Correction for guessing in choice reaction time

Correction for guessing in choice reaction time' JOHN 1. YELLOTT, JR. UNIVERSITY OF MINNESOTA Additional theoretical and experimental results are presented for a choice reaction time performance model described by Oilman (1966). A formula is given for estimating the latency distribution of true recognition responses from the results of a single session; the estimate is invariant with respect to changes in the proportion of "guess" responses and with respect to fluctuations in the latency distribution of guesses. A recent note by OIlman (1966) describes a model for performance in choice reaction time (CRT) sttuations. According to this model the S in a CRT experiment ma,kes two kinds of responses: fast "guess" responses, and slower responses which represent the outcome of a (possibly imperfect) recognition process. The latter will be referred to as "stimulus controlled responses" (SCRs). The relative frequency of guesses and SCRs is controlled by the S, and presl.Ullably reflects his effort to attain an optimal balance between speed and accuracy. OIlman presents results which allow the parameters of the model to be estimated if data are available from several experimental sessions-corresponding to several different observed proportions of correct and incorrect responses. The present note reports an additional result which allows the "trQ.e" reaction time distribution, i.e., the latenoy distribution of the stimulus controlled responses, tobeestimated:(rom the data of a single session. The equation expressing this result can be regarded as a "correction for guessing" formula. Further experimental tests of the model are also reported. Consider a CRT taek involving two stirn-Wi Sl and ~. The SUbJect is instructed to makEl response Ai (i = 1, 2) when st is presented, and we aSSI.UllEl that st and S2 are presented randomly over trials with F( st) =.5. The model supposes that on any trial the s1,1bject makes either a guess ref!ponse (with probabU~ty 1...q) ora SCR (with probability q). If the eubject guesses, he makes response Ai with (bias) probabUity bt regardless of which stimulus was presented. The latency of each guess response is assumed to be a random sample of a random variable Lg with distribution function IlIg [i.e., P(Lg~t)=IlIg(t») and mean fig. If the subject makes a SCR, the response is correct with probability a> .5, and its latency is a random var~able Ls with distribution function Ills and mean fis' The latency of a SCR is independent of which stimulus... response pair actually occurs. It is natural to suppose that guesses are faster than SCRs, i.e., that fig< fis' but the results given below do not depend on such an assumption. In relating the model to observable quantities, the following notation is used: Pc denotes the (marginal) Psychon. Sci., 1967, Vol. 8 (8) probability of a correct response, Pe the probability of an error, Fdt) the probability that a correct response latency is less than or equal to t, F e(t) the probability tba.t an incorrect response latency is ~ t, Me the mean latency 01), correct responses, and Me the mean latency on incorrect responees. The follOwing results can then be derived from the model: 2PcFc(t)=cl(t) [2pc-1] +llIg(t) (1) 2peFe(t)=c2(t) [2pc..1] +1lIg(t) (2) (3) where c (t) = 2allls(t)-llIg (t) 1 2a-1 c2 (t) = 2(1-a) Ills (t) -llIg(t) 2a-1 Equation (3) is the correction for guessing formula that permits estimation of Ills from results of a single session. The following equations are immediate corollaries of (1), (2), and (3), respectively. whe;re 2pc Mc" c1 (2pc-1) + fig (4) 2PeMe;;: c 2 ( 2Pc-1) + fig (5) VcMc - PeM~ = fl. Pc-Pe s (6) 2(1-a) fis - fLg c2 "' _ _."..,;;,..-2.. ~a-i EquatiQns (4) I\lld (5) appear in OllllllUl.' s paper (in somewhat di«erent notation); they describe linear relationI!!hips useful in testing the model. However, the test implied by (4) and (5) couldfa.iUfthe latency distribution o{ guess responses varied from session to session, i.e •• if fi g varied. In such a case the model would remain Elssentially valid, but a plot of 2PcMc and 2PeMe against 2Pc-1 would not reflect this. Consequently, itis of interest to observe that Equation (6) is independent of fig' and thus provides a test of the invariance of fis that is unaffected by changes in guessing latencies, as well as gueesing prob/lbUities. Note that (6) can be obtained by subtracting (5) from (4). Ollman's paper reports a test of the pre<Uction that 2PcMc and 2PeMe will be linearly related to 2pc-1. A somewhat stronger test of the model is afforded by the fact that f;4e relationship between 2PcMe and 2Pc-1 can be predicted using the sioPEl (c2) and intercept (fig) parameters estimated from a plot of 2PeMe against 2Pc"l, the mean of the fis estimates provided by Equation (6), and the fact that c1 -c2 = 2 fis' 321 Method Three hired subjects were run in a two stimulus-two response CRT experiment. There were 16 experimental sessions, each consisting of 480 trials. The choice stimulus on each trial was a red or green illumination of the screen of an lEE Series 10 read-out. In all sessions red and green stimuli occurred randomly over trials with equal probability. Responses were key depressions using the left or right forefinger. Within each session a fixed "deadline" condition prevailed; responses faster than the deadline were rewarded by computing the percentage of all such responses within each session and awarding one half cent for each percentage point. Immediate feedback on response speed was provided on each trial in the form of a 40 msec 800 cps tone burst that occurred if and only if the latency on that trial exceeded the prevailing deadline. Eight deadline conditions were studied: 150, 200, 250, 300,350,400, 500, and 800 msec. Each condition was employed in two sessions, so that 960 responses were madebyeach subject under each deadline. Under all conditions subjects were SUBJECT I SUBJECT 2 • 21l: Me 700 ..lig .203 Jis '297 600 500 2Pe Me 0-.98 x 2PeMe 700 ..lig' 194 ..lis -287 a '.98 • 2Pc Me x 2Pe Me and 400 2Pe Me 300 200 100 o '-';--';;"'..."'""'"~"""'" o ALL SUBJECTS SUBJECT 3 700 )/g.181 .J.'s·292 o· .96 :~~~~ .I 700 .Jig'194 )4,'292 • SUBJ Eel I • SU BJ Eel 2 0 SU BJ Eel 3 .~,Y 0-1.0 ./ ;:''''0 ",M / / • " instructed to be "fast and accurate" and to try not to make more than 5% errors. However, no penalty was imposed for errors. The three Ss had had extensive practice in earlier reaction time tasks, but were naive as to the purpose of the present experiment. Results For purposes of analysis the 960 responses made under each deadline condition were combined and treated as if they had been generated by a Single session. Figure 1 shows the plots of 2PcMc vs 2Pc-1, and 2PeMe vs 2Pc-1, for each subject and for all subjects together. In each graph the lower line of points corresponds to 2PeMe' and the solid line through these points is the best fitting straight line as determined by least (...truncated)