A componential analysis of pacemaker counter timing systems

Thus, subjects engaged in what could be construed as a mediative counting task as they reproduced temporal intervals Getty, 1976, but we did not label it as such, nor did we provide guid

A Componential Analysis of Pacemaker-Counter Timing Systems

J Gregor Fetterman Indiana University-Purdue University at Indianapolis Peter R Killeen

Arizona State University

Why does counting improve the accuracy of temporal judgments? Killeen and Weiss (1987) provided a formal answer to this question, and this article provides tests of their analysis In Experiments 1 and 2, subjects responded on a telegraph key as they reproduced different intervals.

Individual response rates remained constant for different target times, as predicted The variance

of reproductions was recovered from the weighted sum of the first and second moments of the component timing and counting processes Variance in timing long intervals was mainly due to counting error, as predicted In Experiments 3-5, unconstrained response rate was measured and subjects responded at (a) their unconstrained rate, (b) faster, or (c) slower When subjects responded at the preferred rate, the accuracy of time judgment improved Deviations in rates tended to increase the variability of temporal estimates Implications for pacemaker-counter models of timing are discussed.

When asked to estimate the duration of some event without

the aid of a time piece, most people mediate the estimation

by counting (e.g., "one thousand one, one thousand two,"

etc.), and balk if asked to time without counting That this

strategy is so readily adopted suggests that people intuitively

recognize that counting improves the accuracy of time

judg-ments, an intuition supported by long-standing experimental

evidence (e.g., Gilliland & Martin, 1940) The ubiquity of the

practice calls into question experimental psychologists'

at-tempts to prevent or interfere with subjects' counting

strate-gies as a means of eliciting "uncontaminated" temporal

judg-ments

Although it is evident that subdividing a long interval into

subintervals (i.e., counting) improves temporal estimates, it is

not immediately obvious why this should be so Killeen and

Weiss (1987) recently provided an analysis of the timing

process that rationalizes what intuition and data tell us In

this article, we briefly review the "optimal timing" analysis of

Killeen and Weiss, and then present the results of five

exper-iments that confirm some of the predictions that follow from

their analysis

We represent the timing process as a pacemaker-counter

system (elsewhere referred to as a clock-counter system or

pacemaker-accumulator system) In these systems, one

com-ponent (the pacemaker) generates pulses, and another (the

counter) accumulates them and signals when the number of

pulses equals or exceeds a criterion value Pacemaker-counter

systems will improve timing only if the sum of the variances

of the subintervals is less than the variance in estimating the

This research was supported in part by a National Research Service

Award Postdoctoral Fellowship (1 F32 MH09306) from the National

Institute of Mental Health to J Gregor Fetterman, and in part by

National Institute of Mental Health Grant 1 RO1 MH43233 to Peter

R Killeen.

Correspondence concerning this article should be addressed to J.

Gregor Fetterman, Department of Psychology, Indiana

University-Purdue University at Indianapolis, 1125 East 38th Street,

Indianap-olis, Indiana 46205-2810, or to Peter R Killeen, Department of

Psychology, Arizona State University, Tempe, Arizona 85287-1104.

interval as a whole and the counting process does not itself add too much variance to the estimate Traditional treatments

of such systems (e.g., Creelman, 1962; Treisman, 1963) hold that most of the variance in the timing process results from variance in the timing of the subintervals, with zero variance

in the counter This assumption may not generally be correct People make mistakes even with simple counting tasks Thus,

it seems plausible to assume nonzero error in counting, es-pecially under conditions where attention is focused on keep-ing the subintervals constant, and even more so when subjects are distracted or discouraged from counting, as is common practice in timing experiments We shall assume, therefore, that variance in timing an interval results from both variance

in timing the subintervals and variance in counting the sub-intervals We develop and test this notion and its implications more formally here

of the interval generated Then,

Mr =

and

"T f-o "N •

(1)

(2) Equation 1 states that the estimated duration of an interval equals the product of the average subinterval and the average number of subintervals (M«) Equation 2 states that the vari-ance of temporal estimates equals the weighted sum of the variances of the constituent timing and counting processes (Killeen & Weiss, 1987, p 456) Equation 2 is not a "theory"

of timing, but a standard model for random sums (see, e.g., Luce, 1986) It requires that the subintervals by independent and identically distributed, and that error in the counter is independent of error in the pacemaker For notational

Equations 1 and 2 assert that the mean and variance of temporal estimates reflect the joint contribution of the com-ponent process: the means and variances of the pacemaker and counter From Equation 2, we may infer that optimal performance might involve a trade-off between error in

count-766

ing and error in timing For example, as the error involved in

counting increases, it will be to the subjects' advantage to

count more slowly, dividing the interval into fewer and longer

subintervals This trade-off will determine the optimal

dura-tion and number of the subintervals (Killeen & Weiss, 1987,

p 456) '

To implement this analysis and identify the optimal

trade-off, we must specify the functional form that governs the

growth of variance in the component processes We assume

that the growth of these variances corresponds to the following

equations:

a n = at 2 d 2

+ a,d + QO (3)

for the variance of the subintervals, and

for the variance of their number In these equations, all a,

and ft, > 0 These fundamental error equations should not be

taken to imply that quadratic equations with all constants

greater than zero are needed to describe the constituent

vari-ances under all circumstvari-ances On the contrary, these

equa-tions were selected for their generality; any of the coefficients

may be set to zero as appropriate to the model of counting or

timing under consideration.

We assume that a suitably motivated subject will select

values of n and d that yield estimates close to the target time,

t, and that minimize a 2 in Equation 2 Note that given a

mean estimate, t, we need only specify the value of « or d,

and the other follows; the two are interdependent Our

analy-sis proceeds on the assumption that subjects attempt to

min-imize error by optimizing d, because in temporal estimations

and reproductions that is more directly and immediately

under their control than is n.

What value of d should an observer select to minimize <r, 2 7

The answer depends upon the parameter values of the

fun-damental error equations Consider first the case where there

is a fixed component of variability involved in generating the

subintervals, even when the duration of those subintervals

approaches zero («o > 0) This would be the case if there were

intrinsic variability associated with resetting the pacemaker

for the next subinterval Then, assume (8 0 = 0 (which will be

the case if the subject can report when no counts occurred,

with perfect accuracy') Then the optimal duration of the

subintervals is

• [00/02 d* < t (5)

If ft > 0, then Equation 5 will still hold in the limit as t -^

oo, (with one exception, «2 = 0 and 0i > 0, in which case d*

= kt >n ) Because I does not appear in Equation 5, in those

cases where it holds exactly the optimal rate of counting (I/

d*) should be constant and independent of the interval to be

timed We shall return to this prediction when we present the

data.

What of the case where «„ = 0 (as the durations of the

subintervals approach 0, the variance of the subintervals also

approaches zero)? This obtains if the pacemaker is a Poisson

emitter (in which case ct 2 = 0 also) Killeen and Weiss showed

that unless all other relevant parameters were zero (in which

should count as fast as possible In this case, there would be other constraints on accuracy not represented by the funda-mental error equations (i.e., as rates increase beyond some limit, accuracy of counting would be impaired so that the constants in Equation 4 would not stay constant) We subse-quently describe some evidence that this may actually happen Inserting Equation 5 into Equation 2, we obtain

where

and

<T, 2 = At + Bt + c,

B = at + 2[«o(a2 +

(6)

(7) (8)

(9)

Equation 6 states that the variance of temporal estimates will

be a quadratic function of their duration It allows us to predict the accuracy of performance on an experimental task

as a function of t Equation 6 shows that the relative

contri-butions of error in the timing and counting processes should

vary as a function of t Even if subjects do not choose the optimal value of d, Equation 6 remains appropriate, provided that the value of d remains constant (The constituents of the

coefficients A, B, and C would in that case differ from those given by Equations 2-9).

Killeen and Weiss were not the first to formally assess the contribution of counting to the timing process Getty (1976) suggested that the advantage of subdividing a long duration into subintervals derived from the fact that the sum of the variances of the subintervals was less than the variance that results when the interval is estimated without such segmen-tation Getty had subjects count silently at different rates established during a synchronization interval, and had them signal when they had made 5 or (in another condition) 10 counts He found that the variance of the times taken to reach the criterion count was less in the 10-count condition Getty represented the variance of temporal estimates as

<r,2 = naf + a., 2 , (10)

where a 2 is a constant synchronization error, and af was held to be proportional to d 2 (i.e., a/ = kd 1 } Although Getty's

analysis allows that the rate of counting affects timing, it suggests that subjects should count as fast as possible because timing error is held to decrease uniformly as a function of the number of subdivisions of the major interval This prediction, which is counterfactual, results because no provision was made for the contribution of counting error to the timing

1 Although this seems a given for typical subjects, there is an important scenario in which it may fail If the pacemaker is free running rather than synchronized with the onset of the timing process (see, e.g., Kristofierson, 1984), then there will be a synchronization error representable as jS 0 that is proportional to d It is likely that so

small an error will only be apparent in well-practiced subjects, where other sources of variability have been reduced to their minimum A similar error may occur at the end of the interval unless the interval

process If such allowances are made, we find that it is to the

subjects' advantage to count at a moderate pace to achieve a

balance of timing and counting error

The Killeen and Weiss formalism provides a framework for

theories of timing that use pacemaker-counter systems, and it

contains testable predictions about the contributions of the

constituents of such systems to the timing process We shall

attempt to test these predictions by measuring changes in the

means and variances of the component processes (the timing

and counting of subintervals dand «), in addition to standard

measures of time estimation In the literature, these values

are typically inferred from the data as free parameters for

quantitative models of the timing process For example, the

decrease in mean temporal estimates (corresponding to a

lengthening of subjective time) that accompanies the

admin-istration of stimulant drugs (e.g., Doob, 1971) is taken to

imply a decrease in the period (d) of a hypothetical pacemaker

(i.e., an increase in the rate of the clock); but the evidence is

indirect, supported by changes in parameter values recovered

from models fit to overall time estimates

In the following experiments, we attempted direct

measure-ment of the component processes Subjects performed a

tem-poral reproduction task in which they were instructed to

respond on a telegraph key throughout the reproduction The

mean and variance of the successive interresponse intervals

were taken to represent the corresponding statistics on d, and

the number and variance of responses were used as the

statistics on n Thus, subjects engaged in what could be

construed as a mediative counting task as they reproduced

temporal intervals (Getty, 1976), but we did not label it as

such, nor did we provide guidance concerning the rate of

responding during the reproduction (i.e., the rate of counting)

The obtained values of d and « were incorporated in the

equations of the optimal timing model, and predictions were

evaluated against the data

Experiment 1

Method

Subjects The participants were 16 undergraduates enrolled in an

introductory psychology course at Arizona State University All were

right-handed Participation served to partially fulfill course

require-ments.

Apparatus Two telegraph keys were located on the right side of

an Apple He computer and MED Associates interface adjacent to one

another and within easy reach for the subject The durations of events

(stimuli and responses) were recorded to the nearest millisecond.

Procedure Participants were seated in front of the computer, and

the task was explained according to a standardized set of instructions.

The experimenter left the room after ensuring that the subject

under-stood the task.

Each subject made 20 reproductions each of intervals of 4, 10, and

20 s for a total of 60 trials; order of the intervals was counterbalanced

over subjects The intervals were tone durations presented through

the computer; each auditory stimulus was presented just once, on the

first trial of a 20-trial block Subsequent reproductions were guided

by feedback (described later) provided after each estimate

Approxi-mately 3 s elapsed between the presentation of feedback and the

beginning of the next trial, which was signaled by the appearance of

a prompt on the computer monitor Subjects were instructed to begin their reproductions on the appearance of the prompt "GO." Timing

of the reproduction commenced with the subjects' first response on the left telegraph key, and ended with a single response on the right telegraph key.

Participants were instructed to tap the left telegraph key for a period equal to the duration of the tone signal No guidance was provided concerning the rate of tapping, except that subjects were told to tap at whatever rate felt "comfortable." It was emphasized, however, that tapping should continue throughout the reproduction until the subject estimated that a period of time equal to the duration

of the tone signal had elapsed, at which point a single tap on the right telegraph key signified the end of the reproduction All subjects responded throughout the reproduction task, although the rates of tapping varied widely across individuals.

Feedback was provided after each reproduction in the form of a graphic display on the computer screen; the display appeared about

300 ms after subjects signaled the end of their estimate, and remained

on for approximately 3 s The display contained a vertical line down the center of the screen and a horizontal bar, originating at the left edge of the screen The distance of the vertical line from the left edge

of the screen represented the duration of the target interval; the length

of the horizontal bar represented the duration of the subject's repro-duction relative to the duration of the target interval The position of the right edge of the bar relative to the vertical line indicated the direction (underestimate or overestimate) and magnitude of the sub-ject's error of estimation This distance was directly proportional to the percentage error of estimate Reproductions within ±5% of the target time resulted in additional feedback; the words "RIGHT ON" appeared briefly on the screen, immediately following the offset of the graphic display The experimenter ensured that all subjects under-stood the feedback display.

The data from each trial included the times between successive

responses to the telegraph key (d), the total number of responses (ri), and the duration of the reproduction (t) The reproduction data for

each target time were partitioned into blocks of five trials and standard

deviations of t were calculated for each trial block; we inspected the

standard deviations over trial blocks as a measure of stability The analyses that follow are based upon performances averaged over the last five trials at each target time, by which point there were no visible changes in variability All descriptive statistics (over trials and sub-jects) are bimeans (Killeen, 1985) Bimeans are weighted arithmetic means in which the data farthest from the mean are given reduced weights (Mosteller & Tukey, 1977).

Results and Discussion

Subjects were on the average very accurate in reproducing temporal intervals; the best fitting line through the mean estimates had a unit slope and accounted for virtually all of the data variance, and the average data were quite represen-tative of individuals The standard deviation of the estimated duration was an increasing linear function of the required

law Whereas many different functions may be drawn through these points, the linear function provides an accurate and parsimonious summary of the data The coefficient of varia-tion (standard deviavaria-tion divided by the mean) provides an index of the relative acuity of subjects' temporal judgments; for our average data, this measure was 0.106, ranging from 0.028 to 0.212 for individual subjects

Our subjects were instructed to tap as they reproduced temporal intervals, but no directions were given concerning

the rate of tapping The mean tap rate over all intervals was

3.3 Hz, but there were substantial between-subjects differences

in rate (range: 0.8 to 6 Hz) Most important, however, the

majority of subjects (13 of 16) maintained a constant rate of

tapping as they reproduced different intervals in accord with

the predictions of Killeen and Weiss Regressions of tap rate

against T for those individuals indicated that their slopes did

not deviate systematically from zero The mean of all

individ-ual slopes was 0.044, not significantly different from zero,

;(15) = 1.76, p > 05; two-tailed The three exceptions to this

generalization showed monotonic increases in rate with

in-creasing T Most subjects, therefore, behaved in a manner

consistent with predictions from the Killeen and Weiss

analy-sis, which established that the optimal rate of counting should

be independent of T There was no evidence to suggest that

subjects might go slower for long intervals, as some intuitions

(and one possible model of Killeen and Weiss) might suggest

The wide variety of rates indicates that, if the subjects did

select optimal rates of counting, those rates must have been

idiosyncratic Figure 1 expands upon this point by showing

the relation between standard deviations of t and tap rate for

individual subjects at each value of T The standard deviations

in Figure 1 have been normalized to facilitate comparisons

across the different intervals The distribution of points in

Figure 1 indicates little or no relation between tap rate and

accuracy (although there may be a slight tendency for faster

counting to result in lower accuracy) The implication is that

rate of counting either does not matter or that it is optimized

idiosyncratically by each subject We return to this issue in

subsequent experiments

Assume (as before) that the variability of subjects' time

variabil-ity in the timing of subintervals (o-/) and variabilvariabil-ity in the

then we may predict the variability of subjects' reproductions

b

«w

N

«

2

1

0

-A Z(CT4)

* ztrjio)

• Z(U20)

* A A

A M

A

• A

1

-

Response rate (taps/sec)

Figure 1 Normalized (Z-score transforms) standard deviations of

temporal reproductions as a function of response rate (taps per

second) for individual subjects (Negative values indicate less than

average variability, and positive values indicate greater than average

variability.)

by taking the appropriately weighted sum of the constituent variances Combining these variances according to standard techniques, we arrive at Equation 2 Equation 2 states that

from the first and second moments of the component

pro-cesses: the period of the pacemaker (d) and its variance, and the number of counts (ri) and its variance Equation 2 is not

a particular model of timing; it is a standard relation that must hold if we are to proceed with any models that treat timing as a pacemaker-counter system involving the random sum of independent random variables Where the component processes are independent, this includes standard models of timing, where the variance of one of the components (typically the counter) is assumed to be zero

Taking the mean intertap interval for each subject as our

measure of d, and the mean number of subintervals as our measure of n, and inserting them and the appropriate

vari-ances into Equation 2, we obtain the predictions shown in Figure 2, which shows predicted versus obtained variances of

t for each subject It appears that Equation 2 affords a good

approximation for the 20-s interval, a mediocre approxima-tion for the 10-s interval, and a poor approximaapproxima-tion of per-formance for the 4-s interval Linear regressions of predicted versus obtained variance generally confirm the visual display

of Figure 2, but suggest several qualifications The correspon-dence between predicted and obtained variances, as measured

.16 for the 10-s condition (which increases to 92 when the data for two aberrant subjects are dropped), and 92 for the 20-s condition The absolute errors of prediction for the 4-s condition are exaggerated by the logarithmic axes (introduced

to accommodate the large range of data) The slopes of the best fitting lines were close to 1.0 (0.96, 1.1, and 1.1) Figure 2 showed that variance in the sum of individual

intertap intervals (nd) could be approximated from the means

and variances of the number and duration of those intervals That approximation was best for the 20-s interval, where the benefits of count-mediated timing would be greatest, and poorest at the 4-s interval Next, we address the relative contributions of the constituents to overall variance in timing Equation 6 tells us that the contribution of variability in

and will dominate at moderate to long values of T This is

data that bear on this point The table shows the weighted variances of the component processes (averaged across

sub-jects) for each value of T, as expressed in Equation 2 Table

in the variance in number of subintervals; there was very little

change in the variance of d, because the rate of tapping (and

hence the duration and variance of the subintervals) remained

constant over T.

the total number of subintervals (n) The linear relation

portrayed in Figure 3 indicates a Weber-like result for count-ing; the standard deviation is proportional to the mean num-ber of subintervals

10

io

,5,

w

2 10'

Obtained Variance

Figure 2 Obtained variances of temporal reproductions (abscissa)

against predicted (synthesized) variances where the predictions are

based upon Equation 2 (The units are ms2 See text for details.)

Let us review the major findings of this experiment in

which we asked subjects to respond repetitively while

repro-ducing temporal intervals The rate of responding varied

considerably among subjects, but tended to remain constant

for the same subject estimating different intervals Rates of

responding and the standard deviations of temporal

repro-ductions were unrelated; insofar as subjects selected an

opti-mal rate of counting it appeared to be idiosyncratic

Variabil-ity in estimating intervals of time was predicted from a

combination of error in the duration of the subintervals (d)

and error in their number (n); total variability resulted

pri-marily from variability in counting.

Experiment 2 The two-process framework of Killeen and Weiss provided

a first approximation of subjects' performance on a temporal

reproduction task in which the reproduction was

accom-panied by repetitive tapping However, as noted previously,

there were substantial individual differences in tapping; some

subjects tapped rapidly, others slowly, and some at an

inter-mediate pace The large individual differences in tap rates

might indicate that subjects were not performing the task in

the same way A basic question is whether the tapping task,

which we took to represent the constituents of the timing

process, served to mediate temporal reproductions Subjects

were not explicitly instructed to use their tapping to regulate

their time judgments, but only to continue tapping

through-out the reproduction In some cases then, tapping may simply

Table 1

Average Weighted Variances of the Duration (d) and

Number (n) of Subintervals, as Expressed in Equation 2

4 10 20

0.0068 0.0061 0.0066

0.152 0.843 5.039

have been a "filler" activity having nothing to do with subjects' overall estimates of time Experiment 2 was motivated by this basic question about the procedure.

Method

Subjects and apparatus Four Arizona State University students

participated in the experiment Each was paid $10 for their partici-pation The apparatus was the same as described in Experiment 1.

Procedure The basic task was the same as described in

Experi-ment 1 Subjects were instructed to tap on one telegraph key during

a temporal reproduction task and to signal the end of their estimate with a single tap on a second key The intervals were tone durations; each interval was presented directly just once Subsequent reproduc-tions were guided by the graphic feedback display described in Ex-periment 1.

Subjects reproduced intervals of 2, 4, 7, 11, and 16 s Subjects made 50 estimates of the 2- and 4-s intervals, and 40 estimates of the remaining intervals The intervals were presented in a different ran-dom order to each subject The instructions were modified such that each subject was told to "use your tapping to help you judge the passage of time in whatever way seems appropriate to you." In all other respects, the instructions were unchanged from Experiment 1.

Results and Discussion

Figure 4 shows the mean accuracy of temporal reproduc-tions for each subject; the data are averaged over the last 10 trials at each interval value As in Experiment 1, subjects were very accurate in reproducing temporal intervals, although there was a slight tendency toward underestimation at longer intervals.

Figure 5 shows the variances of t for all subjects as a function of t 2 The lines through the points are best fitting lines based upon Equation 6 with B = 0 (a, 2 = At 2 + C) For

only 1 subject (Subject 4) was the goodness of fit to the data slightly improved by the addition of a linear component The

numbers in each panel are the coefficients of variation (CV

= ajt) that provide an index of the relative acuity of subjects'

temporal judgments akin to a Weber fraction The values are impressively low, and considerably less than the average value for Experiment 1 (.106).

Figure 6 shows the average values of d (intertap intervals) for each subject as a function of t All subjects maintained a fairly constant rate of tapping (l/d) as they reproduced

dif-ferent intervals, as predicted by Killeen and Weiss and ob-served in Experiment 1 As before, there were individual differences in tap rate.

Figure 7 shows obtained versus synthesized variances of /, where the synthesized variances are based on a modified version of Equation 2 Inspection of the individual data indicated that the mean of the final intertap interval (when the subject switched from the left to the right key) was, in some cases, quite discrepant from intertap intervals produced

on the left key Accordingly, the data were fit to a modified version of Equation 2:

where n represents the total number of subintervals (including

the final subinterval), (^represents the mean of all subintervals,

6

4

2

-0

ff = 0.118n-0.21 n

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

n

Figure 3 Standard deviations of n (number of taps) as a function

of number of taps emitted.

<jd represents the variance of subintervals on the left key, and

incorporates the time between the last response on the left

key and the terminal response on the right key The

corre-spondence between the parameter-free predicted variances

and obtained variances for individual performances is

reason-able The deviations from predicted performance could have

resulted from drifts in response rates over trials or sequential

dependencies between successive intertap intervals

of n was a linear function of the number of subintervals In

Experiment 2, there was little or no variability in « Subjects

made about the same number of taps for each reproduction

of a given interval, a result that indicates a strategy of explicit

counting For the case where variability in n is zero, Equation

2 reduces to

<r,2 = n<r d (12) The data of Subject 4 (for whom within-interval variance in

20-1

0

(0

t = 0.92T + 0.45

10

T(sec)

20

Figure 4 Mean temporal reproductions as a function of interval

value ( T ) for each subject in Experiment 2.

n over the last 10 trials was zero) in Figure 7 indicate that the

fit of the model equation to this special case was quite good

To what extent were subjects' reproductions mediated by the tapping task? Experiment 2 was motivated by this basic question about the procedure Conceivably, subjects might have performed two tasks that were independent of one another except that the end of the interval caused tapping to stop We believe this to be an unlikely scenario for two reasons First, when debriefed after the experiment, all sub-jects reported that they "counted" their taps as a means of estimating elapsed time, and that they "adjusted" their esti-mates through changes in the duration and number of taps made

Second, suppose t is timed separately and independently of the mechanism that controls tapping, except that the end of t

stops the tapping This scenario would result in a positive

correlation between n and t; more important, however, there should be little or no correlation between d and t because under the hypothesis the estimate of t is held to be indepen-dent of the speed of tapping ( l / d ) If, on the other hand, t is mediated by the duration (d) and number (n) of the subinter-vals, we would expect a positive correlation between d and t.

Table 2 presents the relevant correlational data for all subjects

at all values of T The correlations are based on performance

for the last 10 trials at each interval The pattern of correla-tions is generally supportive of the conclusion that subjects'

overall estimates (t) were mediated by the duration and

number of the subintervals; it provides little or no support for the independent timer scenario Significant positive

correla-tions between d and t were observed in 60% of the cases, as

predicted by the mediation hypothesis but not the

indepen-dent hypothesis; n and t were significantly positively correlated

in only 15% of the cases, further undermining the independ-ent process hypothesis

The results of Experiment 2 corroborate those of Experi-ment 1: Subjects maintained a constant rate of tapping as they reproduced different intervals; the rate of tapping varied among individuals; and here the parameter-free synthetic equation (Equation 2), augmented by the addition of the variance for switching to the second key, more accurately represented the individual obtained variances of subjects' temporal reproductions Counting error was lower in Experi-ment 2, perhaps because all subjects adopted explicit counting strategies; such strategies probably account for the impres-sively low Weber fractions

Experiment 3 Tapping rate varied across subjects in Experiments 1 and

2 We may infer either that subjects did not strive to minimize

error by optimizing d, or that each subject selected what was for him or her the optimal value of d, with large intersubject

differences in the optimum It would seem easy to decide between these alternatives by merely evaluating Equation 5 for each subject, and seeing whether their preferred value of

d was well predicted by that equation However, remember

that the model predicted that rate of tapping should be constant over intervals; the general success of that prediction within individuals precluded a test of Equation 5 This

200 300

Figure 5 Variances of temporal reproductions for individual

sub-jects of Experiment 2 as a function of average estimate squared (?).

(The lines through the data points are best fitting lines based upon

Equation 6 with B = 0.)

22 occurs because Equation 5 requires that we know the

values of the parameters, a 0 and a 2 , of the fundamental error

equation for the duration of the subintervals; but because

there was little within-subject variability in d and thus a highly

restricted range of the predictor, those could not be

estab-lished To establish such a function, in Experiments 3,4, and

5 we forced subjects to count at different rates Furthermore,

by forcing subjects off their preferred rate of counting, we

could potentially assess the extent to which individuals

se-lected optimal rates If subjects chose an optimal rate, then

forced deviations from it should increase the variability of

temporal judgments If, on the other hand, subjects' rates of

counting were arbitrarily selected, forced deviations might be

expected to produce little systematic change in the accuracy

of temporal judgments, increasing it in some cases and

de-creasing it in others.

1200-

1000-^

800-u

S

600-TS

400-

200-0

10

t(sec)

20

Figure 6 Mean intertap intervals (d) for each subject as a function

of mean reproduction (t).

Method

Subjects and apparatus The subjects were 12 undergraduates

enrolled in an introductory psychology course at Arizona State Uni-versity Participation served to partially fulfill course requirements The apparatus was the same as described in Experiment 1.

Procedure The basic task was unchanged from Experiment 1.

Subjects were instructed to tap on one telegraph key during a temporal reproduction task The intervals were tone durations; each interval was presented directly just once Subsequent reproductions were guided by the graphic feedback display described in Experiment 1 The experiment included four phases conducted in two 1-hr ses-sions, with each subject participating in all phases Phases 1 and 2 were scheduled during the first session, and Phases 3 and 4 during the second session, approximately one week later During Phase 1 (free count), subjects made 20 reproductions of a 6-s interval under the instructions that they were to tap during the reproduction at whatever rate felt "comfortable," as in Experiment 1 As before, the experimenter emphasized that the subject should continue tapping throughout the reproduction, and all subjects complied with the instructions During Phase 2 (forced-count control), subjects made

15 reproductions of the intervals 3, 6,12, and 24 s; presentation order was counterbalanced over subjects Subjects were instructed to repro-duce the target time, but also to tap a specified number of times during the course of the reproduction The required number of taps depended upon the subject's rate of counting during Phase 1, such that insofar as the subjects successfully approximated the required number and target time, the rate of tapping would equal that of Phase

1 The key difference in the two conditions was that, in Phase 2, subjects were given explicit instructions concerning the number of taps to make during the reproduction Feedback was provided only with respect to the total duration of the reproduction, not with respect

to the number of taps made.

During Phases 3 and 4, subjects were required to tap at one half (slow condition) or twice (fast condition) their preferred rate As in Phase 2, subjects made 15 reproductions of the intervals 3, 6, 12, and

24 s in each phase; order of the intervals was counterbalanced over subjects Tapping rate was manipulated through changes in the re-quired number of taps for the different intervals and conditions (fast

vs slow) As in Phase 2, feedback was provided for the temporal reproduction, but not for the number of responses made The se-quencing of fast and slow conditions was counterbalanced over subjects.

Data from the last five trials for each target time in Phases 2, 3, and 4 were used in the analyses that follow These data included the

intertap intervals (d), the total number of responses (n), and the duration of the reproduction (t) All averages (over trials and subjects)

are bhneans.

Results and Discussion

The top panel of Figure 8 shows mean accuracy of temporal reproductions over TTor the control, slow, and fast conditions (Phases 2, 3, and 4) As in Experiments 1 and 2, subjects were very accurate in reproducing temporal intervals, and there were no differences in the central tendency of temporal esti-mations for the different experimental conditions.

The bottom panel of Figure 8 shows standard deviations of / for the control condition, where subjects were asked to count

at their preferred rate, and the comparable data set from Experiment 1 Standard deviations increased as a function of

T, as in Experiment 1, but the rate of change was not nearly

as great as in the first experiment The slope of the best fitting

5 10 6 :

e

a

(0

«•* m3

9

(A

10°

10° lO'1 10" 106 10' Obtained Variance

Figure 7 Obtained variances of temporal reproductions (abscissa)

against predicted (synthesized) variances where the predictions are

based upon Equation 11 (The units are ms2 See text for details.)

line through the control data from Experiment 3 (equal to

the coefficient of variation) is 0.03, whereas the comparable

slope from Experiment 1 was 0.106 This difference was

statistically reliable, t(26) = 2.81, p < 01, two-tailed.

We compared subjects' rates of counting against the

re-quired rate as a check on the effectiveness of our

manipula-tion Subjects did not systematically underestimate or

over-estimate their target rate The median absolute percent

devia-tions from the target rates were 14%, 15%, and 12% for the

fast, slow, and control conditions, respectively

Figure 9 shows the variance of t for the different counting

conditions (control, fast, slow) at each rvalue Figure 9, based

on data averaged over all subjects, suggests that pushing

subjects off their preferred rate of counting increased the

variability (though it did not influence central tendency; see

Figure 8) of temporal reproductions, especially in the fast

condition Variability was higher under "nonpreferred"

con-ditions in 36 of 48 comparisons This result is compatible

with the notion that preferred rates were optimal for

individ-ual subjects The pattern portrayed in Figure 9 was not

supported by statistical analysis, however Neither the main

effect of counting condition, F(2, 22) = 1.48), p > 05, nor

the interaction of counting condition and interval value (F <

1) were significant The main effect of interval value was

highly significant, F(3, 33) = 17.31, p < 01, however

Ex-amination of earlier trial blocks led to the same statistical

conclusion

Figure 10 shows changes in the underlying components (n and d) for the different conditions of Experiment 3 The manipulation succeeded in forcing differences in d, thus giving

us a chance to evaluate the parameters in the fundamental error equations The top panel of Figure 10 shows the change

in standard deviation of the intertap interval as a function of

d Variability was well predicted by d (r 2 = 98) The intercept

a0 was not significantly different from zero

Equation 5 (and Killeen & Weiss, 1987, Figure 1) tells us that when constant error (oo) in timing the subintervals is

zero, the optimal value of d is as close to zero as possible;

subjects should normally count at extremely fast rates How-ever, this was not the case, as subjects were able to double their response rates upon command, and accuracy did not get better: If anything, it got worse Although this is what we had expected, the grounds for our expectation are undermined by

the approximately zero value for a 0 , in which circumstances accuracy should uniformly improve as d gets smaller This

mystery is resolved by inspection of the bottom panel of Figure 10, which shows changes in the variability of the

number of subintervals as a function of n The variability of

counting in the fast and slow (i.e., experimental) conditions increased at a much faster rate than it did in the control condition

This analysis shows that when the rate of the pacemaker is extrinsically manipulated, error in timing increases in part because error in counting is affected by the rate at which counting occurs This seems eminently reasonable, but it creates problems for all pacemaker-counter models The basic premises of the Killeen and Weiss model (and a fortiori of all other such pacemaker-counter models) are undermined be-cause the two component processes are not independent

In sum, the manipulations of Experiment 3 improved accuracy by comparison with Experiment 1 (Figure 8) and appeared to influence variability, but those appearances were not statistically reliable (Figure 9) We are not able to con-clude, therefore, that subjects selected optimal rates of re-sponding in Experiments 1 or 2 or in Phase 1 of Experiment

3 The growth of counting error depended not only upon the

value of n but also upon the rate of counting This result is

incompatible with the assumptions of all pacemaker-counter models, wherein the variance in the counting process is held

to be independent of d Experiment 4 serves as a check on

the generality of these results, and afforded the opportunity for a detailed analysis of the data of individual subjects

Experiment 4

Table 2

Pearson Correlations Between d and t

T(s)

2

4

7

11

16

Subject 1 99 1.0 99 -.87 81

Subject 2 25 99 47 88 15

Subject 3 59 19 1.0 -.61 47

Subject 4 99 1.0 99 1.0 99

Note Correlations are based on data from the last 10 trials at each

interval The critical value o f r ( p < 05) is 632.

Method

Subjects and apparatus The 4 subjects had served previously in

Experiment 2 Each was paid $10 for their participation The appa-ratus was the same as described in Experiment 1.

Procedure Subjects engaged in the tapping task while

reproduc-ing different durations As in Experiment 3, the rate of tappreproduc-ing was manipulated through changes in the required number of taps Subjects were "yoked" to their unconstrained rates of Experiment 2 such that the stipulated rate of tapping equaled either that of the unconstrained condition (control), twice the unconstrained rate (fast), or one half

o

9

10

t = 1.0 IT +0.29

10 20

T (sec)

u

•

i»

2

01 = 0.032 t + 0.11

0 10 20 SO

t (sec)

Figure 8 Means (top panel) and standard deviations (bottom panel)

of temporal reproductions as a function of target interval ( T ) for

Experiment 3 (Mean estimates [top panel] are shown for all

condi-tions of Experiment 3 Standard deviacondi-tions are shown for the control

condition of Experiment 3 [filled symbols] and for Experiment 1

[unfilled symbols] The lines through the points are regressions with

parameters as noted.)

the unconstrained rate (slow) Subjects made 40 reproductions of

4-and 11-s intervals under each tap rate condition (control, fast, 4-and

slow), and the conditions were presented in a different random order

to each subject Feedback was provided after each estimate, but only

with respect to the duration of the reproduction The instructions

were as described in Experiment 3, except that the subjects were told

Figure 9 Variances of temporal reproductions as a function of

interval value for the different conditions of Experiment 3.

to "use your tapping to help you judge the passage of time in whatever way seems appropriate to you."

Results and Discusson

The mean accuracy of subjects' estimates was not influ-enced by the experimental manipulations; in all cases subjects were quite accurate in reproducing the target durations, as in Experiment 3.

Figure 11 shows the standard deviations of subjects' repro-ductions for the different intervals and counting conditions The different counting conditions are represented as a

func-tion of the average intertap interval (d), which varied

accord-ing to the individualized required rate of tappaccord-ing It is clear that variations in the rate of tapping affected the variability

of temporal judgments The pattern in this figure suggests an

optimum value of d(in the sense of minimizing variability of

/) in the range of 300 to 400 ms Values above or below this range (but especially above) increased the variability of timing.

As in Experiment 3, changes in the pacing requirement influenced the variability of subjects' temporal reproductions Figure 11 suggests, however, that the optimal rate of tapping was not idiosyncratic as suggested by Experiment 3 Rather the results of Experiment 4 indicate an absolute optimum for

d in the vicinity of 1/3 s.

Experiment 5

In Experiments 3 and 4, we manipulated d through changes

in the required number of taps This approach forced subjects

to attend to the number of taps made, and perhaps facilitated timing for subjects not already predisposed to use counting strategies Almost certainly it served to minimize observed error in counting In Experiment 5, we used a different

technique to force changes in d Subjects were given a

"sam-ple" target rate and asked to match the tempo of their tapping

to the sample The target rate was manipulated over condi-tions, and the effects of variations in the required rate were assessed with respect to the accuracy and variability of sub-jects' reproductions.

Method

Subjects and apparatus The 4 subjects had served previously in

Experiments 2 and 4 They were paid $5 for their participation The apparatus was as described in Experiment 1.

Procedure Subjects made 30 reproductions of an ll-s target

interval by responding on a telegraph key for a period that matched the duration of a tone signal; the signal was presented just once; subsequent reproductions were guided by the feedback described in Experiment 1 Subjects' rate of tapping was manipulated over con-ditions through instructions to match a "sample" rate Before each block of estimates, subjects heard a "sample" consisting of a series of periodic 1000-Hz tones; each tone lasted 15 ms, and the tone series lasted 10 s Under different conditions, the intertone intervals were

150 ms (fast), 350 ms (medium), or 1200 ms (stow); subjects were instructed to practice the sample rate by synchronizing their tapping with the tone and to maintain the sample tempo while reproducing the target interval A reminder of the sample rate was presented after the fifth trial of each block of estimates; the remaining reproductions

0.20 n

0.13-u

in

o.io-

0.05-0.00

103

-J d =0.24d- 0.01

d (sec)

0.6 0.8

•X a n = 0.065n + 0.682

ff n = 0.016n + 0.55

0 50 100 ISO

n

Figure 10 Standard deviations of the duration of intertap intervals

(d; top panel) and number of taps (n; bottom panel) as a function of

d and n, respectively (For the bottom panel, the filled symbols

[experimental] represent performance for the fast and slow

condi-tions, and the unfilled symbols represent performance for the control

condition.)

1000 2000

d (msec)

3000

Figure 11 Standard deviations of temporal reproductions for the

different conditions of Experiment 4 (Data are for each subject as a

function of d, with target interval as the parameter.)

were made without further exposure to the sample rate The sample rates were presented in a different random order to each subject As

in Experiments 3 and 4, subjects were told to "use your tapping to help you judge the passage of time in whatever way seems appropriate

to you."

Results and Discussion

Manipulation of the rate of tapping did not affect the accuracy of subjects' mean temporal reproductions, but the manipulation did produce changes in variability Figure 12 shows the standard deviations of reproductions for individuals for each tap rate condition The patterns are different for different individuals Variability for Subject 1 decreased mon-otonically with increases in the rate of tapping (i.e., decreased

in d), whereas for Subject 4 variability increased

monotoni-cally with increases in the rate of tapping For Subjects 2 and

3 variability was lowest for the medium rate of tapping and higher for faster or slower rates; this result is similar to the pattern portrayed in Figure 1 1 of Experiment 4

The line through the data points shows the predicted

vari-ances of t as a function of d, where the predictions are based

on Equation 2 rewritten with respect to d (Killeen & Weiss,

+ («2 + + (a,t + /3 2 t 2 ) + (13) Estimates of the relevant coefficients of the fundamental error equations were obtained as bimeans of individual subject parameters in regressions corresponding to Equation 3 (data

from this experiment: a 0 = 700 ms, a, = 0, a2 = 0.0025) and Equation 4 (data from Experiment 2: |80 = 0.1, 0, = 0, ft =

0.0005) The predicted variances remain constant over a

substantial range of d with upturns at extremely small (fast

rate of counting) and large (slow rate of counting) values of

d The upturn for large values of d is due primarily to /30, which we interpret as variability resulting from truncation or

rounding error Over a substantial range (250 < d < 500 ms),

changes in the rate of counting have little impact on the variability of timing; this could account for the weak effects

of our manipulations that forced subjects to deviate from their preferred rate of counting Note that these predictions do not take into account the increased variance resulting from coun-ter error at high rates of counting

Figure 1 3 shows the changes in the standard deviation of d

as a function of d for each subject As in Experiment 3, <rrf appears to grow as a linear function of d, and the rate of

change was similar for 3 of the 4 subjects

General Discussion

We asked subjects to tap repeatedly on a telegraph key as they reproduced temporal intervals, and took the duration and number of intertap intervals as measures of the compo-nents of a pacemaker-counter system In Experiments 1 and

2, most subjects did not vary their rate of responding as they reproduced different durations; we interpreted this result as supportive of a fundamental prediction of the optimal timing analysis that, to minimize variability in timing, the rate of counting should remain constant and independent of the