The location and interpretation of the bisection point - Semantic Scholar
KEYED THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY, 2002, 55B (1), 43–60
The location and interpretation of the bisection point Lorraine G. Allan McMaster University, Hamilton, Canada In a temporal bisection task with humans, the observer is required to decide whether a probe duration (t) is more similar to the short referent (S), an RS response, or to the long referent (L), an RL response. Temporal bisection yields 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 with equal frequency, P(RL) = .5, is referred to as the bisection point, T1/2. Bisection models usually interpret T1/2 as identifying the value of t that is equally confusable with S and L, but they differ in their predictions for the location of T1/2. The present paper presents new data relevant to the location and interpretation of T1/2. The data indicate that the empirical values usually are biased, the biases being influenced by duration range, L:S ratio, and probe spacing. Moreover, the biases often are not consistent across observers. It is concluded that empirical values of T1/2 should not be interpreted as indicating the value of t that is equally confusable with S and L.
Creelman (1962) developed a signal detection model for time perception, which specified Poisson perception distributions and therefore proportionality between mean perceived time (mt) and clock time (t, i.e., mt = gt) and also proportionality between variance in perceived time (st) and clock time (st2 = gt). The results of many experiments, however, have indicated that Weber’s Law provides the best description of the relationship between variability in perceived time and clock time (see Allan, 1998). Recently Killeen, Fetterman, and Bizo (1997) developed a signal detection model that incorporates Weber’s Law. In this model, perceived time is normally distributed, mean perceived time is a power function of clock time with an exponent close to 1.0, mt = t and the standard deviation of perceived time is proportional to mean perceived time, st st = =g mt t
1 2
Requests for reprints should be sent to Lorraine G. Allan, Department of Psychology, McMaster University, Hamilton ON, L8S 4K1, Canada. Email: [email protected] The preparation of this paper was supported by a grant to Lorraine G. Allan from the Natural Sciences and Engineering Research Council of Canada. Some of the data were presented at the meeting of the International Society of Psychophysics in Tempe Arizona in October 1999. Ó 2002 The Experimental Psychology Society http://www.tandf.co.uk/journals/pp/02724995.html DOI:10.1080/02724990143000162
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The proportionality constant g is the Weber fraction, and the proportionality relation in Equation 2 is often referred to 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. Superposition in relative time reflects a re-scaling in time, a scale-invariant error distribution for perceived time. Killeen et al. (1997) considered the application of their signal detection model to a number of timing tasks, including temporal bisection. 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 & Gerhardt, 2001; 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 £ t £ L, is presented (where t is clock time), and the observer is required to indicate whether t is more similar to S (RS) or to L (RL). Temporal bisection yields 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 with equal frequency, P(RL) = .5, is often referred to as the bisection point (T 1/2). The scalar property predicts superposition of bisection functions for all referent pairs (i.e., L:S ratios) when P(RL) is plotted against t normalized by the bisection point (i.e., t/T1/2). One interpretation of T1/2 is that it is the value to t that is equally confusable with S and L. The location of T1/2 has generated considerable theoretical interest. Allan and Gibbon (1991) reported that T1/2 was near the geometric mean (GM) of S and L, and that L:S ratio and spacing of the probe values (i.e., linear or logarithmic) had little influence on the location of T1/2. All the L:S ratios in Allan and Gibbon were relatively small, however, resulting in little difference between GM and other measures such as the arithmetic mean (AM) or the harmonic mean (HM) of S and L. Wearden (1991; Wearden & Ferrara, 1995, 1996; Wearden et al., 1997) explored a more extensive set of L:S ratios and concluded that usually T1/2 was closer to AM than to GM and was larger for linearly spaced probes than for logarithmically spaced probes. The main purpose of the present paper is to present new data relevant to the location and interpretation of T1/2. The Killeen et al. (1997) model directly addresses the location of T1/2. In developing their model for bisection, Killeen et al. substituted the logistic distribution for the normal distribution to simplify the mathematical derivation of and the resulting equation for the bisection function.1 For the logistic approximation to the normal, the bisection function is é æ öù ç ÷ ê T -t÷ú ú P(R L )= ê1 + expç 1/ 2 ç 3 ÷ú ê ç s ÷ ê è p t ø úû ë
-1
3
where T1/2, the bisection point, is the signal detection criterion and s t = ( gt ) 2 + pt + c 2 1
See Gibbon (1981) for the bisection function based on the normal distribution.
4
BISECTION POINT
45
Equation 4 is the Weber function derived by Killeen and Weiss (1987), which provides for scalar (g), nonscalar (p), and constant (c) sources of variability. When scalar variability dominates, Equation 4 reduces to Equation 2, and Equation 3 can be rewritten as é æ öù ç ÷ú ê T t ÷ú P(R L )= ê1 + expç 1 / 2 ç 3 ÷ú ê ç gt ÷ ú ê è øû p ë
-1
5
Killeen et al. (1997) noted that the bisection function in Equation 3 is not a logistic function even though it is derived from one, and that it is not a distribution function as it asymptotes at a value less than 1.0. They therefore referred to their bisection function as a pseudo-logistic function and their signal detection model as the pseudo-logistic model (PLM). In the PLM, the perceived value of the probe on each trial is compared to T1/2, and the decision is RL if the perceived value is larger than T1/2. Killeen et al. (1997) suggested that the role of the S and L referents in bisection was to set the value of T1/2. If this were the case, then for unbiased responding, T1/2 would be located at the value of t where the S distribution crosses the L distribution. Killeen et al. provided an approximated solution for unbiased T1/2, which located T1/2 near the HM. Since then P.R. Killeen (personal communication, 1999) has provided an exact solution (see Allan & Gerhardt, 2001): unbiased T1 / 2 =
(
) where r = æ S ö
(S - rL ) + (S - rL ) 2 - (1 - r ) ( gS) 2 ln(r ) 1-r
ç ÷ è Lø
2
6
According to PLM, unbiased T1/2 is located near HM and is dependent on both the L:S ratio and g, but not on probe spacing. Killeen et al. (1997) reported that the Allan and Gibbon (1991) data were consistent with PLM. More recently, Allan and Gerhardt (2001) also concluded that PLM provided an excellent account of their data. As noted earlier, all of the L:S ratios in Allan and Gibbon were relatively small. This was also the case in Allan and Gerhardt. Although Wearden (1991; Wearden & Ferrara, 1995, 1996; Wearden et al., 1997) did examine a broader range of L:S ratios, their analysis was based on between-group comparisons with relatively few observations per observer. Such data are not well suited for the evaluation of quantitative models where one goal is to examine systematic trends among parameters. In the present paper, we present new bisection data for a broad range of L:S ratios and different probe spacings. We adopt the within-subject psychophysical approach of Allan and Gibbon (1991) where individual observers provided data from multiple sessions under many experimental conditions. Of particular interest is the placement of T1/2 relative to unbiased T1/2 specified by PLM (Equation 6).
EXPERIMENTS Subjects The observers were graduate students 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 one hour) per day. Three observers participated in Experiment 1, five in Experiment 2, and seven in Experiment 3.
46
ALLAN
Apparatus Temporal parameters, stimulus presentation, and recording of responses were controlled by a Macintosh computer. The temporal intervals were visual and filled. They were marked by clearly visible geometric forms displayed on an Apple colour 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 Allan and Gibbon (1991), L:S ratios ranged from 1.2 to 2.0. In Experiment 1, the range was broadened to include L:S ratios of 2, 5, and 8.
Procedure L:S ratio (2, 5, 8) and duration range (600 lin, 1000 lin) were varied, resulting in six experimental conditions (see Table 1, 600 lin and 1000 lin). For each condition, there were seven probes, S £ t £ L, spaced in equal arithmetical steps. For 600 lin, the mid t value was constant at 600 ms for the three L:S ratios; for 1000 lin, the mid t value was constant at 1000 ms for the three L:S ratios. Thus, for each duration range, AM was constant across the three L:S ratios, whereas GM and HM decreased as L:S ratio increased. An experimental session consisted of four blocks. L:S ratio and duration range were constant throughout the session. Each block began with eight referent trials. On each referent trial, one of the two referent durations that were in effect for that session was presented. Each referent was presented four times, in a random order. The eight referent trials were followed by 126 probe trials. On each probe trial, TABLE 1 Values of S, L, and t in Experiments 1, 2, and 3 a
L:S ratio
t ——————————————————————– a a S L
600 lin
2 5 8
400 200 133
467 333 289
533 467 444
600 600 600
667 733 755
733 867 911
800 1000 1066
1000 lin
2 5 8
667 333 222
778 555 481
889 777 741
1000 1000 1000
1111 1222 1259
1223 1444 1510
1334 1666 1778
600 log
2 5 8
400 200 133
449 262 188
504 342 266
566 447 377
635 585 533
713 765 754
800 1000 1066
lin
2 4 5.8
100 100 100
117 150 180
133 200 260
150 250 340
167 300 420
183 350 500
200 400 580
log
2 4 5.8
100 100 100
112 126 134
126 159 180
141 200 241
159 252 323
178 317 433
200 400 580
a
In ms.
BISECTION POINT
47
one of the seven t durations was presented. The order of the t values was random with the restriction that each was presented 18 times during a block of 126 probe trials. At the termination of the probe, the observer indicated whether t was more similar to S or to L by pressing S or L, respectively, on the computer keyboard. There was no feedback, and the next probe trial began 1 s after the response was entered. Each observer participated in 24 sessions, 4 at each ratio–range combination. The order of the six ratio–range conditions over the 24 sessions was random with the restriction that each condition occurred an equal number of times before any condition was repeated.
Results and discussion For bisection, the scalar property would be reflected in bisection functions, which superposed across L:S ratio and duration range when plotted against t normalized by the bisection point, T1/2. If T1/2 is located around the GM, as reported by Allan and Gibbon (1991), then the six bisection functions (three L:S ratios and two duration ranges) should superpose when plotted against t/GM. If T1/2 is located around the AM, as suggested by Wearden (1991; Wearden & Ferrara, 1995, 1996; Wearden et al., 1997), then the six bisection functions should superpose when plotted against t/AM. P(RL), averaged over observers, is shown as a function of t/GM in Figure 1A and as a function of t/AM in Figure 1B. It is very clear that superposition of the six functions was much better when t was normalized by AM than when it was normalized by GM. The deviation from superposition in Figure 1A is quite striking. When normalized by GM, the functions do not superpose for either duration range or L:S ratio. As HM < GM, deviation from superposition for t/HM was even greater than that for t/GM. The data presented in Figure 1 suggest that T1/2 was located closer to AM than to GM. Thus the results from Experiment 1 are consistent with those reported by Wearden and Ferrara (1995, 1996) and Wearden et al. (1997)—AM provides a better description of the location of T1/2 than does GM. We obtained PLM estimates of T1/2 for each observer’s data. We assumed, as did Killeen et al. (1997) and Allen and Gerhardt (2001), that the scalar sources of variance dominated (i.e., p = 0 and c = 0 in Equation 4). According to PLM, g should be invariant across L:S ratio and duration range. We fitted Equation 5 to the data of each observer (using the nonlinear fit algorithm from Mathematica) keeping g constant across the six functions and allowing T1/2 to
Figure 1. P(RL) in Experiment 1, averaged over observers, as a function of t/GM (A) and as a function of t/AM (B). The filled symbols and filled lines are for 600 lin, the unfilled symbols and dotted lines are for 1000 lin. The L:S ratios are indicated by shape (diamonds = 2; squares = 5; triangles = 8).
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vary. Table 2 gives, for each observer, the resulting parameter values. PLM provided an excellent fit to the data. Averaged over the six functions, the sum of the squared deviations between the data and the predictions of PLM (w2) was .996, .995, and .996 for O1, O2, and O3, respectively. The excellent fit of PLM to the data is seen in Figure 2 where, for each observer, the six functions are plotted against t/T1/2. For all observers, superpositions of the six functions is seen across both L:S ratio and duration range. Figure 3 plots the PLM estimates of T1/2 for each observer as a function of L:S ratio. Also shown are AM, GM, and unbiased T1/2 (Equation 6).2 Unbiased T1/2 was determined using the average g for the three observers. The values for the 600 lin duration range are shown in Figure 3A, and the values for the 1000 lin duration range are shown in Figure 3B. For both duration ranges, T1/2 for all observers deviated from both GM and unbiased T1/2 and was located around AM. Within the context of PLM, the location of T1/2 around AM implies that responding was biased. PLM is presented schematically in Figure 4 for 600 lin with L:S ratio = 5 and g = .15. Each of the seven distributions represents the distribution of t values generated by one of the seven probe values for L:S ratio = 5 (i.e., 200, 333, 467, 600, 733, 867, 1000). The scalar property (i.e., g = .15) is illustrated by the increasing variability of the distributions as t increases. The leftmost psychometric function was generated by placing T1/2 at unbiased T1/2, and rightmost psychometric function was generated by placing T1/2 at AM. Because of the scalar property, T1/2 located at unbiased T1/2 would result in a preponderance of RL responses. Killeen et al. (1997) recognized this possibility and suggested that in order to balance the frequency of the two response categories, T1/2 might be biased away from unbiased T1/2 towards AM. Given the linear spacing of the probes in Experiment 1, placement of T1/2 at AM would result in P(RL) = P(RS). If the location of T1/2 at AM in Experiment 1 reflects bias, then one would expect the spacing of the probes to influence the location of T1/2. The effect of probe spacing on the location of T1/2 was investigated in Experiment 2. TABLE 2 Values of g and T1/2 in Experiment 1 a
2
T1/2 ———————— 600 lin 1000 lin
O
L:S ratio
g
1
2 5 8
.19
616.8 649.2 622.1
984.9 877.1 963.3
2
2 5 8
.14
572.5 583.5 559.5
944.1 929.5 957.3
3
2 5 8
.16
642.0 685.6 654.7
1049.0 1065.0 1064.0
T1/2 denotes the value estimated from the data, and unbiased T1/2 denotes the value predicted by PLM (Equation 6).
49
Figure 2. P(RL) for the three observers in Experiment 1 as a function of t/T1/2. The filled symbols are for 600 lin, and the unfilled symbols are for 1000 lin. The L:S ratios are indicated by shape (diamonds = 2; squares = 5; triangles = 8).
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ALLAN
Figure 3. T1/2 from Experiment 1 as a function of L:S ratio for 600 lin (A) and for 1000 lin (B). The symbols represent the data from the three observers (squares = O1, diamonds = O2, triangles = O3). The filled line indicates AM, the dashed line indicates GM, and the dotted line indicates unbiased T1/2.
Figure 4. Schematic of PLM for lin600 with L:S ratio = 5 and g = .15. Each of the seven distributions represents the distribution of t values generated by one of the seven probe values for L:S ratio = 5 (i.e., 200, 333, 467, 600, 733, 867, 1000). The scalar property (i.e., g = 1.5) is illustrated by the increasing variability of the distributions as t increases. The leftmost psychometric function was generated by placing T1/2 at unbiased T1/2, and rightmost psychometric function was generated byplacing T1/2 at AM.
EXPERIMENT 2 Procedure The L:S ratios were the same as those in Experiment 1 (2, 5, 8). Only the 600 duration range was used. There were two probe spacings, linear and logarithmic. The values of t for the three L:S ratios and two probe spacings are given in Table 1 (600 lin and 600 log). Each observer participated in 24 sessions, 4 at each ratio–spacing combination.
BISECTION POINT
51
Results and discussion P(RL), averaged over observers, is shown as a function of t/GM in Figure 5A and as a function of t/AM in Figure 5B. As in Experiment 1, superposition of the six functions was better when t was normalized by AM than when it was normalized by GM. In applying PLM to the data, we again assumed that scalar sources dominated (i.e., p = 0 and c = 0 in Equation 4) and that g was invariant. We fitted Equation 5 to the data of each observer, keeping g constant across the six functions and allowing T1/2 to vary. The PLM parameter values are available in Table 3 (see p. 54). As in Experiment 1, the fits of PLM to the data were excellent. Averaged over the six functions, w2 was .995, .999, .994, and .999 for O1 to O5, respectively. The bisection functions are plotted against t/T 1/2 in Figure 6. For each observer, the six functions superposed across both L:S ratio and probe spacing. Figure 7 shows the PLM estimates of T1/2 for each observer and averaged over observers, as a function of L:S ratio for the two probe spacings. Also shown are AM, GM, and unbiased T1/2. For all observers, T1/2 deviated from unbiased T1/2 and tended to be smaller for the logarithmic probe spacing than for the linear probe spacing. For 14 of the 15 comparisons (5 observers and 3 L:S ratios), T1/2 for the logarithmic probe spacing was less than T1/2 for the linear probe spacing. The data from Experiment 2 are consistent with those reported by Wearden and Ferrara (1995, 1996) and Wearden et al. (1997)—probe spacing influences the location of T1/2. Although it is clear from Figure 7 that probe spacing influenced the location of T1/2, the actual placement of T1/2 is less clear. For one observer (O1), T1/2 was close to GM for all L:S ratios; for two observers (O3 and O5) T1/2 decreased with increasing L:S ratio, as does GM, and for two observers (O2 and O4) T1/2 was relatively constant across L:S ratio, as is AM. The data from Experiment 2 indicate that probe spacing influenced the location of T1/2 and that T1/2 for logarithmic probe spacing was smaller than T1/2 for probe linear spacing. However, the actual location of the probe, with regard to AM or GM, varied among observers. The variability among observers seen in this experiment highlights the danger of basing conclusions about the location of T1/2 on average data.
Figure 5. P(RL) in Experiment 2, averaged over observers, as a function of t/GM (A) and as a function of t/AM (B). The filled symbols and filled lines are for 600 lin, the unfilled symbols and dotted lines are for 600 log. The L:S ratios are indicated by shape (diamonds = 2; squares = 5; triangles = 8).
52
Figure 6. P(RL) for the five observers in Experiment 2 as a function of T1/2. The filled symbols and filled lines are for 600 lin, the unfilled symbols are for 600 log. The L:S ratios are indicated by shape (diamonds = 2; squares = 5; triangles = 8).
53
Figure 7. T1/2 from Experiment 2, for each observer and averaged over observers, as a function of L:S ratio for 600 lin (filled symbols) and for 600 log (unfilled symbols). The filled line indicates AM, the dashed line indicates GM, and the dotted line indicates the unbiased T1/2.
54
ALLAN TABLE 3 Values of g and T1/2 in Experiment 2 a
T1/2 ———————— 600 lin 1000 lin
O
L:S ratio
g
1
2 5 8
.24
534.0 445.4 402.6
555.5 441.7 399.2
2
2 5 8
.11
597.3 595.8 608.1
584.5 565.6 546.1
3
2 5 8
.17
541.5 513.0 465.2
518.7 453.6 393.5
4
2 5 8
.22
634.0 601.9 653.2
624.4 566.8 565.3
5
2 5 8
.14
551.7 525.3 498.2
541.4 461.0 472.2
a
In ms.
EXPERIMENT 3 In Experiments 1 and 2, AM was constant across the three L:S ratios, whereas GM and unbiased T1/2 decreased with increasing L:S ratio. In our previous research (Allan & Gibbon, 1991), this was not the case. Rather changes in L:S ratio affected both AM and GM. In Experiment 3, we varied L:S ratio so that both AM and GM increased as L:S ratio increased.
Procedure As in Experiment 2, there were two probe spacings, linear and logarithmic. There were three L:S ratios (2, 4, 5.8). For all L:S ratios, S = 100 ms. The values of t for the three L:S ratios and two probe spacings are given in Table 1 (lin and log). Each observer participated in 18 sessions, 3 at each ratio–spacing combination.
Results and discussion P(RL), averaged over observers, is shown in Figure 8A as a function of t/GM and in Figure 8B as a function of t/AM. Unlike in Experiments 1 and 2, superposition across conditions was better when t was normalized by GM than when it was normalized by AM. As in the previous experiments, we fitted Equation 5 to the data of each observer, keeping g constant across the six functions and allowing T1/2 to vary. The PLM parameter values are available in Table 4. Again the fits of PLM to the data were excellent. Averaged over the six
BISECTION POINT
55
Figure 8. P(RL) in Experiment 3, averaged over observers, as a function of t/GM (A) and as a function of t/AM (B). The filled symbols and filled lines are for 600 lin, the unfilled symbols and dotted lines are for 600 log. The L:S ratios are indicated by shape (diamonds = 2; squares = 4; triangles = 5.8).
functions, w2 was .955, .997, .997, .993, .994, .995, .989 for O1 to O7, respectively. The functions are plotted as a function of t/T1/2 in Figure 9. For each observer, the six functions superposed across both L:S ratio and probe spacing. Figure 10 shows T1/2, for each observer and averaged over observers, as a function of L:S ratio for the two probe spacings. Also shown are AM, GM and unbiased T1/2. As in Experiment 2, T1/2 deviated from unbiased T1/2 and tended to be smaller for the logarithmic probe spacing than for the linear probe spacing. For 17 of the 21 comparisons (7 observers and 3 L:S ratios), T1/2 for the logarithmic probe spacing was less than T1/2 for the linear probe spacing. Averaged over observers, T1/2 for the logarithmic probe spacing was close to GM, and T1/2 for the linear probe spacing was less than AM. However, as in Experiment 2, there was variability among observers with regard to the placement of T1/2.
GENERAL DISCUSSION In all our experiments we obtained excellent fits of PLM to the data of every observer with the restriction that g was invariant across duration range, L:S ratio, and probe spacing. This independence of g from duration range, L:S ratio, and probe spacing is consistent with the data reported by Allan and Gibbon (1991) and Allan and Gerhardt (2001). In contrast, Wearden et al. (1997) and Penney et al. (1998) found that g increased with L:S ratio. There are two major differences between the two groups of experiments. The studies that found independence of g of L:S ratio used relatively short durations, and estimates of g were derived from individual observer data. The studies that found dependence of g on L:S ratio used longer durations and focused on group data. Clearly, further research is needed to determine whether such differences are important. The data from the present experiments indicate that PLM estimates of T1/2 were always larger than those of unbiased T1/2. Due to the scalar property, a consequence of placing T1/2 at unbiased T1/2 is a predominance of RL. With linear probe spacing, placement of T1/2 at AM would result in P(RL) = P(RS), whereas with logarithmic probe spacing, placement of T1/2 at GM would result in P(RL) = P(SS). Thus, if equalizing the number of responses in the
56
ALLAN TABLE 4 Values of g and T1/2 in Experiment 3 a
T1/2 ———————— 600 lin 1000 lin
O
L:S ratio
g
1
2 4 5.8
.39
168.2 232.1 252.4
175.3 202.0 226.5
2
2 4 5.8
.17
158.9 246.3 321.9
163.0 238.2 273.8
3
2 4 5.8
.19
151.9 237.8 291.4
147.8 223.3 240.9
4
2 4 5.8
.20
158.6 250.0 280.2
156.1 231.1 254.5
5
2 4 5.8
.33
136.2 186.6 220.2
135.2 200.2 219.2
6
2 4 5.8
.20
133.2 204.0 233.6
122.1 186.1 209.9
7
2 4 5.8
.34
145.1 229.6 243.4
148.3 183.5 223.6
a
In ms.
two response categories influences the placement of T1/2, we would expect T1/2 to be smaller for the logarithmic probe spacing than for the linear probe spacing. The data from Experiments 2 and 3 were in accord with this exception. Probe spacing influenced the placement of T1/2, and overall T1/2 for logarithmic probe spacing was smaller than T1/2 for linear probe spacing. Although there was consistency with regard to the influence of probe spacing on the placement of T1/2, there was considerable variability among observers in the location of T1/2, both within and between experiments. In Experiment 1, T1/2 was located around AM for all observers. In Experiment 2 and 3, for some observers T1/2 would be best described as located near AM, whereas for other observers T1/2 would be best described as located near GM. In Experiment 1, only linear probe spacing was used, whereas in Experiments 2 and 3, all observers experienced both linear and logarithmic probe spacing. Our data suggest that the total context experienced by an observer influences the location of T1/2. Models other than PLM have been proposed to account for bisection data (e.g., Allan & Gibbon, 1991; Gibbon, 1981; Rodriguez-Girones & Kacelnik, 2001; Wearden, 1991). These
57
Figure 9. P(RL) for the seven observers in Experiment 3 as a function of t/T1/2. The filled symbols are for 600 lin, and the unfilled symbols are for 600 log. The L:S ratios are indicated by shape (diamonds = 2; squares = 4; triangles = 5.8).
58
Figure 10. T1/2 from Experiment 3, for each observer and averaged over observers, as a function of L:S ratio for lin (filled symbols) and for log (unfilled symbols). The filled line indicates AM, the dashed line indicates GM, and the dotted line indicates the unbiased T1/2.
BISECTION POINT
59
models would do no better than PLM with regard to predicting the location of T1/2 for the individual observers in the present experiments. For example, Gibbon (1981) considered the possibility that the decision in bisection is based on a similarity ratio comparison. Specifically, the decision to respond RS or RL is made by comparing the similarity of the perceived value of t to memories of the two referents, S and L. This comparison is based on a ratio of the similarity of t to S relative to the similarity of t to L. The similarity ratio rule places T1/2 at the GM of S and L. The inclusion of a response bias parameter b provides for deviation of T1/2 from GM. T1/ 2 =
GM b
7
Wearden (1991) substituted a similarity difference rule for Gibbon’s (1981) similarity ratio rule. According to the difference rule, the observer responds RL when t is more similar to L than to S and responses RS when t is more similar to S than to L. The similarity difference rule places T1/2 at the arithmetic mean (AM) of S and L, and the inclusion of a response bias parameter allows T1/2 to deviate from AM. Thus, both similarity rules (ratio and difference) involve comparisons of t with both S and L, and both rules provide for deviations of T1/2 from their predicted locations by including a response bias parameter. Neither similarity rule, however, predicts the idiosyncratic patterns exhibited by our observers. All these models interpret unbiased T1/2 as identifying the value of t that is equally confusable with S and L. The models predict different locations for unbiased T1/2—near the HM for PLM, at GM for the Gibbon (1981) similarity ratio rule, and at AM for the Wearden (1991) similarity difference rule. The present data indicate that estimated values of T1/2 are likely biased, and should not be interpreted as identifying the value of t that is equally confusable with S and L. The location of unbiased T1/2 remains a mystery.
REFERENCES Allan, L.G. (1998). The influence of the scalar timing model on human timing research. Behavioural Process, 44, 101– 117. 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. Creelman, C.D. (1962). Human discrimination of auditory duration. Journal of the Acoustical Society of America, 34, 582–593. 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, M.A. (1987). Optimal timing and the Weber function. Psychological Review, 94, 455–468. 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. 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.
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Wewarden, 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. Manuscript received 6 December 2000 Accepted revision received 28 April 2001
The location and interpretation of the bisection point Lorraine G. Allan McMaster University, Hamilton, Canada In a temporal bisection task with humans, the observer is required to decide whether a probe duration (t) is more similar to the short referent (S), an RS response, or to the long referent (L), an RL response. Temporal bisection yields 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 with equal frequency, P(RL) = .5, is referred to as the bisection point, T1/2. Bisection models usually interpret T1/2 as identifying the value of t that is equally confusable with S and L, but they differ in their predictions for the location of T1/2. The present paper presents new data relevant to the location and interpretation of T1/2. The data indicate that the empirical values usually are biased, the biases being influenced by duration range, L:S ratio, and probe spacing. Moreover, the biases often are not consistent across observers. It is concluded that empirical values of T1/2 should not be interpreted as indicating the value of t that is equally confusable with S and L.
Creelman (1962) developed a signal detection model for time perception, which specified Poisson perception distributions and therefore proportionality between mean perceived time (mt) and clock time (t, i.e., mt = gt) and also proportionality between variance in perceived time (st) and clock time (st2 = gt). The results of many experiments, however, have indicated that Weber’s Law provides the best description of the relationship between variability in perceived time and clock time (see Allan, 1998). Recently Killeen, Fetterman, and Bizo (1997) developed a signal detection model that incorporates Weber’s Law. In this model, perceived time is normally distributed, mean perceived time is a power function of clock time with an exponent close to 1.0, mt = t and the standard deviation of perceived time is proportional to mean perceived time, st st = =g mt t
1 2
Requests for reprints should be sent to Lorraine G. Allan, Department of Psychology, McMaster University, Hamilton ON, L8S 4K1, Canada. Email: [email protected] The preparation of this paper was supported by a grant to Lorraine G. Allan from the Natural Sciences and Engineering Research Council of Canada. Some of the data were presented at the meeting of the International Society of Psychophysics in Tempe Arizona in October 1999. Ó 2002 The Experimental Psychology Society http://www.tandf.co.uk/journals/pp/02724995.html DOI:10.1080/02724990143000162
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ALLAN
The proportionality constant g is the Weber fraction, and the proportionality relation in Equation 2 is often referred to 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. Superposition in relative time reflects a re-scaling in time, a scale-invariant error distribution for perceived time. Killeen et al. (1997) considered the application of their signal detection model to a number of timing tasks, including temporal bisection. 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 & Gerhardt, 2001; 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 £ t £ L, is presented (where t is clock time), and the observer is required to indicate whether t is more similar to S (RS) or to L (RL). Temporal bisection yields 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 with equal frequency, P(RL) = .5, is often referred to as the bisection point (T 1/2). The scalar property predicts superposition of bisection functions for all referent pairs (i.e., L:S ratios) when P(RL) is plotted against t normalized by the bisection point (i.e., t/T1/2). One interpretation of T1/2 is that it is the value to t that is equally confusable with S and L. The location of T1/2 has generated considerable theoretical interest. Allan and Gibbon (1991) reported that T1/2 was near the geometric mean (GM) of S and L, and that L:S ratio and spacing of the probe values (i.e., linear or logarithmic) had little influence on the location of T1/2. All the L:S ratios in Allan and Gibbon were relatively small, however, resulting in little difference between GM and other measures such as the arithmetic mean (AM) or the harmonic mean (HM) of S and L. Wearden (1991; Wearden & Ferrara, 1995, 1996; Wearden et al., 1997) explored a more extensive set of L:S ratios and concluded that usually T1/2 was closer to AM than to GM and was larger for linearly spaced probes than for logarithmically spaced probes. The main purpose of the present paper is to present new data relevant to the location and interpretation of T1/2. The Killeen et al. (1997) model directly addresses the location of T1/2. In developing their model for bisection, Killeen et al. substituted the logistic distribution for the normal distribution to simplify the mathematical derivation of and the resulting equation for the bisection function.1 For the logistic approximation to the normal, the bisection function is é æ öù ç ÷ ê T -t÷ú ú P(R L )= ê1 + expç 1/ 2 ç 3 ÷ú ê ç s ÷ ê è p t ø úû ë
-1
3
where T1/2, the bisection point, is the signal detection criterion and s t = ( gt ) 2 + pt + c 2 1
See Gibbon (1981) for the bisection function based on the normal distribution.
4
BISECTION POINT
45
Equation 4 is the Weber function derived by Killeen and Weiss (1987), which provides for scalar (g), nonscalar (p), and constant (c) sources of variability. When scalar variability dominates, Equation 4 reduces to Equation 2, and Equation 3 can be rewritten as é æ öù ç ÷ú ê T t ÷ú P(R L )= ê1 + expç 1 / 2 ç 3 ÷ú ê ç gt ÷ ú ê è øû p ë
-1
5
Killeen et al. (1997) noted that the bisection function in Equation 3 is not a logistic function even though it is derived from one, and that it is not a distribution function as it asymptotes at a value less than 1.0. They therefore referred to their bisection function as a pseudo-logistic function and their signal detection model as the pseudo-logistic model (PLM). In the PLM, the perceived value of the probe on each trial is compared to T1/2, and the decision is RL if the perceived value is larger than T1/2. Killeen et al. (1997) suggested that the role of the S and L referents in bisection was to set the value of T1/2. If this were the case, then for unbiased responding, T1/2 would be located at the value of t where the S distribution crosses the L distribution. Killeen et al. provided an approximated solution for unbiased T1/2, which located T1/2 near the HM. Since then P.R. Killeen (personal communication, 1999) has provided an exact solution (see Allan & Gerhardt, 2001): unbiased T1 / 2 =
(
) where r = æ S ö
(S - rL ) + (S - rL ) 2 - (1 - r ) ( gS) 2 ln(r ) 1-r
ç ÷ è Lø
2
6
According to PLM, unbiased T1/2 is located near HM and is dependent on both the L:S ratio and g, but not on probe spacing. Killeen et al. (1997) reported that the Allan and Gibbon (1991) data were consistent with PLM. More recently, Allan and Gerhardt (2001) also concluded that PLM provided an excellent account of their data. As noted earlier, all of the L:S ratios in Allan and Gibbon were relatively small. This was also the case in Allan and Gerhardt. Although Wearden (1991; Wearden & Ferrara, 1995, 1996; Wearden et al., 1997) did examine a broader range of L:S ratios, their analysis was based on between-group comparisons with relatively few observations per observer. Such data are not well suited for the evaluation of quantitative models where one goal is to examine systematic trends among parameters. In the present paper, we present new bisection data for a broad range of L:S ratios and different probe spacings. We adopt the within-subject psychophysical approach of Allan and Gibbon (1991) where individual observers provided data from multiple sessions under many experimental conditions. Of particular interest is the placement of T1/2 relative to unbiased T1/2 specified by PLM (Equation 6).
EXPERIMENTS Subjects The observers were graduate students 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 one hour) per day. Three observers participated in Experiment 1, five in Experiment 2, and seven in Experiment 3.
46
ALLAN
Apparatus Temporal parameters, stimulus presentation, and recording of responses were controlled by a Macintosh computer. The temporal intervals were visual and filled. They were marked by clearly visible geometric forms displayed on an Apple colour 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 Allan and Gibbon (1991), L:S ratios ranged from 1.2 to 2.0. In Experiment 1, the range was broadened to include L:S ratios of 2, 5, and 8.
Procedure L:S ratio (2, 5, 8) and duration range (600 lin, 1000 lin) were varied, resulting in six experimental conditions (see Table 1, 600 lin and 1000 lin). For each condition, there were seven probes, S £ t £ L, spaced in equal arithmetical steps. For 600 lin, the mid t value was constant at 600 ms for the three L:S ratios; for 1000 lin, the mid t value was constant at 1000 ms for the three L:S ratios. Thus, for each duration range, AM was constant across the three L:S ratios, whereas GM and HM decreased as L:S ratio increased. An experimental session consisted of four blocks. L:S ratio and duration range were constant throughout the session. Each block began with eight referent trials. On each referent trial, one of the two referent durations that were in effect for that session was presented. Each referent was presented four times, in a random order. The eight referent trials were followed by 126 probe trials. On each probe trial, TABLE 1 Values of S, L, and t in Experiments 1, 2, and 3 a
L:S ratio
t ——————————————————————– a a S L
600 lin
2 5 8
400 200 133
467 333 289
533 467 444
600 600 600
667 733 755
733 867 911
800 1000 1066
1000 lin
2 5 8
667 333 222
778 555 481
889 777 741
1000 1000 1000
1111 1222 1259
1223 1444 1510
1334 1666 1778
600 log
2 5 8
400 200 133
449 262 188
504 342 266
566 447 377
635 585 533
713 765 754
800 1000 1066
lin
2 4 5.8
100 100 100
117 150 180
133 200 260
150 250 340
167 300 420
183 350 500
200 400 580
log
2 4 5.8
100 100 100
112 126 134
126 159 180
141 200 241
159 252 323
178 317 433
200 400 580
a
In ms.
BISECTION POINT
47
one of the seven t durations was presented. The order of the t values was random with the restriction that each was presented 18 times during a block of 126 probe trials. At the termination of the probe, the observer indicated whether t was more similar to S or to L by pressing S or L, respectively, on the computer keyboard. There was no feedback, and the next probe trial began 1 s after the response was entered. Each observer participated in 24 sessions, 4 at each ratio–range combination. The order of the six ratio–range conditions over the 24 sessions was random with the restriction that each condition occurred an equal number of times before any condition was repeated.
Results and discussion For bisection, the scalar property would be reflected in bisection functions, which superposed across L:S ratio and duration range when plotted against t normalized by the bisection point, T1/2. If T1/2 is located around the GM, as reported by Allan and Gibbon (1991), then the six bisection functions (three L:S ratios and two duration ranges) should superpose when plotted against t/GM. If T1/2 is located around the AM, as suggested by Wearden (1991; Wearden & Ferrara, 1995, 1996; Wearden et al., 1997), then the six bisection functions should superpose when plotted against t/AM. P(RL), averaged over observers, is shown as a function of t/GM in Figure 1A and as a function of t/AM in Figure 1B. It is very clear that superposition of the six functions was much better when t was normalized by AM than when it was normalized by GM. The deviation from superposition in Figure 1A is quite striking. When normalized by GM, the functions do not superpose for either duration range or L:S ratio. As HM < GM, deviation from superposition for t/HM was even greater than that for t/GM. The data presented in Figure 1 suggest that T1/2 was located closer to AM than to GM. Thus the results from Experiment 1 are consistent with those reported by Wearden and Ferrara (1995, 1996) and Wearden et al. (1997)—AM provides a better description of the location of T1/2 than does GM. We obtained PLM estimates of T1/2 for each observer’s data. We assumed, as did Killeen et al. (1997) and Allen and Gerhardt (2001), that the scalar sources of variance dominated (i.e., p = 0 and c = 0 in Equation 4). According to PLM, g should be invariant across L:S ratio and duration range. We fitted Equation 5 to the data of each observer (using the nonlinear fit algorithm from Mathematica) keeping g constant across the six functions and allowing T1/2 to
Figure 1. P(RL) in Experiment 1, averaged over observers, as a function of t/GM (A) and as a function of t/AM (B). The filled symbols and filled lines are for 600 lin, the unfilled symbols and dotted lines are for 1000 lin. The L:S ratios are indicated by shape (diamonds = 2; squares = 5; triangles = 8).
48
ALLAN
vary. Table 2 gives, for each observer, the resulting parameter values. PLM provided an excellent fit to the data. Averaged over the six functions, the sum of the squared deviations between the data and the predictions of PLM (w2) was .996, .995, and .996 for O1, O2, and O3, respectively. The excellent fit of PLM to the data is seen in Figure 2 where, for each observer, the six functions are plotted against t/T1/2. For all observers, superpositions of the six functions is seen across both L:S ratio and duration range. Figure 3 plots the PLM estimates of T1/2 for each observer as a function of L:S ratio. Also shown are AM, GM, and unbiased T1/2 (Equation 6).2 Unbiased T1/2 was determined using the average g for the three observers. The values for the 600 lin duration range are shown in Figure 3A, and the values for the 1000 lin duration range are shown in Figure 3B. For both duration ranges, T1/2 for all observers deviated from both GM and unbiased T1/2 and was located around AM. Within the context of PLM, the location of T1/2 around AM implies that responding was biased. PLM is presented schematically in Figure 4 for 600 lin with L:S ratio = 5 and g = .15. Each of the seven distributions represents the distribution of t values generated by one of the seven probe values for L:S ratio = 5 (i.e., 200, 333, 467, 600, 733, 867, 1000). The scalar property (i.e., g = .15) is illustrated by the increasing variability of the distributions as t increases. The leftmost psychometric function was generated by placing T1/2 at unbiased T1/2, and rightmost psychometric function was generated by placing T1/2 at AM. Because of the scalar property, T1/2 located at unbiased T1/2 would result in a preponderance of RL responses. Killeen et al. (1997) recognized this possibility and suggested that in order to balance the frequency of the two response categories, T1/2 might be biased away from unbiased T1/2 towards AM. Given the linear spacing of the probes in Experiment 1, placement of T1/2 at AM would result in P(RL) = P(RS). If the location of T1/2 at AM in Experiment 1 reflects bias, then one would expect the spacing of the probes to influence the location of T1/2. The effect of probe spacing on the location of T1/2 was investigated in Experiment 2. TABLE 2 Values of g and T1/2 in Experiment 1 a
2
T1/2 ———————— 600 lin 1000 lin
O
L:S ratio
g
1
2 5 8
.19
616.8 649.2 622.1
984.9 877.1 963.3
2
2 5 8
.14
572.5 583.5 559.5
944.1 929.5 957.3
3
2 5 8
.16
642.0 685.6 654.7
1049.0 1065.0 1064.0
T1/2 denotes the value estimated from the data, and unbiased T1/2 denotes the value predicted by PLM (Equation 6).
49
Figure 2. P(RL) for the three observers in Experiment 1 as a function of t/T1/2. The filled symbols are for 600 lin, and the unfilled symbols are for 1000 lin. The L:S ratios are indicated by shape (diamonds = 2; squares = 5; triangles = 8).
50
ALLAN
Figure 3. T1/2 from Experiment 1 as a function of L:S ratio for 600 lin (A) and for 1000 lin (B). The symbols represent the data from the three observers (squares = O1, diamonds = O2, triangles = O3). The filled line indicates AM, the dashed line indicates GM, and the dotted line indicates unbiased T1/2.
Figure 4. Schematic of PLM for lin600 with L:S ratio = 5 and g = .15. Each of the seven distributions represents the distribution of t values generated by one of the seven probe values for L:S ratio = 5 (i.e., 200, 333, 467, 600, 733, 867, 1000). The scalar property (i.e., g = 1.5) is illustrated by the increasing variability of the distributions as t increases. The leftmost psychometric function was generated by placing T1/2 at unbiased T1/2, and rightmost psychometric function was generated byplacing T1/2 at AM.
EXPERIMENT 2 Procedure The L:S ratios were the same as those in Experiment 1 (2, 5, 8). Only the 600 duration range was used. There were two probe spacings, linear and logarithmic. The values of t for the three L:S ratios and two probe spacings are given in Table 1 (600 lin and 600 log). Each observer participated in 24 sessions, 4 at each ratio–spacing combination.
BISECTION POINT
51
Results and discussion P(RL), averaged over observers, is shown as a function of t/GM in Figure 5A and as a function of t/AM in Figure 5B. As in Experiment 1, superposition of the six functions was better when t was normalized by AM than when it was normalized by GM. In applying PLM to the data, we again assumed that scalar sources dominated (i.e., p = 0 and c = 0 in Equation 4) and that g was invariant. We fitted Equation 5 to the data of each observer, keeping g constant across the six functions and allowing T1/2 to vary. The PLM parameter values are available in Table 3 (see p. 54). As in Experiment 1, the fits of PLM to the data were excellent. Averaged over the six functions, w2 was .995, .999, .994, and .999 for O1 to O5, respectively. The bisection functions are plotted against t/T 1/2 in Figure 6. For each observer, the six functions superposed across both L:S ratio and probe spacing. Figure 7 shows the PLM estimates of T1/2 for each observer and averaged over observers, as a function of L:S ratio for the two probe spacings. Also shown are AM, GM, and unbiased T1/2. For all observers, T1/2 deviated from unbiased T1/2 and tended to be smaller for the logarithmic probe spacing than for the linear probe spacing. For 14 of the 15 comparisons (5 observers and 3 L:S ratios), T1/2 for the logarithmic probe spacing was less than T1/2 for the linear probe spacing. The data from Experiment 2 are consistent with those reported by Wearden and Ferrara (1995, 1996) and Wearden et al. (1997)—probe spacing influences the location of T1/2. Although it is clear from Figure 7 that probe spacing influenced the location of T1/2, the actual placement of T1/2 is less clear. For one observer (O1), T1/2 was close to GM for all L:S ratios; for two observers (O3 and O5) T1/2 decreased with increasing L:S ratio, as does GM, and for two observers (O2 and O4) T1/2 was relatively constant across L:S ratio, as is AM. The data from Experiment 2 indicate that probe spacing influenced the location of T1/2 and that T1/2 for logarithmic probe spacing was smaller than T1/2 for probe linear spacing. However, the actual location of the probe, with regard to AM or GM, varied among observers. The variability among observers seen in this experiment highlights the danger of basing conclusions about the location of T1/2 on average data.
Figure 5. P(RL) in Experiment 2, averaged over observers, as a function of t/GM (A) and as a function of t/AM (B). The filled symbols and filled lines are for 600 lin, the unfilled symbols and dotted lines are for 600 log. The L:S ratios are indicated by shape (diamonds = 2; squares = 5; triangles = 8).
52
Figure 6. P(RL) for the five observers in Experiment 2 as a function of T1/2. The filled symbols and filled lines are for 600 lin, the unfilled symbols are for 600 log. The L:S ratios are indicated by shape (diamonds = 2; squares = 5; triangles = 8).
53
Figure 7. T1/2 from Experiment 2, for each observer and averaged over observers, as a function of L:S ratio for 600 lin (filled symbols) and for 600 log (unfilled symbols). The filled line indicates AM, the dashed line indicates GM, and the dotted line indicates the unbiased T1/2.
54
ALLAN TABLE 3 Values of g and T1/2 in Experiment 2 a
T1/2 ———————— 600 lin 1000 lin
O
L:S ratio
g
1
2 5 8
.24
534.0 445.4 402.6
555.5 441.7 399.2
2
2 5 8
.11
597.3 595.8 608.1
584.5 565.6 546.1
3
2 5 8
.17
541.5 513.0 465.2
518.7 453.6 393.5
4
2 5 8
.22
634.0 601.9 653.2
624.4 566.8 565.3
5
2 5 8
.14
551.7 525.3 498.2
541.4 461.0 472.2
a
In ms.
EXPERIMENT 3 In Experiments 1 and 2, AM was constant across the three L:S ratios, whereas GM and unbiased T1/2 decreased with increasing L:S ratio. In our previous research (Allan & Gibbon, 1991), this was not the case. Rather changes in L:S ratio affected both AM and GM. In Experiment 3, we varied L:S ratio so that both AM and GM increased as L:S ratio increased.
Procedure As in Experiment 2, there were two probe spacings, linear and logarithmic. There were three L:S ratios (2, 4, 5.8). For all L:S ratios, S = 100 ms. The values of t for the three L:S ratios and two probe spacings are given in Table 1 (lin and log). Each observer participated in 18 sessions, 3 at each ratio–spacing combination.
Results and discussion P(RL), averaged over observers, is shown in Figure 8A as a function of t/GM and in Figure 8B as a function of t/AM. Unlike in Experiments 1 and 2, superposition across conditions was better when t was normalized by GM than when it was normalized by AM. As in the previous experiments, we fitted Equation 5 to the data of each observer, keeping g constant across the six functions and allowing T1/2 to vary. The PLM parameter values are available in Table 4. Again the fits of PLM to the data were excellent. Averaged over the six
BISECTION POINT
55
Figure 8. P(RL) in Experiment 3, averaged over observers, as a function of t/GM (A) and as a function of t/AM (B). The filled symbols and filled lines are for 600 lin, the unfilled symbols and dotted lines are for 600 log. The L:S ratios are indicated by shape (diamonds = 2; squares = 4; triangles = 5.8).
functions, w2 was .955, .997, .997, .993, .994, .995, .989 for O1 to O7, respectively. The functions are plotted as a function of t/T1/2 in Figure 9. For each observer, the six functions superposed across both L:S ratio and probe spacing. Figure 10 shows T1/2, for each observer and averaged over observers, as a function of L:S ratio for the two probe spacings. Also shown are AM, GM and unbiased T1/2. As in Experiment 2, T1/2 deviated from unbiased T1/2 and tended to be smaller for the logarithmic probe spacing than for the linear probe spacing. For 17 of the 21 comparisons (7 observers and 3 L:S ratios), T1/2 for the logarithmic probe spacing was less than T1/2 for the linear probe spacing. Averaged over observers, T1/2 for the logarithmic probe spacing was close to GM, and T1/2 for the linear probe spacing was less than AM. However, as in Experiment 2, there was variability among observers with regard to the placement of T1/2.
GENERAL DISCUSSION In all our experiments we obtained excellent fits of PLM to the data of every observer with the restriction that g was invariant across duration range, L:S ratio, and probe spacing. This independence of g from duration range, L:S ratio, and probe spacing is consistent with the data reported by Allan and Gibbon (1991) and Allan and Gerhardt (2001). In contrast, Wearden et al. (1997) and Penney et al. (1998) found that g increased with L:S ratio. There are two major differences between the two groups of experiments. The studies that found independence of g of L:S ratio used relatively short durations, and estimates of g were derived from individual observer data. The studies that found dependence of g on L:S ratio used longer durations and focused on group data. Clearly, further research is needed to determine whether such differences are important. The data from the present experiments indicate that PLM estimates of T1/2 were always larger than those of unbiased T1/2. Due to the scalar property, a consequence of placing T1/2 at unbiased T1/2 is a predominance of RL. With linear probe spacing, placement of T1/2 at AM would result in P(RL) = P(RS), whereas with logarithmic probe spacing, placement of T1/2 at GM would result in P(RL) = P(SS). Thus, if equalizing the number of responses in the
56
ALLAN TABLE 4 Values of g and T1/2 in Experiment 3 a
T1/2 ———————— 600 lin 1000 lin
O
L:S ratio
g
1
2 4 5.8
.39
168.2 232.1 252.4
175.3 202.0 226.5
2
2 4 5.8
.17
158.9 246.3 321.9
163.0 238.2 273.8
3
2 4 5.8
.19
151.9 237.8 291.4
147.8 223.3 240.9
4
2 4 5.8
.20
158.6 250.0 280.2
156.1 231.1 254.5
5
2 4 5.8
.33
136.2 186.6 220.2
135.2 200.2 219.2
6
2 4 5.8
.20
133.2 204.0 233.6
122.1 186.1 209.9
7
2 4 5.8
.34
145.1 229.6 243.4
148.3 183.5 223.6
a
In ms.
two response categories influences the placement of T1/2, we would expect T1/2 to be smaller for the logarithmic probe spacing than for the linear probe spacing. The data from Experiments 2 and 3 were in accord with this exception. Probe spacing influenced the placement of T1/2, and overall T1/2 for logarithmic probe spacing was smaller than T1/2 for linear probe spacing. Although there was consistency with regard to the influence of probe spacing on the placement of T1/2, there was considerable variability among observers in the location of T1/2, both within and between experiments. In Experiment 1, T1/2 was located around AM for all observers. In Experiment 2 and 3, for some observers T1/2 would be best described as located near AM, whereas for other observers T1/2 would be best described as located near GM. In Experiment 1, only linear probe spacing was used, whereas in Experiments 2 and 3, all observers experienced both linear and logarithmic probe spacing. Our data suggest that the total context experienced by an observer influences the location of T1/2. Models other than PLM have been proposed to account for bisection data (e.g., Allan & Gibbon, 1991; Gibbon, 1981; Rodriguez-Girones & Kacelnik, 2001; Wearden, 1991). These
57
Figure 9. P(RL) for the seven observers in Experiment 3 as a function of t/T1/2. The filled symbols are for 600 lin, and the unfilled symbols are for 600 log. The L:S ratios are indicated by shape (diamonds = 2; squares = 4; triangles = 5.8).
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Figure 10. T1/2 from Experiment 3, for each observer and averaged over observers, as a function of L:S ratio for lin (filled symbols) and for log (unfilled symbols). The filled line indicates AM, the dashed line indicates GM, and the dotted line indicates the unbiased T1/2.
BISECTION POINT
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models would do no better than PLM with regard to predicting the location of T1/2 for the individual observers in the present experiments. For example, Gibbon (1981) considered the possibility that the decision in bisection is based on a similarity ratio comparison. Specifically, the decision to respond RS or RL is made by comparing the similarity of the perceived value of t to memories of the two referents, S and L. This comparison is based on a ratio of the similarity of t to S relative to the similarity of t to L. The similarity ratio rule places T1/2 at the GM of S and L. The inclusion of a response bias parameter b provides for deviation of T1/2 from GM. T1/ 2 =
GM b
7
Wearden (1991) substituted a similarity difference rule for Gibbon’s (1981) similarity ratio rule. According to the difference rule, the observer responds RL when t is more similar to L than to S and responses RS when t is more similar to S than to L. The similarity difference rule places T1/2 at the arithmetic mean (AM) of S and L, and the inclusion of a response bias parameter allows T1/2 to deviate from AM. Thus, both similarity rules (ratio and difference) involve comparisons of t with both S and L, and both rules provide for deviations of T1/2 from their predicted locations by including a response bias parameter. Neither similarity rule, however, predicts the idiosyncratic patterns exhibited by our observers. All these models interpret unbiased T1/2 as identifying the value of t that is equally confusable with S and L. The models predict different locations for unbiased T1/2—near the HM for PLM, at GM for the Gibbon (1981) similarity ratio rule, and at AM for the Wearden (1991) similarity difference rule. The present data indicate that estimated values of T1/2 are likely biased, and should not be interpreted as identifying the value of t that is equally confusable with S and L. The location of unbiased T1/2 remains a mystery.
REFERENCES Allan, L.G. (1998). The influence of the scalar timing model on human timing research. Behavioural Process, 44, 101– 117. 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. Creelman, C.D. (1962). Human discrimination of auditory duration. Journal of the Acoustical Society of America, 34, 582–593. 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, M.A. (1987). Optimal timing and the Weber function. Psychological Review, 94, 455–468. 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. 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.
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ALLAN
Wewarden, 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. Manuscript received 6 December 2000 Accepted revision received 28 April 2001
