Central refractoriness in simple reaction time: The deferred processing model

JOURNAL

OF MATHEMATICAL

Central

PSYCHOLOGY:

5,

Refractoriness The Deferred ROBERT

Bell

Telephone

Laboratories,

49-60

(1968)

in Simple Processing

Reaction Model1

Time:

T. OLLMAN

Incorporated,

Whippany,

New

Jersey

07981

The well-known single channel conception of the human operator gives rise very naturally to the deferred processing model for simple reaction time. The predictions of the model principally concern how features of the simple reaction time distribution depend on the foreperiod duration. When developed quantitatively and compared with published data, the predictions seem to be fairly seriously wrong.

INTRODUCTION

If two signals occur in close succession, and the subject is required to respond only to the second member of the pair, then his speed of response to the second signal decreases as the temporal separation between the two signals is decreased (Nickerson, 1965; Davis, 1959). One view is that this reflects the existence of a “Central Refractory the subject cannot begin to deal with the second signal until he has dealt Period”: with the first. If the second signal arrives while the first is still being processed, then treatment of the second signal must be deferred until treatment of the first is complete. The “Refractory Period” is the time required to deal with the first signal. The notion that the human operator can be viewed as a single central processor before which there are storage bins in which sensory inputs queue up until they can be dealt with in turn is a general hypothesis about the organization of human information processing functions. It is the notion which makes the existence of a refractory period sound plausible, and has spawned a wide variety of experiments, recently reviewed by Bertelson (1966). Although the single channel conception says something about a wide variety of situations, it does not say a great deal about any particular one of them. Here we shall be concerned with strengthening the idea so that it says stronger things about a relatively narrow class of experiments: simple reaction time experiments. In view of the considerable variability reaction time data exhibit, it is natural to begin by asking if the refractory period is subject to statistical fluctuation. If it is, 1 Preparation grant NSF GB

of this manuscript 1462 to R. Duncan

was partially Lute.

49 480/5/I-4

supported

by

National

Science

Foundation

50

OLLMAN

then the only intuitively obvious prediction is that relatively long intersignal intervals must give rise to relatively short response times because, with a long interval, it is less likely that the central processor will still be preoccupied with processing the first signal when the second one comes along, and it is relatively likely the processor can begin dealing with it immediately. This is a rather weak prediction. However, there is considerably more predictive power latent in the model: the addition of a few more assumptions, some of which have already been suggested in the literature, lead to much stronger predictions than monotonicity. It is with the impact of the additional assumptions and the less obvious implications of the model that we shall be primarily concerned. The development of the model rests on the observation that the quantity of greatest theoretical importance is the probability that the subject is still refractory when the second signal arrives. Thus, the model must be developed in terms of the distribution function of the refractory period: a development in terms of the mean refractory period will not suffice. THE

DEFERRED

PROCESSING

MODEL

We adopt the terminology of reaction time experiments: the first signal is the warning signal; the second signal is the reaction signal; the foreperiod, the temporal delay between the two, has duration T. It is simplest to act as if the experimenter’s clock were started at the warning signal and to make the appropriate conversion later. The random variable t is the observable delay from warning signal to response; the random variable t, is the duration of the (unobservable) refractory period; the

t

.

.

T

FIG.

postulated

1.

The sequencing by the deferred

1

.

t1

.

.

.

.

of events and processing model.

the

12

relationships

between

l

the

random

variables

CENTRAL

REFRACTORINESS

IN

SIMPLE

REACTION

TIME

51

random variable t, is the (in principle) observable time to process and respond to the reaction signal. These three random variables have cumulative distribution functions F( ), Fi( ), F,( ), respectively. The relationships between the three random variables and T are shown in Fig. 1. The deferred processing model asserts that (i) if processing of the warning signal is complete when the reaction signal arrives, the subject begins working on the reaction signal immediately; but (ii) if processing of the warning signal is not complete when the reaction signal arrives, treatment of the reaction signal is deferred until processing of the warning signal is complete. According to these rules,

It is useful to regard the foreperiod cumulative distribution

This

makes the model

duration

as a dummy

random

variable

having the

succinct: , T}.

t = t, + max{t,

(3)

It might seem desirable to state the distribution function oft in terms of the distribution functions of t, and t, . However, if we content ourselves with expressing the mean and the variance of the reaction time in terms of the mean and the variance of t, , the foreperiod duration T, and the distribution function of the refractory period Fi( ), we can easily find interesting results without making restrictive assumptions about the distribution of t, . We begin with the mean reaction time, which is the expected time interval between presentation of the reaction signal and the observation of a response. Even if they are not independent, the expected value of the sum of two random variables is the sum of their expected values:

E(t) = E(L) + E(max{t, To find E(max{t, , T}), we need the distribution distribution of the maximum of two independent clearly independent) is related to their distributions

Multiplying

by minus one, and then adding 1 -F,&)

, T}).

function of the maximum. The random variables (tl and T are by

one, we obtain

= 1 -F,(x)

(4)

D(x),

the relation

(6)

52 which

OLLMAN

enables us to use the tail formula h-(x”)

Combining

these results,

=

we obtain,

n jI[

for moments sn-l(l

(McGill,

1963):

- H(X)) dx.

(7)

for n = 1,

E(t) = E(h) + jm [l -F,(x) WI dx

(8)

0

which,

because of the way D(x) is defined,

may be rewritten

E(t) = E(h) + j’dx 0

+ j, [l -F,(X)]

dx

= E(&) + T + j, [1 -F&)1 dx. The random variable t - T is the reaction time, relative to the reaction signal. Its expectation is

the time to respond

E(t - T) = E(t) - E(T) = E(t) - T. Therefore,

subtracting

T from both sides of Eq. 9 yields the mean reaction E(t ~ T) = E(t2) + j; [l -F,(x)]

dx.

(9) measured

(10) time:

(11)

To recapitulate: E(t - T) is the mean delay between reaction signal and response; E(t,) is the mean time to process the reaction signal from the time processing actually begins; and 1 -F,(x) is the probability the subject is still in the refractory state x seconds after the warning signal has been presented. Thus, Eq. 11 shows that the subject’s “observable” mean reaction time will exceed his true mean reaction time E(t,) by an amount depending on the probability that he is still refractory when the reaction signal arrives. The demonstration that the foreperiod curve must be monotonic requires the additional assumption that t, does not depend on the foreperiod duration T. From the we point of view from which this model derives, this is a very natural assumption; make it, and retain it in all that follows. Differentiating E(t - T) with respect to T, we obtain, (d/dT) E(t - T) = -(1 - F,(T)). (12) Because F,(X) is a distribution

function, 0
< 1 (x > O),

CENTRAL

REFRACTORINESS

IN

SIMPLE

REACTION

53

TIME

the derivative is never positive; the mean reaction time is a monotone nonincreasing function of the foreperiod duration. If the refractory period were not subject to statistical fluctuation, but rather had some fixed value K, then F1( ) would be a step function:

(0, x < K F1(x) = )I, x > K. In this case, the mean segments:

reaction

time

curve would

(13) be given

by two straight-line

+ K - T, 0 < T < K, T > K.

(14)

That the mean period does not appear explicitly in the development of the model does not mean it is irrelevant. It, too, is related to the function 1 -- F,(X): E(t,) = j”

[l -F,(x)]

dx.

(15)

0

If the refractory period never exceeds the foreperiod interval, F,(T) = E(t,) < T; the converse is not true. Now for the variances. Adding a constant to a random variable does not its variance. Therefore, to obtain the variance of the reaction time, var(t sufficient to find var(t). The variance of the sum of two independent random variables is the sum variances, so var(t - T) = var(t) = var(t,) + var(max{t, , T}), if t, and max{t, , T} are independent. Now employing the relation var(x) and the tail formula

change

T), it is of their

(16)

We assume they are. = E(3)

- Ed,

for the first and second raw moments,

we find:

, T))

var(max(t,

= 2 j;

=

1, then

2 [j)

[l - F,(x) W41C~ldx - rs, P - F,(x) WI dx]’ -~&)Wl[~ldx

+ j, [I -~&4Wl[4

d”]

- [j:dx + j; ~1-W41 dx]’ = 2 j,

[I -F~(x)][x]

dx - 2T j1: [l -F,(x)]

dx - [j,

[l -F,(x)]

dx12. (17)

54

OLLMAN

Differentiating

the reaction

time variance

with

respect to the foreperiod

duration

T, yields &

var(t)

= ‘(‘yp))

+ &

(var(max(l,

, T})).

(18)

The random variable t, does not depend on the foreperiod duration, so its variance does not either; hence the first term on the right is zero. But differentiation of Eq. 17 with respect to T yields

$

(var(max(t,

, T))) = (-2)F,(T)

j;

[l -F,(x)]

dx

(T 2 ‘3,

(19)

which is never positive, so the reaction time variance is a monotonic nonincreasing function of the foreperiod duration. Actually, the assumption of independence between max{t, , T] and t, permits proof of a much stronger result: if the mth raw moment of the reaction time distribution exists, then the mth raw moment is a monotonic nonincreasing function of the foreperiod duration. Such a proof would note that at long foreperiods, long reaction times are less likely than at short foreperiods, and then apply the tail formula for raw moments. The reason for focusing here on means and variances will become clear later.

EFFECTS OF INVARIANCE

FOREPERIOD UNCERTAINTY OF REFRACTORY DELAYS

AND

The mixed foreperiod design is a standard simple reaction time paradigm in which the experimenter selects a particular foreperiod duration Ti(i = 1, 2,..., k) with probability qi on each trial. Although the subject may know what the set of foreperiod durations is, and the probabilities qi , he does not know which foreperiod duration to expect on a particular trial. It has been observed that the distribution of reaction times to a stimulus presented at Ti appears to be influenced by the other values of Tj , j # i, which are being used by the experimenter. That is, not only does the reaction time depend on when the stimulus is presented, but on when it might have been presented (Karlin, 1959). Although it is clear that the refractory mechanism, by itself, cannot account for the foreperiod distribution effect, it is not clear that such an effect is incompatible with the model. It might be proposed that the more uncertain the subject is about how long the foreperiod will be, the longer it takes him to process the second signal, or to execute a response. One thing must not happen: The experimenter’s choice of a foreperiod distribution

CENTRAL

REFRACTORINESS

IN

SIMPLE

REACTION

TIME

55

{(% >Ti), i = 1, L.,

K}, must not affect the time needed by the subject to “process” the warning signal. That is, Fi(x), the distribution of times to process the warning signal, should not depend on when, in the future, the reaction signal may come. This kind of invariance seems basic to the notion of a central refractory period. Examination of Eq. 12 reveals that the slope of the curve relating mean reaction time to the foreperiod duration depends only on the distribution function of refractory delays and does not in any way reflect the distribution of times to process or respond to the second signal. If the distribution of refractory delays is to be unaffected by the experimenter’s choice of a foreperiod distribution, then the slope of the curve relating mean reaction time to foreperiod duration must remain invariant when the experimenter changes foreperiod distributions. Examination of Eq. 19 reveals that the slope of the curve relating response time variance to foreperiod duration also depends only on the distribution of refractory delays. By exactly the same argument, the curve relating reaction time variance to the foreperiod duration must remain invariant when the experimenter changes foreperiod distributions. Nickerson’s (1965) data speak fairly directly to the first of these points. He conducted several different mixed foreperiod experiments, and there is considerable overlap in the ranges of the foreperiod distributions he employed. Instead of reporting means, he reports medians, but the medians give a fairly clear picture of where the means lie. To this viewer, Nickerson’s curves do not appear to differ simply by a constant, as they must if they are to have the same slope. A better summary, but not a good one, is that his curves look like translations of each other. If Nickerson’s data are interpreted in terms of the deferred processing model, then they lead quite unequivocally to the conclusion thatF,(x), the distribution of refractory delays, depends on the experimenter’s choice of a foreperiod distribution {(qi , Ti), i = 1, 2,..., k}. In view of the underlying intuition of what a refractory delay is, this would seem to be an absurd conclusion and constitute grounds for rejecting the deferred processing model and thereby rejecting the single channel operator theory as an explanation of foreperiod effects in simple reaction time.

LIMITATIONS

ON

THE

SIZE

OF

THE

FOREPERIOD

EFFECT

The prediction of the invariance under changes in the foreperiod distribution tests on an interpretation of the deferred processing model. The basic prediction that we have derived from considering its fundamental structure is that the mean and the variance of the reaction time distribution should be monotone nonincreasing functions of the foreperiod duration. Nothing has been said about how strong these dependencies may or may not be. Now we make some assumptions, again founded on an interpretation of the model,

56

OLLMAN

which lead to restrictions on how strongly the mean and the variance of the reaction time distribution may depend on the foreperiod duration. The limitation is this: neither the mean nor the variance may be more than doubled in going from long foreperiods to short ones. The basic substantive assumption that leads to the limitation is that in reacting to the second signal, the subject does the same thing(s) to the second that he does to the first, plus performing the additional operation of making a response. That making a response is a time-consuming operation is not the important part of the assumption; rather, it is that in the statistical sense, he is not faster in processing and responding to the second signal than he is in processing the first. An intuitive grasp of the origin of the limitations is most easily achieved by thinking of the refractory period as constituting a processing unit in time, and the time to process and respond to the reaction signal as constituting a processing unit plus a response unit in time. At long foreperiod durations, it is “likely” that the refractory phase will have passed and the reaction time will average out to one average processing unit plus one average response unit. Going to a short foreperiod will at worst add one processing unit (the time to finish processing the warning signal before going on to the reaction signal) to the time needed to process and respond to the reaction signal. Thus, in passing to a short foreperiod from a long one, the mean reaction time can at most be increased by 100%. With the additional proviso that statistical independence holds twice, a similar conclusion holds for the variances. At a long foreperiod, the reaction time variance will be the sum of the variance of one processing unit plus the variance of a response unit, provided the processing time and the response time are independent. At a short foreperiod, at most the variance of the time needed to process the warning signal can add to the variance of the time to process and respond to the reaction signal, provided the processing times for the first and second signals are independent. Thus, in passing to a short foreperiod from a long one, the reaction time variance can at most be increased by 1OO’A. We show the limitation for the means first. From Eq. 11, we have

E(t -

T) = E(t,) + 1: [l -F,(x)]

dx

< qt,) + j’ [l -F,(x)] dx + 1, [l -~F,(x)1dx 0 = q,) Because t, is supposed

+ E(h).

to represent

(20) how long it takes the subject

to process the

CENTRAL

warning reaction

REFRACTORINESS

IN

SIMPLE

REACTION

57

TIME

signal and t, is supposed to represent how long it takes to (i) process the signal and (ii) make a response, we are led to write

(21)

t, = t, + t, > where t, is the response time random If so, then

variable.

E(h) which

combined

with

(22)

2 E(4),

Eq. 20 implies

E(t - T) < 2.E(t,). On the other hand, Eq. 11 gives E(t, -

(23)

T) > E(t,), so in summary,

we have

E(t,) < E(t - T) < 2E(t,).

(24)

It is already known that the mean is a nonincreasing function of foreperiod duration, so the mean response time obtained at short foreperiods cannot be more than twice that of those obtained at long foreperiod durations. The argument for the variance proceeds in a similar manner. As has been noted, independence between max{t, , T} and t, yields var(t Furthermore,

operating

with

var(max(t, And

because we know

response

, T]) ‘zi

var(t,),

< var(t -

time is independent var(t,)

which

+ var(max{t,

, T}).

(25)

Eq. 17 reveals that

that var(max{t, var(t,)

The

T) = var(t,)

T small (Fi(T)) 6z 0, T large (F1( I’)) e 1.

, T}) is monotone-nonincreasing,

T) < var(t,)

of the central

= var(t,)

+ var(t,)

+ var(t,).

processing

(26) we get

(27) one, so

> var(t,),

implies var(t,)

< var(t -

T) < 2 var(t,).

(28)

Thus, we are led to the conclusion that the variance of the response time distributions obtained at short foreperiod delays cannot be more than twice that of those obtained at long foreperiod delays. Davis (1959) reports data for two individual subjects in particularly good detail. We consider only those sessions in which the subject was required to make only one response (to the second signal) per trial.

58

OLLMAN

For neither subject does the ratio of the longest mean response time to the shortest mean time approach 2, so the data conform to the (weak) requirement of Eq. 24. The monotonicity requirement also appears to be met. The variances are plotted as a function of foreperiod duration in Figs. 2a and b. According to the deferred processing model, the variance should approach var(t,) at the long foreperiod durations; and the variances at short foreperiod durations should not exceed twice that asymptotic value. The horizontal lines are located at twice the height of what seems to be a conservative (on the high side) estimate of the asymptote. In all four cases, this theoretical requirement is drastically violated by the data; indeed, in all four cases there are points which are too large by more than a factor of two. In one case, the largest observed variance is 20 times larger than the smallest. 2400

EXP 2 o---o EXP 4 SUBJECT C

0 “%

7

a

2200 2000

i

::0:

‘“z

2000

SUBJECT

D

w

I 0

100

200

FOREPERIOD

300

400

500

DURATION(msec)

0

100

200

FOREPERIOD

300

400

500

DURATlON(msec)

FIG. 2. (a) Variance of response times as a function of foreperiod delay in a random foreperiod design. The horizontal lines are placed at twice the apparent asymptotes of the curves and are theoretical upper bounds on the variances. Adapted from Davis (1959, pp. 213, 214). (b) Variance of response times as a function of foreperiod delay in a random foreperiod design. The horizontal lines are placed at twice the apparent asymptotes of the curves and are theoretical upper bounds on the variances. Adapted from Davis (1959, pp. 213, 214).

The problem is this: if the variance data are interpreted in terms of the model and the assumption of statistical independence between max{t, , T) and t, , then there is greater variability in the length of time needed to process the warning signal than

CENTRAL

REFRACTORINESS

IN

SIMPLE

REACTION

TIME

59

there is in the total time to process the reaction signal and then make a response. The variance data, the assumption of statistical independence, and the assumption of identically distributed central processing times cannot coexist. Dropping the independence assumption does not buy much: the limitation of a factor of two is raised to a factor of four, still not enough, by assuming a positive correlation between t, and max{t, , T), and it wreaks havoc with the conceptual underpinnings of the model. Dropping the assumption of identically distributed central processing times leaves only the monotonicity prediction, which considering the variability there must be in the estimates of the variances, does not look too bad. But it leaves a rather weak model. There is, of course, the test of slope invariance discussed in the previous section.

DISCUSSION

We have explored two paths along which the deferred processing model might be developed. The first attributed a certain invariance property to the refractory mechanism; this led to a constraint on the relation between foreperiod curves obtained from two or more distinct experiments. The second postulated some commonality between the refractory mechanism and the mechanism whereby the subject responds to the second signal; this led to constraints on the family of reaction time distributions obtained in a single mixed foreperiod experiment. The two sets of assumptions are largely independent; it is not unreasonable to demand that the model conform to both. If both are regarded as legitimate, then the evidence against the model appears to be quite damaging. There will not be unanimity of opinion about the appropriateness of the constraints we have offered. In designing his experiment, Nickerson was guided by the intuition that the refractory period distribution ought to have the kind of invariance property we proposed. Bertelson (1966, p. 161), however, makes a passing comment that indicates he does not regard the invariance requirement as at all reasonable: “intermittency (refractoriness) varies from moment to moment, as a function of prior knowledge about the signal to come.” It is apparent that there are quite discrepant intuitions about what refractoriness is. Hopefully, this development of the deferred processing model will help clarify the issue. The key to the second path of development is the commonality assumption represented by Eq. 21. The intuition which underlies this assumption is that in making a response to the second of two signals, the subject must do everything to the second signal that he must do to the first, plus something else-make an overt response. Examination of the literature suggests that this assumption is not regarded as unreasonable. Because the commonality assumption is the one more likely to be widely regarded as reasonable, the second course of development is the more damaging

60

OLLMAN

to the deferred processing model as an explanation reaction time. The scope of this inquiry has deliberately been deferred processing model account for the various reaction time experiments? When it is described deferred processing model might account for them. this is not so.

of foreperiod

effects in simple

kept narrow: how well does the foreperiod effects found in simple verbally, it seems as though the Closer scrutiny seems to indicate

REFERENCES P. Central intermittency twenty years later. Quarterly Journal of Experimental Psychology, 1966, 18, 153-163. DAVIS, R. The role of “attention” in the psychological refractory period. Quarterly Journal of Experimental Psychology, 1959, 11, 21 l-220. KARLIN, L. Reaction time as a function of foreperiod duration and variability. Journal of Experimental Psychology, 1959, 58, 185-191. MCGILL, W. J. Stochastic latency mechanisms. In R. D. Lute, R. R. Bush, and E. H. Galanter (Eds.), Handbook of mathematical psychology, Vol. I. New York: Wiley, 1963. P. 353. NICKERSON, R. S. Response time to the second of two successive signals as a function of absolute and relative duration of intersignal interval. Perceptual and Motor Skills, 1965, 21, 3-10. BERTELSON,