- Email: [email protected]
Sequential estimations of time
Acta Psychologica 0 North-Holland
40 (1976) 475-487 Publishing Company
SEQUENTIAL Pieter
ESTIMATIONS
OF TIME
A. VROON*
Psychological Institute,
Received
January
UtCw-sity of Lciden,
The Netherlands
1976
Sequential estimations of intervals stored and reproduced by either counting or not counting are compared in experiments with single and repeated presentation of the standard interval. In a non-counting condition the estimates show a negative correlation with the last occurring trial throughout the series after one presentation of the standard. Consequently, in an ongoing process the capacity for (the storage and) the recall of time appears to be limited to the most recent event corresponding with the previous estimate. Correlations are lacking when the standard is presented after every estimate because the most recent information is an external constant. The correlations with the last trial recur when relatively long intervals are used, stored on the basis of various sensory cues. When the difference between the number of these cues is minimized, the correlation disappears. Counting never shows sequential correlations since it favors the coding and storage of time as a number of subjective units. The temporal equivalent of that number is constructed from trial to trial, a representation of the complete interval is not available and the subject does not remember his counting rate.
Research in time psychology is mainly concerned with the experience of duration. The problem is defined by the phenomenon that apparent duration differs from physical time by accelerations and decelerations. Usually, two viewpoints are distinguished: a psychophysiological and a cognitive approach. The first theory says that clock time is reflected in cyclic phenomena such as EEG, heart rate, etc. There is an internal time-keeper, driven by arousal level, metabolism, sensory input and the subject’s cognitive activities. Moreover, subjective time is not infinitely divisible, but consists of quanta, about 100 msec long (Augenstine 1955; Stroud 1955). The research focuses on the nature of the time-keeper and its speed fluctuations as functions of * Requests for reprints should 169, Leiden, The Netherlands.
be sent to P. A. Vroon,
Psychological
Institute,
Rijnsburgerweg
416
P. A. Vroon/Sequetltial estimations of time
various conditions (see Ornstein 1969; Treisman 1963; Vroon and Van Boxtel 1972; Vroon 1974). The cognitive approach says that subjective time is a mental construction, mainly dependent on cognitive operations. This theory is in so far more elegant, that it also explains the effects of higher and lower body temperatures, drugs, and so on (see Hicks et al. 1976; Michon 1965, 1972, 1975, 1976; Ornstein 1969; Vroon 1970, 1976). These theories are not completely incompatible, but the latter has some important methodological consequences. If time is in the first place a dimension of experience, the apparent duration of a period sl~ould not be compared with the clock, but with the subjective duration of another interval. This method was used by Ornstein (1969) who demonstrated that subjective time is closely related to the coding, transmission, storage and retrieval of information. The present study is concerned with the estimation of short intervals, i.e., of a few seconds. It is likely that mechanisms such
Methods
of measuring
subjective
time
An interval map be presented either verbally or not and be estimated in the same ways. A non-verbal presentation consists of a stimulus (tone, pause between clicks, etc.), a non-verbal estimation means that the S reproduces the interval without counting. A verbal presentation means that B mentions a number of time units and in a verbal estimation the S either reports time units or reproduces the interval by counting. Thus, there are four types of stimulus and response combinations. Non-verbal estimation requires that the S has some representation of the complete interval. When the interval is estimated verbally. he mentions time units or builds up the estimate by adding temporal equivalents of time units. In sequential production or reproduction the S is instructed to estimate the standard repeatedly. When a non-verba presentation is combined with a verbal estimation, the results can be interpreted in different ways. (1) Verbalization. Suppose that E presents an interval of 10 set, instructs S to count at a rate of one per subjective second and that S verbally reports a duration of 12 sec. In this case the subjective second was shorter than the objective second or the ‘psychological clock’ ran too fast. The internal and the external clocks show a speed difference. (3) Reproduction. A
P. A. VroonfSequential
estimations
of time
417
10 set standard is presented and E instructs S to count subjective seconds during presentation as well as estimation. Suppose that the S’s estimate is 12 set again. Since it may be assumed that S counted from 1 to n during presentation as well as estimation, a speed fluctuation of the internal clock appears to have occurred in that the time-keeper slowed down between presentation and estimation. Studies by Clausen (1950) and Fraisse (1963) demonstrate that the data of several methods may be inversely related or even incomparable. They report that reproduction without counting (non-verbal estimation) does not correlate with verbalization. In the case of verbalization time is splitted up into units and a too fast or too slow time-keeper leads to systematic over- and underestimations. When the interval is reproduced without counting, the absolute speed of the internal clock is irrelevant. For the same reason there is also no correlation between reproduction without and with counting (Fraisse 1963). He also reports a correlation between verbalization and reproduction with counting, which appears to be likely. It is clear that presence as well as absence of correlations is important in so far that they may reflect fundamental differences between the processes as measured by the various methods.
Hypotheses Two groups of conditions seem to be significant in terms of estimation process variables. (A) Non-verbal presentation - non-verbal estimation serial, non-verbal. The standard is presented once (l)P,t - production, and S is instructed to reproduce it repeatedly. serial, non-verbal. Each standard presentation is (2) Rst - reproduction, followed by one estimation. (B) Non-verbal presentation ~ verbal estimation serial, verbal. The standard is presented once, S is (1) P,, - production, instructed to count subjective time units during presentation and subsequent estimations. (2) R,, - reproduction, serial, verbal. Similar to Rst, but with counting. Within the framework of magnitude estimation models it could be argued that that there are also two stages in time estimation: an input stage in which S experiences and stores the duration of the standard interval, and an output stage in which the interval is estimated. Different effects for each condition are expected. P,t. When the standard is presented, there is an input transformation from the physical magnitude of the standard on the physical continuum to a subjective magnitude on a subjective continuum. When the interval is reproduced the subjective input magnitude is compared with the subjective output magnitude, both being distances on a continuum. Storage and reproduction of time occurs on the basis of some representation of the complete interval. In such an ongoing process a particular estimate may be a copy of the standard, or S forgets the standard and bases himself on (a) previous estimate(s). In other words: the input for a particular estimate has different possible degrees of recency.
478
P. A. Vroon/Sequentiai estimations of time
Rst. Each reproduction is preceded by a constant standard. When, for example, the most recent information defines the input only, no correlations with previous estimates will be observed. Michon (1967) demonstrated that the variability of non-verbal estimations increases considerably as a function of the standard duration. On the basis of these findings one could hypothesize that the ability to represent/store time decreases with increasing standard durations. Consequently, (the) previous estimate(s) may now also define a particular trial. P,,. When S is instructed to count during presentation and estimation, it is likely that the interval is not stored as a distance on some subjective continuum, but as a number. Since misreading of that number is not probable, there will be no errors in the estimation phase. The subjective temporal equivalent of the number has to be reconstructed from trial to trial. If the counting rate varies randomly, there will be no correlations with previous estimates. If such correlations are observed, S must remember the length of his previous subjective time unit, be able to detect a difference and to correct for it. In that case Psv is a mixture of storage of temporal plus numerical units. may present information about such a process. For example, R sv. This condition Psv shows that the last occurring event (trial .Y-- 1) plays a role or even trial (v-2), (x-3), etc. There are several possibilities. (I) When there is only a correlation with (x-l ), there will be no correlation in R,, since event (x-l) is an external standard. (2) If P,, correlates with (~~~1). (x-2), (.x-3), there is evidence for the existence of some internal standard as a (weighted) average of several previous but not with (x- 3). (3) If estimates. Rsv may now correlate with (x-1) and (x-2), is remembered u/zless new Psv shows correlations and Rsv does not, an estimate temporal information is presented. In other words: new information erases memory for previously estimated intervals.
Apparatus The apparatus consisted of a tone generator (sine wave) with a range between 50 and 20,000 Hz, a selector to present intervals between 0.25 and 15 set and an electronic clock with an accuracy of 10 msec. In all experiments the standard consisted of an interval filled with or limited by a 1000 Hz, 73 dB tone, presented binaurally by headphones. Reproduction took place by continuously pressing a microswitch (filled intervals) or by shortly pressing the switch twice (unfilled intervals).
P. A. VroonfSequential
479
estimations of time
-0.40
t I 30
10
20
10
20
30
40
50
10
20
30
40
50
40
50
-0.40 t
I
Q20
0
-a20
-a40
Fig. 1. Autocorrelation functions of three Pst series, each consisting of 200 trials, t = correlation lag (trials), 1 < t < 50. a: subject A, series 2, standard 1 sec. b: subject A, series 2, standard 2 sec. c: subject C, series 2, standard 3 sec.
P. A. Vroon/Sequential estimations of time
480
Sequential effects in P,, Experiment
I
Method Four Ss (A, B, C, D) were prescntecl an interval (tone) of 1, 2 or 3 sec. They were instructed to refrain from counting and to produce the standard interval repeatedly. Every S estimated two series of 200 trials each with a standard duration of 1 set (A), 2 set (B) or 3 SW (C, D). They were not informed about the number of trials required. Results The series of estimates were analyzed with autocorrelation (see Vroon 1974). In autocorrelating the sign and the strength of correlations carries information about the direction and the degree to which an estimate is related to previous events. In order to obtain stationary functions, the curves of the raw data were filtered by fitting polynomials. According to the least squares method described by Lewis (1963) the best fitting polynomial, accompanied by a significant reduction of residual variance, was chosen. Fig. 1 shows the autocorrelation functions of the estimates. Throughout the series, each interval correlates negatively with the last occuring trial which indicates that S tends to produce longer and shorter estimates more or less in alternation. Table 1 presents the first-order autocorrelations (lag t = 1) of all series. The functions of fig. 1 should be considered with caution. The autocorrelations Rt oscillate around the axis Rt = 0 indicating that the trend elimination was sufficient. However, the fact that significant correlations are still observed over many previous trials does not mean that S’s ‘memory space’ for intervals covers a long period of time. The corrclograms indicate the degree estimate (x) is influenced by estimate (X -I), but higher order effects such as as the dependence between (s) and (l-2), (s-3), etc. can only be assessed when the Table 1 6xperirncnt
1, Pearson
autocorrelations
Subject A A B B c C D D
trial (RI).
Scrims
Standard
1
1
2 1 2 1 2
1 2 2 3 3 3 3
1 2 _.~
* p < 0.01, two-tailed.
over the lavt occurring
~_
~.~ ~~ ._~
(xc)
.~~~~__~_~~.-.~~~ .- ~..~
RI m-0.28* -0.35* -0.43* -0.45* -0.36* -0.41* - 0.34* -0.33*
481
P. A. VroonfSequential estimations of time Table 2
Experiment 1, chi-square values as a measure of non-randomness of the estimates. Standard (SK)
in the trial-to-trial fluctuations
Series
x2 1st order
x2 2nd order
x2 3rdOTdeT
1
2
7.i54* 11.24*
1.03 0.93
0.43 0.22
1 2 1 2 1 2
19.84* 18.67* 17.51* 21.03* 16.93* 14.18*
0.39 0.43 5.17 1.15 7&l* 3.22
0.32 1.02 2.29 1.01 3.42 1.33
* p < 0.01. first-order effect is taken into account since the correlations are not independent. If the estimates are described as positive and negative trend deviations without the size of the deviation being taken into account, a binary sequence is obtained. Sequential interrelations may be assessed by studying the randomness of this series. There are three possibilities: (1) the series is random (which means that there are no sequential effects), or the correlations originate from (2) too many alternations (short-long estimates and vice versa), or (3) perseverations (shorter-shorter and longer-longer estimations). The first-order (digrams), second-order (trigrams), and third-order (tetragrams) effects are shown in table 2. The series are not random but the effect is limited to the first-order effect. It is highly significant and due to alternation, which was also clear from the negative autocorrelations over one trial in fig. 1 and table 1. Since there is no higher order non-randomness, autocorrelation is effective for studying the interdependence of time estimates. An advantage over a nonrandomness analysis as shown in table 2 is that a smaller number of measurements is sufficient. If mean autocorrelation functions over Ss are calculated, most of the fluctuations observed in fig. 1 will disappear since it is likely that there are phase shifts in the individual functions. Consequently, it is expected that I& = 0 except for lag t = 1 which will show a negative correlation.
Experiment
2
Method The & (24 freshmen) each in a counterbalanced
estimated a standard (tone) of 0.5, order under the condition Pst.
1 or 2 set 30 times
Results The
mean autocorrelation functions about the individual phase
hypothesis
are shown differences
in fig. 2. It appears that the (see also table 2) is correct:
482
P. A. Vroon/Sequential estimations of time
P,t. 0.5 *cc 0.20
f Rt t -
0.
-
-
/
.
-0.20 -
-0.40 t 5
10
Pst.lSM. 0.20 If%
--
O-
-0.20
-
-0.40 -L 5
10
P st,2rcc. 0.20 -
t Rt
0
I---
-,
-0.m -
-0.40 t 5
10
Fig. 2. Autocorrelation functions of three Pst series with different standard durations, 30 trials, means over 24 Ss, 1 G f G 10. Rt = 0 when t a 2. Thoughout the series there is a significant and negative correlation with the last occurring trial, also observed in experiment 1 (see fig. 1, tables 1, 2). In the case of P,t with standard durations up to 3 set the estimates are independent of each other, except for the last trial. There are two prelimmary conclusions.
P. A. VroonfSequential estimations of time
483
(1) The estimation process under Pst is based on a memory for complete intervals since there is a correlation with the previous trial. (2) It appears that S forgets the standard and bases himself on the most recent event. Thus, the input for a particular estimate has maximum recency. Since only one event is remembered, estimates with an alternating tendency are likely to occur. The S corrects an estimate in one step (see also Michon 1967, who observed the same phenomenon in synchronization experiments). The conclusions can be validated by studying R,t estimates. In this case the last term of the series is not an estimate, but a constant standard. If the memory for the previous estimate is erased by new temporal information, correlations with previous trials will be lacking.
,*f?$ ~
-.20
-
-.LO
-
__iy/
,
t 10
5
7%
.20
0
-. 20
-.bO
Pst
3scc.
7=----T
Fig. 3. Autocorrelation Ss,l
functions of Psl and Rst series, 30 trials, standard 3 set, means over 60
484
Sequential Experirnen
P. A. Vroor~/Sequetztial estimutiom
effects
of time
in R,t
t 3
Method Sixty Ss were split up into two equal groups, each estimating (tone) 30 times under the conditions P,t and R,t in a counterbalanced
a 3 set interval order.
Results Fig. 3 shows the mean autocorrelation functions. A difference between Pst and Rst is clearly observed: in Pst there is a negative and significant correlation with the last occurring trial (p < 0.01, one-tailed), whereas in R,t the estimates are independent. When a short interval is estimated without counting, the most recent information defines the subsequent trial.
Sequential
effects
in P,, and R,,:
Cognitive
transformation
It becomes apparent that non-verbal conditions favor the integral storage of an interval. Counting leads to fragmentizing time on the basis of a subjective time unit. Since this number will not be forgotten, its temporal equivalent has to be reconstructed from trial to trial. If S does not remember the length of his time unit (counting rate), there will be no interdependence between estimates. If he does, P,, involves double storage, i.e., of a number and a time unit. l
4
Method Sixty estimated group
Ss were divided into two equal groups. A 6 XC standard (tone) was under the following conditions. I: (a) P,t, 30 trials (b) P,,, 30 trials group 2: (a) R,t, 30 trials (b) Rsv, 30 trials The order of the conditions per group was fixed since Ss tend to count when longer intervals are to be estimated. If a counting instruction is given, it is difficult to refrain from it in a second series. There were 15 min pauses between the tasks.
P. A. VroonfSequential estimations of time
485
Results The autocorrelograms of both P,v and Rsv are stationary (compare fig. 3, Rst condition). Since there are no significant correlations there is no evidence that previous estimates play a role. Each estimate appears to be constructed separately and (in P,,) the subject does not recall the length of his previous time unit, is able to detect differences and to correct for them. We may conclude that in the case of cognitive coding by counting there is hardly a reason to speak of time experience based on some internal representation of a complete interval. Again, P,t correlates negatively and significantly wjth the last occurring trial (Rr = -0.40; p < O.OS), but the same holds for Rst (Rr = -0.42; p < 0.05). The conclusion that the memory for intervals consists of one event only is valid for periods between 0.5 and 6 set in a Pst condition. It also holds for Rst up to 3 set, but not for 6 set intervals. Data reported by Michon (1967) offer a starting-point for an explanation of this difference. Michon observed that the variability of time estimates increases considerably with the length of the standard. The storage and/or recall of intervals decreases when the standard becomes longer. It is likely that storage is better when more sensory cues are available during imprinting. In the present experiment the previous trial may have been stored better than the standard since S has auditory as well as kinesthetic cues during the estimation phase (pressing the button and hearing the tone). If this is true, the correlation with the last trial will disappear when this difference between standard and estimate is minimized.
Experiment
5
Method Twenty Ss made 30 estimates under condition presented as a pause between two clicks. Each standard a microswitch twice.
R,t. A 6 set standard was was reproduced by pressing
Results The
mean
autocorrelation function shows no correlations (Rt = -0.18, lag pauses are used, S hears two clicks during the presentation of the standard. During the reproduction the sensory cues exceed those of the standard by shortly pressing a microswitch only. Thus, when the sensory cues are more or less comparable the correlation with the previous trial disappears and the system is again updated by the most recent information.
t = 1). When
Conclusions (1) In time estimation experiments, the method should be chosen with care since different techniques appear to measure different processes. The study of the interdependence of estimates within a series
486
P. A. VroonlSequential
estimations
of time
presents information about basic characteristics of the estimation process. Autocorrelations are appropriate for this purpose and preferable to a chi-square analysis of non-randomness since (a) a smaller amount of measurements is sufficient, (b) the interdependence of estimates is restricted to the first-order effect. (2) Direct representation of a complete interval should be distinguished from an indirect representation of time which occurs when S counts subjective time units. Not counting favors a direct representation on some subjective continuum. In this case the most recent information is the input for a trial after one presentation of the standard. The subject tries to copy his previous estimate and, if necessary, corrects in one step so that a tendency to alternate occurs. (3) The effects of repeated standard presentation are not insensitive to interval length. The last event determines the estimates up to 3 set standards so that correlations with estimates are absent. However, a correlation with the previous estimate recurs when longer standards are used and when estimates are accompanied by more sensory cues than the standard. When these cue differences are minimized, the subject bases himself upon the most recent information, i.e., the standard, so that there are 110 correlations with estimates. The cue differences do not appear to play a role when short intervals are used. (4) Counting during presentation and estimation always leads to series without sequential dependence. Cognitive coding by counting implies that the interval is not integrally stored, but as a number of subjective time units. There is no evidence that the length of the subjective time unit is remembered. The temporal equivalent of a number of units is constructed from trial to trial.
References Augenstine, L. G., 1955. Evidences of periodicitics in human task performance. In: 1% Quastler (ed.), Information theory in psychology. Glencoe: The Free Press. Clausen, J., 1950. An evaluation of esperimental methods of time judgment. Journal of Experimental Psychology 40, 756-761. Fraisse, P., 1963. The psychology of time. New York: Harper and Row. Hicks, R. E., G. W. Miller, M. Kinsbourne, 1976 (in press). Prospective and retrospective judgments of temporal duration as a function of information processed. American Journal of Psychology. Lewis, D., 1963. Quantitative methods in psychology. New York: McGraw-llill.
P. A. Vroon/Sequential estimations of time
487
Michon, J. A., 1965. Studies on subjective duration. II: Subjective time measurements during tasks with different information content. Acta Psychologica 24, 205-219. Michon, J. A., 1967. Timing in temporal tracking. Assen: Van Gorcum and Comp. Michon, J. A., 1972. Processing of temporal information and the cognitive theory of time experience. In: J. T. Fraser, F. C. Haber, G. H. Muller teds.), The study of time. Berlin: Springer-Verlag. Michon, J. A., 1975. Time experience and memory processes. In: J. T. Fraser, N. Lawrence (eds.), The study of time, II. New York: Springer-Verlag. Michon, J. A., 1976 (in press). Holes in the fabric of subjective time. Acta Psychologica. Ornstein, R. E., 1969. On the experience of time. Harmondsworth: Penguin. Stroud, J. M., 1955. The fine structure of psychological time. In: H. Quastler ted.), Information theory in psychology. Glencoe: The Free Press. Treisman, M., 1963. Temporal discrimination and the indifference interval: implications for a model of the internal clock. Psychological Monographs 77, whole nr. 576. Vroon, P. A., 1970. Effects of presented and processed information on duration experience. Acta Psychologica 34, 115-121. Vroon, P. A., 1974. Is there a time quantum in duration experience? American Journal of Psychology 87, 237-245. Vroon, P. A., 1976 (in press). On the hemispheric representation of time. In: S. Dornic (ed.), Attention and performance, VI. New Jersey: Erlbaum. Vroon, P. A., and A. Van Boxtel. Testing some implications of the sensory-physiological model of the time sense. Psychologische Forschung 35, 81-92.
40 (1976) 475-487 Publishing Company
SEQUENTIAL Pieter
ESTIMATIONS
OF TIME
A. VROON*
Psychological Institute,
Received
January
UtCw-sity of Lciden,
The Netherlands
1976
Sequential estimations of intervals stored and reproduced by either counting or not counting are compared in experiments with single and repeated presentation of the standard interval. In a non-counting condition the estimates show a negative correlation with the last occurring trial throughout the series after one presentation of the standard. Consequently, in an ongoing process the capacity for (the storage and) the recall of time appears to be limited to the most recent event corresponding with the previous estimate. Correlations are lacking when the standard is presented after every estimate because the most recent information is an external constant. The correlations with the last trial recur when relatively long intervals are used, stored on the basis of various sensory cues. When the difference between the number of these cues is minimized, the correlation disappears. Counting never shows sequential correlations since it favors the coding and storage of time as a number of subjective units. The temporal equivalent of that number is constructed from trial to trial, a representation of the complete interval is not available and the subject does not remember his counting rate.
Research in time psychology is mainly concerned with the experience of duration. The problem is defined by the phenomenon that apparent duration differs from physical time by accelerations and decelerations. Usually, two viewpoints are distinguished: a psychophysiological and a cognitive approach. The first theory says that clock time is reflected in cyclic phenomena such as EEG, heart rate, etc. There is an internal time-keeper, driven by arousal level, metabolism, sensory input and the subject’s cognitive activities. Moreover, subjective time is not infinitely divisible, but consists of quanta, about 100 msec long (Augenstine 1955; Stroud 1955). The research focuses on the nature of the time-keeper and its speed fluctuations as functions of * Requests for reprints should 169, Leiden, The Netherlands.
be sent to P. A. Vroon,
Psychological
Institute,
Rijnsburgerweg
416
P. A. Vroon/Sequetltial estimations of time
various conditions (see Ornstein 1969; Treisman 1963; Vroon and Van Boxtel 1972; Vroon 1974). The cognitive approach says that subjective time is a mental construction, mainly dependent on cognitive operations. This theory is in so far more elegant, that it also explains the effects of higher and lower body temperatures, drugs, and so on (see Hicks et al. 1976; Michon 1965, 1972, 1975, 1976; Ornstein 1969; Vroon 1970, 1976). These theories are not completely incompatible, but the latter has some important methodological consequences. If time is in the first place a dimension of experience, the apparent duration of a period sl~ould not be compared with the clock, but with the subjective duration of another interval. This method was used by Ornstein (1969) who demonstrated that subjective time is closely related to the coding, transmission, storage and retrieval of information. The present study is concerned with the estimation of short intervals, i.e., of a few seconds. It is likely that mechanisms such
Methods
of measuring
subjective
time
An interval map be presented either verbally or not and be estimated in the same ways. A non-verbal presentation consists of a stimulus (tone, pause between clicks, etc.), a non-verbal estimation means that the S reproduces the interval without counting. A verbal presentation means that B mentions a number of time units and in a verbal estimation the S either reports time units or reproduces the interval by counting. Thus, there are four types of stimulus and response combinations. Non-verbal estimation requires that the S has some representation of the complete interval. When the interval is estimated verbally. he mentions time units or builds up the estimate by adding temporal equivalents of time units. In sequential production or reproduction the S is instructed to estimate the standard repeatedly. When a non-verba presentation is combined with a verbal estimation, the results can be interpreted in different ways. (1) Verbalization. Suppose that E presents an interval of 10 set, instructs S to count at a rate of one per subjective second and that S verbally reports a duration of 12 sec. In this case the subjective second was shorter than the objective second or the ‘psychological clock’ ran too fast. The internal and the external clocks show a speed difference. (3) Reproduction. A
P. A. VroonfSequential
estimations
of time
417
10 set standard is presented and E instructs S to count subjective seconds during presentation as well as estimation. Suppose that the S’s estimate is 12 set again. Since it may be assumed that S counted from 1 to n during presentation as well as estimation, a speed fluctuation of the internal clock appears to have occurred in that the time-keeper slowed down between presentation and estimation. Studies by Clausen (1950) and Fraisse (1963) demonstrate that the data of several methods may be inversely related or even incomparable. They report that reproduction without counting (non-verbal estimation) does not correlate with verbalization. In the case of verbalization time is splitted up into units and a too fast or too slow time-keeper leads to systematic over- and underestimations. When the interval is reproduced without counting, the absolute speed of the internal clock is irrelevant. For the same reason there is also no correlation between reproduction without and with counting (Fraisse 1963). He also reports a correlation between verbalization and reproduction with counting, which appears to be likely. It is clear that presence as well as absence of correlations is important in so far that they may reflect fundamental differences between the processes as measured by the various methods.
Hypotheses Two groups of conditions seem to be significant in terms of estimation process variables. (A) Non-verbal presentation - non-verbal estimation serial, non-verbal. The standard is presented once (l)P,t - production, and S is instructed to reproduce it repeatedly. serial, non-verbal. Each standard presentation is (2) Rst - reproduction, followed by one estimation. (B) Non-verbal presentation ~ verbal estimation serial, verbal. The standard is presented once, S is (1) P,, - production, instructed to count subjective time units during presentation and subsequent estimations. (2) R,, - reproduction, serial, verbal. Similar to Rst, but with counting. Within the framework of magnitude estimation models it could be argued that that there are also two stages in time estimation: an input stage in which S experiences and stores the duration of the standard interval, and an output stage in which the interval is estimated. Different effects for each condition are expected. P,t. When the standard is presented, there is an input transformation from the physical magnitude of the standard on the physical continuum to a subjective magnitude on a subjective continuum. When the interval is reproduced the subjective input magnitude is compared with the subjective output magnitude, both being distances on a continuum. Storage and reproduction of time occurs on the basis of some representation of the complete interval. In such an ongoing process a particular estimate may be a copy of the standard, or S forgets the standard and bases himself on (a) previous estimate(s). In other words: the input for a particular estimate has different possible degrees of recency.
478
P. A. Vroon/Sequentiai estimations of time
Rst. Each reproduction is preceded by a constant standard. When, for example, the most recent information defines the input only, no correlations with previous estimates will be observed. Michon (1967) demonstrated that the variability of non-verbal estimations increases considerably as a function of the standard duration. On the basis of these findings one could hypothesize that the ability to represent/store time decreases with increasing standard durations. Consequently, (the) previous estimate(s) may now also define a particular trial. P,,. When S is instructed to count during presentation and estimation, it is likely that the interval is not stored as a distance on some subjective continuum, but as a number. Since misreading of that number is not probable, there will be no errors in the estimation phase. The subjective temporal equivalent of the number has to be reconstructed from trial to trial. If the counting rate varies randomly, there will be no correlations with previous estimates. If such correlations are observed, S must remember the length of his previous subjective time unit, be able to detect a difference and to correct for it. In that case Psv is a mixture of storage of temporal plus numerical units. may present information about such a process. For example, R sv. This condition Psv shows that the last occurring event (trial .Y-- 1) plays a role or even trial (v-2), (x-3), etc. There are several possibilities. (I) When there is only a correlation with (x-l ), there will be no correlation in R,, since event (x-l) is an external standard. (2) If P,, correlates with (~~~1). (x-2), (.x-3), there is evidence for the existence of some internal standard as a (weighted) average of several previous but not with (x- 3). (3) If estimates. Rsv may now correlate with (x-1) and (x-2), is remembered u/zless new Psv shows correlations and Rsv does not, an estimate temporal information is presented. In other words: new information erases memory for previously estimated intervals.
Apparatus The apparatus consisted of a tone generator (sine wave) with a range between 50 and 20,000 Hz, a selector to present intervals between 0.25 and 15 set and an electronic clock with an accuracy of 10 msec. In all experiments the standard consisted of an interval filled with or limited by a 1000 Hz, 73 dB tone, presented binaurally by headphones. Reproduction took place by continuously pressing a microswitch (filled intervals) or by shortly pressing the switch twice (unfilled intervals).
P. A. VroonfSequential
479
estimations of time
-0.40
t I 30
10
20
10
20
30
40
50
10
20
30
40
50
40
50
-0.40 t
I
Q20
0
-a20
-a40
Fig. 1. Autocorrelation functions of three Pst series, each consisting of 200 trials, t = correlation lag (trials), 1 < t < 50. a: subject A, series 2, standard 1 sec. b: subject A, series 2, standard 2 sec. c: subject C, series 2, standard 3 sec.
P. A. Vroon/Sequential estimations of time
480
Sequential effects in P,, Experiment
I
Method Four Ss (A, B, C, D) were prescntecl an interval (tone) of 1, 2 or 3 sec. They were instructed to refrain from counting and to produce the standard interval repeatedly. Every S estimated two series of 200 trials each with a standard duration of 1 set (A), 2 set (B) or 3 SW (C, D). They were not informed about the number of trials required. Results The series of estimates were analyzed with autocorrelation (see Vroon 1974). In autocorrelating the sign and the strength of correlations carries information about the direction and the degree to which an estimate is related to previous events. In order to obtain stationary functions, the curves of the raw data were filtered by fitting polynomials. According to the least squares method described by Lewis (1963) the best fitting polynomial, accompanied by a significant reduction of residual variance, was chosen. Fig. 1 shows the autocorrelation functions of the estimates. Throughout the series, each interval correlates negatively with the last occuring trial which indicates that S tends to produce longer and shorter estimates more or less in alternation. Table 1 presents the first-order autocorrelations (lag t = 1) of all series. The functions of fig. 1 should be considered with caution. The autocorrelations Rt oscillate around the axis Rt = 0 indicating that the trend elimination was sufficient. However, the fact that significant correlations are still observed over many previous trials does not mean that S’s ‘memory space’ for intervals covers a long period of time. The corrclograms indicate the degree estimate (x) is influenced by estimate (X -I), but higher order effects such as as the dependence between (s) and (l-2), (s-3), etc. can only be assessed when the Table 1 6xperirncnt
1, Pearson
autocorrelations
Subject A A B B c C D D
trial (RI).
Scrims
Standard
1
1
2 1 2 1 2
1 2 2 3 3 3 3
1 2 _.~
* p < 0.01, two-tailed.
over the lavt occurring
~_
~.~ ~~ ._~
(xc)
.~~~~__~_~~.-.~~~ .- ~..~
RI m-0.28* -0.35* -0.43* -0.45* -0.36* -0.41* - 0.34* -0.33*
481
P. A. VroonfSequential estimations of time Table 2
Experiment 1, chi-square values as a measure of non-randomness of the estimates. Standard (SK)
in the trial-to-trial fluctuations
Series
x2 1st order
x2 2nd order
x2 3rdOTdeT
1
2
7.i54* 11.24*
1.03 0.93
0.43 0.22
1 2 1 2 1 2
19.84* 18.67* 17.51* 21.03* 16.93* 14.18*
0.39 0.43 5.17 1.15 7&l* 3.22
0.32 1.02 2.29 1.01 3.42 1.33
* p < 0.01. first-order effect is taken into account since the correlations are not independent. If the estimates are described as positive and negative trend deviations without the size of the deviation being taken into account, a binary sequence is obtained. Sequential interrelations may be assessed by studying the randomness of this series. There are three possibilities: (1) the series is random (which means that there are no sequential effects), or the correlations originate from (2) too many alternations (short-long estimates and vice versa), or (3) perseverations (shorter-shorter and longer-longer estimations). The first-order (digrams), second-order (trigrams), and third-order (tetragrams) effects are shown in table 2. The series are not random but the effect is limited to the first-order effect. It is highly significant and due to alternation, which was also clear from the negative autocorrelations over one trial in fig. 1 and table 1. Since there is no higher order non-randomness, autocorrelation is effective for studying the interdependence of time estimates. An advantage over a nonrandomness analysis as shown in table 2 is that a smaller number of measurements is sufficient. If mean autocorrelation functions over Ss are calculated, most of the fluctuations observed in fig. 1 will disappear since it is likely that there are phase shifts in the individual functions. Consequently, it is expected that I& = 0 except for lag t = 1 which will show a negative correlation.
Experiment
2
Method The & (24 freshmen) each in a counterbalanced
estimated a standard (tone) of 0.5, order under the condition Pst.
1 or 2 set 30 times
Results The
mean autocorrelation functions about the individual phase
hypothesis
are shown differences
in fig. 2. It appears that the (see also table 2) is correct:
482
P. A. Vroon/Sequential estimations of time
P,t. 0.5 *cc 0.20
f Rt t -
0.
-
-
/
.
-0.20 -
-0.40 t 5
10
Pst.lSM. 0.20 If%
--
O-
-0.20
-
-0.40 -L 5
10
P st,2rcc. 0.20 -
t Rt
0
I---
-,
-0.m -
-0.40 t 5
10
Fig. 2. Autocorrelation functions of three Pst series with different standard durations, 30 trials, means over 24 Ss, 1 G f G 10. Rt = 0 when t a 2. Thoughout the series there is a significant and negative correlation with the last occurring trial, also observed in experiment 1 (see fig. 1, tables 1, 2). In the case of P,t with standard durations up to 3 set the estimates are independent of each other, except for the last trial. There are two prelimmary conclusions.
P. A. VroonfSequential estimations of time
483
(1) The estimation process under Pst is based on a memory for complete intervals since there is a correlation with the previous trial. (2) It appears that S forgets the standard and bases himself on the most recent event. Thus, the input for a particular estimate has maximum recency. Since only one event is remembered, estimates with an alternating tendency are likely to occur. The S corrects an estimate in one step (see also Michon 1967, who observed the same phenomenon in synchronization experiments). The conclusions can be validated by studying R,t estimates. In this case the last term of the series is not an estimate, but a constant standard. If the memory for the previous estimate is erased by new temporal information, correlations with previous trials will be lacking.
,*f?$ ~
-.20
-
-.LO
-
__iy/
,
t 10
5
7%
.20
0
-. 20
-.bO
Pst
3scc.
7=----T
Fig. 3. Autocorrelation Ss,l
functions of Psl and Rst series, 30 trials, standard 3 set, means over 60
484
Sequential Experirnen
P. A. Vroor~/Sequetztial estimutiom
effects
of time
in R,t
t 3
Method Sixty Ss were split up into two equal groups, each estimating (tone) 30 times under the conditions P,t and R,t in a counterbalanced
a 3 set interval order.
Results Fig. 3 shows the mean autocorrelation functions. A difference between Pst and Rst is clearly observed: in Pst there is a negative and significant correlation with the last occurring trial (p < 0.01, one-tailed), whereas in R,t the estimates are independent. When a short interval is estimated without counting, the most recent information defines the subsequent trial.
Sequential
effects
in P,, and R,,:
Cognitive
transformation
It becomes apparent that non-verbal conditions favor the integral storage of an interval. Counting leads to fragmentizing time on the basis of a subjective time unit. Since this number will not be forgotten, its temporal equivalent has to be reconstructed from trial to trial. If S does not remember the length of his time unit (counting rate), there will be no interdependence between estimates. If he does, P,, involves double storage, i.e., of a number and a time unit. l
4
Method Sixty estimated group
Ss were divided into two equal groups. A 6 XC standard (tone) was under the following conditions. I: (a) P,t, 30 trials (b) P,,, 30 trials group 2: (a) R,t, 30 trials (b) Rsv, 30 trials The order of the conditions per group was fixed since Ss tend to count when longer intervals are to be estimated. If a counting instruction is given, it is difficult to refrain from it in a second series. There were 15 min pauses between the tasks.
P. A. VroonfSequential estimations of time
485
Results The autocorrelograms of both P,v and Rsv are stationary (compare fig. 3, Rst condition). Since there are no significant correlations there is no evidence that previous estimates play a role. Each estimate appears to be constructed separately and (in P,,) the subject does not recall the length of his previous time unit, is able to detect differences and to correct for them. We may conclude that in the case of cognitive coding by counting there is hardly a reason to speak of time experience based on some internal representation of a complete interval. Again, P,t correlates negatively and significantly wjth the last occurring trial (Rr = -0.40; p < O.OS), but the same holds for Rst (Rr = -0.42; p < 0.05). The conclusion that the memory for intervals consists of one event only is valid for periods between 0.5 and 6 set in a Pst condition. It also holds for Rst up to 3 set, but not for 6 set intervals. Data reported by Michon (1967) offer a starting-point for an explanation of this difference. Michon observed that the variability of time estimates increases considerably with the length of the standard. The storage and/or recall of intervals decreases when the standard becomes longer. It is likely that storage is better when more sensory cues are available during imprinting. In the present experiment the previous trial may have been stored better than the standard since S has auditory as well as kinesthetic cues during the estimation phase (pressing the button and hearing the tone). If this is true, the correlation with the last trial will disappear when this difference between standard and estimate is minimized.
Experiment
5
Method Twenty Ss made 30 estimates under condition presented as a pause between two clicks. Each standard a microswitch twice.
R,t. A 6 set standard was was reproduced by pressing
Results The
mean
autocorrelation function shows no correlations (Rt = -0.18, lag pauses are used, S hears two clicks during the presentation of the standard. During the reproduction the sensory cues exceed those of the standard by shortly pressing a microswitch only. Thus, when the sensory cues are more or less comparable the correlation with the previous trial disappears and the system is again updated by the most recent information.
t = 1). When
Conclusions (1) In time estimation experiments, the method should be chosen with care since different techniques appear to measure different processes. The study of the interdependence of estimates within a series
486
P. A. VroonlSequential
estimations
of time
presents information about basic characteristics of the estimation process. Autocorrelations are appropriate for this purpose and preferable to a chi-square analysis of non-randomness since (a) a smaller amount of measurements is sufficient, (b) the interdependence of estimates is restricted to the first-order effect. (2) Direct representation of a complete interval should be distinguished from an indirect representation of time which occurs when S counts subjective time units. Not counting favors a direct representation on some subjective continuum. In this case the most recent information is the input for a trial after one presentation of the standard. The subject tries to copy his previous estimate and, if necessary, corrects in one step so that a tendency to alternate occurs. (3) The effects of repeated standard presentation are not insensitive to interval length. The last event determines the estimates up to 3 set standards so that correlations with estimates are absent. However, a correlation with the previous estimate recurs when longer standards are used and when estimates are accompanied by more sensory cues than the standard. When these cue differences are minimized, the subject bases himself upon the most recent information, i.e., the standard, so that there are 110 correlations with estimates. The cue differences do not appear to play a role when short intervals are used. (4) Counting during presentation and estimation always leads to series without sequential dependence. Cognitive coding by counting implies that the interval is not integrally stored, but as a number of subjective time units. There is no evidence that the length of the subjective time unit is remembered. The temporal equivalent of a number of units is constructed from trial to trial.
References Augenstine, L. G., 1955. Evidences of periodicitics in human task performance. In: 1% Quastler (ed.), Information theory in psychology. Glencoe: The Free Press. Clausen, J., 1950. An evaluation of esperimental methods of time judgment. Journal of Experimental Psychology 40, 756-761. Fraisse, P., 1963. The psychology of time. New York: Harper and Row. Hicks, R. E., G. W. Miller, M. Kinsbourne, 1976 (in press). Prospective and retrospective judgments of temporal duration as a function of information processed. American Journal of Psychology. Lewis, D., 1963. Quantitative methods in psychology. New York: McGraw-llill.
P. A. Vroon/Sequential estimations of time
487
Michon, J. A., 1965. Studies on subjective duration. II: Subjective time measurements during tasks with different information content. Acta Psychologica 24, 205-219. Michon, J. A., 1967. Timing in temporal tracking. Assen: Van Gorcum and Comp. Michon, J. A., 1972. Processing of temporal information and the cognitive theory of time experience. In: J. T. Fraser, F. C. Haber, G. H. Muller teds.), The study of time. Berlin: Springer-Verlag. Michon, J. A., 1975. Time experience and memory processes. In: J. T. Fraser, N. Lawrence (eds.), The study of time, II. New York: Springer-Verlag. Michon, J. A., 1976 (in press). Holes in the fabric of subjective time. Acta Psychologica. Ornstein, R. E., 1969. On the experience of time. Harmondsworth: Penguin. Stroud, J. M., 1955. The fine structure of psychological time. In: H. Quastler ted.), Information theory in psychology. Glencoe: The Free Press. Treisman, M., 1963. Temporal discrimination and the indifference interval: implications for a model of the internal clock. Psychological Monographs 77, whole nr. 576. Vroon, P. A., 1970. Effects of presented and processed information on duration experience. Acta Psychologica 34, 115-121. Vroon, P. A., 1974. Is there a time quantum in duration experience? American Journal of Psychology 87, 237-245. Vroon, P. A., 1976 (in press). On the hemispheric representation of time. In: S. Dornic (ed.), Attention and performance, VI. New Jersey: Erlbaum. Vroon, P. A., and A. Van Boxtel. Testing some implications of the sensory-physiological model of the time sense. Psychologische Forschung 35, 81-92.