarewouldbe verticallystriped, becauseFedericohasfour stripedfiguresandthree blackones(and the sameistrue forsquaresin particular 2 ).Ifinst eadtheyfoll owedthebet hedgingprinciple,theywould haveusedbothFedericoandCyrus to generatetheirprediction.In thatcase,they wouldhavepredicted thatthe squareshouldbesoli dblack eitherbythe kindof computations describedabovefortheNis san/Hyundaiexampl e,orby asimpler computationofcountingupthe colourofthe displayedsquares .Thatis, althoughFederico'sdrawings haveasmalladvantage forstripedfigures, whencombinedwith Cyrus'sdraw ingstheadvantag eswingstoblack figures. Theresultsshowe dthatpeople chosethemultiple categoryresponse(i.e., solidblack)only30%of thetim einsuch problems(Murphy &Ross, 2010b, Exp.1).Anana lysisofind ividu alparticipantsshowedtha tonly7 of47 consistentlyfollowedthisruleacross di ff erentinductions. Instead,most answerswerethesingle categoryresponse(i.e.,vertically striped),and22 participantsconsistentlygave suchanswersacrossdi ff erentproblems. This confirmedthegeneralfindi ngsofearlier workusingsim ilardisplays,aswell asexperiments usingshortstorieswithverbal lystated uncertaintyabouta characterorobject(Maltet al., 1995;Mu rphy&Ross, 1994;Ross& Murphy,1996).Theresultsalso revealedthatthis tendencywasnot monolithic,asaminorityof peopledidseem tofollow somethinglike Anderson's(1991)Bayesianru le. Furthermore,MurphyandRoss's (2010b)Experiment 2foundthat peoplecouldbepersuaded topayattentiontoall therelevant categories 2 Thedisplayswere constructedsothat thepredictions werethesame whetherpeople treated shapeandcolour asindepend ent(asAnderson, 1991,suggests)or insteadrestrictedtheir predictiontoother objectswiththe samegivenfeature (squaresinthis case).There isevidence thatpeopledo somethinglikethe latterinthis task,asdiscussed inExperiment5. REASONINGWITHUNCERT AINCATEGORIES 85 Downloaded by at 06:49 08 March 2012


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withaseemingly simplemethodological change.In theusual paradigm,we askpeople tochoosethemostli kelyca tegoryandtorateit sprobabil ity. Thisisdonein par ttomakesuret hatpe opleagreeonw hichca tegoryis mostlikelyto becorrect. Iftheycho seCyrusas mostlikelytohave drawna square,forexample, theywouldlikely predictthatthenewpicturewas black,butnot becausethey wereusing multiplecategories.Notethatthe probabilityratingisanexplicitind icationthatparticip antsa renotc ertainof theirchoice. IftheyrateFedericoas 65%likely tobethe correctcategory, thiscan onlybebecausetheyhavenotice dthesqu aresthatCyrus made. Therefore,thisratingexplicitlyrequires participantsto acknowledgetheir uncertainty.Surprisingly,thatacknowl edgmentdoesn'tleadthemtoattend toCyr usinthesubseque ntindu ction. IntheirExpe riment2, MurphyandRoss(2010b)askedpeopleto ratethe probabilityofallfourcategories notjustthemostlikelyone.So,a participantmightrateFederico as70%,Cyr usas30%, andtheothertw o categoriesas0%likelyto havedraw nasquare. (Mostresponses wereof this sort.)Nowwhenasked topredict theobject's colour,87%ofthean swers forthiscategor ystructure werethemultiple category response ahuge increasefromthe30% intheearlier experiment.Giving aprobability for eachcategory eliminatedthefocuson asingleoption,overcomin gthe singularityprinciple(Evans,2007). THEPRESENTRESEARCH Thecurren texperimentsinvestig atedinmoredetailjustwhataboutthis taskmakespe opleawarethatth eyshouldbeusingmu ltiplecategories and whypeople appearunawareofth isinthestandardvers ion.Whydoesthe listingofmultipleprob abili tiesleadtoachangeinstrat egy?Thesimplest explanationwouldbethat simplydrawingpeople's attentionto theless likelycategoriesmakes themincludethem intheircompu tations. However,thepresentex perimentswill showthat merelydrawingattention toacat ego rydoesnotnecessarilylea dtoitsbeingco nsidered ,orto correctperformance. Furthermore,itisn'tclearthat whenpeoplegivethe rightanswer theyaredoingsobecause theyareimplement ingtheprinci ple ofbet hedging. Perhapssuccessfulmanipulationsbring thatprinciple to consciousness;orperhapstheyactthrough lessexplicitchannels of activatingcategoriesandbringi ngthemintoworkingmemor ywithoutthe underlyingprinciplesplaying arole.Indeed,implicitmeasur esofbringing analte rnativecategorytomindincreasetheu seofmultiplecategories (Ross&Murphy,1996). Theexperiments alsoinvestigateddi ff erentprocedures thatmightdisrupt thesingularity principleandcausemore accurateresponding.Forexample, ifcate goriesarecompletelyuncertain, thenperha pspeoplewillbemore 86MURPHY,CHEN,ROSS Downloaded by at 06:49 08 March 2012


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likelytoattendtomu ltiple ones.Finally,weinve stiga tedinmoredetailjust howmanype oplespontane ouslyusethemultipl e categorystrategy.Inthe MurphyandRoss(2010a, 2010b)paradigm thatanswercan occurby chanceorguessing.If peoplearenot payingtoomuchattention theymight choosethemultiple category responseevenif theyconsideredonecategory (ornone!).We constructeda controlcondition thatallowsustodiscove r howoftenthis happens,providin gabetter estimateofhowoftenpeople use multiplecategoriesinindu ction. Insummary,the presentexperimentsinvestigate whenandwhy people overcometheir tendencytofocus onasingle category,evenwhen theyare uncertainthatit iscorrect.The goalisto developaprofile ofsituationsthat encourageordiscouragepeoplefromusingmultiplecategories,therebygiving greaterinsightinto theunderlyingcauses ofthisnon normative reasoning strategy. GENERALMETHODOLOGY ParticipantswereNYUstudentsor othermembers ofthecommuni tywho servedinthe experimentsfor coursecredit orpay. Thematerials consistedofdisplaysofthe sortshowninFigure1, except thatcolourwas usedinstead ofthepatte rnsshownthere. Peoplereceived generalinstructions aboutthechildren'sdraw ingprogram,andtheyalso receivedinformationabout the0 100%probabilityscale, bothessentially identicaltothoseusedin ourpastwork (e.g.,Murphy& Ross,1994,2010a, 2010b).Thedispl ayswere printedonpaperandplaced inplasticsheet protectorsinaloose leafbinder. Foreac hca tegorystructure,participan ts turnedapage toseea displayandthen readaseri esofque stionsin a separatebooklet.Thedeta ilsofthequestionsareprov idedlate rineach experiment.Thereweregeneral lythreequestionsabout eachdisplay, oneof themafiller ,whichserved todisguisethepurposeof theexperiment. After completingthatpageinthe booklet,theparticipant turnedtothe nextsetof questionsandpictures. Theexperimentsusuallyhad foursuchdisplaysand setsofque stions,andthey tookabout15minutesto complete.For each displayonecriticalquestioninvolv edpredicting shapegiventhe figure's colour,andthe otherinvolved predictingcolour giventhefig ure'sshape. Therewerealwaystwo versionsofeachform(except inExperiment 5) thatswitched thepredictedfeaturesof thesingle category andmultiple categorystrategies.That is,inFigure1,peoplewhoattendonlyt oFederico wouldpredictasquareto bevertically striped;those whoattendto both categorieswouldpredictsolid black. Inasecondformthestriped andblack patternswouldbeexchanged. Thisway,anypreference foragiven feature wouldbebalanced across thetwoinductionstrategie s.Halftheparticip ants receivedeachformineach experiment. REASONINGWITHUNCERT AINCATEGORIES 87 Downloaded by at 06:49 08 March 2012


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EXPERIMENT1 Asexplained above,inMurphyand Ross's(2010b) Experiment2 people wrotedowntheprobabilities ofeachcategor y,asin (2)below,ratherthan choosingthemostlikelyon eandratin gitsprobabil ity,asin(1).Thisha dan enormouse ff ectonthe inductionrule used,asalmos tallinductionsused multiplecategories. (1)Ihave asquare.W hichch ilddoyou thinkdrewit? Whatistheprobability thatthe childyoujust nameddrewthis? Whatcolourdoyouthink thefigure has? (2)I haveasquare. Whatis theprobability thateachchild drewit? Federico___%Cyrus_ __%George___% Tony___%(mustsum to100) Whatcolourdoyouthink thefigurehas? Thisresultisv eryinteresting,be cau seaseemingly smallmanipulation overcamepeople'sstrongtendency tofocuson asingleca tegory.Logically, whenparticipant sratedFedericoas65%likelytohave drawnthe square, theymusthave beenattributing theremaining35% probabilityto Cyrus, theonlyot herchildwhod rewsquares.How ever,thislogica lfactis apparentlydi ff erentfromactual lywriting downthatCyrushad35% probability,asin(2).Inouroriginalqu estio n,(1), thesecondary categ oryis implicitlyusedindeciding theprobability rating;inthe newerversionthe secondarycategoryisexplicit lyacknowledged.Thisseem sli kelytobe importantinexplainingwhype opleo vercamethesingularityprinciple when answering(2). InMurphy andRoss(2010b),the questionswerecomparedacross experiments,andgiventhe importancewewillattribut etothis e ff ectinour analysisofthetask,it seemswise toreplicate itwithparti cipantsrandoml y assignedtoexperimental forms.Wecompared thestandardquestion,(1) above,totheall categoryquestion,(2),inwhichpeopleestimatedthe likelihoodofallfour categories.The onlydi ff erencewasthat weused thequesti on,''Whichchilddoyouthinkm ostlikelydrewit?''in thestandard conditionratherthan''Whichch ilddoyouthinkdrewit?'', therebyemphasising thattheparticipantwasnot indicatingcerta intyin writingdownaname.This ''mostlikel y'' languagewasused inallthe subsequentexperiments. Method Thisexperiment usedthreedi ff erentdisplaysofthe''childre n'sdrawings'' stimuliasdescribedint heGene ralMethod.Eachdisplayhadtw ocr itical 88MURPHY,CHEN,ROSS Downloaded by at 06:49 08 March 2012


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questionsandonefiller. Atotalof 48participants performedthetask, randomlyassignedtoone ofthetwoquestiontypes justdescribed. Resultsanddiscussion Wefirstestabl ishedthatpeople selectedthetargetcategor yasthe most likelyone. Fortheall categoryquestion, weaccepted anytrialon whichthe targetcategoryhadt hehighestortiedforhighe stprobab ility.One participantinthestandardcon ditionwasde letedformaking morethan one classificationerror.Accuracyof choosingthetarge tcategoryasthe most likely(ortied)was. 97inbothc onditions.Furth ermore,t heestimat ed probabilitythatthiscategor ywascorrect wasvery similaraswell:69.7and 67.7%inthe standardand all categorycond itions.Thusanydi ff erencein theinducti onsmadebythesegroupscann otbeattribut edtodi ff erencesin classification. Themainque stioniswhat inductionsthetwogrou psmade.Ref erringto Figure1,if someonepredicted thatanew squarewoul dbeverticallystriped, thiswouldrefl ectasingle cat egoryinduction;ifhe or shepredictedthe squarewould beblack,thiswouldreflectuse ofmultiplecategor ies.The standardgroupmademultipl e categoryinduc tions34%oft hetime,but theall category groupmadethem69% ofthetime.Althoughnotquite as largeas theoriginale ff ectreportedin Murphyan dRoss(2010b) ,thee ff ectis stillverylarge andis significant,t(45) 3.53,p5.001.Nine subjectsinthe all categorygroupconsistentlyusedmu ltiplecategorie sacrossallproblems, withonlytwoc onsistently usingsingl ecategories.Inthestandardgroup thesenumberswer ethreeand eight.Theseresultsshow thatderivingan d writingdowntheprobabi lityforthele sslikelyc ategoryise ff ectivein encouragingpeopletoattendtoitwhen makinginductions. Thesli ght changeinwordingfromthe earlierprocedure, askingwhi chchild was''most likely'',hadnoapparent e ff ect. EXPERIMENT2 Writingdowntheprobab ilitiesofal lthecateg oriesinExperiment1couldbe havingtwoe ff ects:makingparticipants committo thesecondarycategory beingareal possibility (acommitmentexplanation)orsimply making informationaboutthe secondarycategorymoresalien t(aninformational e ff ect).Whenpeopl ewritedownpr obabilitiesfortwoo rmorecategories in questionset(2)theyare takinganactive steptoconfirm, ''It'snotjust Federico;thereareother possibilities.''By writinganon zero probabilityfor Cyrustheyarecommitting toCyrusbeing afactor intheinduction.This activestepcouldthe nencouraget hemtothinkabout multiplecate gories whenthey gettothe prediction.If theinitial classificationquestio nistotally REASONINGWITHUNCERT AINCATEGORIES 89 Downloaded by at 06:49 08 March 2012


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eliminatedinthisparadigm,peop letend tousemultiple categoriesmore often(Hay es&Newell,2009;for morediscussion ofclassificatione ff ectson laterjudgementss eeBusemeyer,Wang,&Lambert Mogilians ky,2009), whichsuggeststhatcommitm entcouldbeacritical factor. Alternatively,onceonehaswritt enthatFederico andCyrushave 65% and35%of thelikelihood, informationon thequestion nairenow encouragesmultiplecategoryus e.Inversio n(1)ofthequestions thereis nothingaboutCyrus,and sothatinformat ionremainsimplicit,even though Cyrus'sdrawingsmustha vebeen consultedtoderivetheprob abilityfor Federico.Afteranswering(2), theinformation thatCyrushasa35% probabilityissalientandsowillbeta ken intoaccount.Onthe informationalaccount,then,itisnot thewriting downorcommitm ent thathasane ff ectsomuc hasthe mereavailabi lityof anon zero probability forCyrus. Inthepresent experimentwe useatechn ique(suggestedbyHakwan Lau)inwhi chtheinformat ionalcontentofversion (2)ismaint ained withoutanypersonalcommitment. Inthismethod wesimplytell people theprobab ilitythatafigureisineachcategor y.Forexampl e,w emight say,''Ihave afigurethat is65% likelytobe drawnbyFede ricoand 35% likelytobe drawnby Cyrus.What colourdoyouthinkit has?''(To preventpeoplefromderiving theirownprobabilitieswe omittedany informationabouttheoth erfeaturesof theobject.)In termsof informationalsalience,bothFedericoandCyrus havebeenmenti oned, andtheir (correct)probab ilitiesprovided.Therefore,i fitismerelythe availabilityoftheinfor mationaboutboth categoriesthat isessential, peopleshouldchoosethe answerthatuses informationfrom both categoriesmoreoftenthan inacon trolcondition usingtheusual form shownin(1). However,ifpeop lemustthems elvescommitto Cyrusbeing important,thenweshouldnot seeany di ff erencebetweenthe new informationalversionandthestandar dversion. Method Theexperimenta lbookletcontainedthreedi ff erentdisplays.Eachdispl ay hadtwocr iticalques tionsandonefiller;one ofthecriticalquestionsineac h displaywasrelevantto thiscomparison (theotherwillbedescri bedinthe controlexperimentbelow).In theinformationalversion,participantswer e toldtheprob abilitieso fthetwochildren(categories)thatwer emostl ikelyto havedrawn anewfigure.Theprob abilitiesgivenwer ethesame (rounded)as theactual probabilitiesderiv edfromthedisplay.Forexample,ifJorda nhad sixandAdam threeoran gefigures,the informationalquestio nwouldsay,''I haveanew figurethat is67%likel ytohavebeendrawn byJordanand 33% likelytoha vebeen drawnbyAdam.Wh atshapedo youthinkthefigure 90MURPHY,CHEN,ROSS Downloaded by at 06:49 08 March 2012


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