Leek - Adaptive Procedures in Psychophysical Research

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Perception&Psychophysics

2001,63(8),1279-1292

Thestudyofsensationandperceptioninhumansandotheranimalsisexpensiveandinvolvesanumberofpit-fallsanddifficulties.Fechner,in1860,firstrecognizedthatinnerconsciousnessmightbemeasurablebyoutwardbehavior.Itisthemeasurementofthatbehaviordirectly,interpretedasanindirectmeasureofperception,thatisthepurviewofpsychophysics.Overmorethanacentury,meth-odshavebeendevelopedandrefinedthatsupportthesys-tematicexplorationwithinsensorysystemsofthelimitsofdetectionanddiscriminationamongsimilarandconfus-ablephysicalstimuli.

The measurement methodologies developed sinceFechnersrealizationhaveastheirprimarygoalthevalidreflectionofsensoryevents.Inordertohaveconfidenceinthevalidityandreliabilityofsuchmeasures,manysamplesofagivenbehaviortypicallymustbeobservedinastruc-turedandsystematicprocess,inresponsetocarefullycon-structedstimuli.Changesinstimulusstrengthorothercharacteristicsareassociatedwithchangesintheabilitytodetectordiscriminatesuchstimuli.Measuresofperfor-manceonpsychophysicaltasksasafunctionofstimulusstrengthorothercharacteristicsconstituteapsychometricfunction.Acompletecharacterizationofpsychometricperformanceasafunctionofchangesinstimulusstrengthmaybedevelopedbyusingthemethodofconstantstimuli,oneoftheclassicalpsychophysicalmethodologies.Withthisprocedure,asetofstimuliwithstrengthsspanningtherangeofsensationfromimperceptibletoconsistentlyper-ceptibleiscreated.Eachmemberofthestimulussetispre-

sentedtoanobservermanytimes,atrandom,andanob-servationresponseisrequestedaftereachpresentation.Thepsychometricfunctionmaybesampledbyevaluatingthepercentageofpresentationsofeachmemberofthestimulussetthatisdetected.Thefunctionisassumedtobecontinuousalongthestimulusaxis,usuallywithmonoto-nicincreasesinperformancebeingassociatedwithin-creasingstimulusstrength.

Figure 1showsanexampleofapsychometricfunction,measuringthenumberoftimesaparticularauditorystim-uluswasheard,dependingonwhatthestrengthofthestimuluswas.Inthisexample,amethodofdeterminingtheresponsetoagivenstimuluswasusedthatiscalledayesno method:Oneachstimuluspresentation,theob-servergivesoneofthosetworesponses,indicatingwhetherthestimuluswasperceivedornot.Othermethodshavebeendevelopedtomakethisdetermination,suchasthosethatinvolvetwoormoresequentialpresentationsthatdifferalongsomecharacteristicofinterest.Theobserverisaskedtoindicatewhichofthemultiplesequentialpresentationsoneachtrialwasatargetstimulus.Becausethetargetisal-wayspresentinoneandonlyoneofthepresentationsonatrialandtheobservermustselectoneofthepresentationsasaresponse,thesearecalledforced-choicemethods.Per-hapsthemostcommonoftheforced-choicemethodsisthetwo-alternativeforcedchoice(2AFC),althoughaswillbeshownlater,forced-choiceprocedureswiththreeorfouralternativesprovidemoresatisfactorymeasurementofpsychometricperformance.

Althoughsuchafunctionisassumedtounderlietheperceptionofsensorystimuli,oftenonlyoneortwopara-metersofthefunctionwillsufficetosummarizepercep-tion.Themostcommonlydeterminedparameterisamea-sureoflocationofthefunctionalongthestimulusaxis,typicallyspecifiedasathresholdstimulusvalue.Thethresholdisdeterminedasalevelofdetection(ordis-

crimination)performance,andfrequentlythecriterionperformanceforthresholdisselectedtobethemidpointof

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ThisworkwassupportedbyGrantDC00626fromtheNationalIn-stitutesofHealth. Theopinionsorassertionscontainedhereinarethe

privateviewsoftheauthorandarenottobeconstruedasofficialorasreflectingtheviewsoftheDepartmentoftheArmyortheDepartment

ofDefense. CorrespondenceconcerningthisarticleshouldbeaddressedtoM.R.Leek,ArmyAudiologyandSpeechCenter,WalterReedArmy

MedicalCenter,6900GeorgiaAve.,NW,Washington,DC20307-5001(e-mail:[email protected]).

Adaptiveproceduresinpsychophysicalresearch

MARJORIER.LEEK

WalterReedArmyMedicalCenter,Washington,D.C.

Asresearchonsensationandperceptionhasgrownmoresophisticatedduringthelastcentury,newadaptivemethodologieshavebeendevelopedtoincreaseefficiencyandreliabilityofmeasurement.Anexperimentalprocedureissaidtobeadaptiveifthephysicalcharacteristicsofthestimulioneachtrialaredeterminedbythestimuliandresponsesthatoccurredintheprevioustrialorsequenceoftrials.In

thispaper,thegeneraldevelopmentofadaptiveproceduresisdescribed,andthreecommonlyusedmethodsarereviewed.Typically,athresholdvalueismeasuredusingthesemethods,and,insome

cases,othercharacteristicsofthepsychometricfunctionunderlyingperceptualperformance,suchasslope,maybedeveloped.Resultsofsimulationsandexperimentswithhumansubjectsarereviewedtoevaluatetheutilityoftheseadaptiveproceduresandthespecialcircumstancesunderwhichonemightbesuperiortoanother.


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thefunctionspanningtherangefromchanceperformancetoperfectperformance.Asecondsummaryparameterusedtodescribeperformanceistheslopeofthepsychometricfunction,whichisameasureofhowrapidlyperformancechangeswithagivenchangeinstimulusvalue.Often,sen-sorycapabilitiescanbeadequatelydescribedbyathresh-

oldmeasurealonethatis,onesinglepointonthepsy-chometricfunction.However,inthemethodofconstantstimuli,thethresholdisextractedfromafullysampledfunction,makingthemeasurementofthissinglepointonthefunctionveryexpensiveintermsofexperimenttime.Ofnecessity,manytrialsareplacedatstimuluslevelsoftheunderlyingpsychometricfunctionthatarenotinfor-mativeaboutthreshold.Itisnecessarytohaveinforma-tionaboutperformanceattheseoff-thresholdlevelsinordertofullydevelopthepsychometricfunction,butinmanycasesthatadditionalinformationisneitherneces-sarytothegoalsoftheexperimentnorworththeextraex-

perimentaltimeandeffort.Adaptivepsychometricproce-dureshavebeendevelopedtoaddressthismajorprobleminpsychophysicalmeasurementthatis,aninefficientplacementoftrialsalongthestimulusarrayinordertoex-tractarelevantmeasure.Anexperimentalprocedureisadaptiveiftheplacementofeachtrialalongthestimulusarrayisdeterminedbytheresultsofthetrialortrialsthathavegonebefore.Itisacharacteristicofalladaptivepro-ceduresthatknowledgeabouttheoutcome(e.g.,athresh-old)increasessystematicallyastheprocedureisinprogress.Thatis,theselectionofstimuliisdeterminedduringthecourseoftheexperiment,andstimulusplacementisdri-

venbytheadaptivealgorithmtowardthedesiredmea-surementpoint.

Adaptiveproceduresaredesignedtorapidlyextractrel-evantmeasurementsfromapsychometricfunctionthoughttounderlieperformanceonagivensensory/perceptual

task.Generally,twotypesofmeasuresareofinterest:lo-cationalongastimulusaxis(e.g.,threshold)andtheslopeofthefunction(howrapidlyperformancechangeswithchangesinstimulusvalues).Often,onlyalocationmea-sureisrequired,buttherearesomeinvestigationswhosegoalisamorecompletedescriptionoftheunderlying

function,requiringbothlocationandslopemeasures.Theproceduresthemselvesinvolvetwoseparateparts:place-mentoftrialsalongastimulusaxisandanalysisofthedataobtainedtoextractcharacteristicsoftheunderlyingpsy-chometricfunctions.

Thechallengeofadaptivepsychophysicsistomakerel-evantobservationsonthepsychometricfunctionwithmaximumefficiencywithoutsacrificingaccuracy.Adap-tivemethodsofmeasurementhavebeendevelopedwiththegoalofpreservingaccuracyandreliability,whilemaxi-mizingefficiencyandminimizingsubjectandexperimentertime.Thisarticlewilltracethedevelopmentofmodern

adaptiveprocedures,withspecialattentiontothestrengthsandweaknessesofthreeofthemostcommonlyusedmeth-ods.Inaddition,somespecialapplicationswillbediscussed,includingtheuseoftheseprocedurestomonitorchangesinperformancethatareduetolearningorattentionallapsesbysubjects,aswellassomecharacteristicsofadaptivemeth-odsthatbecomeimportantwithapplicationtomorecom-plexapplicationsorinstudyingmultidimensionalstimuli,suchasspeech.ThereaderisdirectedtoTreutwein(1995)foranexcellent,quantitativedescriptionofpsychometricfunctionsandadaptivetechniques.

ORIGINSOFADAPTIVEPSYCHOPHYSICALPROCEDURES

Althoughadaptiveprocedureshavebeeninuseinsomeformformanyyears(seeLevitt,1992,forabriefdiscus-

Figure 1.Exampleofapsychometricfunctionshowingpercentageofcorrectde-tectionsasafunctionofstimuluslevel.Theindividualdatapointsarefittedwithalo-

gisticpsychometricfunction.


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sion),thesystematicapplicationofadaptivealgorithmstothemeasurementofsensoryfunctionmaybetracedtoclinicalroots.ThedevelopmentofthenewdisciplineofaudiologytotesthearingduringandafterWorldWar IIin-cludedamethoddevisedbyHughsonandWestlake(1944)forsearchingquicklyforanauditorythresholdby

startingwithaninaudiblesoundlevelandincreasingtheleveluntilapositiveresponsewasexhibitedbythepatient.HughsonandWestlakeemphasizedtheimportanceofsoundlevelsbeginningbelowalistenersthreshold,thenincreasinguntilthethresholdofsoundwasreached.Theirprocedureunderwentminormodif icationsbyCarhartandJerger(1959),whoproposedthatanauditorytestshouldbeginatarelativelyhighlevelinordertodemonstratethesoundforthelistener.Thelevelwouldthendropinfairlylargestepsuntilthesoundbecameinaudible(signaledbyanegativeresponsefromthesubject),whenthelevelwouldincreaseinasearchforthreshold.Oncethelistener

indicatedthatthesoundwasaudible(apositiveresponse),thelevelwoulddropagain,andanotherincreasingthresh-oldsearchwouldbeinitiated.Thefinalthresholdwastakentobetheaveragelevel,duringanincreasingseries,atwhichthelistenerindicatedthatthesoundwasheard.Althoughbearingasuperficialsimilaritytooneoftheclassicalpsychometricmethodologiescalledthe methodoflimits,thisclinicalmethoddiffersinthatthethresholdsearchalwaysoccursontheascendingtrialsandthede-scendinglevelsareusedonlyassubjectpreparation andtogetintopositionfortheascendingthresholdsearch.Intheclassicalmethodoflimits,bothascendinganddescending

trialsareusedtoidentifywherearesponsechangesfromoneofaudibilitytoinaudibilityorviceversa,andthethresholdisestimatedfromthoseboundarylevels.

Anotherearlyadaptiveprocedurethathasenjoyedwidespreadusebothclinicallyandinresearchapplica-tionsisBekesytracking,originallydevelopedbytheau-ditoryscientistwhoseworkincochlearmodelingwonhimtheNobelprizein1962.Inthisprocedure,amechanicalarmwithapencilattachedisdrivenbyapatientlisteningtotoneschangingcontinuouslyinintensityand,some-times,infrequency.Asthepatientindicatesthatthetoneisheard,thependrawsloweronanintensity-scaledgraph,

andthelevelofthetonedecreases;ifthepatientindicatesthatthetoneisnotheard,tonelevelincreases,accompa-niedbymarkshigheronthegraph.Thethreshold,definedasthatintensityatwhichatoneisjustbarelyheard,istakentobethemidpointoftheup-and-downtracingsofthemechanicalpen.

Themorecommonlyusedadaptiveproceduresinre-searchtodayemergedfromtheseclinicalbeginningsandfrommethodologicalresearchinthe1940stothe1960s.DixonandMood(1948)wereamongthefirsttosystem-aticallyinvestigatethecharacteristicsandstatisticalprop-ertiesofsimpleadaptiveproceduresthatsearchforthresh-

oldthroughacombinationofincreasinganddecreasingstimulussteps,respondingtonegativeandpositivesubjectresponses.Thistypeofprocedurehascometobecalledastaircase andformsthebasisforagreatdealofpsycho-

metrictestingusedtoday.Staircaseproceduresdifferfromearlierclinicaltechniquesinthattheycollectanumberofthresholdestimatesfrombothascendinganddescendingseriesoftrials.Theyrefinethemethodoflimitsbynotre-quiringresponsestoacompletesetoflevelsandbyre-spondingwithchangesindirectionofthestaircaseaftera

changeinthesubjectsresponse.

MODERNADAPTIVEMETHODS

Thecommoncharacteristicsofcurrentlyusedadaptivemethodsarethecollectionofsubjectresponsestoeachtrial,withasystematicmanipulationofthestimuluslevelalongtheexperimentaldimensionofinterest.Eachmethodresultsinaseriesofstimuluslevelspresentedoverthecourseoftheexperiment,alongwiththeassociatedsub-jectresponses.Experimentalvariablesth atmayimpacttheresultsofthemethodologyincludetheamountofdif-

ferencebetweenstimulusvaluespresented(thestepsize),theinitialstartingvalueofthestimulus,theprocessthatguidesthesequenceofpresentationlevelsoneachtrial(thetrackingalgorithm),andthedecisionforendingtheprocess(thestoppingrule).Thegeneralgoalofeachprocedureistomeasurecharacteristicsofthesubjectsperformanceovertheshortestamountoftime,withoutsacrificingac-curacy.Eachmethodmaybemostappropriateinagivenexperimentalsituation,andthereisasubstantialliteraturecomparingtheabilitiesofthesemethodstoprovidebias-freeresultswithhighreliability.

Adaptivemethodologiesthatcurrentlyenjoywide-

spreadusemaybeplacedintothreegeneralcategories,de-finedbytheirsystemsforplacingtrialsalongastimulusarrayandbythemannerinwhicheachestimatesafinaloutcome.Thefirstcategorytobedescribediscalledpa-rameterestimationbysequentialtesting(PEST),anditischaracterizedbyanalgorithmforthresholdsearchingthatchangesbothstepsizesanddirection(i.e.,increasinganddecreasinglevel)acrossasetoftrials.Asecondtypeofadap-tiveprocedureshasbeencalledmaximum-likelihoodpro-ceduresbuttheirmoregeneralcharacteristicisthatsetsofstimulusresponsetrialsarefitwithsometypeofogivalfunctionandsubsequenttrialplacementandthresholdes-

timationistakenfromthosefittedfunctions.Finally,acommonformofadaptiveprocedures,knowngenericallyasstaircase procedures,willbedescribed.Foreachofthesethreecategoriesofprocedures,anexampleofatrackinghistorywillbeshown,inordertounderstandthedifferencesinthemannerinwhichtheadaptiverulesleadtoanestimateofthreshold.

ParameterEstimationbySequentialTestingThePESTprocedure,firstdescribedin1967byTaylor

andCreelman,useschangesinstepsizetofocustheadap-tivetrackevermorefinely,stoppingthetrackwhenthees-

timatehasbeenadequatelydefined.Thefinalestimateissimplythefinalvaluedeterminedbythetrialplacementprocedure.ThePESTalgorithmisdesignedtoplacetrialsatthemostefficientlocationsalongthestimulusaxisin


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ordertoincreasemeasurementprecisionwhileminimiz-ingthenumberoftrialsrequiredtoestimateathreshold.

Figure 2showsatypicalPESTadaptivethresholdtrack,modifiedfromHall(1981),carriedoutaccordingtothesug-gestionsofTaylorandCreelman(1967).Notethatanini-tiallevelandastepsizeareselectedtobeginatrack.After

eachpresentationatafixedlevel,astatisticaltestisap-pliedtoindicatewhetherperformanceatthatlevelisbet-terorpoorerthanthetargetedperformancelevel(e.g.,75%correctdetections).Oncethatdeterminationismade,thelevelmaychangebythecurrentstepsize,andaseriesofpresentationsoccursatthenewlevel,againtestingaftereachpresentationwhetherthelevelshouldbechanged.InFigure 2,aninitialstepsizeof8 dBisused;afterfourpre-sentations,thelevelchanges;ninepresentationsatthenewlevelareneededtodeterminethatthecurrentlevelistoolow,andthetrackmovesbackup,butthistimewithastepsizehalfaslarge.Thenextchangeindirection(occurring

atTrial 21)producesanotherhalvingofstepsize.Furtherchangesinstepsize(alwaysaccordingtoPESTrules)occurthroughoutthetrack,whichterminateswhenacriterionstepsizeisreached.Thelevelspecifiedbythisfinalstepsizeistakenasthefinalthresholdvalue.Theimportantcharacteristicsofthistypeofthresholdtrackarethatthestepsizeschangeaccordingtorulethroughoutthetrack,sothatthetrackexcursionstendtobecomesmallerasathresholdvalueisapproached,andthatthefinalthresholdestimateistakentobethefinalvalueinthetrack,withoutspecificallyconsideringperformanceonprevioustrials.

TherulesforimplementationofPESTwereoriginally

outlinedbyTaylorandCreelman(1967),butmanysubse-quentauthorshaveproposedmodifications.Theoriginalrulesincludewhentochangelevels,aprocessfordecid-ingthenextlevelinvolvingastepsizechangingthrough-outthetrackaccordingtorule,astoppingrulebasedontheapproachofthedecreasingstepsizetoacriterionvalue,andaruleforestimatingthefinalthresholdmea-sure,typicallythelastlevelindicatedbythetrackingrules.

TaylorandCreelmanalsodescribedametric,termedthesweatfactor, thatcouldbeusedtoevaluatetheefficiencyofagivenpsychometricprocedure.Likeningthesweatfactortoameasureoftheamountofworknecessarytoreachacertainlevelofprecisioninthemeasuringalgo-rithm,theydefinedthesweatfactorastheproductofthe

numberof trials andthe variance of the measures.Throughsimulations,theydeterminedthevarianceofthePESTthresholdsandthemeannumberoftrialsnecessarytoachievethatlevelofvariability.Comparingresultsofthoserealisticsimulationswithanidealsweatfactor(gen-eratedfromasimulatedthresholddevicewithcompletestatisticalknowledgeoftheprobabilitiesassociatedwitheachstimuluslevel)producedameasureofefficiencyofthePESTprocedure,whichTaylorandCreelmancalcu-latedtobeabout40%50%.

PESTwasdesignedtobeasefficientaspossibleintheplacementoftrialsalonganarrayofstimuluslevelsand

toforceconvergenceofanadaptivetrackonagivenper-formancelevel(i.e.,threshold)asrapidlyasisconsistentwithaccuracyandreliabilityofmeasurement.Theorigi-nalPESTprocedurecalledfortrialplacementthroughoutthetrackbasedonastatisticaldeterminationofperformanceatthecurrentlevel,incomparisonwithexpectedperfor-manceatatargetedlevel,andathresholdestimatethatwassimplythefinalvalueofthetrack.AlthoughsomemodificationstotheoriginalPESTtrackingrulesweresuggestedbyFindlay(1978)inordertoencouragemorerapidconvergenceofthetrack,laterdevelopmentsemerg-ingfromtheuseofPESThavechangeditstwobasicchar-

acteristicsoftrialplacementandthresholdestimation.Hall(1981)proposedahybridprocedurethatfollowed

PESTrulesfortrialplacementalongthestimulusaxis,butinsteadoftakingthefinalvalueofthetrackasthreshold,attheendofthetrack,heusedperformanceonalltrialstoconstructapsychometricfunction,fromwhichathresholdvaluewasextracted.Thevalueofthishybridprocedurewasthatefficienttrialplacementcouldproceedinaprescribed

Figure 2.AdaptivetrackfollowingthePESTprocedure.Thesedecibelvaluesarerelativetoanarbitrarythresholdof0 dB,shownwiththehorizon talline.Thesedataaremod ifiedfrom Figure 1

inHall(1981).


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manneraccordingtothePESTrules,but,intheend,allthedatagatheredduringtheprocedurewereusedtoconstructthefinalpsychometricfunction.Throughsimulationsandexperimentaltrialswithhumansubjects,Hall(1981)demonstratedthatthehybridprocedurecouldovercomemanyoftheshortcomingsidentifiedinprevioususeof

PEST:Theprocedurewasrelativelytolerantofsubjectlapses,notaffectedsignif icantlybyinappropriatestartinglevelsorstepsizes,andprovidedestimatesofbothathresholdvalueandaslopeofanassumedpsychometricfunction.

Maximum-LikelihoodAdaptiveProceduresAlthoughHalls(1981)hybridprocedurechangedthe

PESTmethodoffinalthresholdestimate,furthermodifi-cationsofPESTchangedtherulesforstimulusplacementaswell.Thesemodificationsmaybeclassifiedintoasec-ondcategoryofadaptiveprocedures,characterizedby

stimulusplacementoneachtrial,drivenbyconsultingthecurrentbestestimateoftheentireunderlyingpsychomet-ricfunctionaftereverystimulusresponsetrial.Astheadaptivetrackgrowsinlength,theestimatedfunctionbe-comesbetterdefinedbythecollectionofdatapointsgen-eratedfromprevioustrials.Aftereachtrial,thesetofstim-uluslevelsandtheproportionofcorrectresponsesassociatedwitheachlevelarecombinedtoformapsychometricfunc-tion,asisshownschematicallyinFigure 1.Theindividualpointsarefittedwithanogivalfunctionofsomekind(Fig-ure 1showsalogisticfunction)andacurrentestimatedthresholdlevelisextracted.Anewpsychometricfunction

isgeneratedaftereachtrialorsetoftrials,andsubsequenttrialsareplacedatatargetedperformancelevelonthemostup-to-datefunction.Amaximum-likelihoodfittingalgorithmistypicallyusedwiththistypeofprocedure.

ThelinkbetweentheoriginalPESTandthemaximum-likelihoodadaptiveproceduresmaybeclearlyseeninpa-persbyPentland(1980)andWatsonandPelli(1983).PentlanddevelopedwhathecalledthebestPEST,totakeadvantageofthestrengthofthemaximum-likelihoodproceduresinthecontextofaPESTadaptivetrack.As-sumingalogisticpsychometricfunctionwithagivenslope,Pentlandsprocedureseekstomaximizetheinfor-

mationprovidedbyeachtrialbyplacinglevelsatthemostcurrentestimateofthe50%pointontheassumedpsy-chometricfunction.InsimulationcomparisonswiththeoriginalPESTandtwootheradaptiveprocedures,Pent-landsmaximum-likelihoodprocedureprovedtobethemostefficient,requiringtheleastnumberoftrialstoreachagivenlevelofprecision.InPentlandsbestPEST,levelsaresetaccordingtoafittedfunctionaftereachtrial(usingallprevioustrials),andafixednumberoftrialsispre-sented.Thethresholdestimateissimplythelastvaluees-timatedasthe50%pointoftheultimatepsychometricfunction.WatsonandPelli,intheirQUESTprocedure,ad-

vocatedtheuseofallinformationavailablefromprevioustrialsinthetrack,supplementedbypriorknowledge(fromtheliterature,previousexperiments,etc.)tosetthenexttestlevel.However,adistinctionwasmadebetweenthe

useofpriorinformationtodrivethetrackandthefinales-timateofathreshold,whichusedonlythedatawithinthetrack.Foranadaptivetrackconsistingof128trials,Wat-sonandPellireportanefficiencyof84%fortheirQUESTprocedure,ascomparedwith40%50%efficiencyfortheoriginalPEST.

Maximum-likelihoodadaptiveproceduresareattractivetoinvestigatorsbecausetheymakefulluseofalltrialsinanexperimentinordertodetermineathreshold,ratherthanestimatingthresholdonlyfromthelevelsvisitedattheendofanadaptivetrack,asintheoriginalPESTpro-cedure.Inmostapplicationsoftheseprocedures,bothafunctionshape(e.g.,alogisticorWeibullfunction)andaslopevaluemustbeassumed,sothatfromtrialtotrial,thefunctionmovesitslocationalongastimuluslevelarrayinordertofindthefunctionleadingtoathresholdestimate.Mostofthedevelopmentoftheseprocedureshasoccurredinthecontextofvisionorauditoryresearch,butLinschoten,

Harvey,Eller,andJafek(2001),inthisissue,havedemon-stratedthevalueofmaximum-likelihoodadaptivemeth-odsinstudyingtasteandsmell.Theyassumedalogisticpsychometricfunctionandreportedthatthemethodsworkedwellinestimatingthresholdswithprecisionandspeed.Althoughforwell-studiedpsychophysicaltasks,informationconcerningthefunctionunderlyingperfor-mancemaybeknownfromtheliterature,additionalnon-adaptivemeasuresmightbenecessarytoestablishthefunc-tionbeforemaximum-likelihoodadaptiveproceduresmaysafelybeimplemented.ThiswastheapproachtakenbySaberiandGreen(1997)toevaluatetheuseofmaximum-

likelihoodadaptiveproceduresinsomemeasuresofaudi-torydiscriminationoftimeandfrequency.

AnillustrationofthistypeofadaptivemethodologymaybetakenfromGreen(1993),whodevelopedamaximum-likelihoodadaptiveprocedureinvolvingayesnopsycho-metrictaskthatpromiseshighlyefficienttrialplacementandthresholdestimation.GreensmethodissimilartotheQUESTprocedureofWatsonandPelli(1983),aswellastosomeotherimplementations,butGreensproceduredoesnotcarryasmanypriorassumptionsasdoesQUEST,andhasalesstheoreticallybasedschemefortheplace-mentoftrials.InGreens procedure,aparticularpsycho-

metricfunctionisassumed(e.g.,alogisticfunction),andarangeofstimulusvaluesthoughttoincludethethresh-oldtobeestimatedisidentified,perhapsthroughpilotworkorconsultingtheliterature.Asetofcandidatepsy-chometricfunctionsiscomputedonthebasisoftheas-sumedformofthefunctionandthepossiblestimulusval-ues.Eachofthecandidatefunctionsisfittedtoallthedatacollectedtothatpointaftereachstimuluspresentation,andthelikelihoodassociatedwitheachfunctioniscom-puted.Themostlikelypsychometricfunctionisthenvis-itedatthetargetperformanceleveltodeterminethestim-ulusleveltobeusedonthenexttrial,followedbyanother

updatingofthecandidatefunctionprobabilities.Thefinalestimateofthresholdisextractedfromthemostlikelypsy-chometricfunctionaftersomenumberoftrialsorinaccordwithsomestoppingrule.Figure 3showsatypicaladaptive


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track,followingGreens(1993)maximum-likelihoodpro-cedure.Thetrackischaracterizedbyawideexcursionoverthefirstfewtrials,butarapidconvergencetoathresholdstimuluslevel.Greenassertedthatareliablethresholdes-timatecouldbegeneratedusingthismethodinasfewas12trials;Leek,Dubno,He,andAhlstrom(2000),describ-

ingastoppingrulebasedonacriterionvariabilityintheadaptivetrack,reportedthat,typically,24trialswouldpro-ducehighlyreliablethresholdestimates.IntheexampledepictedinFigure 3,thethresholdlevelappearstostabi-lizeafterabout2025trials.Theprocedure,developedasayesnotask,hasbeenextendedtoaforced-choicepro-cedurebyDaiandGreen(1992)inastudyofauditoryin-tensitydiscrimination,andforfrequencyandintensitydiscriminationbyHe,Dubno,andMills(1998).Furtherassessmentoftheuseofthisprocedurefordifferentex-perimentaltasksandwithdifferenttypesofsubjectpopu-lationshasalsobeenreportedbyLeeket al.(2000).

StaircaseProcedures

BothPESTandthemaximum-likelihoodproceduresinvolvefairlycomplexstimulusplacementrulesand,insomecases,developmentofthresholdestimatesfromthetrackingdata.Themaximum-likelihoodproceduresalsorequiretheassumptionofaparticularformoftheunder-lyingpsychometricfunction,whichisnotwellestablishedforsomepsychometrictasks.Thesimplicityandflexibil-ityofadaptivestaircaseshaveledtotheiradoptionastheproceduresofchoiceinmanylaboratories.Thesemeth-odsgenerallyusethepreviousoneormoreresponseswithin

anadaptivetracktoselectthenexttrialplacement,thenprovideathresholdestimateinavarietyofways,mostcommonlybyaveragingthelevelsatthedirectionrever-salsintheadaptivetrack(i.e.,theturnaroundpoints).Sim-pleupdownstaircasescallforareductioninstimuluslevelwhenthesubjectsresponseispositive(e.g.,Ihearatone)andanincreaseinstimuluslevelwhenthere-sponseisnegative.Figure 4Ashowsanexampleofasim-pleupdownadaptivetrack.Beginningatalevelabovethreshold,positiveresponsesleadtocontinueddecreasesinstimulusleveluntilanegativeresponseoccurs.This

triggersareversalinthedirectionofthetrack,andlevelsonsubsequenttrialsincreaseuntilthenextchangeinre-sponse.Thesimpleupdownstaircaseproceduretargetsthe50%performancelevelonapsychometricfunctionthatextendsfrom0%correctperformanceatchanceto100%correctperformance.Inotherwords,thetracktargetsthe

stimuluslevelforwhichtheprobabilityofacorrectre-sponseequalstheprobabilityofanincorrectresponseor,equivalently,thelevelatwhichthetrackwouldmoveupordownonthestimulusaxiswithequalprobability.Thevalueofthistypeofprocedureisintheveryfewassump-tionsnecessaryforitsimplementation.Incontrasttothemaximum-likelihoodmethods,noformofthepsychome-tricfunctionneedbeassumed,andthereisnoneedforcomplicatedcomputationandfittingproceduresbetweentrials.Furthermore,incontrasttoPEST,thetrialplace-ment,stepsize,andstoppingdecisionsareallrelativelysim-pleandstraightforward.Theonlynecessaryassumption

foruseofthesemethodsisamonotonicrelationshipbe-tweenstimuluslevelsandperformancelevels.

Levitt(1971)describedageneraltransformationpro-cedurefortargetingspecificlocationsonapsychometricfunction.Inthetransformedmethods,insteadofatracklevelchangeinresponsetoeverytrial,asmandatedforthesimpleupdownproceduretargetingthemidpointofthepsychometricfunction,sequencesofpositiveornegativeresponsesaredeterminedthatresultinanequalprobabil-ityofthetracksmovingineitherdirection.Forthesim-pleupdownprocedure,boththepositiveandthenega-tivesequencesconsistofonetrial,andthetracklevel

movesaftereachresponse,targetingthe50%performancelevel.Totargetahigherperformancelevel,thesequenceforadownwardmovementmaybetwoormorepositiveresponses,andthesequenceforanupwardmovementmayremainatonenegativeresponse.Thisexampleistheex-tensivelyusedtwo-down,one-upprocedure,whichtargetsthe70.7%levelonthepsychometricfunction.Recallingthattheprobabilityofthedownsequencemustequaltheprobabilityofanupsequence,weseethatapositivere-sponsetotwoconsecutivetrialsmustoccurinordertomovethetrackdownward.Ifp istheprobabilityofapos-

Figure 3.Adaptivetrackfollowingamaximum-likelihoodadaptiveprocedure,asdevel-opedbyGreen(1993 ).Decibelvaluesarerelativetothearbitrarythreshold of0 dB.


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itiveresponseonagiventrial,thenp3p mustequal.50,andthereforethetargetprobabilityis.5 5 .707.Simi-larly,athree-down,one-uptransformationleadstoaper-formancetargetof.794(i.e.,p35 .50;the cuberootof.50is.794),asisshownintheexampletrackinFigure 4B.Asinthesimpleupdownstaircase,thethresholdsearchstartsabovethreshold,butinthiscase,adecreaseinstim-uluslevelrequiresthreesequentialpositiveresponses.A

reversalinthetrackoccursafteronenegativeresponse,andagain,threepositiveresponsesarerequiredtobeginanotherdescendingrunoftrials.Inhis1971article,Levittoutlinedanumberofpossibletransformations,alongwiththeirtargetperformancelevels.Oneobviousimplicationis,ofcourse,thatthemorecomplicatedthesequencerule,themoretrialstypicallyrequiredinanadaptivetrackinordertoreachanestimateofthreshold.

Althoughthetransformedupdownmethodsarewidelyused,onerestrictionthathasbeennotedisthatonlyasmallnumberoftargetlevelscanbeestimated.Kaernbach(1991)describedasimpleupdownprocedurethatcould

beusedtoestimateperformanceatmanymoretargetlev-elsthanallowedbythetransformedmethods,byvaryingthestepsizesusedinthetwodifferentstaircasedirections.ThevalueofKaernbachsprocedure,ashedescribedit,

wasinthesimplicityofthealgorithm,relativetothesometimesquiteelaboraterulesnecessaryforthetrans-formedprocedures,anditsabilitytotargetanyperfor-mancelevel,notjustthosethatcouldbeestimatedwithaspecif icsequenceofupanddowntrials.InKaernbachssimpleupdownweightingprocedure,aperformancelevelisanalyzedaccordingtothedesiredratioofuptodownsteps,andthestimuluslevelischangedafterevery

trial.Kaernbachdescribedanexampleoftargeting75%correctperformancewitharatioofuptodownstepsizesof(1 p)/p or,inthiscase,.25/.75,or1/3.Inordertotar-getthatpointonthepsychometricfunction,thestimuluslevelshouldbechangedupwardafteranincorrectre-sponseanddownwardafteracorrectresponse,andthesizeoftheupwardstepshouldbethreetimesthesizeofthedownwardstep.UsingMonteCarlosimulations,Kaern-bach(1991)reportedamodestsavingsofabout10%ofexperimentertimewiththeweightedprocedureoveramoretraditionalstaircase,usingatwo-down,one-uprule(i.e.,targeting70.7%correct).Rammsayer(1992)

evaluatedKaernbachs(1991)weightedmethod,usinghumansubjectsinanauditorytemporaldiscriminationtask.Hereportedthat,atthebeginningofadaptivetracks,theweightedupdownmethodwasmoreefficientthana

Figure 4.Adaptivetracksfollowingastaircaseprocedure.(A)Simpleupdownstaircase;(B)transformedupdownstaircase,followingathree-down,one-upalgorithm.


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transformedfixed-step-sizeprocedurebutthat,fortrackslongerthanabout40trials,therewaslittledifferenceinef-ficiencybetweenthetwoprocedures.Becausetherewasaratherlargedifferenceinstepsizebetweentheupandthedownsteps,Rammsayernotedthathissubjectsre-portedanawarenessofthedirectionofthetrack,which

mightbiastheoutcomeoftheprocedure.Rammsayersug-gestedthatthisproblemcouldbereducedbyinterleavingmorethanonethresholdtrack.However,itmightbepointedoutthatanysavingsinexperimentaltimewouldbelostifmorethanonetrackwerefoundtobenecessaryforrea-sonsotherthanefficiency.Kaernbachs(2001a)articleinthisissueusesthisweightingprocedureinanevaluationofthresholdtracking,usinganunforced-choice method.Hearguesthat,althoughthepsychometricpropertiesofincludingdontknowinthearrayofsubjectresponsesin-dicateonlyasmallimprovementovertheforced-choiceselections,subjectsaregenerallymorecomfortablenot

havingtoindicateananswerwhentheyareveryunsure.

ESTIMATESOFSLOPE

FROMADAPTIVEPROCEDURES

Adaptivemethodshavetendedtofocusonmeasuringonepointonthepsychometricfunctioninordertoesti-mateathresholdorlocationofthefunctionalongastim-ulusaxis.However,inmanycases,theslopeofthefunc-tionmaybeusefulinfullydef iningtheshapeofthefunctionor,fortheoreticalorclinicalreasons,inestablishingtherelationshipbetweenratesofchangeinperformancelevel

andstimuluslevel.Inhisseminalpaper,Levitt(1971)dis-cussedtheoptimalplacementoflevelsalongthestimulusaxis.Ifameasureofthemeanofapsychometricfunctionisdesiredasanestimateofthreshold,itismostefficienttoplacetrialsasnearaspossibletothemidpointofthefunction.However,thestaircaseproceduresmaybeusedtoestimateaslopeofapsychometricfunctionbyplacingtrialsoneachsideofthemeanofthefunction.Therehavebeenanumberofinvestigationsofadaptiveproceduresmeanttoevaluatehowpreciselyandaccuratelytheslope,perhapsinadditiontoathreshold,canbedeterminedfromanadaptiveprocedure.

Threeslightlydifferentprocedureshavebeenproposedrecentlythataredesignedtoefficientlyandpreciselyiden-tifytheslopeandthresholdofthepsychometricfunctionun-derlyingsubjectperformance.Theseproceduresbearsomesimilaritytotheearlierdescribedmaximum-likelihoodprocedures,inthat,startingwithasetofcandidatepsycho-metricfunctions(eitherexplicitlyorimplicitlydescribed),trialplacementoccursatthemostlikelythresholdvalue,aresponseiscollected,andthenthatresponseisincludedwithallotherresponsescollectedintheexperimentalruntogenerateanewsetofcandidatefunctions.Thenextpre-sentationlevelislocatedatthepredictedperformance

levelmostlikely(atthatpointintheexperiment)toreflectthreshold(orsomeothertargetedparameter).Inthethreeproceduresdescribedbelow,meanttofocusonaslopees-timate,thecandidatefunctionsarenotexplicitlydefined,

butratheratwo-dimensionalprobabilitydistributionisup-datedaftereachresponse,andthetwodimensionsrepre-sentthetwoparametersdefiningthepsychometricfunction.

WattandAndrews(1981)describedaprocedurethatwasintendedtomaintaintheadvantagesofamethodofconstantstimuliindevelopingagoodestimationofthe

underlyingpsychometricfunction,usingaprobitfittothedata,butemployinganadaptivechangeinstimuluspre-sentationlevelsinordertoincreaseefficiencyoverthemethodofconstantstimuli.Probit(Finney,1971)isamethodoffittingacumulativenormalfunctiontoasetofpsychometricdata,usingamaximum-likelihoodcriterion.Eachdatapointtobeincludedinthefitisweightedaccord-ingtoitsbinomialvariability,andthenumberoftrialsplacedatthatstimuluslevel.Thus,pointsonthepsycho-metricfunctionthatarecalculatedfrommanytrialsareas-sumedtobemorereliableand,therefore,aremoreimpor-tantinthefittingcalculations.InWattandAndrewss

procedure,afewstimuluslevelsareselectedfortestingfromalargersetofpredeterminedvalues,anumberoftri-alsarepresentedusingonlythoselevels,andthenafunc-tionisfitusingtheprobitmethod.Anewsetofstimuluslevelsisselectedonthebasisoftheprobit-fittedpsycho-metricfunction,andafurtherblockoftrialsispresented.The thresholdandspread(slope)of thecumulative-normalpsychometricfunctionassumedbytheprobitfitconvergeonthevaluesthatbestreflectthesubjectsper-formanceonthesetsoftrials.WattandAndrewsadvo-catedtheuseofthisproceduretoimprovetheefficiencyofmeasuringacompletepsychometricfunction,suchas

onecouldgeneratewiththemethodofconstantstimuli,withouttheneedtoassumeaslopevalue.

Morerecently,King-SmithandRose(1997)reportedamethodspecificallydesignedtomeasuretheslopeofthepsychometricfunction.TheyagreewithLevitt(1971)thattheplacementoftrialsonthepsychometricfunctioncanbeselectedinordertomaximizeefficiencyofmeasure-menteitherofslopeorofthreshold,butnotofboth.Ifthresholdisthetarget,formaximumefficiency,trialsshouldbeplacednearthetargetperformancelevelonthepsychometricfunction,andtheclosertothecorrectper-formancelevel,themoreefficientwillbethemeasure-

ment.However,tomaximizeefficiencyinmeasuringslope,pointsthatbetterdefinethespreadofthepsychometricfunctionareappropriate.Ifthefunctionitselfisasym-metricfunctionaboutitsmidpoint,twopointsequidistantfromthemidpointofthefunctionshouldbeselected.Anunbalancedfunctionwillrequireaslightlydifferentplace-mentoftrials.King-SmithandRosedevelopedanadap-tivemethodformaximumefficiencyinmeasuringeitherslopeorthreshold,makinguseofthebinomialvariabilityassociatedwitheachprobabilitylevelonanassumedpsy-chometricfunction.Stimuliforeachtrialareplacedwiththegoalofmaximizingefficiencybyminimizingthevari-

abilityofestimatesaftereachsetoftrials.Themethodisadaptiveinthatastimuluslevelisdeterminedfromaprobabilitydensityfunctiongeneratedfromprevioustri-als.Thelevelmostlikelytocorrespondtothreshold(ora


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slopevalue)ispresented.Theresponseatthislevelisusedtoupdatetheprobabilityofagivenresponseasafunctionofthetruethreshold,expressedasalikelihoodfunction.TheinformationprovidedbythelikelihoodfunctioniscombinedwiththeinitialprobabilityfunctionbyBayesianmultiplication,resultinginanupdatedfunctiondescrib-

ingtheprobabilitythateachintensityisthethresholdaftertheresponsetothatstimulus.Thisnewprobabilityfunctionisthenusedtobeginthenexttrialandtoestimateathresh-oldforoptimumplacementofthesubsequenttrial.Thisprocesscanbeappliedtosimultaneouslyconvergeonabestthresholdandbestslopebyusingtwo-dimensionalprobabilitydensityfunctionsandlikelihoodfunctionstogeneratethenextstimuluslevel.King-SmithandRosere-portedthatthesemethodsresultinrelativelyhigheffi-ciencyforthemeasurementofboththresholdandslope,whichcanbeimprovedtotheextentthatpriorknowledgeofeitheroftheseparametersmaybeincorporatedorthat

assumptionsconcerningoneofthemmayallowexperi-mentalfocusontheother.

King-SmithandRose(1997)notedthatitissomewhatmoredifficulttogetprecisionintheslopeparameterthaninathresholdparameter.KontsevichandTyler(1999)de-velopedanadaptiveprocesssimilartothatofKing-SmithandRose,reportingthattheirprocedurecouldproducereasonableprecisionintheestimateofthresholdinabout30trialsfora2AFCtaskbutthatabout300trialswerere-quiredinordertoestimateslopewithsimilarprecision.Aswiththeearlierstudy,thekeytoestimatingathresholdandslopeistouseeachtrialtoupdatetheposteriorprob-

abilityofagiventwo-dimensionalprobabilitydistribu-tion.KontsevichandTyleridentifiedapotentialproblemwithselectingtheminimumvarianceofthisdistributioninthatthetwodimensionsofthresholdandslopearein-commensurateand,therefore,someweightingconventionmustbeimposed.Instead,theseauthorsproposedadif-ferentcostfactor,whichcouldbeminimizedinordertodeterminethenextstimuluslevelforpresentation.Thisminimizedfactorcorrespondstomaximuminformationgainaftereachtrial.Usingcomputersimulationsandpsy-chophysicalexperimentswithhumans,thisprocedurewasshowntoconvergetothresholdandslopevalueswithina

relativelysmallnumberoftrials.Leek,Hanna,andMarshall(1992)investigatedtheutil-

ityofusingperformanceonallthetrialsthatmakeupanadaptivethresholdtracktogenerateafullpsychometricfunctionfromwhichathresholdandslopevaluecouldbeextracted,reminiscentofHalls(1981)earliersuggestionofahybridPESTmaximum-likelihoodprocedure.Leeket al.(1992)usedcomputersimulationstodeterminetheprecisionofestimateofslopesthatcouldbeaccomplishedfromastaircaseproceduredesignedtotrackthresholdataparticularperformancelevel.Thesimulationsweremeanttodeterminewhetherbothslopeandthreshold

couldbeobtainedbysimplyreconstructingapsychomet-ricfunctiononthebasisofthetrial-by-trialperformancewithinanupdowntransformedstaircasetrack.Aproce-

durethatisoptimizedtoproducethresholdmeasurementsmightalsoprovideslopeinformation,withlittlelossinprecision.Experimentalrunsweregeneratedfollowingselectedstaircasealgorithmsbyconsultingaknownpsy-chometricfunctiononeachtrialtodeterminearesponse.Attheendofanadaptivetrack,thetrial-by-trialdatawere

summarizedaccordingtoperformanceateachlevelvis-itedbythetrackandthenfittoapsychometricfunctionofthesameformastheoneconsultedinthesimulation.Thresholdsgeneratedbythetrackingalgorithmwerecomparedwiththresholdsextractedfromtheoriginalandreconstructedpsychometricfunctions.Slopeestimatesfromthefittedpsychometricfunctionswerecomparedwiththoseunderlyingperformanceontheadaptivetrials.Thefunctionsreconstructedfromthetrial-by-trialdatawereaccuratereflectionsoftheunderlyingfunctionsaslongasthetrackswereatleast200trialslong.Shortertracksresultedinestimatesofpsychometricfunction

slopethatwerebiasedhigh(i.e.,slopestoosteep).Itwasnotedthatamaximum-likelihoodfitofthedatatothepsy-chometricfunctionprovidedgreaterstabilityofestimatesthandidanearliersetofanalysescarriedoutusingtheprobitfittingprocedure(Finney,1971).Theauthorscau-tionedagainstusingtheprobitprocedurewhenthepsy-chometricfunctionwastransformedsothattherangeofperformancewaslessthan0%100%.Thestatisticalpropertiesofsuchafit,describedfullybyMcKee,Klein,andTeller(1985),alterthevariabilityassociatedwitheachpointinthetransformedfunction,whichiscriticalasaweightingcomponentintheprobitprocedure.Thus,when

thepsychometricfunctionistruncated,asinforced-choiceprocedures,McKeeet al.recommendedthattheprobitfitshouldbeavoided.TheLeeket al.(1992)re-constructionsofpsychometricfunctionsfromtrackingdataresultedinafindingregardingslopeestimatessimi-lartothatfrequentlyreported:Itispossibletoobtainac-curateandreliableslopeestimatesfromadaptiveproce-dures,butthecostismoretrialsandsubsequentlymoreexperimenttime.

Slopesestimatedfromadaptivetrackingproceduresre-portedbyLeeket al.(1992)tendedtobetoohighunlessthetrackswerefairlylong.Thistendencyforanoveresti-

mationofslopefromadaptivemethodshasbeenobservedbeforeandhasusuallybeenattributedtoanasymmetryinthedistributionoftrialplacementsalongtheunderlyingpsychometricfunction.Typically,moretrialsareplacedhigherthanthemidpointofthefunctionthanareplacedlower.However,Kaernbach(2001b),inthisissue,arguesthatthesourceoftheslopeoverestimationisnotthepoordistributionoftrials,but,instead,maybefoundintheadap-tivealgorithmsthemselves.Kaernbachshowsthat,fortri-alsplacedidenticallytotrialplacementwithinanadaptivetrackbutpresentedinrandomorderinsteadofaccordingtothesequenceofthealgorithm,slopesarenotoveresti-

mated,lendingsupporttohisnotionthatthesequentialas-pectsofthetrialplacement,accordingtoadaptiverules,isthesourceoftheslopebias.


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Determinationofsloperequiressomeassumptionsabouttheformofthepsychometricfunction.Thereareanumberofogivalfunctionstochoosefrom,andcriteriaforanexperimenterschoicemaydependontheoreticalorcomputationalissues.Strasburger(2001a),inthisissue,pointsoutthatcomparisonsacrossstudiesandtasksare

hamperedbythevarietyofpsychometricfunctionsthatareselectedandprovidesformulasforconvertingamonganumberofthemostcommonlyusedfunctions.Itissug-gestedinthatpaperthatforcomparisonofresultsacrossstudies,themaximumslopeofafunction,ortheslopeatthepointofinflectionoftheogivalfunction,wouldbeausefulmetric.Inasecondpaperinthisissue(Strasburger,2001b),characterrecognitionismeasuredusinga10-alternativeforced-choiceprocedurewithamaximum-likelihood/PESTprocedure,andthemaximum-slopemet-ricisusedtocompareresultsacrossstudies.

Alsointhisissuemaybefoundthreearticlesaddress-

ingthebestwaytosampleandfitthepsychometricfunc-tion.Althoughprobitfittingiscommonlyused,itisnotal-waysthemostappropriatechoice,becauseofchangesinbinomialvariabilitywithtruncationofthefunctionforforced-choiceproceduresandbecauseitassumesapar-ticularformofthefunctionthatis,thecumulativenor-mal.TwopapersbyWichmannandHill(2001a,2001b)takeuptheissuesofhowbesttosamplethefunctionandhowtodeterminethegoodnessoffitoftheassumedfunc-tion.MillerandUlrich(2001)describeamethodforfit-tingthepsychometricfunctionthatmakesnoassumptionabouttheunderlyingdistribution,asdoesthemorecom-

monlyusedprobitanalysis.

VIOLATIONOFASSUMPTIONS

INADAPTIVEMETHODS

Oneofthestrongestargumentsforusingadaptivepro-ceduresfortherapidandaccurateestimatesofcharacter-isticsofpsychometricperformanceisthatthereareveryfewrestrictionsthatmustbeaccommodated.Twosuchcommonlyacceptedrequirements,however,involvesta-bilityofthemeasurementovertimeandthemonotonicre-lationshipbetweenstimulusstrengthandperformance.

Mostexperimentersacknowledgethatabsolutestabilityoftheunderlyingpsychometricfunctiongenerallyisnotarealisticassumption,sincesubjectstypicallyexperiencesomeperceptuallearningduringthecourseofanexperi-ment,reducingthetruethreshold,oroccasionallyhavelapsesinattentionthatmayservetoincreasethemeasuredthreshold.Changesinthresholdacrossthemeasurementtrackmayalsoresultinshallowercalculatedslopesoftheunderlyingfunction.Violationsofasecondassumption,monotonicityofthepsychometricfunction,mayoccurifmembersofthestimulussetunderexaminationarenotho-mogenousalongagivenstimulusdimension.Stimuliwith

greatercomplexityanddimensionalitysuchasspeech,forexamplearelikelynothomogenousand,therefore,posespecialproblemswhentestedadaptively.Therehavebeenattemptstomonitorviolationsoftheassumptionsof

stabilityofthefunctionsandhomogeneityofthestimuliandtoassessthecostsofsuchviolations.

TrackingThresholdChangesWithMultipleAdaptiveTracks

Althoughapotentialproblemintheimplementationof

adaptiveprocedures,itisalsooneofitsbenefits,asiden-tifiedbyLevitt(1971)andearlierpapers,thatadaptivetrackingproceduresmaybeusedtofollowchangesinthepsychometricfunctionoccurringduringthecourseofanexperiment.Forexample,asperceptuallearningoccursovermanytrials,thethresholdmaybeseentogradually(orsuddenly)improve,asreflectedintheshapesofthethresholdtracks.LeekandWatson(1984)usedthismethodtotracetheimprovementsindetectionofindividualtonesembeddedwithina10-tonepattern.Teninterleavedthresh-oldtracks,onetestingeachofthe10patterncomponents,wereexaminedtodeterminehowthetonefrequencyand

temporalplacementwithinthepatternaffectedtheim-provementsindetectionthresholds.Incontrasttoimprove-mentsinthresholdovertime,adaptivetracksmayalsosig-nalsubjectfatigueordistractionoverthecourseofanexperiment.Hall(1983)suggestedamethodofidentify-ingashiftinasubjectsthreshold,bycomparingthesub-jectsresponseoneachtrialwiththeresponsepredictedfromtheestimatedpsychometricfunctionunderlyingper-formance.Totheextentthatthedifferencebetweenobtainedandpredictedperformanceisclosetozeroforeachtrialinthetrack,theestimatedpsychometricfunctionistakentobestablethroughoutthetrack.Leek,Hanna,andMarshall

(1991)alsoproposedamethodfordeterminingwhetherthetruethresholdofasubjectwasshiftingduringthetimeofitsmeasurement,therebyproducinganinaccuratethresholdestimate.Theirapproachwastoinvestigatethepsychometricpropertiesofanunstableunderlyingpsy-chometricfunctionbysimulatingsystematicchangesinfunctionlocationandcomparingthevariabilitywithinandacrosstwointerleavedadaptivetracks.Thelogicwasthatifapsychometricfunctionwerechangingovertime,thevariabilityinlevelsvisitedwithinasingletrackwouldex-ceedthevariabilityobservedbetweentracksontrialsoc-curringclosetogetherintime.Inadditiontousingthese

twosourcesofvariabilitytomonitorwhetherthefunctionisshiftingintime,Leeket al.(1991)showedhowtheacross-trackvariabilitymaybeusedtogenerateanestimateoftheslopeoftheunderlyingfunction.Incomputersimula-tionsandhumansubjectdata,boththeslopeestimatesandthestability-monitoringprocedurewereshowntoworkwellaslongastheshiftinthresholdsdidnotoccursorapidlythatthetrackscouldnotfollowthechanges.

NonmonotonicPsychometricFunctions

andHeterogenousStimuliEarlyinthecourseofdevelopmentofmodernadaptive

methods,LevittandRabiner(1967)attemptedtoapplyanadaptiveproceduretothemeasurementoflevelsofspeechnecessaryforagivenlevelofperformance.Typically,speechtestingisaccomplishedbypresentingalistof


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speechstimuliatagivenlevelandaskinglistenerstore-peatthestimulitheyheard.Thespeechcommonlycon-sistsofstandardizedlistsofmonosyllables,consonantvowelnonsensesyllables,sentences,orrunningdiscourse.Often,itisdesiredtomeasureperformanceatanumberofdifferentpresentationintensitiesorindifferenttypesand

levelsofcompetingnoise.Speechrecognitionasafunc-tionofincreasinglevelorincreasingsignal-to-noiseratioisanogivalpsychometricfunction.Speechtestingtypi-callyfocusesonsomeportionoftherisingpartofthefunction.LevittandRabinerappliedanupdownstaircaseandblocksofstimuliateachtestedleveltodetermine50%correctidentificationofthespeech.BodeandCarhart(1973)extendedthesefindingsbydevelopingwhattheycalledthedoubletprocedure,usingatransformedupdownadaptivemethod.Theyrantwosequentialtracks,targetingthe29.3%andthe70.7%correctidentificationsignal-to-noiselevel.Theaverageofthefinalthreshold

levelsfromeachtrackconstitutedanestimateof50%cor-rectperformance.Steele,Binnie,andCooper(1978)usedthedoubletadaptiveproceduretostudytheimpactofvi-sualcues(lipreading)ontestsofspeechunderstanding,usingmonosyllabicwordsasstimuli.

Althoughadaptiveprocedurescontinuetobewidelyusedinmeasuringspeechrecognition,thenatureofspeechstimulicreatesgreatervariabilityinmeasurementthanisobservedintestingmorehomogenoussetsofstim-uli,suchastonedetectionordiscrimination.Whenstim-uliarehomogenousinallcharacteristicsexcepttheoneundertest(e.g.,thelevelofapuretoneinafixednoise),

theprocedurecanworkwell.Similarly,ifasetofspeechstimuliishomogenousinallfactorsaffectingintelligibil-ity,thelevelofthestimulieitherinquietorinnoisemaybeusedinanadaptiveprocedure.However,ifthestimuliarenothomogenous,thetrackingproceduremaybecom-promised.Forexample,aseriesofeasilyheardmonosyl-lablesmaydrivethelevelofthetracklow,butasubse-quentpresentationofaninherentlymoredifficultwordwilloccuratalevelthatistoolow,andthetrackmaynottrulyreflectoverallperformance.Heterogeneitywithinstimulussetsthusleadstoinappropriateplacementoftriallevels,greatervariabilityinthetrack,andpossibleconfu-

siontothesubject.Dirks,Morgan,andDubno(1982)notedthisdifficultywhentestingidentificationofmono-syllablesandofspondeewords(i.e.,two-syllablewordswithequalstressoneachsyllable)inagroupofnormalhearingandagroupofhearing-impairedsubjects.Thespeechlevelwasheldconstantthroughoutanadaptivetrack,whereasthelevelofmultitalkerbabblewasvariedaccordingtoanadaptivealgorithmdesignedtotarget29.3%,50%,or70.7%correctwordidentification.Intheseprocedures,asimpleupdownprocedurewasfirstiniti-atedtofindthecorrectrangeofperformancelevelsforagivenindividual,afterwhicheitherthesimpleupdown

algorithmwascontinued(50%correct)oroneofthetrans-formedupdownalgorithmswasimplementedtotargettheothertwoperformancelevels.Dirkset al.establishedthresholdsforeachtargetandstimulussetbutreported

thatperformanceofthetrackswasindeedmorevariablewhenthestimuliweremonosyllables,ratherthanthemorehomogenoussetofspondeewords.Thisincreasedvari-abilitywasespeciallystrikingforthegroupofhearing-impairedlisteners,wholikelyexperiencedothersourcesofvariabilityassociatedwiththeirhearinglossaswell.It

isimportant,therefore,inusingadaptivemethodswithsetsofspeechstimuli,tocontroltheheterogeneityofthestimulitotheextentpossibleandtobealerttoviolationofthemonotonicityassumptionacrosstrialsofanadaptivetrack.

COMPARISONSOFADAPTIVEPROCEDURES

Areanyoftheseadaptiveproceduresbetterthanothers,andhowdotheproceduresinteractwithotherpsychome-tricexperimentalchoices?Therehavebeenanumberof

paperscomparingtheaccuracyandefficiencyofthepro-ceduresdescribedaboveinmeasuringthresholdsandslopesofpsychometricfunctions.Thesecomparisonshavebeenmadebyusingcomputersimulationsofexper-imentaltestsand,insomecases,evaluatingtheperfor-manceofhumanlisteners.

Shelton,Picardi,andGreen(1982)evaluatedanadap-tivestaircase,amaximum-likelihoodprocedure,andPESTincollectingdatafromhumansubjects.Ineachcase,theychoseparametersfortheadaptiveproceduresthatwerecommonlyusedinpractice,measuringhumanperfor-manceonasimultaneous-andaforward-maskingaudi-

torytask.Althoughtheproceduresdidnotproducelargedifferences,thereweresomecharacteristicsofeachthatmightsuggestonechoiceoranotherundercertaincircum-stances.Forshortadaptiveruns(i.e.,lessthan30trials),boththestaircaseandthemaximum-likelihoodproce-duresresultedinslightlybiasedthresholdestimates,al-thoughthebiascouldbemostlyovercomebyrandomiz-ingstartinglevelsforeachadaptiverun.Themaximum-likelihoodprocedure,however,wastheonlyoneofthethreemethodsthatcouldconvergewithinabout10trials,evenwithaslightlybiasedthreshold.Sheltonet al.sug-gestedthatthisproceduremightbemostusefulintesting

infantsandanimals,wherethethresholdsmustbegath-eredrapidly.However,theypointedoutthatthemaximum-likelihoodmethodmaybeparticularlydifficultforinexpe-riencedlisteners,becausethereareveryfewsuprathresholdtrialsaffordedtothesubjects.

Kollmeier,Gilkey,andSieben(1988)usedamathe-maticalmodel,aswellashumandata,tocomparetwostaircaseproceduresandthePESTprocedure,withbotha2AFCandathree-alternativeforced-choice(3AFC)exper-imentaltask.Theyalsoevaluatedbothmodelandhumanlistenersonasetoffixedlevel(nonadaptive)procedures.Theirhumanlistenerswereallexperiencedinthetask,

whichwasdetectionofasignalembeddedinnoisethatis,simultaneousmasking.Themodelpredictedsimilarthresholdsfromadaptiveandnonadaptiveprocedures,butinpractice,humanlistenersactuallyproducedslightlybet-


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terthresholdsinadaptiveprocedures.Themodelpre-dictedthata2AFCprocedureusedinastaircasetargeting71%correctwouldbetheleastefficientprocedure,whereasthe79%3AFCstaircasewouldbethemostefficient.Thehumandataalsosupportedthelatterasthemostefficientprocedure,butresultsweresomewhatvariable.Oneofthe

problemsidentifiedwithhumandata,incontrasttothemodeleddata,wasthatthemodelunderestimatedthevari-abilityproducedbyhumanlisteners.Theauthorssug-gestedthatthereasonforthismightbeanunderlyingpsy-chometricfunctionthatisslowlyvaryingandsuggestedthatvariabilityinthetracksmaybepartitionedintoarapidtrial-to-trialvariability,combinedwithvariabilitycon-tributedfromaslowlyvaryingfunction.Hall(1983)hadearliermadesuchasuggestion,anditlaterwastakenupinsimulationsofunstablepsychometricfunctionsbyLeeket al.(1991).Kollmeieret al.suggestedthatthevariabil-itymightbecontrolledbycombiningthresholdsfromshort

tracks,ratherthanusinglong,perhapsvarying,trackstoestimatethresholds.HicksandBuus(2000)agreedwiththisnotion,findingmoreconsistentthresholdsfromin-terleavingseveralshorttracksthanfromfollowingonelongtrackinvolvingthesametotalnumberoftrials.Insummary,Kollmeieret al.reportedthat,ifnootherex-perimentalconsiderationsdictateotherwise,themostef-ficientcombinationofmethodsisthe79%3AFC.Thisconclusionhasbeenreachedbyanumberofotherauthors.

Kollmeieret al.(1988)alsofoundthatthresholdsfromadaptivetracksinhumanperformancetendedtobebiasedlow(i.e.,betterthresholds)thanwouldbeexpectedby

fixedtrials.Eventhoughsomesimulationstudiessuggestthatthresholdsshouldbesimilarfromfixed-levelandadaptiveprocedures,thereareconsistentreportsofthresh-oldsofhumansubjectsbeingbetterinadaptiveproceduresmeanttotargetthesamelevelasfixed-levelmethods.Tay-lor,Forbes,andCreelman(1983)reportedthisindescrib-ingtheircomparisonsofPESTprocedureswithfixed-levelprocedures.Stillman(1989)comparedthresholdsmeasuredwiththeadaptiveproceduresandthresholdsfromfixed-leveltests,findingthattheadaptivemethodsalwaysproducedthresholdsthatwerelower,justaswasreportedbyKollmeieret al.(1988)andSheltonet al.(1982).

Inotherwords,theadaptiveproceduresalwaysoveresti-matedsubjectsperformance,producingbetter(lower)thresholdsthanwereevidencedinanonadaptivetask.

InStillmans(1989)study,bothinexperiencedandex-periencedsubjectswereused,andcomparisonsweremadeforresultsfromadaptiveandnonadaptiveproceduresandfortwostaircaseproceduresandaPESTadaptiveproce-dure.Thetaskwasa2AFCdetectionofa1-kHztoneinabandpassnoisecenteredat1kHz.Resultsindicatedsim-ilarthresholdsforthe79%staircaseandthePESTproce-dures(targeting80%correct)andsimilarvariabilitywithinthetwostaircaseprocedures.SheltonandScarrow(1984)

alsomeasuredperformanceofhumanlistenerstodeter-minewhethersomeexperimentalchoiceswerebetterthanothers.Alltheirlistenerswereinexperienced,andtheyusedseparategroupsof10listenerseachforeachoffour

conditions,staircaseandmaximumlikelihood,usingbotha2AFCanda3AFCprocedureineach.Thetaskwasde-tectionofatoneinnoise.TogetherwithSheltonet al.(1982),theseauthorsreportedthatthresholdswereessen-tiallyequivalentforallthe2AFCprocedurestested(stair-case,PEST,andmaximum-likelihood)andforthe3AFC

staircaseandmaximum-likelihoodprocedures.Theydidobservesomedifferencesinvariabilityandefficiencyacrosstheprocedures,notingthatthe3AFCstaircasepro-videdthemoststablethresholdsacrossadaptiverunsbutthatthemaximum-likelihoodmethodproducedstabilityearlyinarun.Therefore,accordingtotheseauthors,ifprac-ticetrialsarenotpossibleorthenumberoftrialsislimited,themaximum-likelihoodprocedureshouldbefavored.

SchlauchandRose(1990)primarilyusedsimulationswithasmallsetofhumandatatoinvestigatetheuseofstaircaseprocedureswith2-,3-,and4AFCtasks.Theymea-suredbothefficiency(variability)andthresholdbiasasa

functionofnumberofintervals,stepsize,andthetargetoftheadaptivetrack(equivalently,thedecisionruleforchang-ingstimuluslevels).Theyidentifiedlessvariabilityinthresh-oldmeasurementsasthenumberofintervalsincreasedespecially,from2to3intervals,lesssobetween3and4intervalsandforthehigherperformancetarget(79%vs.71%).Eventakingintoaccountthegreaterexperimentaltimenecessarytopresentthelargernumberofintervals,the3AFCand4AFCprocedureswerestillmoreefficientthanthe2AFC.Theyalsoreportedgreatervariabilityinthresholdestimatesforlargerstepsizes.The2AFC71%targetwasmorebiased(i.e.,identifyingbetterperfor-

mance)thanthe4AFC79%,andtherewasmorebiasforlargerstepsizes,especiallyforthe71%2AFCprocedure.SchlauchandRosesuggestedthatthisbiaswasaresultofbehaviornearchanceperformanceandtheeffectsofguessing.Theyalsofoundnoimprovementsinperformanceofthemethodsforadaptivetrackslongerthan100trials.Byfittingthetrial-by-trialdata,usingaprobitmethod(Finney,1971),thethresholdsrecoveredsomeofthebiasthatwasassociatedwithalltheadaptiveprocedures.Inordertoimproveefficiencyandreducebias,theseauthorsrecommendedfitt ingthetrial-by-trialdataintheadaptivetracktoestimateathresholdandtheuseofsmallstepsizes

inthetrackingprocedure.Althoughthe4AFCproceduregavethebestefficiencyandtheleastbias,thetimetakentopresentfourintervalsoneachtrialmaystrainthemem-oryofsubjectsandmay,intheend,increaseexperimentaltimeeventhoughfewertrialsmightbenecessary.

Insummary,thereislittletorecommendanyofthethreereviewedpsychometricproceduresfromthestand-pointoftheperformanceofthemethodsthemselves.Itseemsclearthatsomeexperimenterselectionsofvariousimplementationsofthemethodsmayincreaseordecreasethebiasandreliabilityoftheprocedures.Inparticular,the2AFCtaskisgenerallyapoorchoice,particularlywhen

pairedwithastaircasetargetof71%.McKeeet al.(1985)providedacleardescriptionoftheimpactoftruncatingthe0%100%psychometricfunctionwhenusingaforced-choiceproceduresuchasthe2AFC.Insteadofthefunc-


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tionsspanningalargerangefromchanceperformanceat0%correcttoperfectperformanceat100%,thesetrun-catedfunctionsresultfromincreasedchancelevels(e.g.,50%for2AFC,33.3%for3AFC),andtherefore,therangeofthepsychometricfunctionisdecreased.Thevariabilityassociatedwitheachpointofthefunction,however,con-

tributestothebiasandvariabilityofmeasurementac-cordingtothebinomialdistribution.Therefore,ingeneral,pointsfallinglowerthanthemidpointofthetruncatedfunctionsgenerallyhavegreatervariability.McKeeet al.suggestedthatmeasurementsarelikelytobemorereli-ableiftheyareontheuppersideofthemidpointofthefunction.Thebinomialvariabilityofthetruncatedpsy-chometricfunctionsmayaccountfortherelativelypoorerpsychometricperformanceofthe2AFC71%combination(targetlowerthanthe75%midpointofthe2AFCfunc-tion),withbetterperformancewhenthecombinationofforced-choicetaskandtargetperformancelevelleadtotri-

alsplacedhigheronthefunction.Green(1990)addressedasimilarpoint,arguingthatthebestplacementofstimulustrialswasnearthetopendofthepsychometricfunction.

SUMMARYANDCONCLUSIONS

Threecategoriesofadaptiveprocedureswerereviewed.PESTproceduresdonotrequireassumptionsabouttheshapeoftheunderlyingpsychometricfunctionandpro-videarapidandsystematicconvergenceonathreshold.Maximum-likelihoodproceduresforplacingtrialsatop-timalstimuluslevelsandforprovidingthresholdandslope

estimatesarecomputationallyintensiveandrequireas-sumptionsregardingtheshapeoftheunderlyingfunction.However,theyconvergeontargetedvaluesveryquicklyandmakegooduseofallthedatacollectedinatrack.Staircasemethodsrequireveryfewassumptionsandhavefairlysimplealgorithmsforplacementofstimuliandestimationofthresholdvalues.Theymaysupportanestimateofslope,solongassufficienttrialsarepresented.

Someofthesemethodshaveslightadvantagesoveroth-ers,givenparticularexperimentalcircumstances.Forex-ample,whentestingmustbeaccomplishedveryquickly,asintestinganimalsorinfants,thefasterconverging

maximum-likelihoodproceduresmightoffersomebene-fitoverlongerstaircaseprocedures.Thereisstrongcon-sensus,however,thatthepopular2AFCproceduresdonothavedesirablestatisticalproperties,particularlywhenpairedwithadaptiveproceduresthattargetrelativelylowperformancelevels(i.e.,belowthemidpointofthepsy-chometricfunction)andshouldbeavoided.Finally,stim-ulitestedinadaptiveproceduresshouldhavethecharac-teristicofhomogeneityandamonotonicrelationshipbetweenstimuluslevelandperformancelevel.Thishasbeenshowntobeproblematic(althoughnotfatal)intest-ingsomekindsofspeechrecognitionadaptively.

Adaptivemethodsofferhighprecisionandreliabilityinpsychometrictesting,atasignificantsavingsintimeovernonadaptivetesting.Overthelast50years,refinementsandevaluationsoftheseprocedureshaveshowntheway

toaselectionofexperimentalvariablesandparametersthatresultinlittlecostforthesavingsintime.Althoughthereareinherentbiasesinsomeofthemethods,thesecanbemostlycompensatedbyathoughtfulconsiderationofexperimentaltechniquesandparameters.

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(ManuscriptreceivedJune22,2001;

revisionacceptedforpublicationSeptember25,2001.)

Leek - Adaptive Procedures in Psychophysical Research

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