Acta Psychologica43 (1979) 431453 0 NorthHollandPublishingCompany
A CRITICAL COMPARISON OF TWO RANDOMWALK MODELS FOR TWOCHOICE REACTION TIME Donald LAMING+ UniQmftY
of Chmbridgi
Dept. of Experimental i?tyc~oIogy, Cambrfdgr , &eat Britain
Received June 1978
Two randomwalk models for twochoice reaction time, being the
[email protected] probability ratio test model initially suggested by Stone and the relative judgment theory recerztlyproposed by Link, are here compared.The comparisonshows the precisemathematical relation between the models, the predictions they have in common, and two points at which they differ. Three experiments concerned with the effect of stimulus probability on twochoice reaction time are examined with respect to three predictions from the models. I~Ithis threeway comparisonthe two models are more like each other than either is to the data; a greater variety of phenomena is revealed than either model can at present accommodate.
1. Introduction
Link (1975) has recently published a model for twochoice reaction time (2CRT) based on his relative judgment theory (MT; Link and Heath 1975), that bears a close mathematical relation to another model, based on information theory, previously published by ?he present author (Laming 1968; also Stone 1960). Both models represent the decision compouent of 2CRT as a random walk between parallel absorbing boundaries. The initial purpose of this paper is to exhibit. me precise relation between the models, to show what properties they have in common, and where their predictions differ. This comparison has previously been considered by Link and Heath (1975), by Swensson and Green (1977), and by Thomas (1975); the present treatment is more succinct. + I thank the Leverhubne Trust Fund for assistance in the prepruationof this paper. I also thank Dr. S. E. G. Lea and the Editor, Dr. A. F, Sanders,for their comments on earlierdrafts of this paper. Requests for reprints should be addressedto Donald Lamii, The University of Cambridge, Departmentof ExperimentalPsychology, Downing Street, Cambridge,CB2 3EB.
D. Lamingflbo mndomwalk models
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In support of RJT Link cited three experiments by different authors, analysed with respect to three predictions from his model. But not every experiment was analysed with respect to every prediction, and the spectrum of results implicitly selected thereby was disproportionately favourable to RJT. So a second purpose of this paper is to complete the analysis of Link’s three experiments, in order to present a more representative spectrum of experimental findings. Taken together the three experiments display a greater variety of phenomena than either model can at present accommodate. In particular, there is a paradox in the behaviour of mean 2CR.Ts that presents a challenge to every contemporary theory of CRT. 2. Relative judgment theory
The two stimuli  let them be S, and .S,  employed in a 2CR experiment are represented by two stochastic processes, XA (t) and Xc(f). When a stimulus is presented for discrimination, the realization of its stochastic process is compared with the realization of a reference process, X,(t), to produce a difference process o&t) = X&j)  XJj),
i=A,B
The difference process is assumed to have stationary and independent increments and it describes a random walk between two parallel absorbing boundaries situated at *A on the decision axis. If the random walk fast reaches the boundary at +A, response A is made; if at A, then response B. The decision component of 2CRT is identified with the passage time of the random walk; to this passage time must be added a ‘constant’ random variable (that is, a random variable independent of both the stimulus and the response), K, representing all the nondecision components of CRT. The random walk begins at a point C, which need not be zero; C is at the disposal of the subject and provides a means whereby the decision process can be adjusted to take account of prior stimulus probabilities and possibly other experimental conditions as well. Thus far, the model derived from RJT (Link 1975) is mathematically equivalent to the model obtained from information theory (Laming 1968), However, in order to derive predictions that can be put to an
D. LominglTivo
randomwoIk mmiek
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experimental test, it is necessary to posit some relation between the two difference processes. It happens that the absorption probabilities and the moments of the passage time distributions for an arbitrary random walk between parallel absorbing boundaries can always be obMned from Wald’s Identity (see Wald 1947; Bartlett 1966: 17). But the free parameters of the process are such that there is no testable prediction that can be compared with CR data unless there are two random walks (one for each stimulus) between the same absorbmg boundaries and some particular relation is posited between the two. At this point the models differ. Without any loss of generality the difference processes, DA(~) and D&j, may be taken as the immediate representations of the stimuli. It is convenient to think of the random walk as evolving step by step, where each step has probability density function fA (x) or f&c) according as S, or S, is presented. Initially Link does not impose any constraint on the difference processes, but in practice he assumes that DA(t) and Do are mirrorimages of each other, so that
f&
= fA (xl.
(la)
Information theory, on the other hand, requires that f&) = exfAWI.
(lb)
These two relations between the representations of the stimuli have been proposed for mathematical, rather than psychological, reasons, and there is no simple characterization of their psychological significance. Relation (lb) makes the random walk analogous to a sequential probabilityratio test (SPRT), which represents an optimal use of all the information available (Wald and Wolfowitz 1948). This property was in Stone’s mind when he first proposed this test procedure as a possible basis for a CR model (Stone 1960). But in RJT, with its underlying structure >f a reference process and two difference processes, any particular relation between the difference processes might be considered plausible. Relation (la) has the virtue, shared with (1b), that it leads to simple distributionfree predictions that may be readily compared with experimental data. But only these two relations seem to have this property. Any other relation one might consider is likely to prove intractable. This means that for practical purposes only these two
D. LamingfTwo randomwalk models
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randomwalk models are worth investigating, It will be well to look a little deeper into their properties and their interrelation to see why this is so. ?he behatiour of random walks is most conveniently studied by means of the moment generating functions of the step distributions, fA (x) and fs(x). The moment generating function is defined as M&z) =_cea fi(x, dx,
i=A,
B,
(2)
where z is a real variable. The continuous curve in fig. 1 is an example of such a function; it has equation M&)
= (1  z)22k
and is the moment generating function of a gamma distribution of parameter 2, suitably centred. Provided the step may go in either direction, positive or negative, and the moment generating function exists, the equation MA(z) = 1 has two real roots, one at the origin and one at, say 8~. Without loss of generality the difference process DA(t) can be scaled such that 0 A = 1 (as in fig. _l).
Fii 1. Moment g~ncmting functions of cmtain gamma distrhhons of parameter2, illustrating tha interrelationsimplicit in the SPRT and RJT models.
D. Lamingho
rundovtwalk models
Substitution Mg(Z) = M&
435
of (1 b) into (2) shows that, for the SPRT model,  1);
(3b)
in this case MB(z) is the dotted curve in fig. 1. Since M,(z) = 1 at z = 1 and z = 0, so MB(z) = 1 at z = 0 and z = 1; and, under an arbitrary scaling, if MA(2) = 1 at 2 = (?A, MB(z) = 1 at z = 8~. This property of the moment generating function is instrumental in the derivation of distributionfree predictions; the mathematical significance of this property is explained in the appendix. Substitution of (la) in (2) shows that, for RJT, M&)
= M/l tz),
(W
the broken curve in fig. 1. It is obvious from (3a) that if M,(z) = 1 at H = B,, MB(Z) = 1 at z = BA, the same property that is found in the SPRT model. Therefore it is to be expected that the two models will often give formally identical predictions. In particular, if M,(z) is a symmetric function  that is, if the distribution of the unit step, fA (x), is symmetric (e.g., a normal distribution)  relations (3a) and (3b) each imply the other and the two models are mathematically equivalent (see Swensson and Green 1977: Thomas 1975). The fundamental difference, then, between the two models lies in the relation, (la) or (1 b), that is posited between the random walk representations of the two stimuli. The estimators of the boundary values (see appendix) are the same in each case, so that the two models give identical predictions of the relation between mean CRT, contingent on the stimulus, and the probabilities of error. They differ, however, in the following ways: (1) In RJT D&) is distributed as DA(t) and, as a consequence, predictions are symmetric with respect to S, and S,. I shall say that the decision process is symmetric in this case, distinguishing this carefully from the property of symmetry of the random walks D&t). The SPRT model, on the other hand, admits asymmetric decision processes (see Laming 1968: 133). (2) The SPRT model predicts that the distribution of 2CRT conditional on a given response will be independent of the stimulus (Stone 1960: App, 1); RJT, on the other hand, allows errors to be either faster or slower than the same response given correctly depending on the
436
D.Lam&z&vomndomwdkmodels
direction of the asymmetry in the moment generating function of an incremental step in the random walk (Link and Heath 1975: 88). Herein lies the motivation of RJT as a model for CRTs. In 2CR experiments errors are usually faster than the same response given correctly and an additional subprocess has to be added, post hoc, to the SPRT model to accommodate this finding (Laming 1968: 8085). Relative judgment theory, on the other hand, takes the difference between errors and correct RTs to be a property of the stimulus, whilst retaining many of the other attractive features of randomwalk models. But this accommodation is achieved at a price: to obtain testable prel dictions from RJT it is necessary to assume that the decision process is symmetric. Therefore, in any critique of RJT two questions are of particular importance: (i) Does the difference in RT between an error and the same response given correctly, and especially the sign of the difference, depend only on the stimuli employed, or are other factors involved as well? (ii) Are the data from 2CR experiments always symmetric with respect to the two stimuli?
3. Three experiments In support of RJT Link (1975) cited three experiments: A. Experiment 2 from Laming (1968). B. An original experiment designed explicitly to test the theory. C. An otherwise unpublished experiment by Kadlac and Theios. Hereafter these experiments will be referred to as Experiments A, B, and C respectively. All three experiments compared different stimulus presentation probabilities in a visual 2CR task. Experiment A (Laming 1968) used, as stimuli, vertical lines, 0.5 in. wide and either 4.0 or 2.83 in. high, presented in a common spatial location, so that only the length of the lmet was available as a discriminative cue. Stimulus probabilities were changed by steps of 0.125 from one series of trials to the next, from 0.25 to 0.75 for half the subjects and in the contrary direction for the remaining half. The responses were made by pressing on keys with the left! and right forefmgers. Twentyfour subjects were employed and a
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431
total of 4800 CRTs were recorded under each pair of presentation probabilities. In most 2CRT experiments (including Experiments A and C here) diffe: ’ pairs of stimulus probabilities are realized in separate series of trials, being defined for the subject by the relative frequency with which each stimulus is presented. But in Experiment B (Link’s experiment) the stimulus probabilities were randomly arranged within a common series of trials. This was accomplished by presenting the subject with the value of P(S,) (the displayed value ranging from 0.125 to 0.875 in steps of 0.125) at the beginning of each trial, followed one second later by either S, or S,, selected randomly according to the displayed probability. The stimuli were two square figures, distinguished solely by the orientation of a diagonal which connected either the lower left and upper right comers, or the lower right and upper left. The responses were made by lifting the left or right forefinger from one of two keys, which otherwise had to be kept depressed. Four subjects were employed and a total of 400 CRTs were recorded for each displayed stimulus probability. Experiment C (by Kadlac and Theios) looked not only at stimulus presentation probabilities, but at the effect of different responsestimulus intervals (RSIs) as well. The stimuli were the numerals 1 and 2 and the responses were made by depressing one of two response keys. Within each series of trials the stimulus probabilities were constant (being either 0.3, 0.4, 0.6 or 0.7), but the RSI varied, being either 400 or 600 msec at random. Kadlac and Theios recorded about 1400 CRTs from each of 24 subjects, who were instructed to respond prior to a deadline set at 380 msec after stimulus onset. Link analyzed the data from these three experiments with respect to three predictions from his model concerning: (1) The choice of starting point in the random walk. (2) The difference in mean RT between an error and the same response given correctly. (3) The relation between mean RT conditional on the stimulus and the probabilities of error. I shall reconsider each of Link’s predictions with respect to each experiment in turn, either summarizing the already published analysis, or presenting the results afresh where Link’s treatmeut is incomplete.
D. Lamhtg/Two randomwalk models
438
4. The choice of startingpoint in the random walk The starting point of the random walk is at the subject’s disposal and is chosen, one might expect, to match in some way the stimulus presentation probabilities. Link (1975) examined two hypotheses concerning the choice of starting point: (i) The starting point is chosen to minimize the total number of errors (‘error minlmization’j. (ii) An ‘expectancy’ hypothesis, specifically, C/A = (Q  l/2),
(4)
oundaries of the random walk are situated at +A, the starting point at C, and nA is the prior probability of SA. There is also a third hypothesis which I have proposed elsewhere: (iii) The starting point is chosen to minimize the unconditional mean CRT subject to a given proportion of errors (‘optimization’; Laming 1968: 38). It should be realized that each of these hypotheses may be combined with either model, RJT or SPRT; moreover, the predictions concerning error probabilities which follow therefrom are the same for both models. This is so because the models differ solely in the choice of statistical representation of S, to be paired with a given SA, and the particular choices, (la) and (1 b), which characterize the two models lead to identical marginal absorption probabilities. The following discussion will not discriminate between the two models; it is intended to supplement Link’s (1975) examination of hypotheses (i) and (ii). 4.1. Error minimization The distances of the boundaries from the starting point may be estimated from the proportions of errors. Let pfi be the probability of response i to stimulus i (i, j = A, B). The estimation equations, valid for both models, are (Link 1975: 118; without loss of generality 8 can be set equal to 1 in Link’s Eqs. 1 and 2).
A  C= log (PAAPAB), A + C = log (PBB/PBA j.
D. LamingfTwo randomwalk models
439
If S, is presented with probability ?rA, the expected probability of error is
which, for fixed A and variable C, is a minimum when
c=; 1% bA/(l  flA))*
0)
This is hypothesis (i) (error minimization). It follows readily by substituting from (5) into (6) and differentiating with respect to C. Link (1975: tables 2 and 3) shows that (7) is inconsistent with parameter estimates from Experiment A. 4.2, Optimization It is implicit in hypothesis (i) that the distance between the boundaries (2A) is fixed and only the starting point (C) is at the subject’s disposal; otherwise the expected probability of error can be made as small as one pleases by taking A sufficiently large. But the value of A determines not only the probability of error, bul also the mean CRT, so that, ifA be fixed, there can be no tradeoff between speed and accuracy. This is contrary to the findings in many experiments; Pew (1969) has reviewed eight experiments which showed a specific trading relation, and Laming (1968: Experiment 3) and Swensson (1972) have explicit counterexamples. If, on the other hand, A bet at the subject’s disposal as well as C, the principle of ‘error minimization”, by itself, does not make sense. By increasing the distance between the boundaries of the random walk, the probability of error can be made as small as desired at the expense of a concomittant increase in mean RT, and a tradeoff between speed and accuracy is generated thereby. In this situation (iii) is a more plausible hypothesis, namely, that the starting point is chosen to minimize the unconditional mean CRT subject to a given proportion of errors, with the location of performance on the speedaccuracy operating characteristic being determined by other factors such as payoffs and instructions to the subject. Alternatively, one might hypothesise that the starting point is chosen to minimize the proportion of errors subject to a given mean CRT; in the context of the present two models this is equivalent
D. Lamin&wo random~wafkmodels
440
to (iii) (see Laming 1968: Theorems A3 and A8). Contrary to what one might expect, hypothesis (iii) does not lead to iZq. (7), but, iustead (see Laming 1968: App. A), to C=
log IQ/ (1  %4)).
(8)
Tabk 1 Posterior probabilities of error from Experiment A (Laming 1968: Experiment 2). =A
flen‘d,:
P(errorlRg)
0.250 0.375 0.500 0.625 0.750
0.029 0.028 0.025 0.027 0.023
0.022 0.024 0.029 0.033 0.024
Tabk 2 Posterior probabilities of error from Experiment B (Link’s experiment). i
=A 0.125 0.250 0.375 0.500 0.625 0.750 0.875 
’
fyerroru?~)
&enorlR~)
0.083 0.158 0.029 0.059 0.106 0.088 0.052
0.066 0.086 0.135 0.035 0.031 0.040 0
I
If the decision protess is symmetric (as in RJT), hypothesis (iii) is equivalent to the assertion that each response is made to the same accwracy, that is, with the same posterior probability of error. The proof, which is valid for both models, is in Laming (1968: 123). This result leads to a simple, modelfree, test. Table 1 shows the posterior proportions of errors for each response aId, each value of UA in Experiment A. The proportions range from 0.022 to 0.033 and the differences between them are not significant (Laming 1968: 55). Table 2 shows the corresponding results from Ex
D. Lamhg/7bo
mndomwalk mod&
441
pediment B (Link’s experiment) and in this case hypothesis (iii) is not supported. An unusual feature of Link’s experiment was that the different stimulus probabilities were randomized, and at the beginning of each trial the subject was shown the value of nA effective on that trial. Since the presentation probabilities were thus known exactly, subjects may have been tempted, on some trials at least, to make a fast guess in favour of the more probable stimulus. The posterior error probabilities in table 2 reflect a relatively large number of errors in favour of the more probable stimulus in all conditions except nA = 0.500 and the mean error RTs shown in table 5 of Link (1975) are likewise consistent with the suggestion that most errors were fast guesses. Table 3 Posterior probabilities of error from Experiment C (Kadlac and l’heios’ experiment). RS Interval = 400
RS Interval = 600
*
PyerrorV?)
Equivalent w
p(error IR)
Equivalent n
0.3 0.4 0.6 0.7
0.121 0.119 0.098 0.072
0.43 0.41 0.53 0.57
0.165 0.173 0.139 0.117
0.45 0.47 0.53 0.55
The corresponding results for Experiment C (by Kadlac and Theios) are shown in table 3. Here there is a systematic bias away from the predicted equality, the posterior proportions of errors are greater for small values of II. One can calculate stimulus probabilities for which the obtained proportions of error would accord exactly with hypothesis (iii), and these values are shown as ‘Equivalent 8’ in table 3. It is interesting that although the proportions of errors are greater for the longer RSI (this is to be expected  see Laming 1979a), the equivalent Rvalues do not change. It is possible that the hypothesis fails here because the subjects were incompletely adjusted to the bias in the stimulus series (cp. Laming 1968: experiment 1). 4.3. Expectancy As an alternative to (i), Link (1975) preferred hypothesis (ii) (expectancy), which states that the relativeposition of the starting point is
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442
governed by (4). ThJs looks counterintuitive because, as Link points out, as A + 1, the probability of error does not tend to zero and is ultimately greater than (1  n). However, this hypothesis works with Experiment A (as Link showed) because, for this particular experiment, it provides a close approximation to (8). Expanding (8) in powers of (n  i). C=4(n  i) + 16(x  $3/3 + 64(r  $)“/5 + . . . In Experiment A the mean estimate of A was 3.59, so that,
C/A= 1.110  +)+U((n  i,“,, which is numerically close to (4). Link did not compare hypothesis (ii) with the data from either Experiment B or C and simple calculations on his tab&ted results show that it does not fit. 5. The difference in RT between errors and correct responses
The SPRT model asserts that the RT associated with an error has the same distribution as that associated with the same response given correctly (Stone 1960: App. l), and this is manifestly untrue. Relative judgment theory says that the difference in mean RT between an error and the same response given correctly may be positive or negative, depending on the direction of the asymmetry in the distribution of the difference process, but is always proportional to the distance from the starting point to the boundary at which the response is uttered (A* C).
Let 7 = _MA’(O)r’MA’(+l), where MA’(z) is the differential coefficient of fifA(z). For any MA (2) such that MA (1) = 1 = MA (0) (this condition can always be satisfied by an appropriate scaling of the random walk) it can be seen by referring to fisp 1 that the gradients of MA (z) at 1 and at 0 are of opposite sign, so that y is alwr!: J positive. Link and Heath (1975) show that R+TAA
 RT,~B
= (A
Klr,,

= (A +0
RTSIA

0
(7 
1MPY,
(7  l)/pr,
(9)
D. Laming/Tivo randomwalk models
443
where RTq is the mean RT contingent on response i and stimulus~ and P = MA '(0). If 7 > 1, as in fig. 1, correct responses take longer than errors. The differences in mean RT between an error and the same response given correctly from Experiment B are displayed in fig. 3 of Link (1975). It is sufficient for the present argument to state that the data  RT,.r) proportional to (A f C) !;on form nicely t:o (9) with (RT,,, over a range of RT differences from 0 to 300 msec. But this is accompanied by another, less satisfactory, feature, namely, RTWOr‘Yconstant, independent of the starting point in the random walk (Link 1975: table 5). Using the formulae in the appendix to Link’s paper, it can be shown that the decision component of RT,,,,
to the first order of approximation. The least square regression of RTerro*(in msec) on (A f C) in Link’s experiment is
RTemx=3240.1(4&C), the correlation between RTr and (A f C) being negligible. Strictly zero correlation between these quantities corresponds to 7 = 03 or MA '(1) = 0.which does not represent any possible choice of difference process. (If MA‘( 1) = 0, the equation MA (zj1 = 0 has a multiple root at z = 1 and, in addition, a root at z = 0, by detinition. But the second derivative
MA “0) = ? x2 ezx f’ (xl dx DD
is strictly positive defmite so that MA (z)1 = 0 can have at most two real roots.) It seems more likely that a majority of the errors in Link’s experiment were fast guesses, and this is also suggested by ,:he posterior error probabilities in table 2. The differences in mean RT between an error and the same response given correctly from Experiments A and C are shown in figs. 2 and 3 respectively. These data do not conform to (9) because the intercepts on the ordinate am clearly greater than zero. In all three experiments cited by Link errors are faster than correct responses, but in some experiments, especially those employing a difti
D. Lamln&vo
444
1:o
2'0
370
4%
nmdomwalkmotiela
60
(AX) Fii. 2. Differences in mean RT between en error and the same re&ponaegiven correctly in Experiment A (Laming, 1968, Experiment 2).
cult perceptual discrimination, errors may be slower than correct responses (Pike 1968). Relative judgment theory accommodates this fmding by positing a different kind of difference process  the sign of the relation between error and correctRT (9) depends on the direction of the asymmetry in the difference distribution. This makes the
Fig. 3.,E;KTertacesin mean RT between an error and the same response given correctly in Experhen t C (ICad& and Tbeios’ experiment).
D. LamingfTwo randomwalk mcuiek
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sign of the difference a property of the stimuli employed. Thus, while the magnitude of the difference between error RT and RT for the same response given correctly may vary with certain experimental conditions, its sign should be invariant (the stimuli being fixed). There are two experiments which pose problems to this idea. Firstly, Laming (1968: Experiment 7) obtained both positive and negative differences between error and correctRTs in an experiment which compared different numbers of alternatives (up to 5) at different levels of discriminability. Errors were slower than correct responses in the more difficult discriminations, even though these discriminations had a stimulus in common with easier discriminations in which errors were faster. Secondly, Swensson (1972), working with a difficult twochoice discrimination, showed that the relation between error and correctRT may change sign, depending on the level of accuracy required of the subject. Errors were slower than correct responses (for a fixed pair of stimuli) at high levels of accuracy. 6. Mean CRT The SPRT model and RIT give formally identical predictions for mean CRT, conditional on the stimulus, in relation to the probabilities of error. In the notation of Link (1975): RT_r= E(K) + Zi/cc,
(10)
where RT*i is the mean RT conditional on stimulus i, K is a ‘constant’ random variable representing the nondecision components of RT, and ZA = PAA
log (PAAIPAB)  PBA
‘B
lOg tpBB/pB~)
= ‘BB
log (PBB,PBA ),
 PAB log
(11)
(PAA/PAB).
Mean CRTs from Experiment C are displayed in fig. 4 of Link (1975); it is sufficient for the present discussion to state that they conform satisfactorily to (10). Mean CRTs from Experiment A are given by Link, and compared with numerical predictions, in tabular form only. The comparison looks satisfactory, but there are important properties of the data that fail to stand out in a table. The data from Experiment A are
446
D. Lamhgfm~
randomwalk models
therefore reproduced here in tig. 4, and the corresponding results from Experiment B are shown in fig. 5. Both experiments appear to conform satisfactorily to (10); however there is a technical difficulty in the statistical evaluation of the equation because there are errors of estimation in both RI’, and Zi. The least squares regression of RT., on Zi, as used by Link, is not necessarily correct. In fact, all that one can say is that the maximum likelihood estimate of the functional relation between these variables is intermediate between the regression of RT_ion Zi and that of Zi ou RT.i (see Kendall and Stuart 1967: Ch. 29). Both regression lines are shown in fig. 4 and fer each stimulus the maximum likelihood estimate of the unknown functional relation lies somewhere within the shaded pencil. Fortunately, the pencils in fig. 4 are narrow enough to lead to some important con
Fig. 4. Mean CRTs in Experiment A (Laming, 1968, Experiment 2).
D. Laming/Tbo randomwalkmodels
0.5
447
1.0
1.5
2.0
2.5
3.0
6 Fig. 5. MeanCRTs in Experiment B (Link*sexperiment).
elusions. (As the data stand in fig. 4 the difference between the two pencils is not quite significant. But if the quadratic trend in the data be removed, the pencils differ significantly at the 0.025 level. The test employed depends on a comparison of the correlation matrices for S, and Sg (Kullback 1959: 321). If the RT, Zi axes be rotated so as to produce zero correlation in the combined sample, the regression pencils differ according as there is a difference in the rotated correlation matrices of the subsamples.) (i) The coefficient of 2, in (10) (i.e., the slope of the pencil) is greater for S, than for SA, so that Experiment A appears asymmetric with respect to the stimuli, contrary to the assumption of RJT. The SPRT model, on the other hand, requires no stipulation about symmetry (see Laming 3968: 135138).
D. Lamb&v0 tat&mwlk
448
models
(ii) The maximum likelihood estimate of E(K), the mean of the ‘nondecision’ components of RT, cannot be less than 327 msec for S, and 272 msec for S,. But simple RT measured under iden:ical conditions to Experiment A has a mean within the range 220.9 to 227.4 msec, depending on the particularcombination of stimulus and response (Laming 1968: Experiment 4). It is natural to expect that CRT, extrapolated to a condition where one of the stimulus probabilities is 1 and all the others 0, will give a value comparable to simple RT (without prejudice to the question whether simple RT contains only nondecision components). But the estimates from Experiment A are too great in comparison by 50 to 100 msec. Swensson (1972) also has found extrapolated CRT to exceed simple RT by a ‘deadtime’ in this range. (iii) The difference between the estimates of E(K) for the two stimuli in Experiment A is of a greater order of magnitude than the corresponding difference in simple RT. Therefore some part of the intercept at 2 = 0 has to be attributed to the decision process; that is, a model is required which gives a theoretical relation of the form (cf 10) RT.i
=
E(K) +
i[zi+ B)/Pi,
(12)
where B is a constant, presumably independent of the stimulus.
7. Diiussion The purpose of this paper IS twofold: first, to exhibit the precise mathematical relation between the 2CRT randomwalk models derived from RJT and from information theory, and, second, to present a more complete and incisive analysis of the three experiments cited by Link in support of the former. These two models appear to be the only cases in which a random walk will yield distributionfree predictions for CRT, so the comparison is of more than ordinary interest. But it is clear, by nc:v, that the two models are more like each other than either is to the *data; there is no point in contriving an experiment which purports to discriminate betwe en them.
D. LamingfTwc randomwalk models
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7.1. Choice of starting point
The possible assumptions concerning the choice of starting point in the random walk are common to both models. No one assumption can be picked out as ‘correct’ because the evidence on this point from different experiments is not consistent. The relevant evidence consists of the relative proportions of the two kinds of error in relation to the stimulus probabilities. It is known that the relative proportions of error depend on the order in which the different stimulus probabilities are presented to the subject (Laming 1968: Experiment 1); they may also depend on other incidental features of the experiment, not least on the instructions given to the subjects (e.g. Laming 1968: Experiment 3). This means that no great consistency can be expected between different experiments by different authors. 7.2. Difference in RT between errors and correct responses The raison d’etre of RJT is, ultimately, the possibility it offers of explaining the difference between RT for an error and for the same response given correctly. This difference poses a problem for CR theory, chiefly because the SPRT model requires it to be zero, and it almost never is. Relative judgment theory allows errors to be either faster or slower than correct responses, with the sign of the difference depending only on the particular stimuli to be discriminated. But in none of the experiments examined here does the pattern of results conform entirely to that predicted by RJT, and in two other experiments the difference in RT has been shown to change sign while the stimuli remain fixed. I think a more likely explanation of the difference in RT between an error and the same response given correctly concerns the subject’s difficulty in locating the onset of the reaction stimulus in real time. Originally (Laming 1968: 8085) I suggested that CR subjects typically begin sampling the stream of stimulus information before the reaction stimulus has been presented  that is, they begin with stimulation from the preexposure field, trying to decide whether it emanates from stimulus A or stimulus B ’ and, in consequence, the preex’posure field interferes in the decision process. In a recent development of this idea (Laming 1979a and b) it appears to provide a unification of several diverse CRT phenomena, relating ,the changes in CR performance (in both RT and proportion of errors) follcwing an error, the pattern of
490
D. LamingjTbvo randomwalk Ktodels
autocorrelation in RTs, and the difference in RT between an error and the same response gi=jen correctly when errors are faster than correct responses. For those cases in which errors are slower than correct responses a fresh principle has to be invoked (see Laming 1968: 85). But in view of Swensson’s (1972) finding that for a difficult twochoice discrimination errors are faster or slower than correct responses according as l>owor high accuracy is required of the subject, I doubt if a more parsimonious account of the phenomena is possible. ?.3. Mean CR T The prediction relating mean CRT, conditional on a particular stimulus, to the two probabilities of error is common to both random walk models. The predicted relation (10) fits the data well provided only one stimulus is considered at a time (though a curvilinear relation might fit the data in fig. 4 better). But if it is required that the relation should fit data from both stimuli with common parameters, some unexpected difficulties arise. These difficulties are apparent in Experiment A (fig. 4), but not in many other experiments (e.g. Experiment B, fig. S), precisely because the discrimination employed in the former experiment.is psychophysically asymmetric. It is clear from tig. 4 that in the predicted relation (10) there must not only be a different gradient (l/p) for each stimulus, but also a different intercept at Z= 0; moreover, the intercept is between 50 and 100 msec greater than the corresponding simple RT. This immediately suggests the following explanation which, as it turns out, will not do: The probabilities of error (of which Zi is a nonlinear function) fluctuate from trial to trial dependent on several different factors (see Laming 1968: Ch. 8) and in the present case they have been averaged over subjects. Since Zj (11) is a positively accelerated function, averaging the probabilities of error first and then entering them into relation (11) gives a smaller value of Zi than would be obtained by calculating Zi separately for each trial (assuming that to be possible) and averaging afterwards. This means that the values of Z plotted in lig. 4 are underestimates of the true information collected to which the mean CRTs properly relate. The extent of the bias in estimation is unknown, but if the relation (10) has a different gradient for each stimulus, the underestimation of Z will produce different intercepts as a matter of statistical artifact.
D. Laming/7Mo random.waik models
451
It needs to be emphasised that although such an artifact undoubtedly exists in the data, it cannot possibly explain the different intercepts found in fig. 4 because the two regression pencils intersect at a positive value of Zp The statistical artifact would provide a positive constant B in (12) whereas a large negative value is required to fit the data. Moreover, any psychological assumption implying only a partial use of the information collected or a limit on the efficiency of processing would function like the statistical artifact, requiring the regression pencils to intersect at a negative value of Zi. Herein lies the importance of the argument: those modifications to either model that one would naturally propose to explain why relation (10) extrapolated to Z = 0 gives an intercept much greater than simple RT (fig. 5 here; Swensson 1972) will not work with the data in fig. 4. The problem is more intractable than it at tirst appears, and one might expect its ultimate resolution to be correspondingly profound. A systematic examination of the three experiments cited by Link (1975) has revealed a greater variety of CR phenomena than either randomwalk model can at present comprehend. There is a danger in confining attention to just those properties of the data to which the models speak; one might, and probably will, overlook the critical result which points the direction in which theory needs to be developed. I suspect that the configuration of the data in fig. 4 constitutes such a result. Appendix: Boundaryvalue estimators for randomwalk models Let M(,) be the moment generating function of a unit step in an arbitrary randomwalk between absorbing boundaries at (2and b, a < 0 < b. Of necessity, M(0) = 1 and Bartlett (1966: 17) shows that M(z) = 1 has exactly one other real root, B0 (see fig. 1). Substitution of this value into Wald’s Identity gives expressions for the absorption probabilities (Bartlett 1966: 18): Pa
=
Pb =
(AlI
e
The quantity Pa = 1  Pb may be determined by experiment. But (Al) contains two free parameters, a0, and be,, and experihnent yields only
D. Lami&Tlvo
IS2
randomwalk models
one datum, P4’ This explains why any CR model must relate to two randomwalks between the same absorbing boundaries. Let M*(z) be the moment generating function of a second randomwalk between the same absorbing boundaries, having its second real root at 8,“. Then Pa* =
IfB,
_po* a*,* _e%* e 1
042)
*=OO,then
P,*lP, = e”“o
(A3)
The quantities Pa and Pa* can be determined by experiment and adO estimated by log (p,*Jp,) (see Link 1975: 118). If 8,* # 8,, then the elimination of b from (Al) and (A2) gives
(
1 _
pa*&%*
1 pa*
,)l’“*e.eb=.(l z;‘) lleo
(A4)
Given 00* as a function of 8,) one can always estimate a0, from experimental determinations of Pa and Pa*. But unless 8,* = 8, (note that if 8,* = 8,. Pa* = PO and no estimation is possible) the relation (A4) will not be tractable. The condition 6,* ‘= 19, is ensured by either (la) or (lb) and possibly by other relations as well. But in general (Le. excluding the RJT and SPRT models) the relation posited between B,,,* and 19O will imply some specific, and probably abstruse, assertion about the distributions
fA @x)and f&e. Rederences
Bartlett. M. S., 1966. An introduction to stochastic proaues. 2nd ed. Cambridge: Cambridge ihiversity Press. Kendall, M. G. and A. Stuart, 1967. The advanced theory of rbtistics. Vol. 2: lnfexnce and n4ationship. 2nd ed. London: Griffin.
D. LamingfTwo randomwalk models
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Ktdlback, S., 1959. Information theory and statistics. New York: Wiley. Laming, D. R. J., 1968. Information theory of choicereaction times. London: Academic Press. Laming, D. R. J., 1979a. Autocorrelation of choice reaction times. Acta Psychologice 43, 381412. Laming, D. R. J., 1979b. Choice reaction performance following an error. Acta Psychologica 43, 199224. Liik, S. W., 1975. The relative judgment theory of two choice response time. Journal of Mathematical Psychology 12,114135. Link, S. W. and R. A. Heath, 1975. A sequential theory of psychologzical discrimination. Psychometrika 40,77105. Pew, R. W,, 1969. The speedaccuracy operating characteristic. In: W. G. Koster fed.), Attention and performance II. Acta Psychologica 30,1626. Pie, A. R., 1968. Latency and relative frequency of response in psychophysical discrimination. British Journal of Mathematical and Statistical Psychology 21,161182. Stone, hf., 1960. Models for choicereaction time. Psychometrika 25,251260. Swensson, R. G., 1972. The elusive tradeoff: speed vs accuracy in visual diirimination tasks. Perception and Psychophysics 12,1632. Swensson, R. G. and D. hi. Green 1977. On the relations between random walk models for twochoice response times. Journal of Mathematical Psychology l&282391. Thomas, E. A. C., 1975. A note on the sequential probability ratio test. Psychometrika 40. 107111. Wald, A., 1947. Sequential analysis. New York: Wiley. Wald, A. and J. Wolfowitz, 1948. Optimum character of the rmquential probability ratio test. Annals of Mathematical Statistics 19,326339.