Unequal Weber fractions for the categorization of brief temporal intervals

Transcription

1 Attention, Perception, & Psychophysics 21, 72 (5), doi:1.3758/app Unequal Weber fractions for the categorization of brief temporal intervals SIMON GRONDIN Université Laval, Québec, Québec, Canada How constant is the Weber fraction (WF) for brief time intervals? This question was assessed in three experiments with two base durations (BDs), and 1 sec, and with different ways of estimating the WF. In Experiment 1, the psychometric functions were drawn on the basis of 4, 8, or 12 comparison intervals with the shortest to longest duration ranges being kept constant. The results revealed no effect of the number of intervals, but the WF (threshold/bd) was significantly lower at sec. In Experiment 2, the comparison intervals were distributed over three duration ranges. There was no range effect, and the WF was generally lower at sec than at 1 sec. In Experiment 3, one condition allowed a comparison of the BD with the same range between the shortest and longest comparison intervals. Once again, the WF was lower at sec than at 1 sec. Overall, the results reveal (1) that increasing the number of comparison intervals or the duration range does not seem to affect the value of the WF and (2) that the WF is lower at sec than at 1 sec, which is inconsistent with the scalar property of some timing models. Although the question remains open (Ivry & Schlerf, 28), many time perception researchers believe that there is an internal clock that is, a unique central device dedicated to measuring the passage of time (see, e.g., Grondin, 28; Meck, 23). This clock is often described as a pacemaker that emits pulses, whose accumulation in a counter constitutes the basis for the experience of time (Grondin, 21; Killeen & Weiss, 1987). The presence of this clock counter device underpins the scalar expectancy theory (SET), which is a very influential theory in timing research (Wearden, 23). A basic tenet of SET is that the variability of temporal estimates increases proportionally with the magnitude of time (see Gibbon, 1977, 1991; Gibbon, Church, & Meck, 1984). From a psychophysics perspective, this means that the Weber fraction (WF) should remain constant over a wide range of durations. In the information-processing version of SET, variance in a timing task is reported as also belonging to memory and, to some extent, to decisional processes (Gibbon et al., 1984; Meck, 23). There are time perception models that are not based on a pacemaker counter perspective (see Grondin, 21). On the one hand, some models are referred to as being dependent specifically on a modality (Johnston, Arnold, & Nishida, 26) or as coordination-dependent systems (Jantzen, Steinberg, & Kelso, 25). For instance, in the state-dependent networks model, judging duration means being able to recognize spatial patterns of activity (Buonomano, 27; Karmarkar & Buonomano, 27). On the other hand, there is a model, the dynamic attending theory, based on the capacity to synchronize (attunement) the internal, attentional rhythmicity (an oscillatory process) with the temporal structure of environmental stimuli (Jones, 1976; Jones & Boltz, 1989; Large & Jones, 1999). The variability of temporal estimates is often assessed on the basis of some variation of the method of constant stimuli the bisection or the temporal generalization method where intervals of different lengths have to be categorized as short or long. In the bisection task, a short standard and a long standard (shortest and longest of a distribution of intervals) are presented before the presentation of comparison intervals. In the generalization task, only one standard (midpoint of a distribution of comparison intervals) is presented, and a participant decides whether a given interval is identical to the standard (or, as in the present experiments, whether the interval is shorter or longer than the standard). The results can be used to obtain a psychometric function, where the probability of responding long is plotted as a function of the length of the intervals. The steepness of the function indicates the degree of variability. According to SET, when psychometric functions drawn for different duration ranges are normalized by their mean, they superimpose (see Church, 23; Wearden & Lejeune, 28). Some authors have reported that the choice of time intervals that is, the spacing between the data points on the function influences the estimation of variability (Wearden & Ferrara, 1995, 1996; but see Church & Gibbon, 1982). Moreover, this effect may depend on the range of durations under investigation (Grondin, Bisson, & Gagnon, 29). Larger differences between the longest and shortest intervals are reported as leading to higher estimates of sensitivity. Therefore, manipulating the spacing S. Grondin, simon.grondin@psy.ulaval.ca 21 The Psychonomic Society, Inc. 1422

2 TIME INTERVAL CATEGORIZATION 1423 between data points when the method of constant stimuli is used might have an impact on the verdict regarding the suitability of the Weber function for time. In the present series of experiments, the purpose was to test the constancy of the WF for brief temporal intervals in conditions involving various spacings of intervals and various numbers of points on the psychometric functions. Increasing the number of points was expected to increase the difficulty of the task, especially in conditions in which more points were distributed over a larger range of duration. It is known, for instance, that the perceived duration of intervals is influenced by the global distributional properties of the series of intervals used within a session (Jones & McAuley, 25; McAuley & Jones, 23). In the present study, assessments were based on individual estimates of temporal sensitivity in a very simple basic condition that is, one in which a temporal judgment followed the presentation of a single time interval. Sensitivity was estimated for two very distinct standard durations (2 and 1, msec) and within a range where explicit counting was not expected to provide any benefits ( 1 sec; Grondin, Meilleur-Wells, & Lachance, 1999). In Experiments 1 and 2, the distance between the intervals was kept proportional in the 2- and 1,-msec durations (i.e., it was multiplied by 5). In Experiment 3, one of the spacing conditions was the same in the 2- and 1,-msec conditions. EXPERIMENT 1 Method Participants. Twelve volunteers from Université Laval (5 women and 7 men) took part in the experiment. The average age was 25.3 years. The participants received $3 Canadian for taking part in the experiment. The study was approved by the Comité d éthique de la recherche de l'université Laval. All the participants gave written informed consent prior to the experiment. Apparatus and Stimuli. The intervals to be discriminated were marked by two 1-kHz 2-msec auditory signals. The signals were produced by an IBM PC and were presented binaurally through headphones (Sony MDR-V6). Each participant was seated at a computer in a dimly lit room and was asked to respond either short or long by pressing 1 or 3, respectively, on the computer keypad. Procedure. The experiment consisted of six experimental sessions of 3 4 min for each participant, one session for each experiment condition. A minimum of 1 h was required between sessions. The experiment included two independent variables: standard interval durations, or base durations (2 and 1, msec), and number of comparison intervals (4, 8, or 12). Comparison intervals were separated by 22, 1, or 6 msec for the 2-msec standard, ranging from 167 to 233 msec in the 4-point condition, from 165 to 235 msec in the 8-point condition, and from 167 to 233 msec in the 12-point condition. For the 1,-msec standard, comparison intervals were five times longer that is, separated by 11, 5, or 3 msec, ranging from 835 to 1,165 msec in the 4-point condition, from 825 to 1,175 msec in the 8-point condition, and from 835 to 1,165 msec in the 12-point condition (see Table 1 for a summary). Each session consisted of six blocks, with a 2-sec pause between blocks. Each block contained 18 (4-point condition), 9 (8-point condition), or 6 (12-point condition) randomly ordered presentations of the comparison intervals, for a total of 72 trials per block. Thus, results from the six blocks were combined to trace one psychometric function per condition, where each point is represented by 18, 54, or 36 observations in the 4-, 8-, and 12-point conditions, respectively. Table 1 Shortest and Longest Comparison Intervals in Each Condition in Experiments 1 3 Comparison Intervals (msec) Number of Intervals 2-msec Standard 1,-msec Standard Experiment 1 Different Numbers of Comparison Intervals; Same Proportional Range , , ,165 Experiment 2 Unique Number of Comparison Intervals; Same Proportional Ranges 8 (narrow) ,7 8 (middle) ,14 8 (large) ,21 Experiment 3 Unique Number of Comparison Intervals; Nonproportional Ranges 6 (narrow) ,5 6 (large) ,1 The single-stimulus method was employed (Morgan, Watamaniuk, & McKee, 2). Each trial consisted of the presentation of one interval. At the beginning of each session, the standard (2 or 1, msec) was presented 1 times. The participants were informed that they would have to judge whether later intervals were longer or shorter than the standard (temporal generalization). After the presentation of the standard, the experimental trials began. Each trial consisted of one comparison interval marked by two brief auditory signals (2 msec). The participants had to judge whether the interval was shorter or longer than the standard presented at the beginning of the session. After the participant entered a response, the correct response was indicated on the computer screen. Data analysis. For each participant and for each experimental condition, a 4-, 8-, or 12-point psychometric function was traced, plotting the 4, 8, or 12 comparison intervals on the x-axis and the probability of responding long on the y-axis. The cumulative normal distribution was fitted to the resulting curves. Two indices of performance were estimated for each psychometric function, one for sensitivity and one for the perceived duration. As an indicator of temporal sensitivity, which was the most important issue in the present study, estimates of the standard deviation on the psychometric function were determined. Using one SD (or variance) is a common procedure for expressing temporal sensitivity (Grondin, 28; Killeen & Weiss, 1987). The other dependent variable is the bisection point (BP). The BP can be defined as the x value corresponding to the.5 probability of long responses on the y-axis. Longer perceived durations are reflected by smaller BP values. Results Figure 1 provides an illustration of the grouped data in each experimental condition. Six psychometric functions were constructed for each individual, one for each of the six experimental conditions. For the 4-point conditions, 2 or 1, msec, the goodness of fit was highly satisfactory (mean R 2 values.98). For the 8-point conditions, the mean R 2 value was above.95 at 2 msec and above.92 at 1, msec. For the 12-point conditions, the mean R 2 value was above.92 at 2 msec and above 5 (range from.71 to.96) at 1, msec.

3 1424 GRONDIN R 2 = R 2 = R 2 = Probability of Responding Long R 2 = R 2 = R 2 = ,55 1, ,25 1,75 1,125 1, ,15 1,45 1,75 1,15 1,135 1,165 Comparison Intervals (msec) Figure 1. Psychometric function (pooled results) for the 4- (left), 8- (middle), and 12-point (right) conditions in Experiment 1. Upper panels, standard 2 msec; lower panels, standard 1, msec.

4 TIME INTERVAL CATEGORIZATION 1425 In order to compare directly the sensitivity values at and 1 sec, the WF was computed: WF SD/base duration. Table 2 provides the mean results in each condition. A 2 3 ANOVA with repeated measures revealed that there was a significant duration effect [F(1,11) 157, p, 2.591], with the WF being lower in the -sec than in the 1-sec condition; however, the number of comparison intervals exerted no significant effect [F(2,22).598, p.554, 2.52], and the interaction effect was not significant [F(2,22) 35, p 53, 2.38]. In order to compare directly the perceived duration values at and 1 sec, the constant error (CE) was computed: CE BP base duration. The 2 3 ANOVA with repeated measures revealed no significant effect. EXPERIMENT 2 In this experiment, the spacing between comparison intervals was kept proportional from to 1 sec. However, there were eight comparison intervals in each of three conditions, but the spacing between these intervals differed. In other words, the duration range covered in each condition was now varied. Method Participants. Twelve volunteers from Université Laval (8 women and 4 men) took part in the experiment. The average age was 25.5 years. The participants received $3 Canadian for taking part in the experiment. Apparatus and Stimuli. The materials were the same as those in Experiment 1. Procedure. The participants followed the same procedure as that in Experiment 1, but the comparison intervals in each of the six experimental conditions (2 3) were different. First, there were 8 comparison intervals in each condition. For the 2-msec standard, each comparison interval was separated by 4, 8, or 12 msec (ranging from 186 to 214 msec, from 172 to 228 msec, and from 158 to 242 msec, respectively, as indicated in Table 1). For the 1,-msec standard, comparison intervals were multiplied by 5 that is, were separated by 2, 4, or 6 msec (ranging from 93 to 1,7 msec, from 86 to 1,14 msec, and from 79 to 1,21 msec, respectively). Table 2 Weber Fraction in Each Condition in Experiments 1 3 Number of Intervals 2 msec 1, msec Experiment 1 Different Numbers of Comparison Intervals; Same Proportional Range Experiment 2 Unique Number of Comparison Intervals; Same Proportional Ranges 8 (narrow) (middle) (large) Experiment 3 Unique Number of Comparison Intervals; Nonproportional Ranges 6 (narrow) (large) In other words, the range between the briefest and longest comparison intervals in the narrowest condition is multiplied by two and by three, respectively, in the other two conditions. Results As in Experiment 1, one psychometric function for each of the six experimental conditions was constructed for each individual (see the pooled data in Figure 2). For the 4-msec spacing conditions, the goodness of fit was highly satisfactory (mean R 2 values.96 at 2 msec and.93 at 1, msec). For the 8-msec spacing conditions, the mean R 2 values were above.98 at 2 and 1, msec. For the 12-msec spacing conditions, the mean R 2 values were above.99 at 2 and 1, msec. Table 2 provides the mean WF in each condition. Once again, the WFs were lower at sec than at 1 sec; this observation applies to each of the three spacing conditions. A 2 3 ANOVA with repeated measures revealed that no effect was significant [duration, F(1,11) 29, p.117, 2 8; spacing of intervals, F(2,22) 87, p.594, 2 2; interaction, F(2,22) 189, p 8, 2.133]. Note, however, that the lowest WF at sec was significantly lower than the lowest WF at 1 sec [t(11) 26, p.5]. Finally, the 2 3 ANOVA with repeated measures revealed no significant effect for the CE. EXPERIMENT 3 In this experiment, the spacing of the comparison intervals from to 1 sec was not kept proportional. The parameters were selected in such a way that in one condition, the spread of the distribution of comparison intervals was the same at and 1 sec. Method Participants. Twelve volunteers from Université Laval (6 women and 6 men) took part in the experiment. The average age was 24 years. The participants received $24 Canadian for taking part in the experiment. Apparatus and Stimuli. The materials were the same as those in the previous experiments. Procedure. The experiment consisted of four sessions of 3 4 min for each participant, one session for each experimental condition. A minimum of 1 h was required between sessions. The experiment included two independent variables: base duration (2 and 1, msec) and the distribution of comparison intervals (narrow vs. large). At 2 msec, the comparison intervals lasted 175, 185, 195, 25, 215, and 225 msec (narrow range) and 15, 17, 19, 21, 23, and 25 msec (large range). At 1, msec, the comparison intervals lasted 95, 97, 99, 1,1, 1,3, and 1,5 msec (narrow range) and 9, 94, 98, 1,2, 1,6, and 1,1 msec (large range). Note that there were 1 msec between the shortest and longest comparison intervals in the large condition at 2 msec and in the narrow condition at 1, msec. In all the conditions, there were six comparison intervals. Each session consisted of six blocks, with a 2-sec pause between blocks, and each block contained 72 trials that is, 12 repetitions of each comparison interval. Results One psychometric function for each of the four experimental conditions was constructed for each individual (see the pooled data in Figure 3). At 2 msec, the goodness

5 1426 GRONDIN R 2 = R 2 = R 2 = Probability of Responding Long R 2 = R 2 = R 2 = ,1 1,3 1,5 1, ,2 1,6 1,1 1, ,3 1,9 1,15 1,21 Comparison Intervals (msec) Figure 2. Psychometric function (pooled results) for the narrow (left), middle (middle), and large (right) spacing conditions in Experiment 2. Upper panels, standard 2 msec; lower panels, standard 1, msec.

6 TIME INTERVAL CATEGORIZATION Probability of Responding Long 1. R 2 = R 2 = R 2 = R 2 = ,1 1,3 1, , 1,5 1,1 Comparison Intervals (msec) Figure 3. Psychometric function (pooled results) for the narrow (left) and large (right) spacing conditions in Experiment 3. Upper panels, standard 2 msec; lower panels, standard 1, msec. of fit was highly satisfactory (mean R 2 values.98 in the narrow- and the large-range conditions). At 1, msec, the mean R 2 values were 6 and.96 in the narrow- and large-range conditions, respectively. Table 2 provides the mean WF in each experimental condition. A 2 2 ANOVA with repeated measures revealed that there was a significant duration effect [F(1,11) 945, p.11, 2 62], with the WF being once again lower in the -sec than in the 1-sec condition; the range of comparison intervals exerted no significant effect [F(1,11) 164, p 85, 2.13], and the interaction effect was not significant [F(1,11) 3.128, p.15, 2 21]. Moreover, a t test indicated that the best performance at sec was significantly better than the best performance at 1 sec [t(11) 3.163, p.1; see Figure 4]. Finally, the 2 2 ANOVA with repeated measures revealed no significant effect for the CE. DISCUSSION The present series of experiments generated findings regarding not only the potential spacing effect of temporal sensitivity on estimates, but also the impact of using different numbers of comparison intervals when thresholds

7 1428 GRONDIN Weber Fraction Large Narrow 1 1 Standard Duration (sec) Figure 4. Mean Weber fraction in the narrow- and large-range conditions for each standard duration in Experiment 3. have to be estimated. Most important, the experiments offer a critical look at Weber s law for the discrimination of brief temporal intervals. Method Effects In these experiments, the purpose was to adopt a classical method for estimating sensitivity (threshold) to time intervals, as well as to manipulate different parameters in order to assess the extent to which the threshold estimates varied and, thus, the impact of the experimental manipulations on Weber s law. The first step involved checking whether the number of intervals chosen for building the psychometric functions would influence the threshold estimates. Depending on the nature of time representation in memory, different impacts might be expected at and 1 sec. Indeed, increasing the number of comparison intervals might have a greater impact in the 1-sec range than in the -sec range, since the intervals would be longer. However, the results do not support this hypothesis. Moreover, both the effect size for the number-of-interval condition and the duration number-of-interval interaction were very small. Of course, this conclusion regarding the number-of-interval issue is not definitive, for it is based on the acceptance of a null hypothesis. Nevertheless, it is reasonable to believe that the number of intervals on psychometric functions has a negligible effect on temporal sensitivity estimates, for increasing the number of intervals in an experimental session does not seem to cause any overload in memory, even with longer intervals. Strictly speaking, the data from Experiments 2 and 3 indicate that the spacing of comparison intervals has no effect on estimates of temporal sensitivity, at either or 1 sec. This finding is inconsistent with the data in Wearden and Ferrara (1995, 1996). Moreover, preliminary lab results indicate some spacing effect as well, with a narrower range leading to a lower threshold at sec (Grondin et al., 29). In Experiment 3, the tendency of the results indicates that, at 1 sec, narrower spacing decreases sensitivity (higher threshold). Indeed, in this experiment, the individual R 2 values were lowest (and the thresholds highest) in the 1-sec and narrow condition. It is not excluded that the dissimilarities between the present results and those in Wearden and Ferrara (1995, 1996) rely on subtle methodological differences. (1) Wearden and Ferrara used bisection, not the generalization variant used here; (2) the spacing effects in their experiments involved different spacings of comparisons between the short and long standards (e.g., linear vs. logarithmic), not different numbers of linearly spaced stimuli, as in the present investigation, or different linear spacings; and (3) their theoretical focus was on whether people were averaging all the stimuli in the bisection experiment together or using the short and long standards. As well, although a kind of generalization task was used here and in Ferrara, Lejeune, and Wearden (1997; see also Wearden & Grindrod, 23, for similar results), the method in each set of experiments differed since, on the one hand, the task required saying (yes or no) whether the intervals was like the standard, whereas it was a shorter/ longer response that was requested from the participants in the present experiments. 1 This distinction may account for the different findings, considering that a part of variance in the information- processing version of SET is actually located at the decision process level (Wearden & Grindrod, 23). The Weber Fraction Apart from highlighting a potential spacing effect, the results in all of the conditions in the present experiments point in the same direction: There were unequal WFs for the different duration ranges, and the WF values were smaller at sec than at 1 sec. This is a violation of Weber s law, which states that the ratio of threshold to magnitude should remain constant. Of critical importance is the fact that the inequality of the WF is due not to a higher value at sec, but to a higher value at 1 sec. Indeed, what is usually observed in psychophysics is an increase of the WF for lower magnitudes of a sensory continuum, a phenomenon easily accounted for by the generalized form of Weber s law. The present findings are therefore totally inconsistent with this generalized version. If anything, they are exactly the opposite. Such a result is consistent with the one reported by Lavoie and Grondin (24), where the Weber fraction was higher at 2 sec than at sec. The latter result might have been due to an increase of the Weber fraction with an interval longer than 1 sec. Several studies have shown an increase of the ratio when durations were longer than 2 sec (Getty, 1975; see Fraisse, 1978, for a brief overview, and Bizo, Chu, Sanabria, & Killeen, 26, for animal timing data). The literature on the WF for the discrimination of tempi which involves the discrimination of empty time intervals indicates that Weber s law holds when intervals range from 4 to 1,45 msec (Ehrlé & Samson, 25; Halpern & Darwin, 1982). Beyond this range, when single intervals are presented, the WF can be reduced if an explicit counting strategy is adopted (Grondin et al., 1999). In the present experiments, the WF inequality occurred within the narrow - to 1-sec range.

8 TIME INTERVAL CATEGORIZATION 1429 Why the WFs are unequal at and 1 sec remains a difficult question. One may argue that longer intervals occupy a larger space in working memory, which would exceed the temporal capacity of working memory (Gilden & Marusich, 29; Lavoie & Grondin, 24). However, in such a case, increasing the number of comparison intervals, as in Experiment 1, should have led to more damage (increased WF) at 1 sec than at sec, which was not the case. On the other hand, the unequal WF might be due not to a high value at 1 sec, but to a low value at sec, which implies an interval range that is closer to the one that one has to deal with in normal speech. Perhaps the greater amount of experience with such intervals in the speech context leads to better discrimination in this range. 2 Implications for the Study of Time That Weber s law does not hold is not specific to the study of psychological time. Somewhat along the line proposed by Holway and Pratt (1936), Masin (29) reported a large series of results, based on multiple sensory contin ua, showing different sorts of violation of Weber s law (see also Cobb, 1932). A failure of Weber s law has an important impact on the study of time. As was noted in the introduction, one popular contemporary theory of time perception, SET, is founded on the scalar property that is, on the fact that the variability increases proportionally with the magnitude of time (Wearden, 23). The pacemaker counter device described earlier is the basis of SET and is reported to conform to some form of Weber s law (Killeen & Weiss, 1987). The proportional view of SET, or of Weber s law, allows positing that a common timekeeping mechanism (a single internal clock) is used for different values on the temporal continuum. Breaking the continuum might mean abandoning this unifying principle. This possibly opens the door to a fragmented and less simple view in which multiple timing mechanisms could be argued to be at play. Indeed, such a view is consistent with pharmacological evidence supporting the hypothesis that there are two distinct timing mechanisms underlying the processing of temporal intervals, one in the subsecond and one in the second range (see Rammsayer, 28). The first one would be a subcortical (or sensory-based or automatic) timing mechanism like Karmarkar and Buonomano s (27) state-dependent network model, for instance whereas the second one requires the support of more cognitive resources. The present data do not indicate what the appropriate mechanism for very brief intervals is, or what the mechanisms for intervals in the - and 1-sec range would be. However, they do indicate that the viability of the singleclock hypothesis requires the identification of an additional, nontemporal source of variance that exerts more influence at 1 sec than at sec. It is usually recognized that the sensory effects associated with the marking process cause more variance at sec than at 1 sec. So, this could not be the source of that additional variance. If it exists, therefore, the additional variance source is probably located at a higher, cognitive level of time processing. Returning to SET s information-processing framework, it is not excluded that the decisional processes are the correct location. AUTHOR NOTE This research was made possible by a research grant awarded by the Natural Sciences and Engineering Research Council of Canada. This study was presented at the 5th Annual Meeting of the Psychonomic Society, held in Boston in November 29. I thank Nicolas Bisson for his help in this project and two anonymous reviewers for their very helpful comments on an earlier version of the manuscript. Correspondence should be addressed to S. 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