journal of experimental psychologyrdluce/bio/pre1990/1967/... · cal nature of this system...

Journal of

Experimental Psychology

VOL. 75, No. 1 SEPTEMBER 1967

SOME EXPERIMENTS ON SIMPLE AND CHOICEREACTION TIME l

JOAN GAY SNODGRASS R. DUNCAN LUCENew York University University of Pennsylvania

AND

EUGENE GALANTER

University of Washington

4 experiments in simple and choice reaction time (RT) are reported.Experiment I examines the effect of monetary payoffs on the accu-racy and variability of time estimation. Experiment II examines theeffect of moving the position of a narrow payoff band along the timeaxis on the variability of observed RTs. This appears to alter theproportion of bona fide reactions (of low variability) and of morevariable time estimates of the foreperiod duration. Experiment IIIis designed to assess the factors responsible for the increased meanand variability of choice compared with simple RT distributions. Itit concluded mat information processing rather than motor factors isprimarily responsible for the difference between simple and choiceRT. Experiment IV studies the relation between RT of correct anderror responses as a function of variations in stimulus probability in a2-choice RT paradigm. Finally, several theoretical distributions areevaluated by the empirical distributions obtained in Experiments II,III, and IV; none seems wholly satisfactory, but those with roundedmodes and an exponential tail (e.g., the gamma) are clearly notadequate.

A common problem in simple RT normally unrecorded, or after the re-experimentation is that of anticipations, action signal, in which case they areor voluntary time estimates of the fore- recorded as particularly fast RTs. In-period. These may occur prior to the vestigators use various techniques toreaction signal, in which case they are circumvent the problem of anticipa-

1 The authors gratefully acknowledge the puter time was provided by National Insti-following sources of support for this work: tutes of Health Grant FR-1S to the John-Theoretical and experimental support was son Research Foundation, School of Medi-provided by National Science Foundation cine, University of Pennsylvania. A shortGrant GB 1462 to the University of Penn- report of this work was read to the Fifthsylvania and by Office of Naval Research, Scientific Meeting of the Psychonomic So-Contract No. NONR 477(34) to the Uni- ciety, Niagara Falls, Ontario, Canada, Oc-versity of Washington. The extensive com- tober 1964.

1

J. G. SNODGRASS, R. D. LUCE, AND E. GALANTER

tions. These include elimination of re-sponses prior to some time, usuallyapproximately 100 msec., the so-called"irreducible minimum" (Woodworth &Schlosberg, 1954); employing variablerather than constant foreperiods; andintroducing catch trials on some pro-portion of the trials (Drazin, 1961).Because it is questionable whetherthese techniques completely eliminateanticipations, the first two experimentswere run. The first was designed toestimate the variability and accuracyof time estimation when payoffs withfeedback are employed and the secondto devise a payoff scheme which wouldsimultaneously reward the fastest pos-sible true reactions to the signal andpenalize anticipations and slow re-spones.

GENERAL METHODSubjects.—Ten university undergraduates,

three of whom were female, (CP, EM, andSR), served as 5s. The 5s BW and GCparticipated in Exp. I; 5s AB, ES, KM, andCP in Exp. II; AH, MO, and EM in Exp.Ill; and AH and SR in Exp. IV. Note thatAH participated in both Exp. Ill and IV.All 5s had normal hearing and were right-handed, except for ES who nonetheless usedhis right hand to respond. They were paidas a result of their performance in the ex-periments, and in some cases they receivedadditional payments for their time. Datafor each .9 are analyzed separately.

Apparatus.—The 5s sat in an IndustrialAcoustics Company acoustic chamber whichattenuated outside noise by 30-65 db. overthe audible-frequency range from 100 hz.The warning and reaction signals were puretones generated by an oscillator and pre-sented binaurally through Permoflux PDR-8earphones mounted in semiplastic cushions.Except in Exp. I and for 5 AB in Exp. II,the tones were turned on and off by a sine-zero switch when the sine wave crossed zeroamplitude with positive slope.

Tone duration was controlled by countingcycles of a 1,000-hz. tone generated by asecond oscillator. In Exp. I, III, and IV,and for AB in Exp. II, the tone duration was143 msec.; it was 124 msec, for remaining 5sin Exp. II. The warning signal (So) pre-

ceded either reaction signal (Si, Ss) by 2,000msec. (±5 msec.). The interval betweenthe two signals was controlled by a synchron-ous motor-driven cam switch. The mechani-cal nature of this system prevented us fromcontrolling the foreperiod interval moreexactly. Tone frequency was controlled bya Krohn-Hite programmable oscillator. InExp. I and II, So and Si were 1,000-hz.tones; in Exp. Ill, So and 1,100 hz., Si was1,000 hz., and Ss was 1,200 hz.; in Exp. IV,50 was 1,000 hz., Si was 900 hz., and Sa was1,100 hz.

The intensity of all tones was .030 v., orabout 30 db. above .001 v. However, be-cause the tones were not filtered, there wasan unmeasured acoustical click at the be-ginning and end of the tones. All 5s inExp. I and II, and SR in Exp. IV used apair of hand-held joined microswitches actu-ated by typewriter-type keys. All 5s inExp. Ill and AH in Exp. IV used two 6 in.X 1 in. Plexiglas keys which were set flushin a response board and were separated bya short partition. These keys were adjustedto require a small amount of pressure anda very short travel. The typewriter keysrequired somewhat more travel.

Times were measured to the nearest msec,by a Hewlett-Packard electronic counter andwere recorded on an associated printer. InExp. I, the counter was actuated by So.This could have been done in the later ex-periments and would have enabled us torecord the times of responses prior to Si, butit had the disadvantage of degrading theaccuracy of measured RTs from Si to ±5msec, because of the inaccuracy of the camcontrolling the onset of Si. To increase theaccuracy of measured RTs in Exp. II throughIV, the timer was actuated by a 30-v. pulsethat accompanied the onset of the reactiontone; this reduced the measurement error to±1 msec. In these experiments, the occur-rence, but not the time, of responses prior to51 was recorded.

The presentation of Si and Ss in the choiceexperiment was controlled by a punchedtape read by a Western Union Teletype tapereader. Tapes of SOO random choices withP = .1, .3, .5, .7, and .9 were prepared sub-ject to the condition that the proportion ofSi signals within each block of 100 trialsexactly equalled P.

A display panel in front of 5 provided in-formation feedback after every trial. Theform of this information is described foreach experiment. The amount of moneywon or lost on a trial was presented by

ACCURACY AND VARIABILITY OF REACTION TIMES

means of projection units in the panel. InExp. IV, a bulb above the correct key wasilluminated as soon as S responded to thereaction signal.

Procedure.—Explicit payoffs were pre-sumed to fulfill part of the function of theusual instructions; therefore, the instructionsconsisted primarily of a detailed explanationof the payoffs to be used. The 5" was givenas much information as possible about theexperimental condition in the form of graphs,payoff matrices, information about the pre-sentation probabilities of the signals, etc.In both the simple and choice situations,bands of latencies were differentially re-warded. If 5" responded either before orafter the band he either lost money or failedto win anything. In the simple reactionconditions, payoffs were for latency alone,and in the two-choice conditions, for bothlatency and accuracy. The 5" was en-couraged to earn as much money as he could.

After each trial, 5" was informed of theamount of money won or lost by that re-sponse. In Exp. Ill this information wassufficient for 5" to know whether he was toofast, too slow, or his relative position withinthe band. For Exp. II and IV, in whichthe payoff band was symmetric, the positionon the display panel of the amount of moneylost informed S whether he was too fastor too slow relative to the payoff band. InExp. Ill, where payoffs for speed and ac-curacy were separate, the payoff for latencywas displayed first, followed by that foraccuracy.

The 5" was allowed as much practice as hewished before each new experimental condi-tion. Brief rest periods were given aftereach block of approximately 100 trials or atS's request During each rest, 5" was toldthe total amount of money that he had won

or lost during the preceding block. Ex-perimental sessions consisted of approxi-mately 500 responses, excluding practicetrials, and lasted between 1 and 14 hr.

EXPERIMENT I

Previous Es have investigated therelation between the variability andconstant error of time estimates andthe duration of the interval being esti-mated. The standard deviation of timeestimates was found by Woodrow(1930, 1933) to be approximately 10%of the mean duration for intervals be-tween 1 and 2 sec., in the absenceof feedback and payoffs. Klemmer(1957) analyzed the joint effect offoreperiod (FP) variability and timeestimation variability within the frame-work of information theory. He foundmean RT to be a linear function ofstimulus uncertainty, denned as thesum of the variance of 5"s time esti-mates and the variance of the FP dis-tribution.

Procedure.—The two S's, each of whomhad previously made about 20,000 RTs, wereeach instructed to try to estimate times by re-sponding when they felt that a prescribedtime period had elapsed from the onset of a1,000-hz. tone. The actual elapsed timeswere shown to 6" in milliseconds on thedigital readout of the Hewlett-Packard timer.The 6" was paid off for accuracy as follows:.9^ for responses with ±20 msec, of the goal,,B4 for responses in the adjoining 20-msec.bands, and so on, with no payoffs for re-

TABLE 1MEAN (M) AND STANDARD DEVIATION (o-) IN MSEC. AND THEIR RATIO

FOR ESTIMATES OF VARIOUS ELAPSED TIMES

TargetIntervalin msec.

60010001900350050001900

BW

M

601.7991.4

1914.13504.34853.71937.5

ff

75.2119.3154.8401.8

1536.8160.0

<r/M

0.1250.1200.0810.1150.3160.083

GC

u

601.41060.51894.83464.94923.51898.0

<r

49.597.3"

209.7335.4275.3160.6

a/H

0.0830.0920.1110.0970.0560.085

• The first response was excessively long and has been omitted.


sponses more than 180 msec, from the goal.Each run consisted of 500 estimates of agiven time; these times were 1.0, .6, 1.9,3.5, 5.0, and 1.9 sec., in that order.

Results and discussion.—The distri-butions of responses were sufficientlysymmetric so that, for our purposes, itis adequate to report only the meansand standard deviations, which areshown in Table 1. Except, perhaps,for the longest time, for which BWseems excessively variable and GC ex-cessively consistent, these data arestable and systematic. The stability ofthe repeated value, 1.9 sec., seems mostsatisfactory. In addition, plots of thetrial-by-trial estimates do not exhibitany apparent time trends within a run.The distribution for the shortest time,.6 sec., is highly peaked at the mode,similar to the RT distributions shownlater; however, unlike them, it is sym-metric about the mode.

The primary conclusion is that,within the range from .6 to 5.0 sec., "̂scan time estimate quite accurately, butthat the standard deviation of their re-sponses is of the order of 10% of themean, even when payoffs are employed.This figure is comparable to that ob-tained by Woodrow (1930, 1933) forintervals of comparable duration ob-tained without payoffs or feedback.As we shall see, this relative variabilityis considerably in excess of that forRT when time is measured from thewarning signal; therefore, however 5"smay use time estimation in our laterexperiments when we ask them to re-act, it cannot be that they simply esti-mate the length of the foreperiod onevery trial.

EXPERIMENT IIAs noted previously, RT investi-

gators have used various techniques,among them variable FPs and catchtrials, to experimentally eliminate timeestimates of the FP duration. We con-

sider here an alternative scheme—thatof rewarding 5s for responding withina latency band centered at T, where t(the foreperiod duration) is fixed andT is varied over a range from timesless than the (apparent) true reactiontime to times considerably greater.The S's task under the band payoffthen becomes one of reacting con-sistently within the band, for he is onlyrewarded for those responses occurringwithin the latency band and punishedfor responses which are too fast or tooslow for it.

If we assume that two responsemechanisms are available, namely, 6"can either time estimate from S0 or Sxor react to Sj and if further we assumeas the data of Exp. I indicate that thevariability of the time estimates in-creases with the time to be estimated,then we can predict qualitively how thevariability of the RT distribution mustchange with T. For explicitness, al-though it is not essential to the argu-ment, suppose that, as is approximatelytrue (see Table 1), the standard devi-ation of time estimates is a fixed pro-portion k of the mean estimate; ac-cording to our data k is approximately.1. When T is very short—so shortthat if 5" reacts to S± he will necessarilybe slower than the payoff band—thenhe has no reasonable option but to timeestimate from S0, and so the standarddeviation of the RT distribution shouldbe approximately k(t + T). As T isincreased sufficiently, reactions to Sjbecome feasible. Since the variabilityof the reactions is less than that of theestimates the observed variability shoulddecrease, reaching a minimum when allresponses are reactions. With furtherincreases in T, the reactions will fallshort of the payoff band, and 5" is againforced to time estimate, but now it isto his advantage to estimate from Sjrather than S0 (because the variability


is less and so a positive payoff is morelikely). The standard deviation of theRT distribution should be approxi-mately kT. According to this argu-ment, then, a plot of the variability ofthe RT distribution vs. T should be Ushaped with (for all T < t) a mini-mum located approximately at the"true" reaction time.

Procedure.—Three .9s were run under eachof six different payoff bands: (Band 1) 85-104 msec., (Band 2) 105-124 msec., . . .(Band 6) 185-204 msec. A fourth 5", AB,was run only on Bands 2-5. The S" won2^ for responding within the band and lostl«f for responding outside it. For AB,the bands were presented in random order;for the other three 6"s, the order of presenta-tion of bands was constrained as follows:no band was followed by an adjacent band;difficult bands (1, 2, and 6) alternated witheasy ones; and the first band in the serieswas always easy. These three 5"s had threepractice sessions, consisting of 500 trialseach, during which the 6 bands were used.Their order of use was different from thatof the experimental sessions, but subject tothe same restrictions. Each daily experi-mental session consisted of 500 trials undera single band.

Analyses are based upon distributions ob-

tained from individual S"s under a singleband. Each is based upon 500 trials, exceptfor AB under Band 2, where only 190 ob-servations are included in the analysis be-cause the remaining RTs were mistimed.

Results and discussion.—The pro-portion of responses prior to Sj washighest for Band 1 (8.8%, 4.0%, and3.0% for Ss KM, CP, and ES, re-spectively), next highest for Band 2(.5%, 2.8% .0%, and 1.2% for AB,KM, CP, and ES, respectively), andwas negligible for Bands 3-6.

Figure 1 presents the means andthe proportion of reactions occurringwithin the payoff band for each 5" as afunction of band position. The latterseems to be a better measure of vari-ability in this situation than the em-pirical standard deviation because, aswe show later, the RT distributionsmay have high tails—falling off as anegative power of t rather than as anexponential—in which case the em-pirical variance is a rather variablestatistic.

It is clear from Fig. 1 that the meanRT follows the band position closely.

Q K MA ESO C PX AB

I'

I'

jBAND

FIG. 1. Mean RT (left) and the proportion of responses in the payoff band (right)for the distributions of RTs of individual 5s under band-payoff conditions. (The verticalbar to the left of each group of mean RTs represents the range of the band.)


Except for the two fastest bands, allmeans are well within the band range.In terms of our measure of variability—the proportion of responses withinthe band—the least variable band forCP is 2, for ES and KM is 3, and forAB is 4. With the empirical variance,the minimum band is shifted from 3to 5 for KM and the others are un-changed. Note that the maximumseems fairly clearly defined and that,for the best bands, a surprisingly highproportion (50-65%) of the responseslie within its narrow, 20-msec. width.Moreover, pressing for short timesseems to force 5"s increasingly to timeestimate from S0 as judged both bythe increased frequency of unambiguousanticipations (responses before Sj)and by the marked increase in vari-ability. This shift to much greatervariability appears to occur in theneighborhood of 90-100 msec., whichis near the value of the irreducibleminimum defined by others. In sum-mary, then, although the observed RTdistribution is affected by outcomes,this may be due entirely to replacingreactions by time estimations withoutin any way affecting the magnitude ofthe "true" RT.

EXPERIMENT III

We consider next the source of thedifference between simple and choiceRT. Not only are choice RTs longeron the average than simple ones, theyare also more variable (Woodworth &Schlosberg, 1954). It seems desirableprogressively to complicate the simplereaction until we reach the choice re-action and to measure both the meanand variability of the RTs for each ofthe successive experimental steps todetermine which difference or differ-ences between the two are crucial.

Procedure,—The 6"s were run in situa-tions of increasing complexity, the simplestbeing a simple reaction to a single tone and

TABLE 2CONDITIONS USED IN EXP. Ill

Condition

Simple-1

Simple-2

Recognition

Choice

Response Key

Left

(1) Sr-F

(3) S2— S

(Si— F(S)

iSr-F

fSr— F(7) [s2— s

Right

(2) S2-F

(4) S,— S

fSr-F(«> L .,l.S2— F

fSr-S(8)

S2— F

Sl_F... (9)..-S2— F

S2— S-- - (10) - - -S i— S

Note.—Si = 1,000-hz. tone, Ss = 1,200-hz. tone.F =• fast band payoff, S = slow band payoff, numbersin parentheses are names for conditions.

the most complex, a two-choice reaction toone of two tones. Two relatively wide, non-overlapping, adjacent payoff bands wereused. A "fast" band differentially re-warded responses between 100 and 200 msec.,and a "slow" band differentially 'rewardedresponses between 200 and 300 msec. Re-sponses prior to the band were fined 5tf;those in the first third won 3^, in the sec-ond, 24, and in the last third, I t f ; thoseslower than the band received no payoffat all.

Distributions of RTs were collected underthe following procedures:

1. Simple—1. One signal, one response,a single payoff band rewarding all re-sponses.

2. Simple—2. Two signals, one response,a single band rewarding all responses re-gardless of which signal was presented.

3. Recognition. Two signals, one re-sponse, the fast payoff band rewardingresponses to one signal and the slowband rewarding responses to the other.

4. Choice. Two signals, two responses,a single band rewarding all responses,regardless of accuracy. In addition, 5"won 1$ for each correct response andlost 5$ for each incorrect response.

Recognition is so named because in order towin money, S1 must first recognize whetherthe signal presented is the fast one (i.e., the


one to which the fast payoff applies) or theslow one, for if he reacts faster than 200msec, to the slow one, he loses 5tf. TheS indicates that he has recognized the sig-nal by making differential temporal responseswith the same key. We define recognitiontime as the time to respond to the fast signal.

With two signals, one low (1,000 hz.)and one high (1,200 hz.) frequency, twopayoff bands, fast and slow, and two re-sponse keys, right and left, each proceduregenerates a variety of conditions. The 10shown in Table 2 were run. Of these, 6result in two distributions of RTs, brokendown by the signal presented and/or theresponse made; thus, there are a total of 16distributions which are available for com-parison. Of these, only the 8 run under thefast payoff band are relevant to our originalquestion about differences between simpleand choice RT. Conditions using the slowpayoff band were included for practice inthe recognition condition, where both payoffswere used simultaneously, and to answeithe general question of what, if any, changein the form of the distribution occurs undera simple shift of the payoff function alongthe time axis.

Each condition was run during each ofeight experimental sessions, resulting in 16distributions (two for each stimulus) of 240RTs each. Certain restrictions were im-posed upon the order of presentation of the10 conditions within a session: any recogni-tion or choice condition was preceded byboth of the relevant simple reaction condi-tions, e.g., Cond. 7 was always preceded by

Q CMA AH0 MO

FIG. 2. Mean RTs of individual 5"s forconditions under the fast and slow payoffbands. (Conditions are: S-l = Simple-1;S-2 = Simple-2; R = Recognition; and C =Choice.)

0 CMA AH0 MO

5-1 S-2 Ft C S-t R C

CONDITION

FIG. 3. Standard deviations of RT dis>tributions under fast and slow payoff bandsfor the same conditions as in Fig. 2.

both 1 and 3; and the recognition and choiceconditions were always preceded by ap-proximately equal numbers of fast and slowpayoff conditions. Because both Simple-2conditions were run with fast payoffs only,these always appeared at the end of thesession. With these restrictions, 16 order-ings of the conditions were possible, ofwhich 8 were selected. These were pre-sented in a random order to the 3 5"s.

Results and discussion.—A session-by-session analysis of median RTs forall conditions indicated that the datafor AH and MO were stable over ses-sions, but the median RT for EM de-clined appreciably from Day 1 to Day2 after which it was stable. Therefore,the results reported for EM are basedupon only her last seven sessions, andso her RT distributions consist of 210,rather than 240, observations.

Figure 2 shows the mean RT forthe four conditions run under the fastpayoff band and the three conditionsrun under the slow payoff band. Re-sults for the fast payoff are unequi-vocal: mean RTs for both recognitionand choice conditions are much longerthan those for either of the simple con-ditions ; indeed, recognition and choice-mean times are not even within the fastpayoff region. The position of the pay-off band affected RT means for thesimple conditions markedly, but it


changed those for the recognition andchoice conditions little, if at all. Thechoice condition results in somewhatlonger mean RTs than recognitionunder both payoff conditions.

The standard deviations for the sameconditions exhibit a similar pattern tothat of the means (Fig. 3). Again,the two simple conditions are verysimilar under fast payoffs, both havingrelatively low variances, whereas rec-ognition and choice are more variableand do not appear to differ much fromone another. Location of payoff bandhas little effect on variances of recog-nition and choice, but it affects thevariance of the simple distributionsmarkedly. In fact, variances of thesimple condition under the slow payoffare not much smaller than those of therecognition distribution for two 5"s, andthe variance for one 5" in the simplecondition is actually larger than thecorresponding choice variance underthe slow payoff. Thus, it appears thatlengthening simple RTs by payoffsadds almost as much variability as doesincreasing the complexity of the de-cision.

We conclude from the results of thisexperiment that the major part of theincrease in time and variability betweensimple and choice RT is contributed bythe process of recognition.

EXPERIMENT IV

We have already seen that increasingthe number of responses from one totwo, i.e., from simple to choice RT, in-creases both the mean and variabilityof the RT distribution. We next con-sider the effects of different presenta-tion probabilities of the two signals onthe mean and variability of both thecorrect and error distributions.

Several investigators have found thatchanges in presentation probabilities offixed sets of signals had effects on thelatencies of correct responses (Cross-

man, 1953; Fitts, Peterson, & Wolpe,1963; Hyman, 1953). Specifically, themean RT of the correct response to asignal decreases as the probability ofthat signal being presented is increased.Typically, the results have been inter-preted within the theoretical frame-work of information theory. Onecharacteristic of the method of dataanalysis usually employed has been toaverage across signal-response pairs;when this is done, a measure of re-dundancy or information transmittedby the selection of a signal is linearlyrelated to the mean RT over all signals.However, the information measuredoes not accurately predict the meanRT of responses to specific signals(Hyman, 1953). Various models ofchoice RT predict that stimulus pre-sentation probability and latencies ofcorrect responses will be inversely re-lated and sometimes they specify theform of the latency distribution (e.g.,Audley, I960; LaBerge, 1962; Luce,1960; Stone, 1960). However, it isoften difficult to determine what theypredict about the latencies of error re-sponses.

One problem in evaluating thesemodels, or of developing new ones, isthe lack of detailed empirical informa-tion about the relation between the la-tencies of erroneous and correct re-sponses and about the forms of theresulting empirical distributions.

The purpose of the present experi-ment is to obtain data for which de-tailed relationships between correct anderror latencies can be determined, andsome attributes of the distributionsobserved.

Procedure.—Five presentation probabilities,.1, .3, .5, .7, and .9, were used with the twoS's. Only one probability was used duringa single experimental session. The Swas informed of its value at the beginning ofthe session. The same payoff function wasused with each probability. The 6" won Itffor responding correctly from ISO to 200


msec., lost Itf for responding incorrectly inthe same band, and received no payoff forall other responses. This relatively difficultpayoff function was chosen to produce higherror rates. When 5 failed to respondwithin the payoff band, he received informa-tion about the direction of his error.

Results and discussion.—The datamay be summarized in a 2 X 2 table inwhich each cell has two entries, theconditional response probability Pr(RySi) (here denoted pif} and theRT distribution R t i ( f ) . The relationbetween />u and />21 that is generated byvarying presentation probability (P),the isosensitivity plot, is shown in Fig.4 for each 5". These data are similarto those reported in the signal-detectionliterature where Pr(Yes/Signal) andPr(Yes/Noise) covary with the pres-entation probability and payoffs. Incontrast to the signal-detection results,however, the points generated by our6"s do not appear to lie on a single iso-sensitivity curve. In particular, the P= .5 point for AH and the P = .7point for SR lie closer to the chancediagonal than do the others. Thisanomaly presumably arises because thetwo reaction signals are, in fact,perfectly discriminable when 6" takesenough time to respond, and the errorsarise because 5" decides to take less in-formation than is necessary for moreaccurate responding. If this interpre-tation is valid, the discriminability ofthe signals is essentially under 5"s con-trol, rather than E's, and our 5s ap-parently varied the discriminabilityfrom probability to probability by ob-serving for shorter or longer times,thus causing a shift from one isosensi-tivity curve to another.

Next, we examine mean RT as afunction of several probabilities. Agiven RT distribution Ry has at leastthree probabilities associated with it:(a) the presentation probability,Pr(S(), (b) the unconditional response

probability, Pr(/?/), and (c) the condi-tional response probability, py. In theordinary choice RT experiment inwhich errors are negligible, only thecorrect latencies are shown, usually asa function of the presentation proba-bility. Because in nearly error-freeperformance response and stimulusprobability are nearly identical, a plotof RT vs. response probability wouldbe about the same. In this experiment,presentation and response probabilitiesfor the correct responses are almostidentical, because our 5"s tended tomatch response probabilities with pre-sentation probabilities. However, forincorrect responses, response and pre-sentation probabilities are distinctlynot the same. Figure 5 shows meanRTs (filled circles) and error RTs(open circles) as a function of each ofthe three probabilities. It is clear thatthe most nearly identical functions forboth error and correct latencies occurwhen they are plotted against uncondi-tional response probability.

(.9)

0.5-

• U)

0(.5> 0(,7I

• (.5) /

/

O ('31 /

• (.I) /X

/

•1.9) /

/

/

• AHO S R

0.5Pr(R,

FIG. 4. Iso-sensitivity plots relating,Pr(Ri/Si) to Pr(Ri/S2) as a function ofpresentation probability, P, of Si, for each 5"separately. (The value of P is shown inparentheses beside each point. The dottedline indicates chance performance.)

10 J. G. SNODGRASS, R. D. LUCE, AND E. GALANTER

FIG. 5. Mean error and correct RT plotted against (a) stimulus probability, (b) con-ditional response probability, and (c) unconditional response probability. (Filled circlesare means of correct and unfilled circles of incorrect distributions.)

However, even across a fixed re-sponse probability there seem to benontrivial differences between the errorand correct latencies. Figure 6 pre-sents in greater detail the relation be-

tween the means and the standard devi-ations of correct and error distributionsfor the nonsymmetric probabilities P= .1, .3, .7, and .9 for AH. The dataare plotted in terms of the more prob-

250

<no

I 200

<rzUJ

ISO

inazo

540

tuo

'20

r_Tii TJ! TiJ Tj] Tli Tji Tij T|j

FIG. 6. Means and standard deviations of correct and incorrect choice RT distributionsfor AH for P = .1, .3, .7, and .9. (The more probable stimulus and its associated responseare both denoted », and the less probable is denoted /. TIJ refers to that distribution ofRTs of response / to stimulus i, where Pr (Si) > Pr(Sj).

ACCURACY AND VARIABILITY OF REACTION TIMES 11

able stimulus, Si or 82, depending on P.For this S, the ordering of the meansand standard deviations is consistentbut different: if we define i and / sothat Pr(Si) > Pr(Sj), then the orderis it, ji, ij, and // for the means andij, ii, ji, and // for the standard devia-tions. These relations for the other S,although in the same general direction,are not as consistent and they are notshown.

Two aspects of these data are ofrelevance in evaluating or developingmodels of the choice process. The firstis that latencies of error and correct re-sponses are more closely related toresponses than to stimuli. This seemsto be true whether the errors are gen-erated by time pressure, as in this RTexperiment, or by low stimulus dis-criminability, as the experiments ofCarterette et al. (1965) suggest. Thesecond finding is that error latenciestend to lie intermediate to correctlatencies.

FORM OF THE RT DISTRIBUTION

A problem which has remained un-solved in spite of its long history is theform or forms of the underlying RT dis-tribution. Perhaps the earliest proposalwas the normal distribution; however, RTdistributions, especially simple ones, arenotoriously skewed and are bounded frombelow. These facts led Woodworth andSchlosberg (1954) to propose the log nor-mal as a substitute.

Conceptions of the reaction process as aseries of intervening, nonobservable stepsin a chain have been the basis of many re-cent theoretical proposals. In particular,if one assumes that the overall RT resultsfrom a finite sum of independent, iden-tically distributed exponential randomvariables, the gamma distribution results.If these random variables are not iden-tically distributed, but still exponential,the general Erlang or generalized gammadistribution results (McGill & Gibbon,1965). Under fairly general conditions,

the normal distribution results if a verylarge (infinite) number of independent,identically distributed components aresummed (central-limit theorem).2

The gamma distribution has been sug-gested for RT by Bush and Mosteller(1955), Christie (1952), Luce (1960),and Restle (1961). A discrete analogof the gamma, the negative binomial, wasderived by LaBerge (1962) in his theoryof RT.

As a preliminary step, the empirical RTdistributions collected in Exp. II, III, andIV were classified as to Pearson type bythe procedures described in Appendix A.As noted there, none were classified asordinary gamma and only 13 were clas-sified as generalized gamma. A strikingaspect of the results of the analysis is thatthe simple RT distributions in particularare far outside the limits for the gamma.In fact, by inspecting the simple distribu-tions themselves, it becomes evident thatany Pearson distribution is unlikely to ac-count for them: they are both too peakedand have too high tails (which are usuallyfound in RT distributions and have some-times been attributed to experimental er-ror) to be fit by distributions withrounded modes and exponential tails.

In lieu of any suitable theoretical al-ternative, we attempted to fit the doublemonomial (DM) to our RT distributions.The DM was derived by Luce and Galan-ter (1963, p. 288) as the form of a gen-eralization function in magnitude estima-tion. It consists of two power functionsthat meet at the mode. As a result it hastwo qualitative features of our empiricaldistributions—peakedness at the mode andhigh tails (in the sense that a negativepower approaches zero much more slowlythan an exponential). The density func-

2 It should be pointed out that if the com-monly accepted assumptions of the central-limit theorem are altered, a variety of otherlatency distributions can arise (Gnedenko &Koltnogorov, 19S4) ; however, the results arestated in characteristic function form and arenot readily translatable back into distribu-tion terms. This, however, is an importantdirection that must ultimately be clarifiedby those interested in the form of the RTdistribution.


tion of the DM is:

(5 + i ) (e- l )€ +6

0 if I < 0

if 0 < t<

where t0 is the mode.In addition, we have fit these data by

the ordinary gamma distribution with itsorigin located at a value t0, not neces-sarily 0, which we here call the displacedgamma. The density function of the dis-placed gamma is:

A \t ~~~ VQJ O

r(a)

Procedures for estimating the three pa-rameters for each distribution are givenin Appendix B.

Table 3 presents the results of thecomparison between likelihood values for

TABLE 3NUMBER OF DISTRIBUTIONS FOR WHICH THE

LIKELIHOOD is LARGER

Exp.

II

III

IV

Type

simplesimplerecognitionchoicechoice8

Total

Number of Cases Favoring

DoubleMonomial

2020

715

53

Gamma

245

1118

40

•Only distributions of more than 50 observationswere analyzed; these include some distributions of in-correct responses.

the two theoretical distributions. TheDM fits the simple RT distributions andthe gamma the choice distributions moreoften by this test. We do not know, how-ever, how good a test of goodness-of-fitthe likelihood comparison is.

Even though Table 3 indicates that thegamma distribution does better than the

DOUBLE MONOMIAL

g*£

fB «

1007 200 300 400 900 100 200 300 <00 900

FIG. 7. Two empirical RT distributions (solid lines) are plotted twice, each time witha best fitting theoretical density function (dotted lines). (The left pair are fitted by theDM, the right pair by the gamma. The theoretical mode for the DM and the theoreticalstarting point for the gamma are indicated by fo on the abscissas.)


DM on almost half of the empirical dis-tributions, plots of the fits of the theoret-ical distributions to the empirical datapoints, such as those shown in Fig. 7, aredisturbing. The distribution in the lowerhalf of the figure is best fit (by the likeli-hood criterion) by the gamma and yet tothe eye the DM seems to be the betterrepresentation. The difference in evalua-tion arises from the details of the tails,which are difficult to see by eye and whichmay be of secondary importance in manycontexts.

Table 4 shows the results of x2 tests on

the DM. In the overwhelming majorityof the narrow band, simple RT distribu-tions from Exp. II, the DM is rejectedat the .05 level, whereas the wide bandsimple distributions from Exp. Ill werefit more often than not.

Although the DM distribution is notcompletely satisfactory as judged by ^2>the source of its failure is interesting andmakes clear that the classically proposeddistributions, in particular the gamma,would fare even worse.8 The maximumlikelihood estimates of the parameterstake into account both the tails and themode of the distributions, and to get thetails of the DM sufficiently high (8 and esufficiently small) the mode is not suffi-ciently peaked. In addition, the empiricaldistributions tend to be rather morejagged than would be predicted by asmooth, continuous density function andsuch large sample sizes. However, inlight of the two processes in simple RT—reaction and time estimation—for whichevidence was obtained by Exp. II, thisfact is not surprising. Perhaps if wewere able to obtain RT distributions un-contaminated by time estimates, the DMwould fit better.

CONCLUSIONThe most striking conclusion we draw

from these studies is that, even though ourdata are highly systematic, we do not havean adequate theoretical understanding of

3 We did not calculate xa for the bestfitting gamma distributions because, in thelight of this analysis, the added computercost did not seem warranted.

TABLE 4RESULTS OF x! TESTS BETWEEN DM

AND EMPIRICAL DISTRIBUTIONS

Type

Exp. IIsimple

Exp. Illsimplerecognitionchoice

Exp. IVchoice"

Total

Number of DistributionsWhich are

Not Rejected

2

1766

«5

36

Rejected*

20

766

10

49

• Only distributions of more than 100 observationswere analyzed; these include some distributions ofincorrect responses.

*t <.OS.

them. It is clear that Ss respond sensiblyand effectively to differential temporalpayoffs and that some use is probablymade of time estimation as well as of re-actions, but we do not understand the de-tailed mechanisms very well. This may bea result of faulty experimental procedures;for example, some 5s probably used handposition to control their RTs in some pay-off conditions. However, we cannot hideall of our theoretical difficulties behindexperimental problems. The form of theindividual empirical RT distribution isanother indication that existing theoriesare probably inadequate. These distribu-tions appear simultaneously to have hightails and a peaked mode. If this is areliable finding, summary statistics basedon the moments are much less satisfactorythan those, such as median, interquartilerange, and percentage of responses in aninterval, that are less sensitive to the oc-casional very long RTs.

In addition to the regularity that RTexhibits as a function of several experi-mental manipulations—including the loca-tion of the payoff bands and the type ofreaction required—there are also sys-tematic trading relations between pairs ofpsychological measures, such as RT andresponse probability. Since comparableresults have been found by other in-vestigators and since similar trading rela-


tions have proved most illuminating indetection work, they probably should bepursued much more intensively within theRT context.

REFERENCES

AUDLEY, R. J. A stochastic model for indi-vidual choice behavior. Psychol. Rev.,I960, 67,1-15.

BUSH, R. R., & MOSTELLER, F. Stochasticmodels for learning. New York: Wiley,1955.

CARTERETTE, E. C., FRIEDMAN, M. P., &COSMIDES, R. Reaction-time distributionsin the detection of weak signals in noise./. Acoust. Soc. Amer., 1965, 38, 531-542.

CHRISTIE, L. S. The measurement of dis-criminative behavior. Psychol. Rev., 1952,59, 443-452.

GROSSMAN, E. R. F. W. Entropy and choicetime; the effect of frequency unbalanceon choice-response. Quart. J. exp. Psy-chol., 1953, 5, 41-51.

DRAZIN, D. H. Effects of foreperiod, fore-period variability, and probability of stim-ulus occurrence on simple reaction time./. exp. Psychol., 1961, 62, 43-50.

FITTS, P. M., PETERSON, J. R., & WOLFE, G.Cognitive aspects of information pro-cessing: II. Adjustments to stimulusredundancy. /. exp. Psychol., 1963, 65,423-432.

GNEDENKO, B. V., & KOLMOGOROV, A. N.Limit distributions for sums of independentrandom variables. Reading, Mass.: Addi-son-Wesley, 1954.

HYMAN, R. Stimulus information as adeterminant of reaction time. /. exp. Psy-chol, 1953, 45, 188-196.

KENDALL, M. G., & STUART, A. The ad-vanced theory of statistics, Vol. 1. Lon-don: Charles Griffin, 1961.

KLEMMER, E. T. Simple reaction time asa function of time uncertainty. /. exp.Psychol., 1957, 54,195-200.

LABERGE, D. A recruitment theory ofsimple behavior. Psychometrika, 1962, 27,375-396.

LUCE, R. D. Response latencies and prob-abilities. In K. J. Arrow, S. Karlin, & P.Suppes (Eds.), Mathematical methods inthe social sciences, 1959. Stanford: Stan-ford Univer. Press, 1960. Pp. 298-311.

LUCE, R. D., & GALANTER, E. Psychophysi-cal scaling. In R. D. Luce, R. R. Bush,& E. Galanter (Eds.) Handbook of mathe-matical psychology, Vol. 1. New York:Wiley, 1963. Pp. 245-308.

McGiLL, W. J., & GIBBON, J. The general-gamma distribution and reaction times./. math. Psychol., 1965, 2, 1-18.

RESTLE, F. Psychology of judgment andchoice. New York: Wiley, 1961.

STERNBERG, S. Estimating the distributionof additive reaction-time components. Pa-per read at Psychometric Society, NiagaraFalls, Ontario, Canada, October 1964.

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

WOODROW, H. The reproduction of tem-poral intervals. /. exp. Psychol., 1930, 13,473-499.

WOODROW, H. Individual differences inthe reproduction of temporal intervals.Amer. J. Psychol., 1933, 45, 271-281.

WOODWORTH, R. S., & SCHLOSBERG, H. Ex-perimental psychology (Rev. ed.), NewYork: Holt, 1954.

APPENDIX A

PEARSONIAN CLASSIFICATION OF EMPIRICAL DISTRIBUTIONS

From the moments of the empirical distributions, values of K were calculated asdescribed in Kendall and Stuart (1961), where

+ 3)']

- 300(202 - 30! - 6)]and

The value of K is used to classify the empirical distributions as to type of Pearsondistribution. For K less than 0 or greater than 1, the distributions are classified as beta(of the first and second kind, respectively); for K between 0 and 1, the distribution


TABLE A

TYPE OF PEARSON DISTRIBUTION AS DETERMINED BY VALUE OF STATISTIC

Type

Exp. IIsimple

Exp. Illsimplerecognitionchoice

Exp. IVchoice"

Total

Type IV

14

1474

14

S3

Beta I, II

8

1058

24

55

* All distributions, including error responses, were analyzed.

is classified as Type IV; and for «-»<», the distribution is gamma. Table A sum-marizes the results of this analysis.

Instances of each of the three distributions other than the gamma were found.Approximately equal numbers of our distributions were classified as Type IV and betaand instances of both types of betas appeared. There is a suggestion in Table 5 thatmore simple distributions are of Type IV and more choice distributions are beta.

It is not surprising that no instances of the gamma distributions were foundbecause K — oo (the gamma distribution) corresponds to an exact relation among themoments. Unfortunately, however, there is no way of determining, from the value ofK alone, how close a distribution approaches the gamma. Sternberg (1964) hasrecently shown that certain relations must hold among the moments for a distribu-tion to be gamma. Specifically, for the ordinary gamma (equal time constants foreach of c exponential steps where c is an integer)

= 4/c and 3 + 6/c.

This describes a series of points lying on the line satisfying

ft = 3

For nonintegral c and equal time constants, the distributions lie between these points,on the same line. Sternberg's results are consistent with Pearson's in that when therelation is satisfied, 2/32 — 3ftt — 6 = 0 and K = oo *

For the generalized gamma (unequal time constants), he has shown that thefollowing inequalities must be satisfied :

0 < ft < 43 + (3/2)13, < 02 < 3 + (27/2)1/3!*.

Figure Al shows the relation between p± and j82 for the empirical distributionscollected in Exp. II, III, and IV. The insert covers the region near and withinthe limits for the generalized gamma. The normal distribution lies at the pointj8± = 0, /32 = 3. It is clear that although a number of distributions lie within or nearthe region of the generalized gamma, none of them falls on the line for the ordinarygamma. By Sternberg's criteria, only 13 distributions are generalized gamma distribu-

* It is also, of course, true that if 4/3a — 3/3i = 0, then K = °o ; however this relation be-tween /3i and /3a satisfied neither Sternberg's nor Pearson's restrictions (Kendall & Stuart,1961, p. 175).


120-

• SIMPLE• RECOGNITION» CHOICE

180 200

FIG. Al. Relation between & and fa for Exp. II, III, and IV. (Insert is an enlargementof the shaded region. The diagonal line is the relation between ft and /3a that must holdfor the gamma distribution and also the lower bound on /Sa for the generalized gamma.Points lying between the lower and upper bounds for the generalized gamma are unfilledand those lying outside such bounds are filled.)

tions. Of these, two are simple band distributions from Exp. II, six are choicedistributions and two are recognition distributions from Exp. Ill, and three are choicedistributions from Exp. IV.

APPENDIX B

ESTIMATION PROCEDURES

1. Double monomial. If N is the number of observations, the maximum likelihoodestimates of the three parameters N0 (rank of the RT corresponding to t0), 8 and eare easily shown to satisfy:

= Ne- I1

log/O - -Tf E

- 1,1

+ 1logk — log/o

where the logarithms are natural ones. We used N/2 as a first estimate of N0, andthe three formulas were iterated by computer until the estimate of N0 converged or20 iterations were completed. In very case in which N0 did not converge, it oscillatedbetween two integers. In these cases we calculated maximum likelihood and x

2 valuesfor the two sets of three parameter values and their means, and that set of valuesyielding the highest value of likelihood was chosen.


2. Displaced Gamma. Maximum likelihood of t0 A, and a satisfy:

j- + X - l o g i L ( f c - « - 4" - '

X - ( « -

where 5

To solve for a, we iterated the first formula, where the (n— l)st value of « wasinserted in K. The procedure was terminated when the difference between the »thand (n— l)st value of a was less than 1(M.

In contrast to the DM, in all cases the three equations failed to converge on f0.Therefore, we explored the possible range of reasonable values of tot calculated alikelihood value for each set of parameters, and chose that t0 and corresponding valuesof a and A. for which the likelihood was a maximum. The possible range of ta wastaken to be from — 2,000 msec, (i.e., the onset of S0) to *t — 1 msec., where t1 is thevalue of the shortest RT in the empirical distribution.

(Received August 19,1966)5 An abbreviated form of Stirling's approximation for T (a) was used to obtain K:

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