Perception & Psychophysics

19 76. 1'01.19 ill. 9:·101.

Speed-accuracy tradeoff functionsin choice reaction time: Experimental designs

and computational procedures

CHARLES G. WOOD and J. RICHARD JENNINGSWalter Reed Army Institute ofResearch, Washington, D. C. 20012

The existence of tradeoffs between speed and accuracy is an important interpretative problem in choicereaction time (RTl experiments. A recently suggested solution to this problem is the use of completespeed-accuracy tradeoff functions as the primary dependent variable in choice RT experiments instead ofa single mean RT and error rate. This paper reviews and compares existing procedures for generatingempirical speed-accuracy tradeoff functions for use as dependent variables in choice RT experiments. Twomajor types of tradeoff function are identified, and their experimental designs and computationalprocedures are discussed and evaluated. Systematic disparities are demonstrated between the twotradeoff functions in both empirical and computer-simulated data. Although all existing procedures forgenerating speed- accuracy tradeoff functions involve empirically untested assumptions, one procedurerequires less stringent assumptions and is less sensitive to sources of experimental and statistical error.This procedure involves plotting accuracy against RT over a set of experimental conditions in whichsubjects' criteria for speed vs. accuracy are systematically varied.

Despite the widespread use of choice reaction timeI RTl as a dependent variable in contemporaryinformation processing research. there remain anumber of methodological and interpretativeproblems associated with RT experiments (Pachella.19-4). The present paper concerns a potentialsolution to one of the more important interpretativeproblems in choice RT data. the possibility oftradeotfs between speed and accuracy.

The term "speed-accuracy tradeoff' refers to theobservation that subjects are capable of tradingincreases in speed for decreases in accuracy and viceversa oyer a substantial range (e.g .. Fitts. 1966:Lappin &: Disch. 19-2a: Pachella &: Fisher. 1969.19-2: Swensson , 19-2a). In a typical RT experiment.subjects are instructed to respond as rapidly andaccura tely as possible. However. instead ofresponding at a rate that will produce zero errors.subjects virtually always perform at somewhat lessthan perfect accuracy. Thus. the RT and error ratevalues obtained in typical RT experiments represent acompromise between the incompatible demands formaximum accuracy and minimum RT. The particularcompromise between speed and accuracy adopted in agiven situation (i.e .. the speed-accuracy criterion) canbe manipulated by a number of variables, includinginstructions or explicit payoffs favoring speed oraccuracy re.g.. Fins. 19bb: Lappin &: Disch. 1972a:

Swensson, 19-2a).

The authors wish to thank R. T. OHman for making available acopy of his unpublished manuscript (~Ole 1). Requests for reprintsshould be sent to Charles C. Wood. Division of NeuropsychiatryWalter Reed Army Institute of Research. Washington. D.C. 20012.

92

Two im portant limitations on the interpretation ofRT data imposed by the possibility of speed-accuracytradeoffs have been discussed by Pachella (1974).First. subjects' criteria for speed vs. accuracy mayvary across experimental conditions. renderingimpossible a comparison of RT between conditionsunconfounded with changes in speed-accuracycriteria. The second and more subtle limitation iscaused by the typical form of the tradeoff functionrelating average proportion correct to average RT.Because such functions are negatively accelerated.very small changes in error rate at high levels ofaccuracy may produce large changes in RT (see. forexample. Figure 1 and Pachella, 1974. Figure 4>.This is precisely the region of the speed-accuracytradeoff function (i.e .. error rates less than lQ<rol inwhich reponed data from many choice RTexperiments fall.'

The difficulties created by variable error rates inRT experiments have led to suggestions that thecomplete tradeoff function between speed andaccuracy be used as a measure of RT performance inwhich changes in bias for speed vs. accuracy can bedirectly assessed (e.g .. Lappin &: Disch. 19""2a. b:Pachella, 1974: Pew. 1969). Lappin and Disch(19'2a) summarize the logic and rationale for suchsuggestions in the following way: "This technique isbased upon the proposition that a decision processdetermines the point in time at which perceptualprocessing is terminated and an overt response isselected. Thus. a given pair of RT and error ratevalues reflects performance under a specific decisioncriterion: but the complete tradeoff function betweenRT and error rate is presumed to reflect performance

SPEED-ACCURACY TRADEOFF FUNCTIONS IN CHOICE REACTION TIME 93

REACTION TIME

Figure I. An idealized speed-accuracy tradeoff function in whichaccuracy is plotted as a function of reaction time. The value It is theRT at which accuracy hegins to exceed chance, while t2 is the RT atwhich accuracy reaches maximum.

under varying decision criteria applied to a constantperceptual process. The functional relation betweentwo dependent variables is thereby used to obtain adecision-free measure of the perceptual process. Thesame approach is used in signal detection theory: Thefunctional relation between hit rate and false-alarmrate. the receiver' operating characteristic (ROC), isused as the decision-free measure of detection, ratherthan any specific value of hit rate or false-alarm rate"(p.4l0).

Pursuing the analogy with signal detection theory,the rationale for computing speed-accuracy tradeofffunctions provides a means of distinguishing threedifferent sources of a change in mean RT from oneexperimental condition to another: (a) a change insubjects' speed-accuracy criteria; (b) a change inprocessing efficiency independent of changes inspeed-accuracy criteria; or (c) some combination ofthe two.! Identical tradeoff functions acrossexperimental conditions would imply that differencesin RT or accuracy were produced by changes inspeed-accuracy criterion. In contrast, tradeofffunctions that differ significantly across conditionswould imply differences in processing efficiencyindependent of changes in criterion for speed vs.accuracy. This rationale has proven useful in anumber of recent experiments. For example, variableswhose RT effects have been associated with changes inthe complete tradeoff function include stimulusdiscrirninability (Pachella & Fisher, 1972), stimulusintensity (Lappin & Disch. 1972b; Rabbitt & Vyas,1970), amount of practice in the task (Rabbitt &Vyas, 1970), and graded doses of alcohol (Jennings,Wood, & Lawrence, 1976). In contrast, variableswhose RT effects have been primarily associated withchanges along a single tradeoff function includestimulus probability (Harm & Lappin, 1973; Lappin& Disch, 1972a) and "repetition effects" in serialchoice reactions (Swensson, 1972b).

1.0

(1)A = m(RT -c),

An idealized version of an empirical speed-accuracytradeoff function is presented in Figure 1, in which ageneralized measure of accuracy, A (0 ~ A ~ 1), isplotted as a function of RT. This idealized functionwill be used to illustrate some basic propertiescommon to all empirical speed-accuracy tradeofffunctions. regardless of the procedures by which theyare derived. Specific procedures for deriving suchfunctions are considered in detail in subsequentsections.

The idealized function in Figure 1 may be dividedinto three distinct regions: (a) a region in whichRT < t 1 and A ~ 0; (b) a region in whicht 1 ~ RT ~ t 2 and A is some increasing function ofRT; and (c) a region in which RT > t 2 and A ~ 1.

The first region represents the average intervalnecessary for accuracy to exceed levels expected bychance. Some models of the tradeoff process assumethat the upper bound of this region, t 1, approximatesthe mean RT required for simple detection of thestimuli involved, where no choice response is required(e.g .. Edwards, 1965; Stone, 1960). However, datapresented by Swensson (l972a) suggest that theduration of the first region may be significantly longerthan the "simple RT" for the same stimuli. Pendingadditional data. a strictly operational interpretationof the region RT < t 1 appears preferable tointerpretations in terms of specific processing stages.

The second region of the idealized function inFigure 1 is the region that has received most empiricaland theoretical attention. In this region, A is anincreasing function ofRT. Although the exact form ofthis increasing function remains in question, it hasbeen typically approximated by the linear equation

The purpose of the present paper is to review andcompare existing procedures for generating empiricalspeed-accuracy tradeoff functions. First some basiccharacteristics of empirical tradeoff functions areoutl ined. Second, the major experimental designs andcomputational procedures used to generate suchfunctions are reviewed and their assumptions arediscussed and evaluated. Third, empirical compari­sons of the two major types of tradeoff function arepresented based on both experimental andcomputer-simulated data. Throughout this paper, theprimary emphasis will be upon the use of tradeofffunctions as dependent variables which do not dependupon any particular model of the tradeoff process.However, a number of the empirical issues to beconsidered may have implications for tradeoff models.

in which A and RT represent measures of accuracy

Basic Characteristics of EmpiricalSpeed.Accuracy Tradeoff Functions

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94 WOOD AND JENNINGS

and reaction time. respectively; c corresponds to t 1 inFigure 1. the intercept of the function with the RTaxis at A = 0; and III is the slope of the functionreflecting the amount of increase in A for each unitincrease in RT. The linear relationship between A andRT expressed by Equation I has typically accountedfor greater than 80% of the empirical variance inthese two variables (e.g.. Lappin & Disch. 1972a).

Under conditions of adequate linear fit. theparameters m and c may be used as summarystatistics for an analysis of the effects of variousindependent variables on the tradeoff function. Forexample. Lappin and Disch (1972b) reported thatincreases in stimulus intensity produced both anincrease in the slope and a decrease in the intercept ofempirical tradeoff functions. At the present stage ofdevelopment. it is unclear whether the slope andintercept parameters represent completely inde­pendent or partially correlated aspects of perfor­mance. Therefore. in addition to separatecomparisons of the slope and intercept parametersacross experimental conditions, it may be useful tocombine the slope and intercept into a single measurerepresenting the overall level of performance in eachcondition. One way of combining the slope andintercept parameters is to insert fixed values ofaccuracy into Equation 1 for each condition and solvefor the corresponding values of RT. In this way, an"equal-accuracy contour" may be derived whichrepresents the effect of the different experimentalconditions on RT at a constant level of accuracy.Similarly. "equal-R'I contours" may be derived byinserting fixed values of RT into Equation 1 andsolving for corresponding values of accuracy. Asecond way to combine the slope and interceptparameters is to compute the integrated. area underEquation 1 between the intercept and some arbitraryvalue of RT near the upper end of the range ofempirically obtained RTs. resulting in a measurereflecting the total area under the tradeoff function.This procedure is analogous to computing the areaunder an ROC function in a signal detectionexperiment (Green & Swets, 1966).

The parameter estimates for m and c in Equation Ias well as the other measures just discussed all dependupon the particular scale of measurement selected forA. Although different transformations of proportioncorrect. PIC). are specified as linearly related to RTby different models and theories. to date no singletransformation has been shown to provide consistentlybetter fits to empirical data than various alternatives.For example. Lappin and Disch (1972a) reported noconsistent differences in least squares goodness of fitamong measures of A including: (a) d', thedetectability measure of signal detection theory;(b) (d ')2 discussed by Taylor. Lindsay. and Forbes(1967); (c) Ht. the information transmitted betweenstimuli and responses; and (d) -l nn, the sensitivity

parameter of Luce's (1963) choice theory. The averageproportion of variance accounted for by Equation 1ranged between .86 and .91 for the different Ameasures. Roughly equivalent fits have been reportedby Swensson (1972a) for (d')2. Ht. and "odds"transformations derived from stimulus samplingmodels. At the present time. therefore, a choiceamong alternative scales for A need not implyacceptance of a particular model or theory over otheralternatives.

The asymptotic third region of the idealizedfunction in Figure 1. in which RT > t2 and A ~ 1,has received only limited empirical attention. Thisregion deserves additional study, however. since itmay provide an important link between the study ofdiscrimination under time limitations and the moretraditional approaches to discrimination in whichaccuracy is the primary performance limitation (seeSwensson, 1972a).

Experimental Designs andComputational Procedures

Two major methods have been used to generatespeed-accuracy tradeoff functions from empiricaldata. Ollman (Note 1) has recently discussedtheoretical aspects of these two types of tradeofffunction. and his definitions and terminology areemployed in the present paper. The first function istermed the speed-accuracy tradeofffunction (SATF).This function is defined as the relationship betweenaccuracy and RT across a set of experimentalconditions designed to manipulate subjects' speed­accuracy criteria systematically over a wide range. Forexample. consider an experiment in which subjectsare rewarded for correct responses which occur beforea specified RT "deadline" on each trial. Bysystematically varying the length of the deadlineinterval in different experimental conditions. a widerange of speed and accuracy of performance may beobtained. The SATF for such an experiment would becomputed simply by plotting PIC) (or someappropriate transform) against mean RT (or othermeasure of central tendency) for each of the differentdeadline conditions.

The type of design in which a number of differentexperimental conditions are used to vary subjects'speed-accuracy criteria will be referred to as amultiple-condition design. At a minimum, two suchconditions are necessary to generate data for anSATF. although several conditions are typically usedin practice to provide a more fine-grained estimate ofthe function. In addition to the RT deadlineprocedure described above (cf. Fitts. 1966; Green &Luce, 1973; Link. 1971; Pachella & Fisher, 1969.1972; Pachella & Pew. 1968). a second type ofmultiple-condition design involves no explicitdeadline intervals but relies instead upon a

SPEED-ACCURACY TRADEOFF FUNCTIONS IN CHOICE REACTION TIME 95

continuous payoff scheme that systematically rewardsdifferent speed-accuracy criteria in differentexperimental conditions (e.g., Swensson, 1972a). Athird procedure which combines aspects of both thedeadline and continous payoff methods is the "bandpayoff' procedure introduced by Snodgrass, Luce,and Galanter (1967). In this procedure, payoffsdecrease according to some specified function as theobtained RT deviates further in either direction froma specified target value. Thus, a variety of differentprocedures can be used to manipulate subjects'speed-accuracy criteria in a multiple-conditiondesign. For convenience, such conditions will bereferred to as speed-emphasis conditions, regardlessof the specific form of the criterion manipulation.While there are a number of interesting psychologicalquestions concerning different speed-emphasismanipulations (e.g., Swensson & Thomas, 1974),such questions are beyond the scope of the presentpaper and will not be considered further.

The second major type of speed-accuracy tradeofffunction is termed the conditional accuracy function(CAF).3 This function is formally defined as theconditional probability of a correct response giventhat RT equals a particular value t. computed at allvalues of t. In practice, the CAF is computed byranking the RTs in a given experiment from low tohigh, partitioning the data into equal-N categories,and plotting accuracy against mean RT for each RTcategory (e.g., Lappin & Disch, 1972a; Rabbitt &Vyas, 1970).

In contrast to the SATF, the CAF does not requirea multiple-condition design and can be computedfrom a single RT distribution. In principle, therefore,the CAF can be computed from any choice RTexperiment in which both speed and accuracy ofperformance are measured. In practice, the CAF hasoften been computed from an experimental designwhich shall be termed a single-condition design todistinguish it from the multiple-condition designrequired for the SATF. Instead of using differentspeed-emphasis conditions to manipulate subjects'speed-accuracy criteria, the single-condition designinstructs subjects to vary their criteria over a widerange in a single experimental condition. An exampleof this design is the experiment of Lappin and Disch(1972a). in which "well-practiced subjects were askedto respond at a speed that would result in an error rateof about 25%. They were instructed to avoid guessing.but to adjust their RTs from trial to trial in order tomaintain the appropriate speed and error rate."(p, 422). Other examples of CAFs derived fromsingle-condition designs have been reported by Harmand Lappin (1973), Lappin and Disch (1972b, 1973),and Rabbit and Vyas (1970).

Despite their superficial similarities, the SATF andCAF differ in a number of important respects. Somedifferences are associated with the experimental

designs typically used to generate each type offunction, while other differences are associated withthe computational procedures used to derive the twofunctions. Since the CAF is not limited tosingle-condition designs, it is particularly importantin comparing the SATF and CAF to distinguishbetween factors associated with the experimentaldesigns and factors associated with the computationalprocedures per se.

The multiple-condition design required for theSATF and the single-condition design typically usedfor the CAF differ in three major ways. ~e firstdifference concerns the specific means used to varysubjects' speed-accuracy criteria in the two designs.The multiple-condition design employs explicitexperimental manipulations to induce systematicvariations in subjects' criteria, while the single­condition design relies on subjects to vary theircriteria appropriately over the course of a singleexperimental condition. These procedural differencesmay significantly influence both the range andstability of criterion variations actually obtained in agiven experiment. The second difference betweenmultiple- and single-condition designs concerns theexperimenter's ability to determine whether subjects'criteria have varied appropriately in a givenexperiment. Although there are no means presentlyavailable for determining the exact range of criteriaemployed in a given experiment, the multiple­condition design at least permits the experimenter todetermine whether RT and accuracy have variedsystematically across speed-emphasis conditions. Incontrast, the single-condition design. provides nomeans other than the single RT distribution and errorrate for empirically assessing whether subjects variedtheir criteria according to instructions. The thirdmajor difference between multiple- and single­condition designs concerns the interaction ofexperimental variables with subjects' speed-accuracycriteria. Both designs assume that the variations incriteria do not interact with performance in otherways. Although no systematic investigations of thisassumption have been reported, the single-conditiondesign seems less likely to produce interactionsbetween criterion variations and other aspects ofperformance than the multiple-condition design inwhich subjects' criteria are explicitly manipulated.

In addition to the differences in experimentaldesigns just described, the computational proceduresused to derive the SATF and CAF also differsignificantly. This difference concerns the way inwhich the data are partitioned into categories prior tocomputing RT and accuracy values, and hassignificant implications for the assumptions requiredby the two procedures. The SATF partitions the datainto categories based on the different speed-emphasisconditions in a multiple-condition experiment. Themean RT and accuracy values for each

96 WOOD AND JENNINGS

speed-emphasis condition are then used as summarystatistics for performance over the range ofspeed-accuracy criteria which actually occurred inthat condition. Note that this procedure requires onlyrelativelv weak assumptions concerning variations insubjects; speed-accuracy criteria. First. the SATFrequires that subjects' criteria vary systematicallyacross the different speed-emphasis conditions. Asnoted above. this assumption can be assessed in theRT and accuracy data from any given experiment.Second. it is not necessary to assume that a singlecriterion or even a small range of criteria areemployed within a given speed-emphasis condition.Rather. the mean RT and accuracy values for eachcondition are assumed to be reasonable summarystatistics for the bivariate speed-accuracy distributionassociated with the average speed-accuracy criterionemployed in that condition.t

In contrast, the CAF partitions the data intocategories based on obtained RT instead of on someexternal basis as does the SATF. This fact makes theCAF dependent upon very specific assumptionsconcerning variability in subjects' speed-accuracycriteria. By computing accuracy conditional uponobtained RT. the CAF implicitly assumes either:(a) that the CAF is invariant across changes inspeed-accuracy criteria; or alternatively (b) that agiven subject's speed-accuracy criterion is constant inthe data from which a CAF is computed. If neither ofthese conditions is met. then computing accuracyconditional upon obtained RT would mix togethertrials which have similar RTs but which arise fromhigh-accuracy and low-accuracy criteria.

The assumption of a constant speed-accuracycriterion in the data contributing to a given CAF isdifficult to justify on logical grounds and cannot betested empirically. In the single-condition designtypically used to generate a CAF. subjects areinstructed to vary their speed-accuracy criteria fromtrial to trial in order to maintain a specified averageRT and error rate as noted above. Indeed, Lappin andDisch (1972a) explicitly noted that their design"virtually guarantees that the decision criteria did notremain fixed" (p.423). A possible way to reducecriterion variability is to compute a CAF from dataobtained in a single speed-emphasis condition of amultiple-condition design. However. the a priorilikelihood that a given empirical RT distributionresults from a constant criterion is very low. even inthe data from a single speed-emphasis condition.More importantly. the degree of criterion variabilityin a single RT distribution cannot be empiricallydetermined regardless of the experimental designused to generate the data. That is. it is impossible todistinguish RT variability associated with variationsin speed-accuracy criteria from RT variability arisingfrom a single speed-accuracy criterion. Thus,regardless of whether a CAF is computed from a

single-condition design like that of Lappin and Disch(l972a) or from a single speed-emphasis condition ina multiple-condition design. the constant criterionassumption is probably unjustified and. in any case,cannot be empirically tested.

The alternative assumption. that the CAF isinvariant across changes in speed-accuracy criterion.may be more reasonable a priori than the constantcriterion assumption. If the CAF were invariantacross changes in criterion. variability in subjects'criteria would be immaterial since the CAFs fordifferent criteria would be the same. Unlike theassumption of a constant speed-accuracy criterion.the question of CAF invariance is amenable to directempirical test. The nature of this empirical test.together with theoretical aspects of the CAFinvariance question. have been recently discussed byOHman (Note 1) in the context of the data reported bySchouten and Bekker (1967).

Schouten and Bekker (1967) employed amultiple-condition design with different pacingsignals ("intended RTs") to manipulate subjects'speed-accuracy criteria. Subjects were instructed torespond "in coincidence" with the pacing signal oneach trial. The interval between the stimuli and thepacing signals was constant within a block of trialsand varied systematically across blocks from 100 to800 msec. Separate CAFs were computed for eachindividual speed-emphasis condition. Based on theirresults. Schouten and Bekker (1967) made thefollowing important observation: "The fractions oferrors obtained from different enforced TS [i.e., thedifferent speed emphasis conditions] reasonably fitone standard curve. An actual reaction time of, say.250 msec can result from both an induced reactiontime of T = 200 msec and T = 300 msec. Theexperimental fact that all fractions of errors fit onestandard curve then means that this fraction isdetermined by the actual rather than the intendedreaction time" (p. 146).

Based on the Schouten and Bekker (1967) data.OHman (Note 1) introduced an important theoremconcerning the relation between the CAF and SATF,In effect. this theorem states that if the CAF isinvariant across changes in speed-accuracy criteria.then the CAF is a close approximation to the SATFfor the same set of data. However, as Oilman (Note 1)points out. the empirical utility of this theoremdepends entirely upon the assumption of CAFinvariance across changes in speed-accuracy criterion.At present. the only available data in which thisassumption may be tested are those of Schouten andBekker (1967). Moreover. although Schouten andBekker (1967) concluded that their functions fit "onestandard curve, statistical support for thisconclusion was not presented and close inspection oftheir data reveals a small but systematic departurefrom invariance (see especially their Figure 4). These

SPEED-ACCURACY TRADEOFF FUNCTIONS IN CHOICE REACTION TIME 97

Comparison of Empirical Conditional AccuracyFunctions from Different Speed-Emphasis Conditions

Figure 2. Conditional accuracy functions derived from differentspeed. emphasis (RT deadline) conditions. The nominal deadlinesin milliseconds for each condition are shown at the right of eachfunction. The line at P(C) = .S represents chance performance.

experience with the alcohol manipulation.Separate CAFs were computed for each deadline

condition for each subject in the following manner.The RTs for each condition were tanked from low tohigh, divided into five equal-N categories, and PIC)and mean RT for each category were computed. Thisprocedure resulted in five separate CAFs for eachsubject, each CAF consisting of five jointspeed-accuracy values (i.e., mean RT and PIC) foreach quintile of each RT distribution). Mean CAFsacross the five subjects are shown in Figure i In thisfigure, PIC) is plotted as a function of mean RT foreach RT quintile in each of the five deadlineconditions.

In marked contrast to the apparent invariance inthe Schouten and Bekker (1967) data, the resultsshown in Figure 2 reveal systematic differences in theCAFs for the different speed-emphasis conditions.For example, in the RT range between 225 and275 msec, mean PIC) across the five subjects rangedfrom approximately .67 in the 175-msec deadlinecondition to .94 in the 375-msec condition. Forintermediate deadline conditions. PIC) was anincreasing function of the nominal deadline interval.

Statistical comparison of the CAFs for differentdeadline conditions is complicated by the varyingdegree ofCAF overlap along the RT axis. To evaluatethe statistical reliability of the results shown inFigure 2, the RT interval between 225 and 275 msecwas selected as having a sufficient number of trials inall CAFs for each subject to permit reliablecomparison. Accordingly, the PIC) values for thisinterval were computed for each subject and deadlinecondition and were compared using the Friedmannonparametric analysis of variance procedure (Siegel,1956). The results of this analysis indicated that theaverage PIC) values for 225 < RT < 275 differedsignificantly as a function of deadline conditionX2r = 16.20, df = 4, p < .01.

These results clearly demonstrate that the CAF wasnot invariant across changes in speed-accuracycriteria in the present experiment. That is, accuracydepended not only upon obtained RT but also uponthe speed-emphasis condition in which a given trialoccurred. This conclusion should not, however, beovergeneralized. The statistical considerations dis­cussed by Ollman (Note 1) indicate that a sensitivetest of CAF invariance requires far more trials in eachCAF than were available for the present comparison.Nevertheless, most of the sources of statistical andexperimental error in such a comparison are biased infavor of accepting the hypothesis of CAF invariance.Therefore, the clear failure of CAF invariance underthe relatively insensitive conditions of the presentcomparison suggests that CAF invariance should notbe assumed in a given experiment without directempirical support. 6

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The present comparison of CAFs across differentspeed-emphasis conditions was derived from a recentexperiment concerning the effects of graded doses ofalcohol on speed-accuracy tradeoff functions in choiceRT (Jennings, Wood, & Lawrence, 1976). Amultiple-condition design was employed in which fivesignaled deadline intervals were used to manipulatesubjects' speed-accuracy criteria. The deadlineintervals were 175,225, 275, 325, and 375 msec, eachpresented in a separate block of 100 trials. A dailysession consisted of one block of trials for eachdeadline condition. preceded by a block with450-msec deadline presented for practice at thebeginning of each session.

Two pure tone stimuli (1,000 and 1.100 Hz, 70 dBSl.) were assigned to two alternative response buttons.On each trial. the acoustic stimulus was followed by avisual stimulus which signaled the end of the deadlineinterval. Five subjects received at least two sessions ofpractice before beginning a five-dose alcoholschedule. The data presented here were derived fromthe last practice session for each subject prior to any

observations raise the possibility that the apparentsupport for CAF invariance in the data of Schoutenand Bekker (1967) resulted not from an invariantCAF across speed-emphasis conditions but fromsignificant variability in subjects' speed-accuracycriteria within each condition. 5 In light of thispossibility, an independent test ofCAF invariance wasdeemed highly desirable. The results of this test arepresented in the following section, which comparesempirical CAFs from different speed-emphasisconditions in a manner analogous to that described bySchouten and Bekker (1967).

98 WOOD AND JENNINGS

Comparison of Conditional Accuracy Functionsand Speed.Accuracy Tradeoff Functions

in Empirical and Computer.Simulated Data

The results presented in the preceding section raisesignificant questions concerning the suitability of theCAF for use as a bias-controlled dependent variable inchoice RT experiments. Together with the evidenceagainst CAF invariance just presented. the inability todetermine the degree of criterion variability in a givenRT distribution suggests that empirical CAFs may beinfluenced significantly by variations in subjects'speed-accuracy criteria. In order to assess theexpected magnitude of criterion variability effects onthe CAF, this section first compares CAFs and SATFsderived from empirical data containing variability insubjects' speed-accuracy criteria. Second. computer­simulated data are presented in which criterionvariability effects are directly investigated bysystematically varying the RT distributionsrepresenting different speed-accuracy criteria.

It should be emphasized that both the empiricaland computer-simulated comparisons in this sectionare based on data corresponding to a multiple­condition design. Thus. these comparisons illustratethe consequences of computing CAFs over data whichare presumed to contain systematic criterionvariability based on the demonstrated effectiveness ofthe speed-emphasis conditions. The inference is madethat similar effects can occur when CAFs arecomputed from data in which the magnitude ofcriterion variability cannot be directly assessed.

Figures 3a and 3b compare SATFs (filled circles)and CAFs (open circles) for two typical subjects fromthe experiment discussed in the preceding section.The solid and dotted lines are the least-squaressolution to Equation 1 with A = P(C) for the SATFand CAF. respectively. The consequence of mixingdata from different speed-accuracy criteria in theCAFs for these data is that both the slope and theintercept of the CAFs are lower than those of thecorresponding SATFs. The reasons for this disparitymay be seen intuitively by reexamination of Figure 2.An SATF for the data in Figure 2 would plot P(C)against mean RT for each of the five deadline

conditions. resulting in a function connecting the"centroids" of the within-condition CAFs represent­ing differing average criteria. In contrast. the overallCAF for these data would first rank the RTs across alldeadline conditions, then compute P(C) and mean RTbased on equal-N categories across the entire range ofRT. The net result of mixing data representingdifferent criteria is to increase the accuracy reflectedby the overall CAF relative to the SATF at short RTs.while this effect is reversed at long RTs. In effect.then. a CAF computed on data having substantialcriterion variability yields a function which reflectsneither the SATF nor the individual CAFsrepresenting each speed-accuracy criterion. but somecombination of the two.

A related consequence of the ranking procedurerequired by the CAF is that the magnitude of criterionvariability is directly related to the amount of overlapbetween the RT distributions representing differentspeed-accuracy criteria. This relationship was firstnoted in empirical comparisons of CAFs and SATFslike those shown in Figure 3. Subjects showingsmaller disparities between CAFs and SATFs also hadless variable RT distributions in the differentspeed-emphasis conditions. For example, the meanstandard deviation (SO) over the five deadlineconditions for the subject represented in Figure 3awas 62.49 msec. The corresponding average SD forthe subject with the smaller disparity between CAFand SATF (Figure 3b) was 52.42 msec.

In order to illustrate the range of disparitiesbetween SATF and CAF which can be produced bydifferent amounts of overlap. computer-simulateddata are presented below in which two variablesrelated to such overlap were systematically varied.The first variable is simply the standard deviation(SO) of the RT distributions associated with eachspeed-accuracy criterion (SD = 10. 40. and 70 msec,respectively). The second variable is the difference inmean RT between correct and error responsesassociated with each criterion (RT c - RTe = 25. O.and -25 msec, respectively). 7 Thomas (1974) andOilman (Note 1) have independently demonstratedthat RT c - RTe for a given distribution is directlyrelated to the form of the CAF for that distribution:

1.0

.9

.8

Figure 3. Speed. accuracy tradeoff func­tions (mled circles and soUd lines) and .7

conditional accuracy functions (open circlesand dotted lines) derived from the same sets of .6

data for each of two subjects. The line at P(C) .5

= .5 represents chance performance.

a

150 200 250 300 350 400

REACTION TIME (rnsec)

b

150 200 250 300 350 400

REACTION TIME (rnsec)

SPEED-ACCURACY TRADEOFF FUNCTIONS IN CHOICE REACTION TIME 99

v-l-·

Figure 4. Speed-accuracy tradeoff func­tions (filled circles and sond lines) andconditional accuracy functions (open circlesand dotted nnes) derived from computer­simulated data. The three columns differ Inthe varlabUlty of RT dlstributioqs for eachspeed-accuracy criterion (SD = lB, 40, and70 msec from left to right). The three rowsdiffer In the relationship between the meanRTs for correct and error responses(RTc - RTe= 25,0, and - 25 msec from top tobottom). The line at P(C) = .5 rep_ntlchance performance.

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ISO 200 250 300 350 400 150 200 250 300 350 400 "cT:~-c--~~-=-~

REACTION TME (msec)

positive values of RTc - RTe indicate CAFs withpositive slopes, negative values ofRTc - RTe indicateCAFs with negative slopes, and RT distributions withRTc - RTe = 0 indicate CAFs with zero slope. Thus,in addition to influencing the amount of overlapbetween RT distributions from different criteria, thevariations in RTc - RTe also illustrate the effects ofvariation in the form of the CAFs for individualcriteria on the overall CAF computed acrossvariations in criteria.

The three values of SO and the three values ofRTc - RTe were combined factoriaUy to produce thenine pairs of functions presented in Figure 4.Functions in different columns differ in SO (l0, 40,and 70 msec from left to right), while functions indifferent rows differ in RTc - RTe (25, 0, and-25 msec from top to bottom). Figure 4f, forexample, shows the functions with SO = 70 msecand equal mean RTs for correct and error responses.

The pairs of functions shown in Figure 4 illustratehow the magnitude of the disparity between a CAFand a SATF derived from the same data variessystematically as a function of the variability of theindividual RT distributions and the relation betweencorrect and error distributions. Consider firstFigure 4d, in which the mean RTs for correct anderror responses are equal (RTc = RTe) and the RTvariability is relatively low (SO = 10 msec). In thiscase, there is very little overlap in the RT distributionsfor different speed-emphasis conditions and,consequently, very little effect of plotting accuracycontingent upon obtained RT in the CAF. However,increases in the overlap between adjacent RTdistributions caused by increasing SO (center andright columns of Figure 4) are associated withincreasing disparity between CAFs and SATFs (cf.Figures 4d, 4e, and 4{).

The effect of variations in the relationship betweenRT distributions for correct and error responses (andtherefore the slope of the individual CAFs) may beillustrated by comparing the different rows inFigure 4. When errors are faster on average thancorrect responses (top row, RTc - RTe = 25 msec),the CAFs are steeper than the CAFs in the middlerow which have the same SO but equal correct anderror RTs. In contrast, when errors are slower onaverage than correct responses (bottom row,RTc - RTe = -25 msec), CAFs are much less steer;than the corresponding CAFs with equal correct anderror RTs. The CAFs in the bottom row arenonmonotonic functions, with accuracy firstdecreasing and then increasing with increasing RT.The least squares linear solutions for these CAFsaccount for little ofthe total variance and are includedonly for consistency with the remainder of the figure.Like the effect of increases in SO, these effects of therelative speed of correct and error responses are alsoproduced by mixingresponses from different averagespeed-accuracy criteria.

The comparisons of CAFs and SATFs presented inthis section should serve to increase suspicion of theCAF as a suitable function for general use as adependent variable in choice RT experiments. Thecomputer simulations demonstrate that the effects ofcriterion variability are highly variable and dependupon the RT distributions for individual speed­accuracy criteria which cannot be empiricallymeasured. Both the degree of variability associatedwith each criterion and the form of the CAFs forindividual criteria can exert substantial influences ona CAF computed over different speed-accuracycriteria. In contrast, the SATF is by definition lesssensitive to criterion variability than the CAF, and theresults of the computer simulations verify this

100 WOOD AND JENNINGS

difference between the two functions. While the SATFremained relatively stable over large changes in thevariability of RT distributions representing differentspeed-accuracy criteria. the CAFs ranged fromessentially flat when RTc - RTe <0 to very steepwhen RT c - RTe >O. and varied systematically withchanges in SD. Thus, unless it becomes possible todetermine the degree and form of criterion variabilityin the RT distribution from a single condition, itappears to be essentially impossible to estimate themagnitude of criterion variability effects in a givenempirical CAF.

Since a number of variables other than thoseillustrated here can potentially affect the form andmagnitude of criterion variability effects on the CAF,the results of this section should be viewed asillustrative rather than definitive in nature.Nevertheless, the disparities between CAFs andSATFs illustrated here are not atypical, since thoseshown in Figures 4c, 4e. and 4f conform closely tothose obtained in empirical data (cf, Figure 3).

Conclusions

The purpose of this paper has been to review andevaluate alternative procedures for generatingempirical speed-accuracy tradeoff functions for use asbias-controlled dependent variables in choice RTexperiments. Although the primary focus has beenupon empirical functions which do not depend uponany particular model of the tradeoff process, theresults also have implications for the use of tradeofffunctions in testing alternative models.

From an empirical perspective, an analysis of theexperimental designs and computational proceduresfor the SATF and CAF suggests that the two functionsdiffer both in terms of complexity of experimentaldesign and also in the stringency of their underlyingassumptions. In terms of experimental complexity.the CAF would appear to be far superior. Theprocedure of computing accuracy conditional uponobtained RT allows the CAF to be computed from anyRT distribution in which both RT and accuracy arerecorded on each trial. Thus, the CAF requires onlyminimal experimental effort over that required in atraditional choice RT experiment. In contrast, theSATF requires a specialized experimental designinvolving multiple speed-emphasis conditions.

The simplicity of experimental design required forthe CAF is not without significant cost, however, sincecomputing accuracy conditional upon obtained RTrequires stringent assumptions which are either lessjustitiable or less testable than those of the SATF.Specifically, the CAF requires either: (a) that theCAF is invariant across changes in subjects'speed-accuracy criteria; or (b) that a given subject'scriterion is constant in the data from which a CAF iscomputed. An empirical test of the CAF invariance

assumption was presented in which this assumptionclearly failed. The constant-criterion assumption isdifficult to justify on logical grounds and cannot betested empirically. In contrast, the SATF requiresrelatively weak assumptions concerning variability insubjects' speed-accuracy criteria. First, subjects'criteria must vary systematically across the differentspeed-emphasis conditions. This requirement can beassessed in the sr and accuracy data from any givenexperiment. Second, the mean RT and accuracyvalues for each speed-emphasis condition are assumedto be reasonable summary statistics for the bivariatespeed-accuracy distribution associated with theaverage criterion in that condition. This assumption isreasonable a priori and is not contradicted byavailable data.

The empirical consequences of computing CAFsover data with variability in subjects' speed-accuracycriteria were examined by comparing CAFs andSATFs in: (a) empirical data containing criterionvariability, and (b) computer-simulated data in whichthe form and magnitude of criterion variability weresystematically varied. If the consequences ofcomputing CAFs over data with criterion variabilitywere either small or predictable, then theexperimental simplicity of the CAF might outweigh itsmore restrictive assumptions. However. the results ofthe computer simulations demonstrated that theeffects of criterion variability on the CAF were bothlarge and variable, and were dependent uponvariables which cannot be directly measured inempirical RT distributions. In contrast. the SATF isby definition relatively unaffected by the samechanges in criterion variability which produced largealterations in the CAF.

Despite these reservations concerning the generaluse of the CAF as a dependent variable. the CAF is animportant theoretical tool and may have empiricalutility in certain circumstances. Although both theCAF and SATF have been used to test various theoriesand models of the speed-accuracy tradeoff process(e.g .. Green & Luce, 1973; Lappin & Disch. 1972a;Swensson, 1972a) , the CAF appears to be morediscriminating among alternative models than theSATF (d. Pachella, 1974; Thomas. 1974). However,theoretical predictions about the CAF are limited inthat they have typically been derived under theassumption of a constant speed-accuracy criterion.The incorporation of criterion variability effects intothe predictions of existing models would appear to bea particularly useful theoretical advance. The CAFmay also be useful in a strictly empirical sense when amultiple-cond irion design is not experimentallyfeasible or when the potential effects of criterionvariability are inconsequential for the specificapplication involved. In addition, if criterionvariability can be assumed to be equal in differentconditions of a choice RT experiment. then the CAF

SPEED-ACCURACY TRADEOFF FUNCTIONS IN CHOICE REACTION TIME 101

might be a reasonable means of comparingperformance across conditions.

In summary, the function relating accuracy toRT over multiple speed-emphasis conditions (theSATF) would appear to be the best availableprocedure for generatin? empiri~al.speed.-accuracytradeoff functions. This function IS simple tocompute, is dependent upon less stringentassumptions than the CAF, and is less sensitive tovariability in subjects' speed-accuracy criteria.Although the multiple-condition design required tocompute the SATF is clearly more complex than thedesign required for the CAF, its additional benefitsappear to outweigh the additional cost. Thisconclusion must be tempered by the as yet untestedpossibility t~at the payoff or ~eadline ~anipulati~ns

associated with the SATF may interact with the choiceRT task in ways other than simply manipulatingsubjects' speed-accuracy criteria. For example, themultiple-condition design may be thought of as a"secondary task" paradigm of the type discussed byKerr (1973) in the sense that adhering toexperimenter-imposed speed-emphasis conditionsmay represent an additional demand on processingcapacity over and above the choice RT task beinginvestigated. In view of the potential utility of theSATF in a wide variety of experimental contexts, thispossibility merits systematic investigation.

REFERENCE NOTE

I. Oilman. R. T. On the similarity between two types ofspeed/accuracy tradeoff. Unpublished manuscript. 1974.

REFERENCES

EDWARDS. W. Optimal strategies for seeking information:Models for statistics. choice reaction times. and humaninformation processing. Journal of Mathematical Psychology.1965. 2. 312-329.

FITTS. P. M. Cognitive aspects of information processing: Ill.Set for speed versus accuracy. Journal of ExperimentalPsychology. 1966. 71,849-857.

GREEN, D. M., & LUCE, R. D. Speed-accuracy tradeoffin auditory detection. In S. Kornblum (Ed.), Attention andperformance IV. New York: Academic Press, 1973.

GREEN, D. M .. & SWETS, J. A. Signal detection theory andpsychophysics. New York: Wiley, 1966.

GRICE. G. R.. HUNT. R. L., KUSHNER, B. A.. & MORROW. C.Stimulus intensity, catch trial effects, and the speed-accuracytradeoff in reaction time: A variable criterion theory inter­pretation. Memory & Cognition. 1974, 2. 758-770.

HARM. O. 1.. & LAPPIN. J. S. Probability. compatibility.speed. and accuracy. Journal of Experimental Psychology,1973. 100.416-418.

JENNINGS, J. R., WOOD. C. c., & LAWRENCE, B. E. Effectsof graded doses of alcohol on speed-accuracy tradeoff inchoice reaction time. Perception & Psychophysics. 1976,19, 85-91.

KERR, B. Processing demands during mental operations. Memory& Cognition. 1973. 1. 401-412.

LApPIN. 1. S.. & DISCH, K. The latency operating characteristic:I. Effects of stimulus probability on choice reaction time.Journal ofExperimental Psychology, 1972,92.419-427. (a)

LApPIN, 1. S., & DISCH, K. The latency operating characteristic:II. Effects of visual stimulus intensity on choice reaction time.Journal 0.1 Experimental Psychology, 1972, 93, 367-372. (b)

LApPIN. J. S .. & DISCH, K. Latency operating characteristic:Ill. Temporal uncertainty effects. Journal of ExperimentalPsychology, 1973, 98, 279-285.

LINK. S. W. Applying RT deadlines to discrimination reactiontime. Psychonomic Science, 1971, 25, 355-358.

LUCE. R. D. Detection and recognition, In R. D. Luce, R. R. Bush.& E. Galanter (Eds.), Handbook of mathematical psychology(Vol. O. New York: Wiley, 1963.

PACHELLA, R. G. The interpretation of reaction time ininformation processing research. In B. Kantowltz (Ed.),Human information processing: Tutorials in performance andcognition. Potomac, Md: Erlbaum Associates, 1974.

PACHELLA, R. G., & FISHER, D. F. Effect of stimulusdegradation and similarity in the trade-off between speed andaccuracy in absolute judgments. Journal of ExperimentalPsychology. 1969, 81, 7-9.

PACHELLA, R. G., & FISHER, D. F. Hick's law and the speed­accuracy trade-off in absolute judgment. Journal of Experi­mental Psychology, 1972, 92, 378-384.

PACHELLA. R. G .. & PEW. R. W. Speed-accuracy tradeoffin reaction time: Effect of discrete criterion times. JournalotExperimental Psychology, 1968,76. 19-24.

PEw. R. W. The speed-accuracy operating characteristic. InW. G. Koster (Ed.), Attention and performance II.Amsterdam: North Holland, 1969.

RABBITT. P. M. A., & VYAS, S. M. An elementary preliminarytaxonomy for some errors in laboratory choice RT tasks.In A. F. Sanders (Ed.), Attention and performance II/.Amsterdam: North Holland, 1970.

SCHOUTEN, 1. F., & BEKKER, J. A. M. Reaction time andaccuracy. In A. F. Sanders (Ed.), Attention and performance.Amsterdam: North Holland, 1967.

SIEGEL, S. Nonparametric statistics. New York: McGraw-Hili,1956.

SNODGRASS, J. G., LUCE, R. D., & GALANTER, E. Someexperiments on simple and choice reaction time. JournalofExperimental Psychology, 1967, 27, 143-153.

STONE. M. Models for choice-reaction time. Psychometrika,1960, 25. 251-260.

SWENSSON, R. G. The elusive tradeoff: Speed versus accuracyin visual discrimination tasks. Perception & Psychophysics,1972. 12. 16-32. (a)

SWENSSON, R. G. Trade-off bias and efficiency effects inserial choice reactions. Journal of Experimental Psychology,1972. 95. 397-407. (b)

SWENSSON. R. G., & THOMAS. R. E. Fixed and optionalstopping models for two-choice discrimination times. Journalof Mathematical Psychology, 1974, 11, 213-236.

TAYLOR, M. M .. LiNDSAY, P. H., & FORBES. S. M. Quantifica­tion of shared capacity processing in auditory and visualdiscrimination. In A. F. Sanders (Ed.), Attention andperformance, Amsterdam: North Holland, 1967.

THOMAS. E. A. C. The selectivity of preparation. PsychologicalReview. 1974. 81. 442-464.

WICKELGREN. W. A. Dynamics of retrieval. In D. Deutsch andJ. A. Deutsch (Eds.), Short-term memory. New York:Academic Press. in press.

NOTES

I. Even further. Pachella (1974) notes: "In the extreme, ifsubjects actually produced zero errors in all conditions (as thegeneral reaction time instructions ask of them), the reaction timeswould be essentially un interpretable. This is because an infinitenumber of average reaction times can result in zero errors ... Thus.very low error rates. while often the mark of a careful experiment,may also result in artifactual differences in reaction time" (p, 63).

102 WOOD AND JENNINGS

2. The term "efficiency" used to refer to the bias-free componentof RT performance in the present paper was introduced bySwensson (1972b). This component has been referred to as"capacity" by Thomas (1974) and "perceptual processing" byLappin and Disch (1972a). Other terms for "speed-accuracycriterion" used in the present paper have been "trade-off bias"(Swensson, 1972b) and "decision criterion" (Lappin & Disch.1972a).

3. A number of different terms have been used to refer toempirical functions relating speed and accuracy of performance.Functions analogous to the CAF have been termed "latencyoperating characteristics" by Lappin and Disch (1972a). "Tfunctions" by Rabbit and Vyas (1970), and "micro-tradeoff"functions by Pachella (1974) and Thomas (1974). Functionsanalogous to the SATF have been termed "speed-accuracyoperating characteristics" by Pew (1969) and "macro-tradeofffunctions" by Pachella (1974) and Thomas (1974). The use ofOilman's (Note I) terminology for the two computationalprocedures in the present paper is intended to be both a meaningfuldescription of how each function is derived but also theoreticallyneutral with respect to alternative models of the tradeoff process.

4. This assumption is quite reasonable when the data from agiven speed-emphasis condition represent the region of thespeed-accuracy tradeoff in which accuracy is an increasing functionof RT (i.e., the region between t, and t, in Figure I). However,when performance outside the region between t , and t, is heavilyrepresented in the data from a given speed-emphasis condition, theresulting mean RT may be significantly influenced by RTs outsidethe tradeoff region. Therefore, when generating an empiricalSATF: (a) the manipulations of subjects' speed-accuracy criteriashould be designed with a viewtoward minimizing the range of RTsin each speed-emphasis condition; and (b) speed-emphasisconditions with accuracies very near chance or perfect performancemay be omitted from the process of fitting the data with linear (oralternative) equations. A very similar problem occurs in fitting

empirical CAFs with linear equations. Lappin and Disch (l972a)suggest a method for minimizing the effect of data outside thetradeoff region for the CAF.

5. Oilman (Note I) raises this possibility explicitly, and a numberof other investigators have expressed similar concerns concerningthe possibility of distortion in CAFs computed over trials withvarying speed-accuracy criteria (e.g .. Pachella, 1974, p. 79;Wickelgren, in press, pp. 15-17; and from a slightly differentperspective by Thomas. 1974. pp. 447-449).

6. Two additional sources of evidence are also inconsistent withthe invariance assumption. A recent experiment from thislaboratory has replicated and extended the test of CAF invariancereported here. Within-condition CAFs very similar to those inFigure 2 were obtained using a band payoff instead of anRT-deadline design. Second. although their experiment was notexplicitly addressed to the invariance assumption, Grice, Hunt,Kushner. and Morrow (1974) showed that functions analogous tothe CAF from an "accuracy emphasis" condition differedsubstantially from those from a "speed emphasis" condition (cf.their Figures 11 and 12).

7. In order to be comparable to the empirical functions presentedin Figure 3, the CAFs and SATFs derived from thecomputer-simulated data were also based on five speed-emphasisconditions with 100 trials per condition. For simplicity, normaldistributions were used for both the correct and error response RTdistributions in all comparisons. The means of the correct responsedistributions were held constant at 200. 225, 250, 275, and300 msec, respectively. The means of the error responsedistributions were varied relative to these values to produce thevariations in RTc - RTe-

(Received for publication July 7, 1975;revision received October 6, 1975.)