Are the Referents Remembered in Temporal Bisection?€¦ · the referents might be more appropriate...

Learning and Motivation 33, 10–31 (2002)

doi:10.1006/lmot.2001.1097, available online at http://www.idealibrary.com on

Are the Referents Remembered in Temporal Bisection?

Lorraine G. Allan

McMaster University, Hamilton, Ontario, Canada

In roving-referent bisection, three duration values are presented sequentially oneach trial, two referents and then a probe, and the observer is instructed to indicatewhether the probe is more similar to the duration of the first or second trial referent.Data from three experiments are reported which consistently indicate that humanobservers do not compare the probe to the trial referents in roving-referent bisection.Rather, they base their judgment on the absolute value of the probe rather than onits relationship to the trial referents. The data are consistent with Gibbon’s (1981)formulation of the decision process in bisection which specifies that the probe iscompared to a criterion duration value. Although many studies have used bisectionto study temporal memory, the data from the present experiments suggest that refer-ent memory is not a dominant source of variability in human temporal bisection. 2002 Elsevier Science (USA)

John Gibbon’s (1981) now classic paper, ‘‘On the form and location ofthe bisection function for time,’’ launched temporal bisection as a popularpsychophysical procedure for the study of time perception. In the prototypictemporal bisection task with humans, two referents, one short (S) and theother long (L), are explicitly identified by familiarizing the observer withthe referents either at the beginning of a block of trials (e.g., Allan, 1999;Wearden, 1991; Wearden & Ferrara, 1995, 1996; Wearden, Rogers, &Thomas, 1997) or periodically throughout a block of trials (e.g., Allan &Gibbon, 1991; Penney, Allan, Meck, & Gibbon, 1998). On probe trials, atemporal interval t, S # t # L, is presented, and the observer is required toindicate whether t is more similar to S (RS) or to L (RL). Temporal bisectionyields a psychometric function relating the proportion of long responses,P(RL), to probe duration t. The value of t at which RS and RL occur withequal frequency, P(RL) 5 0.5, is often referred to as the bisection point(T1/2). One interpretation of T1/2 is that it is the value of t that is equallyconfusable with S and L.

The research reported in this paper was supported by a grant to Lorraine G. Allanfrom the Natural Sciences and Engineering Research Council of Canada. Correspondenceand reprint requests should be addressed to Lorraine G. Allan, Department of Psychology,McMaster University, Hamilton ON, L8S 4K1, Canada. Fax: (905) 529-6225. E-mail:[email protected].

100023-9690/02 $35.00 2002 Elsevier Science (USA)All rights reserved.

REFERENTS AND BISECTION 11

Since the referents are not available on probe trials, it was often assumedthat stored memories of S and L were established and that on each trial theprobe was compared to these stored memories (e.g., Allan & Gibbon, 1991;Gibbon, 1981; Wearden, 1991). That is, the decision to respond RS or RL

was made by comparing the similarity of the perceived value of t on a giventrial to memories of the two referents, S and L. In fact, temporal bisectionbecame a favored psychophysical procedure for the study of temporal mem-ory in humans (see Allan, 1998). More recent reports, however, have sug-gested that the probe might not be compared to memories of the referents(Allan, 1999; Allan & Gerhardt, 2001; Rodriguez-Girones & Kacelnik, 2001;Wearden & Ferrara, 1995, 1996). A specific goal of the present experimentswas to obtain a better understanding of the role of the referents in temporalbisection. A more general goal was to further our understanding of temporalmemory.

Wearden and Ferrara (1995, 1996) modified the prototypic bisection taskso that no member of the set of probes was identified as a referent. Rather,observers were instructed to partition the values of t into two categories, RS

or RL. To distinguish the two tasks, Wearden and Ferrara (1995, 1996) la-beled the task in which the referents are identified and observers are in-structed to compare the similarity of t to the referents as ‘‘similarity’’ andthe task in which the referents are not identified and observers are instructedto partition the t values into two categories as ‘‘partition.’’1 Wearden andFerrara (1995) compared the similarity and partition tasks and concludedthat they yielded similar data. Allan (1999) and Allan and Gerhardt (2001)suggested that a decision rule that does not involve a direct comparison withthe referents might be more appropriate than the similarity rule for both thesimilarity and the partition bisection tasks. They suggested that the bisectiontask be considered within a signal detection context. That is, the observerin bisection establishes a criterion duration value which is determined by thevalues of the referents. On each trial, the observer compares t to this singlecriterion rather than to memories of the two referents.

In fact, Gibbon (1981) had derived a signal detection bisection functionbased on the comparison of the probe to a criterion duration. In his derivation,he assumed that perceived time is normally distributed, that mean perceivedtime is a power function of clock time with an exponent close to 1.0,

µt 5 t , (1)

and that the standard deviation of perceived time is proportional to meanperceived time,

1 While the partition task is similar to the many-to-few task described by Allan (1979) inthat there are multiple stimuli and two responses, the two tasks differ in a fundamental way.In partition, there is no trial-by-trial feedback, whereas in many-to-few, there is.

12 LORRAINE G. ALLAN

σ t

µ t

5σt

t5 γ . (2)

The proportionality constant γ is the Weber fraction. Gibbon (1981) re-ferred to the proportionality relation in Eq. (2) as scalar variability or thescalar property. This scalar property between the standard deviation and themean results in distributions of perceived time that superpose when thetemporal axis is normalized with respect to the mean of the distribution.For bisection, the scalar property predicts superposition of psychometricfunctions for all r values when plotted against t normalized by the bisectionpoint (i.e., t/T1/2).

Recently, Killeen, Fetterman, and Bizo (1997) simplified the Gibbon(1981) derivation of, and the resulting equation for, the bisection functionby substituting the logistic distribution for the normal distribution. For thelogistic approximation to the normal, the bisection function is

P(RL) 5 31 1 exp1T1/2 2 t√3π

σt 2421

,(3)

where T1/2, the bisection point, is the criterion and

σt 5 √(γt)2 1 pt 1 c2 . (4)

Equation 4 is the Weber Function derived by Killeen and Weiss (1987) whichprovides for scalar (γ), nonscalar (p), and constant (c) sources of variability.When scalar variability dominates, Eq. (4) reduces to Eq. (2), and Eq. (3)can be rewritten as

P(RL) 5 31 1 exp1T1/2 2 t√3π

γt 2421

.(5)

Killeen et al. (1997) noted that the bisection function in Eq. (3) is not alogistic function even though it is derived from one and that it is not a distri-bution function since it asymptotes at a value less than 1.0. They, therefore,referred to their bisection function as a Pseudo-Logistic function and theirmodel as the Pseudo-Logistic Model (PLM).

Rodriguez-Girones and Kacelnik (2001) also modified the prototypic bi-


section task. In their modification, three duration values were presented se-quentially on each trial: the two referents, counterbalanced for order, andthen the probe. Observers were instructed to indicate whether the last dura-tion on the trial (i.e., t) was more similar to the duration of the first or secondtrial referent. Rodriguez-Girones and Kacelnik (2001) examined two ver-sions of trial-referent bisection. In fixed-referent bisection, the referent pairwas constant during a block of trials, whereas in roving-referent bisection,the referent pair varied from trial to trial. They argued that stored memoriesof S and L could not be established in roving-referent bisection since S andL varied from trial to trial. They suggested that the probe on each trial wascompared directly to each of the trial referents presented on that trial. Thus,in the Rodriguez-Girones and Kacelnik (2001) account of temporal bisection,although the trial referents were involved in the comparison on each trial,long-term memories were not established and stored.

Allan and Gerhardt (2001) directly compared the role of the referents inprototypic (i.e., no trial referents) and trial-referent bisection. They con-cluded that the decision to respond RS or RL was not based on a direct compar-ison of the probe to the referents either in prototypic or in trial-referent bi-section. They suggested that t was compared to a criterion duration value inboth prototypic and trial-referent bisection, and they showed that PLM pro-vided an excellent account of their data.

Thus, while the Rodriguez-Girones and Kacelnik (2001) analysis of theirroving-referent data suggested that the probe was compared to the trial refer-ents, the Allan and Gerhardt (2001) analysis of their roving-referent datasuggested that the probe was not compared to the trial referents. The numberof S and L referent pairs in the roving task in Rodriguez-Girones and Kacel-nik (2001) was much larger than in Allan and Gerhardt (in press). It mightbe that the decision process in roving-referent bisection depends on the num-ber of referent pairs. In the present experiments we show that the resultsreported by Allan and Gerhardt (2001) are not attributable to the limitednumber of their S and L referent pairs.

If the probe is not directly compared to the trial referents, then the lengthof the delay between the termination of the referent pair and the presenta-tion of the probe (the referent–probe interval, RPI) should not affect the slopeof the bisection function. In Allan and Gerhardt (2001), RPI was constant at1000 ms. In Experiment 2, we vary RPI.

Often, bisection experiments have been conducted to determine the loca-tion of T1/2, the value of t that is equally confusable with S and L (see Allan,1999). Therefore, trial feedback was not given. In Experiment 3, we providetrial feedback to examine its influence on the decision process. Specifically,we were interested in determining whether feedback in roving-referent bi-section would result in t being compared to the trial referents rather than toa criterion value.


GENERAL METHOD

Observers

The observers were graduate students or research assistants in the Psychol-ogy Department at McMaster University who were paid for their participa-tion. They were required to complete a minimum of five sessions per week,with the restriction of a maximum of two sessions (separated by at least1 h) per day. Six observers participated in Experiment 1, four observers par-ticipated in Experiment 2, and four observers participated in Experiment 3.Of the four observers in Experiment 2, one (VL) had previously participatedin Experiment 1. All observers in Experiment 3 had participated in eitherExperiment 1 (KG and NC) or Experiment 2 (LH and CN).

Apparatus

Temporal parameters, stimulus presentation, and recording of responseswere controlled with a Macintosh computer. The temporal intervals werevisual and filled. They were marked by clearly visible geometric forms dis-played on an Apple color monitor. S referent values were always marked bya red circle, L referent values by a green circle, and probe values by a blacksquare.

EXPERIMENT 1

In the Rodriguez-Girones and Kacelnik (2001) experiment, S was uni-formly distributed between .5 and 2.0 s. For any value of S, L 5 (r)(S) for1.5 # r # 8, with an upper limit of L 5 5.5 s. For example, for S 5 1,L was equally likely to take on any value between 1.5 and 5.5 s. The durationof t was a multiple τ of the geometric mean of the trial referents (t 5τ√SL), where .5 # τ # 2.0. These algorithms for determining S, L, andt allowed for values of t less than S and greater than L. In the typical bisectiontask, S # t # L, and there are data which indicate that the bisection functionobtained when t is not between S and L is different from that obtained whent is between S and L (e.g., Siegel & Church, 1984). In Experiment 1, werestricted the range of t so that S # t # L.

In order to avoid explicit counting, Allan and Gerhardt (2001) used dura-tion values which were considerably shorter than those in Rodriguez-Gironesand Kacelnik (2001). In Experiment 1, we also used these shorter values.

Method

A session consisted of four blocks of 120 trials. On each trial S was ran-domly selected from a uniform distribution bounded by 400 and 700 ms,and L was the integer value of (r)(S), for r 5 1.50, 1.75, or 2.00. The valueof r was constant during a session and varied between sessions. Five probecategories were associated with every referent pair: S, S1, GM, L2, and L,


where GM was the integer millisecond value of the geometric mean of theS and L referents on that trial, S1 was the value of the mean of S and GM,and L2 was the value of the mean of GM and L. The value of t on eachtrial was randomly selected from the five probe categories with the restrictionthat each category occurred 24 times during a block of 120 trials. Thus theactual value of t for a particular category depended on the values of S andL on that trial. For example, suppose the category on a particular trial wasGM and r 5 2.00. If S 5 480 were selected for that trial, then L 5 960,GM 5 678, and t 5 678. However, if S 5 630 were selected for that trial,then L 5 1260, GM 5 890, and t 5 890.

On each trial, t was preceded by the two referents, counterbalanced fororder: S, L, t or L, S, t. The interval between the three successive trial dura-tions was 1 s. The observer’s task on each trial was to decide whether t wasmore similar to S or to L. At the termination of the probe, the observer indi-cated the decision by pressing S or L on the computer keyboard. There wasno feedback, and the next trial began 1 s after the response was made.

The experiment consisted of 12 sessions, four at each of the three r values.The order of r across sessions was random, with the restriction that eachvalue of r occurred once before any value of r was repeated. A session lastedabout 45 min, depending on the self-pacing of the observer’s responding.

Results

Rodriguez-Girones and Kacelnik (2001) stated that

If the standards have a different duration at each trial, it does not make any senseto calculate the probability of responding L for a given probe duration. Indeed, classi-fying a 2-s probe as L reflects different processes if the standards were 1-s and 2-slong or if they were 2-s and 4-s long (p. 534).

This assumption that t was compared to the trial referents resulted inRodriguez-Girones and Kacelnik (in press) normalizing t by the geometricmean of the trial referents, GM 5 √SL. As a first step in the analysis of ourdata, we also normalized t by GM. For each r value, normalization of t byGM collapses the many values of t to five values, one for each of the fiveprobe categories (S, S1, GM, L2, and L).

For each observer, P(RL) for each r value was based on all four blocks fromall four sessions. Figure 1 shows P(RL), averaged over observers, plotted asa function of t/GM, separately for each value of r. The bisection functionsare clearly monotonic with t/GM and superpose for the three values of r.A closer look at the data, however, suggests that normalizing by t/GM isinappropriate. Figure 2 again shows P(RL), averaged over observers, plottedas a function of t/GM. Each panel illustrates the data for one of the three rvalues. In each panel, there are four functions. One of the functions repre-sents the data collapsed across all S values (i.e., the data from Fig. 1). Theother three functions represent the data for three ranges of S values: 400 #


FIG. 1. P(RL), averaged over observers, as a function of t/GM for each r value (Experi-ment 1).

S # 499, 500 # S # 599, and 600 # S # 700. If t were being comparedto the trial referent pair, the functions for the three S ranges should be similar.It is clear from Fig. 2, however, that the bisection function depends on therange of S. Specifically, for every value of t/GM, P(RL) increased as S in-creased. Since for a constant t/GM as S increased so did t, it appears thatRL was determined by t rather than t/GM.

The data in Fig. 2 suggest that t, rather than being compared to the trialreferents, is judged independently of the referents. The values of t rangedfrom 400 to (r)(700) ms, in 1-ms steps. We grouped the t values in 100-msbins (i.e., 400–499, 500–599, etc.) and determined P(RL) for each bin. Figure3 shows P(RL) for each observer as a function of t, where t is the midpointof a 100-ms bin. For every observer, P(RL) is a monotonic function of t, andas r increases the functions are displaced to the left. The data in Fig. 3 suggestthat the roving-referent bisection task was transformed by the observers intothe partition bisection task. As t increased, so did P(RL) regardless of thetrial referent pair. While the lower boundary for S was the same for all valuesof r, the upper boundary increased with r, resulting in P(RL) for a particularvalue of t decreasing as r increased. For example, the observer was less likelyto categorize t 5 500 ms as long when r 5 2 than when r 5 1.5.

The data from Experiment 1 are consistent with the conclusion reachedby Allan and Gerhardt (2001) that t is not compared to the trial referents inroving-referent bisection. They suggested that t is compared to a criterionduration value, and they showed that their data were consistent with PLM.Killeen et al. (1997) developed PLM for the prototypic (i.e., no trial refer-ents) bisection task and suggested that the role of the referents is to set thevalue of the criterion T1/2. On each trial, the perceived value of the probe iscompared to T1/2, and the decision is RL if the perceived value is larger thanT1/2. In roving-referent bisection, however, there are multiple referent pairs.


FIG. 2. P(RL), averaged over observers, as a function of t/GM. Each panel presents thedata for one of the r values. The filled symbols represent the data for the three ranges of Svalues and the open symbols represent the data collapsed over all S values (Experiment 1).

Allan and Gerhardt (2001) suggested that in roving-referent bisection, likein partition, the value of the T1/2 is set by the range of t values.

In applying PLM to the present data, we assumed, as did Allan and Ger-hardt (2001), that the scalar sources of variance dominated (i.e., p 5 0 andc 5 0 in Eq. (4)). The similar slopes for the three values of r in Fig. 3suggested that γ might be constant across the three functions. Thus, we fitEq. (5) to the data of each observer (using the nonlinear fit algorithm fromMathematica), allowing T1/2 to vary but keeping γ constant across the threefunctions. Table 1 gives, for each observer, the resulting parameter values.The lines in Fig. 3 are the values of P(RL) predicted by PLM. It is clear thatPLM provided an excellent fit to the data. Averaged over the three functions,the sum of the squared deviations between the data and the predictions ofPLM (ω2) ranged from .992 to .999 for the six observers.


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TABLE 1Values of γ and T1/2 in Experiment 1

Observer r γ T1/2

BM 1.50 .17 6801.75 7402.00 781

SH 1.50 .14 6881.75 7642.00 836

KE 1.50 .22 4711.75 4762.00 536

VL 1.50 .18 5981.75 6192.00 646

NC 1.50 .17 6601.75 7172.00 742

KG 1.50 .21 7121.75 7792.00 822

Figure 4 plots, for each observer, the three psychometric functions againstt normalized by the bisection point (t/T1/2). For all observers, the three func-tions superposed across r. The superposition across r is consistent with thescalar property.

In PLM, if responding is unbiased, T1/2 should be located at the value oft where the S distribution crosses the L distribution. Killeen et al. (1997)provided an approximated solution for Unbiased T1/2, but since then, Killeen(personal communication) has provided an exact solution (see Allan & Ger-hardt, 2001),

Unbiased T1/2 5(S 2 rL) 1 √(S 2 rL)2 2 (1 2 r)((γS)2 ln(r))

1 2 r, (6)

where S is the shortest value of t, L is the longest value of t, and r 5(S/L)2. In Experiment 1, the range of S values was the same for the threevalues of r, 400 to 700 ms. The effect of increasing r was to increase theupper limit for L, and therefore t, from 1050 ms for r 5 1.5 to 1400 ms forr 5 2. Thus, as the possible values of t increased, so would the value ofUnbiased T1/2. T1/2 estimated from the data (Table 1) is plotted in Fig. 5,averaged over observers, as a function of r. Also shown are the UnbiasedT1/2 values predicted by Eq. (6). T1/2, like Unbiased T1/2, increased with r.An examination of Table 1 indicates that this was the case for all observers.There was, however, between-observer variability in the placement of T1/2


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FIG. 5. T1/2, averaged over observers, as a function of r. The filled symbols represent thedata and the open symbols represent Unbiased T1/2 (Experiment 1).

relative to Unbiased T1/2, with some observers setting T1/2 larger than Unbi-ased T1/2 and other observers setting T1/2 smaller than Unbiased T1/2. Thisbetween-observer variability has been found in our previous applications ofPLM to bisection data (Allan, 1999; Allan & Gerhardt, 2001).

Discussion

The data from Experiment 1 indicate that the probe in trial-referent bi-section is not compared directly to the trial referents. Rather the trial-referenttask appears to be transformed into the partition task. The present experimentshows that this transformation is not limited to the roving task used by Allanand Gerhardt (2001) where the variability in the trial referent pairs was mini-mal. Rather, the transformation of roving-referent bisection into partitionbisection also occurs when the variability in referent pairs is comparableto that in Rodriguez-Girones and Kacelnik (2001). It is plausible that thistransformation also occurred in the Rodriguez-Girones and Kacelnik (2001)roving task but was not observed because it was hidden by their data analysiswhich collapsed over referent pair. As was the case in Allan and Gerhardt(2001), PLM provided an excellent description of the data.

EXPERIMENT 2

Psychophysical models developed to incorporate sensory memory oftenpredict that memory variability should increase as the temporal interval be-tween the two events to be compared is increased. For example, in theirrandom-walk model for the comparison of two trial events, Kinchla and Al-lan (1969) suggested that the variability in the sensory representation of thefirst event increased during the interval separating the two events. Data existthat indicate that as the interval is increased from 500 to 2000 ms, discrimina-


bility for dimensions such as intensity and movement decreases (see Allan,Kristofferson, & Rice, 1974). Thus, if the probe in trial-referent bisectionwere compared to the trial referents that must be remembered over RPI, onemight expect γ to increase as RPI is increased. In contrast, if the probe iscompared to a criterion rather than to the trial referents, then one would ex-pect γ to remain constant across variations in RPI. In Experiment 1, RPI wasconstant at 1000 ms. In Experiment 2, RPI was varied from 500 to 2000 ms.

Method

Unlike Experiment 1, where RPI was 1000 ms, in Experiment 2, RPI wasvaried among the four blocks. There were four RPI values (500, 1000, 1500,and 2000 ms), and the order of the RPI values in a session was random. Ther values were the same as in Experiment 1 (1.50, 1.75, and 2.00), and againr was constant within a session and was varied between sessions. The experi-ment consisted of 24 sessions, 8 at each of the three r values. A sessionlasted about 45–60 min, depending on the value of RPI and the self-pacingof the observer’s responding. In all other respects, the procedure was thesame as in Experiment 1.

Results and Discussion

In Experiment 2, 12 psychometric functions (3 r values and 4 RPI values)were generated for each observer. Figure 6 shows P(RL), averaged over ob-servers, as a function of t. As in Experiment 1, each t value is the midpointof a 100-ms bin. Each panel of Fig. 6 plots the four RPI functions for eachvalue of r. For each r value, the functions are remarkably similar across thefour RPI values.

If RPI had no effect on γ, then one should be able to fit PLM to the 12functions with a single value of γ. Table 2 gives the parameter values ob-tained by fitting Eq. (5) to the data of each observer, keeping γ constantacross the 12 functions. Averaged over the 12 functions, ω2 ranged from.995 to .998 for the four observers. Figure 7 plots, for each observer, the 12psychometric functions against t normalized by the bisection point (t/T1/2).For all observers, the 12 functions superposed across r and RPI. The superpo-sition across r is consistent with the scalar property. The superposition acrossRPI indicates that γ is unaffected by RPI.

T1/2 estimated from the data (Table 2) is plotted in Fig. 8 as a functionof RPI for each value of r. As in Experiment 1, T1/2 increased with r.For each r value, T1/2 decreased with increasing RPI. An examination ofTable 2 reveals that this decrease was mainly due to two observers (LH andVL). For the other two observers (CH and CN), T1/2 was relatively constantacross RPI.

The data from Experiment 2 clearly indicate that γ remained constantacross variations in RPI. This result is consistent with the view that t is notcompared to the trial referents.


FIG. 6. P(RL), averaged over observers, as a function of t for the four RPI values. Eachpanel presents the data for one of the r values (Experiment 2).


T1/2

Observer r γ 500 1000 1500 2000

LH 1.50 .15 746 713 682 6331.75 765 758 725 6942.00 797 795 770 733

CH 1.50 .12 645 666 658 6541.75 687 690 694 6922.00 735 742 730 735

CN 1.50 .14 666 670 650 6391.75 708 710 696 6932.00 749 762 753 752

VL 1.50 .17 660 656 621 5921.75 691 688 660 6592.00 718 711 701 682


FIG. 7. P(RL), for each observer, as a function of t/T1/2. There are 12 functions (3 r val-ues 3 4 RPI values) for each observer (Experiment 2).

EXPERIMENT 3

The data from Experiments 1 and 2 indicate that the probe is not comparedto the trial referents in roving-referent bisection. As is typically the case inbisection, trial feedback was not provided in Experiments 1 and 2. In Experi-ment 3, we examine whether trial feedback, based on comparing the probeto the trial referents, would influence performance. Specifically, we were


FIG. 8. T1/2, averaged over observers, as a function of RPI for each r value (Experi-ment 2).

interested in determining whether such feedback would induce the observerto compare the probe to the trial referents.

Method

There were 12 sessions, 6 at each of two r values, 1.5 and 2.0. Each sessionconsisted of four blocks of 96 trials. Six probe categories were associatedwith every referent pair: S, S1, S11, L22, L2, and L. The spacing betweenthe six probe categories was linear (40 ms for r 5 1.5, and 80 ms for r 52.0). The value of t on each trial was randomly selected from the six probecategories with the restriction that each category occurred 16 times duringa block of 96 trials. Auditory computer feedback, ‘‘correct’’ or ‘‘error,’’ wasprovided immediately after the observer entered the response. The next trialbegan 1.5 s after the feedback terminated. In all other respects, the procedurewas the same as in Experiment 1.

Feedback was determined by the relationship between t and the arithmeticmean (AM) of the trial referents. Specifically, t , AM (i.e., S, S1, and S11)was considered ‘‘short’’ and t . AM (i.e., L22, L2, and L) was considered‘‘long.’’ Thus, the feedback was ‘‘correct’’ if RS occurred to S, S1, or S11or RL occurred to L22, L2, and L, and the feedback was ‘‘error’’ if RS

occurred to L22, L2, or L or RL occurred to S, S1, and S11.

Results and Discussion

Figure 9 shows P(RL), averaged over observers, plotted as a function oft/GM. Each panel illustrates the data for one of the two r values. In eachpanel, there are four functions, one for trials where 400 # S # 499, one fortrials where 500 # S # 599, one for trials where 600 # S # 700, and onecollapsed across all S values. The pattern of results in Fig. 9 is very similar


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Observer r γ T1/2

CN 1.50 .14 674 6702.00 762 762

KG 1.50 .21 683 7122.00 818 822

LH 1.50 .15 688 7132.00 765 795

NC 1.50 .17 681 6602.00 759 742

to that seen in Fig. 2, where the data were generated without feedback. Thus,even with feedback, the bisection function depends on the range of S.

Figure 10 shows the data for each observer plotted as a function of t. Eachof the observers in Experiment 3 had participated in either Experiment 1 orExperiment 2. The comparable data from the earlier experiments are alsoshown. It is remarkable how similar the feedback data are to the earlier no-feedback data. The presentation of explicit feedback, based on the relation-ship of t to the trial referents, had little influence on performance.

PLM was fit to the data using the value of γ estimated previously for eachobserver. Table 3 gives, for each observer, the resulting parameter values.Even with the constraint imposed by using a value of γ estimated from adifferent set of data, PLM provided an excellent fit to the data, with ω2 rang-ing from .990 to .997 for the four observers. Figure 11 plots, for each ob-server, the four psychometric functions against t normalized by the bisectionpoint (t/T1/2).

CONCLUDING COMMENTS

The data from all three experiments consistently indicate that the probeis not compared to the trial referents in roving-referent bisection. Rather thejudgment appears to be based on the absolute value of t rather than on itsrelationship to the trial S and L values. The reliance on the absolute valueof t occurred even when explicit feedback, based on the trial referents, wasprovided. Even though this feedback would often indicate that the responsewas an ‘‘error,’’ the observers did not modify their responding to be consis-tent with the feedback. In fact, performance with and without feedback wasremarkably similar for all observers. Although many studies have used bi-section to study temporal memory (see Allan, 1998), the data from the pres-ent experiments suggest that referent memory is not a dominant source ofvariability in bisection.

The present data are consistent with PLM [and therefore with Gibbon’s(1981) formulation of the decision process in bisection], which specifies that


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the probe is compared to a criterion duration value. Killeen et al. (1997)originally developed PLM for the prototypic (i.e., no trial referents) bisectiontask, and suggested that the role of the referents is to set the value of thecriterion T1/2. In roving-referent bisection, however, there are multiple refer-ent pairs. Allan and Gerhardt (2001) suggested that in roving-referent bi-section, like in partition, the value of the T1/2 is set by the range of t values.

There are other data in the literature that are consistent with our conclusionthat the probe is compared to a criterion value rather than to other trial values.For example, Allan et al. (1974) reported data from a forced-choice (FC)duration discrimination task. In FC, two duration values, one short (S), theother long (L), are presented sequentially on each trial, counterbalanced fororder (SL or LS), and the task is to indicate the order, RSRL or RLRS. These dataindicated that variability remained constant across variations in the intervalbetween the two trial durations. Allan et al. (1974; see also Allan, 1977)concluded that the two trial durations were not directly compared to eachother, but rather that each was compared to a criterion value and each wascategorized as RS or RL.

The intriguing, and thus far unanswered, question is why is the probe notcompared to the trial referents. Our present data, as well as our earlier data,suggest that direct comparisons of trial durations are difficult, if not impossi-ble, for human observers.

REFERENCES

Allan, L. G. (1977). The time-order error in judgments of duration. Canadian Journal ofPsychology, 31, 24–31.

Allan, L. G. (1979). The perception of time. Perception & Psychophysics, 26, 340–354.Allan, L. G. (1998). The influence of Scalar Timing on human timing. Behavioural Processes,

44, 101–117.

Allan, L. G. (1999). Understanding the bisection psychometric function. In W. Uttal and P.Killeen (Eds.), Fechner Day 99: Proceedings of the fifteenth annual meeting of the Inter-national Society for Psychophysics. Tempe, AZ: Arizona State University.

Allan, L. G., & Gerhardt, K. (2001). Temporal bisection with trial referents. Perception &Psychophysics, 63, 524–540.

Allan, L. G., & Gibbon, J. (1991). Human bisection at the geometric mean. Learning & Motiva-tion, 22, 39–58.

Allan, L. G., Kristofferson, A. B., & Rice, M. E. (1974). Some aspects of perceptual codingof duration in visual duration discrimination. Perception & Psychophysics, 15, 83–88.

Gibbon, J. (1981). On the form and location of the bisection function for time. Journal ofMathematical Psychology, 24, 58–87.

Killeen, P. R., Fetterman, J. G., & Bizo, L. A. (1997). Time’s causes. In C. M. Bradshaw &E. Szabadi (Eds.), Time and behaviour: Psychological and neurological analyses (pp. 79–131). Amsterdam: Elsevier Science.

Killeen, P. R., & Weiss, N. A. (1987). Optimal timing and the Weber function. PsychologicalReview, 94, 455–468.

Kinchla, R. A., & Allan, L. G. (1969). A theory of visual movement perception. PsychologicalReview, 76, 537–558.


Penney, T. B., Allan, L. G., Meck, W. H., & Gibbon, J. (1998). Memory mixing in durationbisection. In D. A. Rosenbaum & C. E. Collyer (Eds.), Timing of behavior: Neural,psychological, and computational perspectives (pp. 165–193). Cambridge, MA: MITPress.

Rodriguez-Girones, M. A., & Kacelnik, A. (2001). Relative importance of perceptual andmnemonic variance in human temporal bisection. Quarterly Journal of Experimental Psy-chology, 54A, 527–546.

Siegel, S. F., & Church, R. M. (1984). The decision rule in temporal bisection. In J. Gibbon &L. G. Allan (Eds.), Timing and time perception (pp. 643–645). New York: Annals ofthe New York Academy of Sciences.

Wearden, J. H. (1991). Human performance on an analogue of an interval bisection task.Quarterly Journal of Experimental Psychology, 43B, 59–81.

Wearden, J. H., & Ferrara, A. (1995). Stimulus spacing effects in temporal bisection by hu-mans. Quarterly Journal of Experimental Psychology, 48B, 289–310.

Wearden, J. H., & Ferrara, A. (1996). Stimulus range effects in temporal bisection by humans.Quarterly Journal of Experimental Psychology, 49B, 24–44.

Wearden, J. H., Rogers, P., & Thomas, R. (1997). Temporal bisection in humans with longerstimulus durations. Quarterly Journal of Experimental Psychology, 50B, 79–94.

Received April 1, 2001, revised May 7, 2001

Are the Referents Remembered in Temporal Bisection?€¦ · the referents might be more appropriate...

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