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Are the Referents Remembered in Temporal Bisection?
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 on each trial, two referents and then a probe, and the observer is instructed to indicate whether 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 human observers 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 on its 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 is compared to a criterion duration value. Although many studies have used bisection to study temporal memory, the data from the present experiments suggest that referent 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 of the bisection function for time,’’ launched temporal bisection as a popular psychophysical procedure for the study of time perception. In the prototypic temporal bisection task with humans, two referents, one short (S) and the other long (L), are explicitly identified by familiarizing the observer with the 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, a temporal interval t, S ⱕ t ⱕ L, is presented, and the observer is required to indicate whether t is more similar to S (R S) or to L (R L). Temporal bisection yields a psychometric function relating the proportion of long responses, P(R L), to probe duration t. The value of t at which R S and R L occur with equal frequency, P(R L) ⫽ 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 equally confusable with S and L. The research reported in this paper was supported by a grant to Lorraine G. Allan from the Natural Sciences and Engineering Research Council of Canada. Correspondence and 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]. 10 0023-9690/02 $35.00 2002 Elsevier Science (USA) All rights reserved.
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Since the referents are not available on probe trials, it was often assumed that stored memories of S and L were established and that on each trial the probe was compared to these stored memories (e.g., Allan & Gibbon, 1991; Gibbon, 1981; Wearden, 1991). That is, the decision to respond R S or R L was made by comparing the similarity of the perceived value of t on a given trial to memories of the two referents, S and L. In fact, temporal bisection became a favored psychophysical procedure for the study of temporal memory in humans (see Allan, 1998). More recent reports, however, have suggested 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 experiments was to obtain a better understanding of the role of the referents in temporal bisection. A more general goal was to further our understanding of temporal memory. Wearden and Ferrara (1995, 1996) modified the prototypic bisection task so 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, R S or R L. To distinguish the two tasks, Wearden and Ferrara (1995, 1996) labeled the task in which the referents are identified and observers are instructed to compare the similarity of t to the referents as ‘‘similarity’’ and the task in which the referents are not identified and observers are instructed to partition the t values into two categories as ‘‘partition.’’ 1 Wearden and Ferrara (1995) compared the similarity and partition tasks and concluded that they yielded similar data. Allan (1999) and Allan and Gerhardt (2001) suggested that a decision rule that does not involve a direct comparison with the referents might be more appropriate than the similarity rule for both the similarity and the partition bisection tasks. They suggested that the bisection task be considered within a signal detection context. That is, the observer in bisection establishes a criterion duration value which is determined by the values of the referents. On each trial, the observer compares t to this single criterion rather than to memories of the two referents. In fact, Gibbon (1981) had derived a signal detection bisection function based on the comparison of the probe to a criterion duration. In his derivation, he assumed that perceived time is normally distributed, that mean perceived time is a power function of clock time with an exponent close to 1.0, µ t ⫽ t,
(1)
and that the standard deviation of perceived time is proportional to mean perceived time, 1 While the partition task is similar to the many-to-few task described by Allan (1979) in that 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.
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LORRAINE G. ALLAN
σt σt ⫽ ⫽ γ. µt t
(2)
The proportionality constant γ is the Weber fraction. Gibbon (1981) referred to the proportionality relation in Eq. (2) as scalar variability or the scalar property. This scalar property between the standard deviation and the mean results in distributions of perceived time that superpose when the temporal axis is normalized with respect to the mean of the distribution. For bisection, the scalar property predicts superposition of psychometric functions for all r values when plotted against t normalized by the bisection point (i.e., t/T 1/2). Recently, Killeen, Fetterman, and Bizo (1997) simplified the Gibbon (1981) derivation of, and the resulting equation for, the bisection function by substituting the logistic distribution for the normal distribution. For the logistic approximation to the normal, the bisection function is
P(R L) ⫽
冤 冢 冣冥 1 ⫹ exp
T 1/2 ⫺ t √3 σt π
⫺1
,
(3)
where T 1/2 , the bisection point, is the criterion and σ t ⫽ √(γt) 2 ⫹ pt ⫹ c 2 .
(4)
Equation 4 is the Weber Function derived by Killeen and Weiss (1987) which provides 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(R L) ⫽
冤 冢 冣冥 T ⫺t 1 ⫹ exp 1/2 √3 γt π
⫺1
.
(5)
Killeen et al. (1997) noted that the bisection function in Eq. (3) is not a logistic function even though it is derived from one and that it is not a distribution function since it asymptotes at a value less than 1.0. They, therefore, referred to their bisection function as a Pseudo-Logistic function and their model as the Pseudo-Logistic Model (PLM). Rodriguez-Girones and Kacelnik (2001) also modified the prototypic bi-
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section task. In their modification, three duration values were presented sequentially on each trial: the two referents, counterbalanced for order, and then the probe. Observers were instructed to indicate whether the last duration on the trial (i.e., t) was more similar to the duration of the first or second trial referent. Rodriguez-Girones and Kacelnik (2001) examined two versions of trial-referent bisection. In fixed-referent bisection, the referent pair was constant during a block of trials, whereas in roving-referent bisection, the referent pair varied from trial to trial. They argued that stored memories of S and L could not be established in roving-referent bisection since S and L varied from trial to trial. They suggested that the probe on each trial was compared 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 in prototypic (i.e., no trial referents) and trial-referent bisection. They concluded that the decision to respond R S or R L was not based on a direct comparison of the probe to the referents either in prototypic or in trial-referent bisection. They suggested that t was compared to a criterion duration value in both prototypic and trial-referent bisection, and they showed that PLM provided an excellent account of their data. Thus, while the Rodriguez-Girones and Kacelnik (2001) analysis of their roving-referent data suggested that the probe was compared to the trial referents, the Allan and Gerhardt (2001) analysis of their roving-referent data suggested that the probe was not compared to the trial referents. The number of S and L referent pairs in the roving task in Rodriguez-Girones and Kacelnik (2001) was much larger than in Allan and Gerhardt (in press). It might be that the decision process in roving-referent bisection depends on the number of referent pairs. In the present experiments we show that the results reported by Allan and Gerhardt (2001) are not attributable to the limited number of their S and L referent pairs. If the probe is not directly compared to the trial referents, then the length of the delay between the termination of the referent pair and the presentation of the probe (the referent–probe interval, RPI) should not affect the slope of the bisection function. In Allan and Gerhardt (2001), RPI was constant at 1000 ms. In Experiment 2, we vary RPI. Often, bisection experiments have been conducted to determine the location of T 1/2, 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 provide trial feedback to examine its influence on the decision process. Specifically, we were interested in determining whether feedback in roving-referent bisection would result in t being compared to the trial referents rather than to a criterion value.
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LORRAINE G. ALLAN
GENERAL METHOD
Observers The observers were graduate students or research assistants in the Psychology Department at McMaster University who were paid for their participation. They were required to complete a minimum of five sessions per week, with the restriction of a maximum of two sessions (separated by at least 1 h) per day. Six observers participated in Experiment 1, four observers participated in Experiment 2, and four observers participated in Experiment 3. Of the four observers in Experiment 2, one (VL) had previously participated in Experiment 1. All observers in Experiment 3 had participated in either Experiment 1 (KG and NC) or Experiment 2 (LH and CN). Apparatus Temporal parameters, stimulus presentation, and recording of responses were controlled with a Macintosh computer. The temporal intervals were visual and filled. They were marked by clearly visible geometric forms displayed on an Apple color monitor. S referent values were always marked by a red circle, L referent values by a green circle, and probe values by a black square. EXPERIMENT 1
In the Rodriguez-Girones and Kacelnik (2001) experiment, S was uniformly distributed between .5 and 2.0 s. For any value of S, L ⫽ (r)(S) for 1.5 ⱕ r ⱕ 8, with an upper limit of L ⫽ 5.5 s. For example, for S ⫽ 1, L was equally likely to take on any value between 1.5 and 5.5 s. The duration of t was a multiple τ of the geometric mean of the trial referents (t ⫽ τ√SL), where .5 ⱕ τ ⱕ 2.0. These algorithms for determining S, L, and t allowed for values of t less than S and greater than L. In the typical bisection task, S ⱕ t ⱕ L, and there are data which indicate that the bisection function obtained when t is not between S and L is different from that obtained when t is between S and L (e.g., Siegel & Church, 1984). In Experiment 1, we restricted the range of t so that S ⱕ t ⱕ L. In order to avoid explicit counting, Allan and Gerhardt (2001) used duration values which were considerably shorter than those in Rodriguez-Girones and 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 randomly selected from a uniform distribution bounded by 400 and 700 ms, and L was the integer value of (r)(S), for r ⫽ 1.50, 1.75, or 2.00. The value of r was constant during a session and varied between sessions. Five probe categories were associated with every referent pair: S, S⫹, GM, L⫺, and L,
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15
where GM was the integer millisecond value of the geometric mean of the S and L referents on that trial, S⫹ was the value of the mean of S and GM, and L⫺ was the value of the mean of GM and L. The value of t on each trial was randomly selected from the five probe categories with the restriction that each category occurred 24 times during a block of 120 trials. Thus the actual value of t for a particular category depended on the values of S and L on that trial. For example, suppose the category on a particular trial was GM and r ⫽ 2.00. If S ⫽ 480 were selected for that trial, then L ⫽ 960, GM ⫽ 678, and t ⫽ 678. However, if S ⫽ 630 were selected for that trial, then L ⫽ 1260, GM ⫽ 890, and t ⫽ 890. On each trial, t was preceded by the two referents, counterbalanced for order: S, L, t or L, S, t. The interval between the three successive trial durations was 1 s. The observer’s task on each trial was to decide whether t was more similar to S or to L. At the termination of the probe, the observer indicated the decision by pressing S or L on the computer keyboard. There was no 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 each value of r occurred once before any value of r was repeated. A session lasted about 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 sense to calculate the probability of responding L for a given probe duration. Indeed, classifying a 2-s probe as L reflects different processes if the standards were 1-s and 2-s long or if they were 2-s and 4-s long (p. 534).
This assumption that t was compared to the trial referents resulted in Rodriguez-Girones and Kacelnik (in press) normalizing t by the geometric mean of the trial referents, GM ⫽ √SL. As a first step in the analysis of our data, we also normalized t by GM. For each r value, normalization of t by GM collapses the many values of t to five values, one for each of the five probe categories (S, S⫹, GM, L⫺, and L). For each observer, P(R L) for each r value was based on all four blocks from all four sessions. Figure 1 shows P(R L), averaged over observers, plotted as a function of t/GM, separately for each value of r. The bisection functions are 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 is inappropriate. Figure 2 again shows P(R L), averaged over observers, plotted as a function of t/GM. Each panel illustrates the data for one of the three r values. In each panel, there are four functions. One of the functions represents the data collapsed across all S values (i.e., the data from Fig. 1). The other three functions represent the data for three ranges of S values: 400 ⱕ
16
FIG. 1. ment 1).
LORRAINE G. ALLAN
P(R L), averaged over observers, as a function of t/GM for each r value (Experi-
S ⱕ 499, 500 ⱕ S ⱕ 599, and 600 ⱕ S ⱕ 700. If t were being compared to 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 the range of S. Specifically, for every value of t/GM, P(R L) increased as S increased. Since for a constant t/GM as S increased so did t, it appears that R L was determined by t rather than t/GM. The data in Fig. 2 suggest that t, rather than being compared to the trial referents, is judged independently of the referents. The values of t ranged from 400 to (r)(700) ms, in 1-ms steps. We grouped the t values in 100-ms bins (i.e., 400–499, 500–599, etc.) and determined P(R L) for each bin. Figure 3 shows P(R L) for each observer as a function of t, where t is the midpoint of a 100-ms bin. For every observer, P(R L) is a monotonic function of t, and as r increases the functions are displaced to the left. The data in Fig. 3 suggest that the roving-referent bisection task was transformed by the observers into the partition bisection task. As t increased, so did P(R L) regardless of the trial referent pair. While the lower boundary for S was the same for all values of r, the upper boundary increased with r, resulting in P(R L) for a particular value of t decreasing as r increased. For example, the observer was less likely to categorize t ⫽ 500 ms as long when r ⫽ 2 than when r ⫽ 1.5. The data from Experiment 1 are consistent with the conclusion reached by Allan and Gerhardt (2001) that t is not compared to the trial referents in roving-referent bisection. They suggested that t is compared to a criterion duration value, and they showed that their data were consistent with PLM. Killeen et al. (1997) developed PLM for the prototypic (i.e., no trial referents) bisection task and suggested that the role of the referents is to set the value of the criterion T 1/2 . On each trial, the perceived value of the probe is compared to T 1/2 , and the decision is R L if the perceived value is larger than T 1/2. In roving-referent bisection, however, there are multiple referent pairs.
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FIG. 2. P(R L), averaged over observers, as a function of t/GM. Each panel presents the data for one of the r values. The filled symbols represent the data for the three ranges of S values and the open symbols represent the data collapsed over all S values (Experiment 1).
Allan and Gerhardt (2001) suggested that in roving-referent bisection, like in partition, the value of the T 1/2 is set by the range of t values. In applying PLM to the present data, we assumed, as did Allan and Gerhardt (2001), that the scalar sources of variance dominated (i.e., p ⫽ 0 and c ⫽ 0 in Eq. (4)). The similar slopes for the three values of r in Fig. 3 suggested that γ might be constant across the three functions. Thus, we fit Eq. (5) to the data of each observer (using the nonlinear fit algorithm from Mathematica), allowing T 1/2 to vary but keeping γ constant across the three functions. Table 1 gives, for each observer, the resulting parameter values. The lines in Fig. 3 are the values of P(R L) predicted by PLM. It is clear that PLM provided an excellent fit to the data. Averaged over the three functions, the sum of the squared deviations between the data and the predictions of PLM (ω 2) ranged from .992 to .999 for the six observers.
FIG. 3. P(R L), for each observer, as a function of t for each r value. The symbols represent the data for the three values of r and the lines are the predictions of PLM (Experiment 1).
18 LORRAINE G. ALLAN
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TABLE 1 Values of γ and T 1/2 in Experiment 1 Observer BM
SH
KE
VL
NC
KG
r
γ
T 1/2
1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00
.17
680 740 781 688 764 836 471 476 536 598 619 646 660 717 742 712 779 822
.14
.22
.18
.17
.21
Figure 4 plots, for each observer, the three psychometric functions against t normalized by the bisection point (t/T 1/2). For all observers, the three functions superposed across r. The superposition across r is consistent with the scalar property. In PLM, if responding is unbiased, T 1/2 should be located at the value of t where the S distribution crosses the L distribution. Killeen et al. (1997) provided an approximated solution for Unbiased T 1/2 , but since then, Killeen (personal communication) has provided an exact solution (see Allan & Gerhardt, 2001), Unbiased T 1/2 ⫽
(S ⫺ rL) ⫹ √(S ⫺ rL) 2 ⫺ (1 ⫺ r)((γS) 2 ln(r)) , 1⫺r
(6)
where S is the shortest value of t, L is the longest value of t, and r ⫽ (S/L) 2. In Experiment 1, the range of S values was the same for the three values of r, 400 to 700 ms. The effect of increasing r was to increase the upper limit for L, and therefore t, from 1050 ms for r ⫽ 1.5 to 1400 ms for r ⫽ 2. Thus, as the possible values of t increased, so would the value of Unbiased T 1/2 . T 1/2 estimated from the data (Table 1) is plotted in Fig. 5, averaged over observers, as a function of r. Also shown are the Unbiased T 1/2 values predicted by Eq. (6). T 1/2 , like Unbiased T 1/2 , 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 T 1/2
FIG. 4.
P(R L), for each observer, as a function of t/T 1/2 for each r value. The symbols represent the data for the three values of r (Experiment 1).
20 LORRAINE G. ALLAN
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21
FIG. 5. T 1/2, averaged over observers, as a function of r. The filled symbols represent the data and the open symbols represent Unbiased T 1/2 (Experiment 1).
relative to Unbiased T 1/2 , with some observers setting T 1/2 larger than Unbiased T 1/2 and other observers setting T 1/2 smaller than Unbiased T 1/2. This between-observer variability has been found in our previous applications of PLM to bisection data (Allan, 1999; Allan & Gerhardt, 2001). Discussion The data from Experiment 1 indicate that the probe in trial-referent bisection is not compared directly to the trial referents. Rather the trial-referent task appears to be transformed into the partition task. The present experiment shows that this transformation is not limited to the roving task used by Allan and Gerhardt (2001) where the variability in the trial referent pairs was minimal. Rather, the transformation of roving-referent bisection into partition bisection also occurs when the variability in referent pairs is comparable to that in Rodriguez-Girones and Kacelnik (2001). It is plausible that this transformation also occurred in the Rodriguez-Girones and Kacelnik (2001) roving task but was not observed because it was hidden by their data analysis which 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 often predict that memory variability should increase as the temporal interval between the two events to be compared is increased. For example, in their random-walk model for the comparison of two trial events, Kinchla and Allan (1969) suggested that the variability in the sensory representation of the first event increased during the interval separating the two events. Data exist that indicate that as the interval is increased from 500 to 2000 ms, discrimina-
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LORRAINE G. ALLAN
bility for dimensions such as intensity and movement decreases (see Allan, Kristofferson, & Rice, 1974). Thus, if the probe in trial-referent bisection were compared to the trial referents that must be remembered over RPI, one might expect γ to increase as RPI is increased. In contrast, if the probe is compared to a criterion rather than to the trial referents, then one would expect γ to remain constant across variations in RPI. In Experiment 1, RPI was constant 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 was varied 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. The r values were the same as in Experiment 1 (1.50, 1.75, and 2.00), and again r was constant within a session and was varied between sessions. The experiment consisted of 24 sessions, 8 at each of the three r values. A session lasted about 45–60 min, depending on the value of RPI and the self-pacing of the observer’s responding. In all other respects, the procedure was the same 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(R L), averaged over observers, as a function of t. As in Experiment 1, each t value is the midpoint of a 100-ms bin. Each panel of Fig. 6 plots the four RPI functions for each value of r. For each r value, the functions are remarkably similar across the four RPI values. If RPI had no effect on γ, then one should be able to fit PLM to the 12 functions with a single value of γ. Table 2 gives the parameter values obtained by fitting Eq. (5) to the data of each observer, keeping γ constant across the 12 functions. Averaged over the 12 functions, ω 2 ranged from .995 to .998 for the four observers. Figure 7 plots, for each observer, the 12 psychometric functions against t normalized by the bisection point (t/T 1/2). For all observers, the 12 functions superposed across r and RPI. The superposition across r is consistent with the scalar property. The superposition across RPI indicates that γ is unaffected by RPI. T 1/2 estimated from the data (Table 2) is plotted in Fig. 8 as a function of RPI for each value of r. As in Experiment 1, T 1/2 increased with r. For each r value, T 1/2 decreased with increasing RPI. An examination of Table 2 reveals that this decrease was mainly due to two observers (LH and VL). For the other two observers (CH and CN), T 1/2 was relatively constant across RPI. The data from Experiment 2 clearly indicate that γ remained constant across variations in RPI. This result is consistent with the view that t is not compared to the trial referents.
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FIG. 6. P(R L), averaged over observers, as a function of t for the four RPI values. Each panel presents the data for one of the r values (Experiment 2).
TABLE 2 Values of γ and T 1/2 in Experiment 2 T 1/2 Observer LH
CH
CN
VL
r
γ
500
1000
1500
2000
1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00
.15
746 765 797 645 687 735 666 708 749 660 691 718
713 758 795 666 690 742 670 710 762 656 688 711
682 725 770 658 694 730 650 696 753 621 660 701
633 694 733 654 692 735 639 693 752 592 659 682
.12
.14
.17
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LORRAINE G. ALLAN
FIG. 7. P(R L), for each observer, as a function of t/T 1/2. There are 12 functions (3 r values ⫻ 4 RPI values) for each observer (Experiment 2).
EXPERIMENT 3
The data from Experiments 1 and 2 indicate that the probe is not compared to the trial referents in roving-referent bisection. As is typically the case in bisection, trial feedback was not provided in Experiments 1 and 2. In Experiment 3, we examine whether trial feedback, based on comparing the probe to the trial referents, would influence performance. Specifically, we were
REFERENTS AND BISECTION
FIG. 8. ment 2).
25
T 1/2 , averaged over observers, as a function of RPI for each r value (Experi-
interested in determining whether such feedback would induce the observer to 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 session consisted of four blocks of 96 trials. Six probe categories were associated with every referent pair: S, S⫹, S⫹⫹, L⫺⫺, L⫺, and L. The spacing between the six probe categories was linear (40 ms for r ⫽ 1.5, and 80 ms for r ⫽ 2.0). The value of t on each trial was randomly selected from the six probe categories with the restriction that each category occurred 16 times during a block of 96 trials. Auditory computer feedback, ‘‘correct’’ or ‘‘error,’’ was provided immediately after the observer entered the response. The next trial began 1.5 s after the feedback terminated. In all other respects, the procedure was the same as in Experiment 1. Feedback was determined by the relationship between t and the arithmetic mean (AM) of the trial referents. Specifically, t ⬍ AM (i.e., S, S⫹, and S⫹⫹) was considered ‘‘short’’ and t ⬎ AM (i.e., L⫺⫺, L⫺, and L) was considered ‘‘long.’’ Thus, the feedback was ‘‘correct’’ if R S occurred to S, S⫹, or S⫹⫹ or R L occurred to L⫺⫺, L⫺, and L, and the feedback was ‘‘error’’ if R S occurred to L⫺⫺, L⫺, or L or R L occurred to S, S⫹, and S⫹⫹. Results and Discussion Figure 9 shows P(R L), averaged over observers, plotted as a function of t/GM. Each panel illustrates the data for one of the two r values. In each panel, there are four functions, one for trials where 400 ⱕ S ⱕ 499, one for trials where 500 ⱕ S ⱕ 599, one for trials where 600 ⱕ S ⱕ 700, and one collapsed across all S values. The pattern of results in Fig. 9 is very similar
FIG. 9. P(R L), averaged over observers, as a function of t/GM. Each panel presents the data for one of the r values. The filled symbols represent the data for the three ranges of S values and the open symbols represent the data collapsed over all S values (Experiment 3).
26 LORRAINE G. ALLAN
FIG. 10. P(R L), for each observer, as a function of t. The filled symbols represent the data for the two values of r in Experiment 3 and the open symbols represent the appropriate data from a previous experiment.
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LORRAINE G. ALLAN
TABLE 3 Values of γ and T 1/2 in Experiment 3 Observer CN KG LH NC
r
γ
T 1/2
1.50 2.00 1.50 2.00 1.50 2.00 1.50 2.00
.14
674 762 683 818 688 765 681 759
.21 .15 .17
670 762 712 822 713 795 660 742
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. Each of the observers in Experiment 3 had participated in either Experiment 1 or Experiment 2. The comparable data from the earlier experiments are also shown. It is remarkable how similar the feedback data are to the earlier nofeedback data. The presentation of explicit feedback, based on the relationship 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 each observer. Table 3 gives, for each observer, the resulting parameter values. Even with the constraint imposed by using a value of γ estimated from a different set of data, PLM provided an excellent fit to the data, with ω 2 ranging from .990 to .997 for the four observers. Figure 11 plots, for each observer, the four psychometric functions against t normalized by the bisection point (t/T 1/2). CONCLUDING COMMENTS
The data from all three experiments consistently indicate that the probe is not compared to the trial referents in roving-referent bisection. Rather the judgment appears to be based on the absolute value of t rather than on its relationship to the trial S and L values. The reliance on the absolute value of t occurred even when explicit feedback, based on the trial referents, was provided. Even though this feedback would often indicate that the response was an ‘‘error,’’ the observers did not modify their responding to be consistent with the feedback. In fact, performance with and without feedback was remarkably similar for all observers. Although many studies have used bisection to study temporal memory (see Allan, 1998), the data from the present experiments suggest that referent memory is not a dominant source of variability 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
FIG. 11. P(R L), for each observer, as a function of t/T 1/2 . The filled symbols represent the data for the two values of r in Experiment 3 and the open symbols represent the appropriate data from a previous experiment.
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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) bisection task, and suggested that the role of the referents is to set the value of the criterion T 1/2 . In roving-referent bisection, however, there are multiple referent pairs. Allan and Gerhardt (2001) suggested that in roving-referent bisection, like in partition, the value of the T 1/2 is set by the range of t values. There are other data in the literature that are consistent with our conclusion that 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), the other long (L), are presented sequentially on each trial, counterbalanced for order (SL or LS), and the task is to indicate the order, R SR L or R LR S. These data indicated that variability remained constant across variations in the interval between the two trial durations. Allan et al. (1974; see also Allan, 1977) concluded that the two trial durations were not directly compared to each other, but rather that each was compared to a criterion value and each was categorized as R S or R L. The intriguing, and thus far unanswered, question is why is the probe not compared to the trial referents. Our present data, as well as our earlier data, suggest that direct comparisons of trial durations are difficult, if not impossible, for human observers. REFERENCES Allan, L. G. (1977). The time-order error in judgments of duration. Canadian Journal of Psychology, 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 International 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 & Motivation, 22, 39–58. Allan, L. G., Kristofferson, A. B., & Rice, M. E. (1974). Some aspects of perceptual coding of 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 of Mathematical 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. Psychological Review, 94, 455–468. Kinchla, R. A., & Allan, L. G. (1969). A theory of visual movement perception. Psychological Review, 76, 537–558.
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Penney, T. B., Allan, L. G., Meck, W. H., & Gibbon, J. (1998). Memory mixing in duration bisection. In D. A. Rosenbaum & C. E. Collyer (Eds.), Timing of behavior: Neural, psychological, and computational perspectives (pp. 165–193). Cambridge, MA: MIT Press. Rodriguez-Girones, M. A., & Kacelnik, A. (2001). Relative importance of perceptual and mnemonic variance in human temporal bisection. Quarterly Journal of Experimental Psychology, 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 of the 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 humans. 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 longer stimulus durations. Quarterly Journal of Experimental Psychology, 50B, 79–94. Received April 1, 2001, revised May 7, 2001
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 on each trial, two referents and then a probe, and the observer is instructed to indicate whether 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 human observers 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 on its 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 is compared to a criterion duration value. Although many studies have used bisection to study temporal memory, the data from the present experiments suggest that referent 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 of the bisection function for time,’’ launched temporal bisection as a popular psychophysical procedure for the study of time perception. In the prototypic temporal bisection task with humans, two referents, one short (S) and the other long (L), are explicitly identified by familiarizing the observer with the 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, a temporal interval t, S ⱕ t ⱕ L, is presented, and the observer is required to indicate whether t is more similar to S (R S) or to L (R L). Temporal bisection yields a psychometric function relating the proportion of long responses, P(R L), to probe duration t. The value of t at which R S and R L occur with equal frequency, P(R L) ⫽ 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 equally confusable with S and L. The research reported in this paper was supported by a grant to Lorraine G. Allan from the Natural Sciences and Engineering Research Council of Canada. Correspondence and 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]. 10 0023-9690/02 $35.00 2002 Elsevier Science (USA) All rights reserved.
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Since the referents are not available on probe trials, it was often assumed that stored memories of S and L were established and that on each trial the probe was compared to these stored memories (e.g., Allan & Gibbon, 1991; Gibbon, 1981; Wearden, 1991). That is, the decision to respond R S or R L was made by comparing the similarity of the perceived value of t on a given trial to memories of the two referents, S and L. In fact, temporal bisection became a favored psychophysical procedure for the study of temporal memory in humans (see Allan, 1998). More recent reports, however, have suggested 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 experiments was to obtain a better understanding of the role of the referents in temporal bisection. A more general goal was to further our understanding of temporal memory. Wearden and Ferrara (1995, 1996) modified the prototypic bisection task so 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, R S or R L. To distinguish the two tasks, Wearden and Ferrara (1995, 1996) labeled the task in which the referents are identified and observers are instructed to compare the similarity of t to the referents as ‘‘similarity’’ and the task in which the referents are not identified and observers are instructed to partition the t values into two categories as ‘‘partition.’’ 1 Wearden and Ferrara (1995) compared the similarity and partition tasks and concluded that they yielded similar data. Allan (1999) and Allan and Gerhardt (2001) suggested that a decision rule that does not involve a direct comparison with the referents might be more appropriate than the similarity rule for both the similarity and the partition bisection tasks. They suggested that the bisection task be considered within a signal detection context. That is, the observer in bisection establishes a criterion duration value which is determined by the values of the referents. On each trial, the observer compares t to this single criterion rather than to memories of the two referents. In fact, Gibbon (1981) had derived a signal detection bisection function based on the comparison of the probe to a criterion duration. In his derivation, he assumed that perceived time is normally distributed, that mean perceived time is a power function of clock time with an exponent close to 1.0, µ t ⫽ t,
(1)
and that the standard deviation of perceived time is proportional to mean perceived time, 1 While the partition task is similar to the many-to-few task described by Allan (1979) in that 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.
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σt σt ⫽ ⫽ γ. µt t
(2)
The proportionality constant γ is the Weber fraction. Gibbon (1981) referred to the proportionality relation in Eq. (2) as scalar variability or the scalar property. This scalar property between the standard deviation and the mean results in distributions of perceived time that superpose when the temporal axis is normalized with respect to the mean of the distribution. For bisection, the scalar property predicts superposition of psychometric functions for all r values when plotted against t normalized by the bisection point (i.e., t/T 1/2). Recently, Killeen, Fetterman, and Bizo (1997) simplified the Gibbon (1981) derivation of, and the resulting equation for, the bisection function by substituting the logistic distribution for the normal distribution. For the logistic approximation to the normal, the bisection function is
P(R L) ⫽
冤 冢 冣冥 1 ⫹ exp
T 1/2 ⫺ t √3 σt π
⫺1
,
(3)
where T 1/2 , the bisection point, is the criterion and σ t ⫽ √(γt) 2 ⫹ pt ⫹ c 2 .
(4)
Equation 4 is the Weber Function derived by Killeen and Weiss (1987) which provides 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(R L) ⫽
冤 冢 冣冥 T ⫺t 1 ⫹ exp 1/2 √3 γt π
⫺1
.
(5)
Killeen et al. (1997) noted that the bisection function in Eq. (3) is not a logistic function even though it is derived from one and that it is not a distribution function since it asymptotes at a value less than 1.0. They, therefore, referred to their bisection function as a Pseudo-Logistic function and their model as the Pseudo-Logistic Model (PLM). Rodriguez-Girones and Kacelnik (2001) also modified the prototypic bi-
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section task. In their modification, three duration values were presented sequentially on each trial: the two referents, counterbalanced for order, and then the probe. Observers were instructed to indicate whether the last duration on the trial (i.e., t) was more similar to the duration of the first or second trial referent. Rodriguez-Girones and Kacelnik (2001) examined two versions of trial-referent bisection. In fixed-referent bisection, the referent pair was constant during a block of trials, whereas in roving-referent bisection, the referent pair varied from trial to trial. They argued that stored memories of S and L could not be established in roving-referent bisection since S and L varied from trial to trial. They suggested that the probe on each trial was compared 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 in prototypic (i.e., no trial referents) and trial-referent bisection. They concluded that the decision to respond R S or R L was not based on a direct comparison of the probe to the referents either in prototypic or in trial-referent bisection. They suggested that t was compared to a criterion duration value in both prototypic and trial-referent bisection, and they showed that PLM provided an excellent account of their data. Thus, while the Rodriguez-Girones and Kacelnik (2001) analysis of their roving-referent data suggested that the probe was compared to the trial referents, the Allan and Gerhardt (2001) analysis of their roving-referent data suggested that the probe was not compared to the trial referents. The number of S and L referent pairs in the roving task in Rodriguez-Girones and Kacelnik (2001) was much larger than in Allan and Gerhardt (in press). It might be that the decision process in roving-referent bisection depends on the number of referent pairs. In the present experiments we show that the results reported by Allan and Gerhardt (2001) are not attributable to the limited number of their S and L referent pairs. If the probe is not directly compared to the trial referents, then the length of the delay between the termination of the referent pair and the presentation of the probe (the referent–probe interval, RPI) should not affect the slope of the bisection function. In Allan and Gerhardt (2001), RPI was constant at 1000 ms. In Experiment 2, we vary RPI. Often, bisection experiments have been conducted to determine the location of T 1/2, 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 provide trial feedback to examine its influence on the decision process. Specifically, we were interested in determining whether feedback in roving-referent bisection would result in t being compared to the trial referents rather than to a criterion value.
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GENERAL METHOD
Observers The observers were graduate students or research assistants in the Psychology Department at McMaster University who were paid for their participation. They were required to complete a minimum of five sessions per week, with the restriction of a maximum of two sessions (separated by at least 1 h) per day. Six observers participated in Experiment 1, four observers participated in Experiment 2, and four observers participated in Experiment 3. Of the four observers in Experiment 2, one (VL) had previously participated in Experiment 1. All observers in Experiment 3 had participated in either Experiment 1 (KG and NC) or Experiment 2 (LH and CN). Apparatus Temporal parameters, stimulus presentation, and recording of responses were controlled with a Macintosh computer. The temporal intervals were visual and filled. They were marked by clearly visible geometric forms displayed on an Apple color monitor. S referent values were always marked by a red circle, L referent values by a green circle, and probe values by a black square. EXPERIMENT 1
In the Rodriguez-Girones and Kacelnik (2001) experiment, S was uniformly distributed between .5 and 2.0 s. For any value of S, L ⫽ (r)(S) for 1.5 ⱕ r ⱕ 8, with an upper limit of L ⫽ 5.5 s. For example, for S ⫽ 1, L was equally likely to take on any value between 1.5 and 5.5 s. The duration of t was a multiple τ of the geometric mean of the trial referents (t ⫽ τ√SL), where .5 ⱕ τ ⱕ 2.0. These algorithms for determining S, L, and t allowed for values of t less than S and greater than L. In the typical bisection task, S ⱕ t ⱕ L, and there are data which indicate that the bisection function obtained when t is not between S and L is different from that obtained when t is between S and L (e.g., Siegel & Church, 1984). In Experiment 1, we restricted the range of t so that S ⱕ t ⱕ L. In order to avoid explicit counting, Allan and Gerhardt (2001) used duration values which were considerably shorter than those in Rodriguez-Girones and 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 randomly selected from a uniform distribution bounded by 400 and 700 ms, and L was the integer value of (r)(S), for r ⫽ 1.50, 1.75, or 2.00. The value of r was constant during a session and varied between sessions. Five probe categories were associated with every referent pair: S, S⫹, GM, L⫺, and L,
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where GM was the integer millisecond value of the geometric mean of the S and L referents on that trial, S⫹ was the value of the mean of S and GM, and L⫺ was the value of the mean of GM and L. The value of t on each trial was randomly selected from the five probe categories with the restriction that each category occurred 24 times during a block of 120 trials. Thus the actual value of t for a particular category depended on the values of S and L on that trial. For example, suppose the category on a particular trial was GM and r ⫽ 2.00. If S ⫽ 480 were selected for that trial, then L ⫽ 960, GM ⫽ 678, and t ⫽ 678. However, if S ⫽ 630 were selected for that trial, then L ⫽ 1260, GM ⫽ 890, and t ⫽ 890. On each trial, t was preceded by the two referents, counterbalanced for order: S, L, t or L, S, t. The interval between the three successive trial durations was 1 s. The observer’s task on each trial was to decide whether t was more similar to S or to L. At the termination of the probe, the observer indicated the decision by pressing S or L on the computer keyboard. There was no 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 each value of r occurred once before any value of r was repeated. A session lasted about 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 sense to calculate the probability of responding L for a given probe duration. Indeed, classifying a 2-s probe as L reflects different processes if the standards were 1-s and 2-s long or if they were 2-s and 4-s long (p. 534).
This assumption that t was compared to the trial referents resulted in Rodriguez-Girones and Kacelnik (in press) normalizing t by the geometric mean of the trial referents, GM ⫽ √SL. As a first step in the analysis of our data, we also normalized t by GM. For each r value, normalization of t by GM collapses the many values of t to five values, one for each of the five probe categories (S, S⫹, GM, L⫺, and L). For each observer, P(R L) for each r value was based on all four blocks from all four sessions. Figure 1 shows P(R L), averaged over observers, plotted as a function of t/GM, separately for each value of r. The bisection functions are 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 is inappropriate. Figure 2 again shows P(R L), averaged over observers, plotted as a function of t/GM. Each panel illustrates the data for one of the three r values. In each panel, there are four functions. One of the functions represents the data collapsed across all S values (i.e., the data from Fig. 1). The other three functions represent the data for three ranges of S values: 400 ⱕ
16
FIG. 1. ment 1).
LORRAINE G. ALLAN
P(R L), averaged over observers, as a function of t/GM for each r value (Experi-
S ⱕ 499, 500 ⱕ S ⱕ 599, and 600 ⱕ S ⱕ 700. If t were being compared to 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 the range of S. Specifically, for every value of t/GM, P(R L) increased as S increased. Since for a constant t/GM as S increased so did t, it appears that R L was determined by t rather than t/GM. The data in Fig. 2 suggest that t, rather than being compared to the trial referents, is judged independently of the referents. The values of t ranged from 400 to (r)(700) ms, in 1-ms steps. We grouped the t values in 100-ms bins (i.e., 400–499, 500–599, etc.) and determined P(R L) for each bin. Figure 3 shows P(R L) for each observer as a function of t, where t is the midpoint of a 100-ms bin. For every observer, P(R L) is a monotonic function of t, and as r increases the functions are displaced to the left. The data in Fig. 3 suggest that the roving-referent bisection task was transformed by the observers into the partition bisection task. As t increased, so did P(R L) regardless of the trial referent pair. While the lower boundary for S was the same for all values of r, the upper boundary increased with r, resulting in P(R L) for a particular value of t decreasing as r increased. For example, the observer was less likely to categorize t ⫽ 500 ms as long when r ⫽ 2 than when r ⫽ 1.5. The data from Experiment 1 are consistent with the conclusion reached by Allan and Gerhardt (2001) that t is not compared to the trial referents in roving-referent bisection. They suggested that t is compared to a criterion duration value, and they showed that their data were consistent with PLM. Killeen et al. (1997) developed PLM for the prototypic (i.e., no trial referents) bisection task and suggested that the role of the referents is to set the value of the criterion T 1/2 . On each trial, the perceived value of the probe is compared to T 1/2 , and the decision is R L if the perceived value is larger than T 1/2. In roving-referent bisection, however, there are multiple referent pairs.
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FIG. 2. P(R L), averaged over observers, as a function of t/GM. Each panel presents the data for one of the r values. The filled symbols represent the data for the three ranges of S values and the open symbols represent the data collapsed over all S values (Experiment 1).
Allan and Gerhardt (2001) suggested that in roving-referent bisection, like in partition, the value of the T 1/2 is set by the range of t values. In applying PLM to the present data, we assumed, as did Allan and Gerhardt (2001), that the scalar sources of variance dominated (i.e., p ⫽ 0 and c ⫽ 0 in Eq. (4)). The similar slopes for the three values of r in Fig. 3 suggested that γ might be constant across the three functions. Thus, we fit Eq. (5) to the data of each observer (using the nonlinear fit algorithm from Mathematica), allowing T 1/2 to vary but keeping γ constant across the three functions. Table 1 gives, for each observer, the resulting parameter values. The lines in Fig. 3 are the values of P(R L) predicted by PLM. It is clear that PLM provided an excellent fit to the data. Averaged over the three functions, the sum of the squared deviations between the data and the predictions of PLM (ω 2) ranged from .992 to .999 for the six observers.
FIG. 3. P(R L), for each observer, as a function of t for each r value. The symbols represent the data for the three values of r and the lines are the predictions of PLM (Experiment 1).
18 LORRAINE G. ALLAN
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TABLE 1 Values of γ and T 1/2 in Experiment 1 Observer BM
SH
KE
VL
NC
KG
r
γ
T 1/2
1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00
.17
680 740 781 688 764 836 471 476 536 598 619 646 660 717 742 712 779 822
.14
.22
.18
.17
.21
Figure 4 plots, for each observer, the three psychometric functions against t normalized by the bisection point (t/T 1/2). For all observers, the three functions superposed across r. The superposition across r is consistent with the scalar property. In PLM, if responding is unbiased, T 1/2 should be located at the value of t where the S distribution crosses the L distribution. Killeen et al. (1997) provided an approximated solution for Unbiased T 1/2 , but since then, Killeen (personal communication) has provided an exact solution (see Allan & Gerhardt, 2001), Unbiased T 1/2 ⫽
(S ⫺ rL) ⫹ √(S ⫺ rL) 2 ⫺ (1 ⫺ r)((γS) 2 ln(r)) , 1⫺r
(6)
where S is the shortest value of t, L is the longest value of t, and r ⫽ (S/L) 2. In Experiment 1, the range of S values was the same for the three values of r, 400 to 700 ms. The effect of increasing r was to increase the upper limit for L, and therefore t, from 1050 ms for r ⫽ 1.5 to 1400 ms for r ⫽ 2. Thus, as the possible values of t increased, so would the value of Unbiased T 1/2 . T 1/2 estimated from the data (Table 1) is plotted in Fig. 5, averaged over observers, as a function of r. Also shown are the Unbiased T 1/2 values predicted by Eq. (6). T 1/2 , like Unbiased T 1/2 , 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 T 1/2
FIG. 4.
P(R L), for each observer, as a function of t/T 1/2 for each r value. The symbols represent the data for the three values of r (Experiment 1).
20 LORRAINE G. ALLAN
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FIG. 5. T 1/2, averaged over observers, as a function of r. The filled symbols represent the data and the open symbols represent Unbiased T 1/2 (Experiment 1).
relative to Unbiased T 1/2 , with some observers setting T 1/2 larger than Unbiased T 1/2 and other observers setting T 1/2 smaller than Unbiased T 1/2. This between-observer variability has been found in our previous applications of PLM to bisection data (Allan, 1999; Allan & Gerhardt, 2001). Discussion The data from Experiment 1 indicate that the probe in trial-referent bisection is not compared directly to the trial referents. Rather the trial-referent task appears to be transformed into the partition task. The present experiment shows that this transformation is not limited to the roving task used by Allan and Gerhardt (2001) where the variability in the trial referent pairs was minimal. Rather, the transformation of roving-referent bisection into partition bisection also occurs when the variability in referent pairs is comparable to that in Rodriguez-Girones and Kacelnik (2001). It is plausible that this transformation also occurred in the Rodriguez-Girones and Kacelnik (2001) roving task but was not observed because it was hidden by their data analysis which 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 often predict that memory variability should increase as the temporal interval between the two events to be compared is increased. For example, in their random-walk model for the comparison of two trial events, Kinchla and Allan (1969) suggested that the variability in the sensory representation of the first event increased during the interval separating the two events. Data exist that indicate that as the interval is increased from 500 to 2000 ms, discrimina-
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bility for dimensions such as intensity and movement decreases (see Allan, Kristofferson, & Rice, 1974). Thus, if the probe in trial-referent bisection were compared to the trial referents that must be remembered over RPI, one might expect γ to increase as RPI is increased. In contrast, if the probe is compared to a criterion rather than to the trial referents, then one would expect γ to remain constant across variations in RPI. In Experiment 1, RPI was constant 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 was varied 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. The r values were the same as in Experiment 1 (1.50, 1.75, and 2.00), and again r was constant within a session and was varied between sessions. The experiment consisted of 24 sessions, 8 at each of the three r values. A session lasted about 45–60 min, depending on the value of RPI and the self-pacing of the observer’s responding. In all other respects, the procedure was the same 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(R L), averaged over observers, as a function of t. As in Experiment 1, each t value is the midpoint of a 100-ms bin. Each panel of Fig. 6 plots the four RPI functions for each value of r. For each r value, the functions are remarkably similar across the four RPI values. If RPI had no effect on γ, then one should be able to fit PLM to the 12 functions with a single value of γ. Table 2 gives the parameter values obtained by fitting Eq. (5) to the data of each observer, keeping γ constant across the 12 functions. Averaged over the 12 functions, ω 2 ranged from .995 to .998 for the four observers. Figure 7 plots, for each observer, the 12 psychometric functions against t normalized by the bisection point (t/T 1/2). For all observers, the 12 functions superposed across r and RPI. The superposition across r is consistent with the scalar property. The superposition across RPI indicates that γ is unaffected by RPI. T 1/2 estimated from the data (Table 2) is plotted in Fig. 8 as a function of RPI for each value of r. As in Experiment 1, T 1/2 increased with r. For each r value, T 1/2 decreased with increasing RPI. An examination of Table 2 reveals that this decrease was mainly due to two observers (LH and VL). For the other two observers (CH and CN), T 1/2 was relatively constant across RPI. The data from Experiment 2 clearly indicate that γ remained constant across variations in RPI. This result is consistent with the view that t is not compared to the trial referents.
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FIG. 6. P(R L), averaged over observers, as a function of t for the four RPI values. Each panel presents the data for one of the r values (Experiment 2).
TABLE 2 Values of γ and T 1/2 in Experiment 2 T 1/2 Observer LH
CH
CN
VL
r
γ
500
1000
1500
2000
1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00 1.50 1.75 2.00
.15
746 765 797 645 687 735 666 708 749 660 691 718
713 758 795 666 690 742 670 710 762 656 688 711
682 725 770 658 694 730 650 696 753 621 660 701
633 694 733 654 692 735 639 693 752 592 659 682
.12
.14
.17
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FIG. 7. P(R L), for each observer, as a function of t/T 1/2. There are 12 functions (3 r values ⫻ 4 RPI values) for each observer (Experiment 2).
EXPERIMENT 3
The data from Experiments 1 and 2 indicate that the probe is not compared to the trial referents in roving-referent bisection. As is typically the case in bisection, trial feedback was not provided in Experiments 1 and 2. In Experiment 3, we examine whether trial feedback, based on comparing the probe to the trial referents, would influence performance. Specifically, we were
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FIG. 8. ment 2).
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T 1/2 , averaged over observers, as a function of RPI for each r value (Experi-
interested in determining whether such feedback would induce the observer to 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 session consisted of four blocks of 96 trials. Six probe categories were associated with every referent pair: S, S⫹, S⫹⫹, L⫺⫺, L⫺, and L. The spacing between the six probe categories was linear (40 ms for r ⫽ 1.5, and 80 ms for r ⫽ 2.0). The value of t on each trial was randomly selected from the six probe categories with the restriction that each category occurred 16 times during a block of 96 trials. Auditory computer feedback, ‘‘correct’’ or ‘‘error,’’ was provided immediately after the observer entered the response. The next trial began 1.5 s after the feedback terminated. In all other respects, the procedure was the same as in Experiment 1. Feedback was determined by the relationship between t and the arithmetic mean (AM) of the trial referents. Specifically, t ⬍ AM (i.e., S, S⫹, and S⫹⫹) was considered ‘‘short’’ and t ⬎ AM (i.e., L⫺⫺, L⫺, and L) was considered ‘‘long.’’ Thus, the feedback was ‘‘correct’’ if R S occurred to S, S⫹, or S⫹⫹ or R L occurred to L⫺⫺, L⫺, and L, and the feedback was ‘‘error’’ if R S occurred to L⫺⫺, L⫺, or L or R L occurred to S, S⫹, and S⫹⫹. Results and Discussion Figure 9 shows P(R L), averaged over observers, plotted as a function of t/GM. Each panel illustrates the data for one of the two r values. In each panel, there are four functions, one for trials where 400 ⱕ S ⱕ 499, one for trials where 500 ⱕ S ⱕ 599, one for trials where 600 ⱕ S ⱕ 700, and one collapsed across all S values. The pattern of results in Fig. 9 is very similar
FIG. 9. P(R L), averaged over observers, as a function of t/GM. Each panel presents the data for one of the r values. The filled symbols represent the data for the three ranges of S values and the open symbols represent the data collapsed over all S values (Experiment 3).
26 LORRAINE G. ALLAN
FIG. 10. P(R L), for each observer, as a function of t. The filled symbols represent the data for the two values of r in Experiment 3 and the open symbols represent the appropriate data from a previous experiment.
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TABLE 3 Values of γ and T 1/2 in Experiment 3 Observer CN KG LH NC
r
γ
T 1/2
1.50 2.00 1.50 2.00 1.50 2.00 1.50 2.00
.14
674 762 683 818 688 765 681 759
.21 .15 .17
670 762 712 822 713 795 660 742
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. Each of the observers in Experiment 3 had participated in either Experiment 1 or Experiment 2. The comparable data from the earlier experiments are also shown. It is remarkable how similar the feedback data are to the earlier nofeedback data. The presentation of explicit feedback, based on the relationship 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 each observer. Table 3 gives, for each observer, the resulting parameter values. Even with the constraint imposed by using a value of γ estimated from a different set of data, PLM provided an excellent fit to the data, with ω 2 ranging from .990 to .997 for the four observers. Figure 11 plots, for each observer, the four psychometric functions against t normalized by the bisection point (t/T 1/2). CONCLUDING COMMENTS
The data from all three experiments consistently indicate that the probe is not compared to the trial referents in roving-referent bisection. Rather the judgment appears to be based on the absolute value of t rather than on its relationship to the trial S and L values. The reliance on the absolute value of t occurred even when explicit feedback, based on the trial referents, was provided. Even though this feedback would often indicate that the response was an ‘‘error,’’ the observers did not modify their responding to be consistent with the feedback. In fact, performance with and without feedback was remarkably similar for all observers. Although many studies have used bisection to study temporal memory (see Allan, 1998), the data from the present experiments suggest that referent memory is not a dominant source of variability 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
FIG. 11. P(R L), for each observer, as a function of t/T 1/2 . The filled symbols represent the data for the two values of r in Experiment 3 and the open symbols represent the appropriate data from a previous experiment.
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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) bisection task, and suggested that the role of the referents is to set the value of the criterion T 1/2 . In roving-referent bisection, however, there are multiple referent pairs. Allan and Gerhardt (2001) suggested that in roving-referent bisection, like in partition, the value of the T 1/2 is set by the range of t values. There are other data in the literature that are consistent with our conclusion that 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), the other long (L), are presented sequentially on each trial, counterbalanced for order (SL or LS), and the task is to indicate the order, R SR L or R LR S. These data indicated that variability remained constant across variations in the interval between the two trial durations. Allan et al. (1974; see also Allan, 1977) concluded that the two trial durations were not directly compared to each other, but rather that each was compared to a criterion value and each was categorized as R S or R L. The intriguing, and thus far unanswered, question is why is the probe not compared to the trial referents. Our present data, as well as our earlier data, suggest that direct comparisons of trial durations are difficult, if not impossible, for human observers. REFERENCES Allan, L. G. (1977). The time-order error in judgments of duration. Canadian Journal of Psychology, 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 International 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 & Motivation, 22, 39–58. Allan, L. G., Kristofferson, A. B., & Rice, M. E. (1974). Some aspects of perceptual coding of 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 of Mathematical 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. Psychological Review, 94, 455–468. Kinchla, R. A., & Allan, L. G. (1969). A theory of visual movement perception. Psychological Review, 76, 537–558.
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Penney, T. B., Allan, L. G., Meck, W. H., & Gibbon, J. (1998). Memory mixing in duration bisection. In D. A. Rosenbaum & C. E. Collyer (Eds.), Timing of behavior: Neural, psychological, and computational perspectives (pp. 165–193). Cambridge, MA: MIT Press. Rodriguez-Girones, M. A., & Kacelnik, A. (2001). Relative importance of perceptual and mnemonic variance in human temporal bisection. Quarterly Journal of Experimental Psychology, 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 of the 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 humans. 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 longer stimulus durations. Quarterly Journal of Experimental Psychology, 50B, 79–94. Received April 1, 2001, revised May 7, 2001