Stevens' power law and the problem of meaningfulness

Frequently, it is postulated that the results of a ratio production (resp., ratio estimation) experiment can be summarized by Stevens' power law psi=alphaphi(beta). In the present article, it is argued that the power law parameters depend, among other things, on the standard stimulus presented as a reference point, and the physical stimulus scale by which the physical intensities are measured. To formalize this idea, a new formulation of Stevens' power law is presented. We show that the exponent in Stevens' power law can only be interpreted in a meaningful way if the stimulus scale is a ratio scale. Furthermore, we present empirically testable axioms (termed invertibility and weak multiplicativity) which are both necessary and sufficient for the power law exponent to be invariant under changes of the standard stimulus. Finally, invertibility and weak multiplicativity are evaluated in a ratio production experiment. Ten participants were required to adjust the area of variable circles to prescribed ratio production factors. Both axioms are violated for all participants. The results cast doubts on the well-established practice of comparing power law exponents across different modalities.     

The perception of empty and filled time intervals by pigeons

Pigeons were trained in a within-subjects design to discriminate durations of a filled interval (2 s and 8 s of light) and durations of an empty interval (2 s and 8 s bound by two 500-ms light markers). Filled intervals required a response to one set of comparisons (e.g., blue vs. yellow), whereas empty intervals required a response to a different set of comparisons (e.g., red vs. green). Psychophysical testing indicated that empty intervals were judged to be longer than equivalent durations of a filled interval. This finding was replicated when the anchor durations used during training were changed to 1 s and 4 s, or 4 s and 16 s. The difference between the point of subjective equality (PSE) for the empty intervals and the PSE for filled intervals increased as the magnitude of the anchor duration pairs increased. In addition, the difference limens (DL) for empty intervals were smaller than those for filled intervals, and they also increased as the magnitude of anchor duration pairs increased. An analysis of the Weber fractions (WF; i.e., DL/PSE) provided evidence for superimposition of the empty and filled timing functions across the different sets of anchor durations. These results suggest that the accumulation of subjective time was greater for empty intervals than for filled intervals. Within the framework of scalar timing theory, this difference in timing appeared to be the result of a clock rate difference rather than a switch latency difference.     

Judgments of the duration of visually marked empty time intervals: Linking perceived duration and sensitivity

The capability of subjects to categorize (as short or long) visually marked empty time intervals was investigated in three experiments. Two visual signals, located 18° to the left (L) and to the right (R) of a fixation point in the visual field, established four marking conditions, two unilaterally presented (L-L and R-R) and two bilaterally presented (L-R and R-L). In Experiments 1 and 2, the results show that discrimination is better with unilateral sequences than with bilateral sequences and that the perceived duration is longer with an L-R than with an R-L sequence. In addition, Experiment 2 shows that, in comparison with a condition in which Markers 1 and 2 remain identical for a complete session, varying the markers from trial to trial does not decrease discrimination. Also, Experiment 2 shows that discrimination is better when both visual markers are presented at fovea than it is in the unilateral conditions. Experiment 3 shows that bilateral intervals are perceived as being longer and are better discriminated than are intervals marked by an intermodal sequence (auditory-visual or visual-auditory). The general discussion reports the implications of having different perceived duration and sensitivity levels, in various marker-type conditions, for an internal-clock hypothesis. Some implications of these results for a lateralized-timer hypothesis are also discussed.     

The perception of brief temporal intervals: power functions for auditory and visual stimulus intervals

Two experiments were performed to examine the role of method of estimation and the employment of a standard stimulus on the judged duration of auditory and visual stimuli presented for brief temporal intervals (0.25 to 5.0 s). The results indicate that the relationship between judged and physical duration is nearly direct and linear. Psychophysical methodology and stimulus modality exerted little influence on the obtained power functions.     

Differences in duration discrimination of filled and empty auditory intervals as a function of base duration

In the present experiments, participants were presented with two time intervals that were marked by auditory signals, and their task was to decide which of the two was longer in duration. In Experiment 1, the base durations were 50 and 1,000msec, whereas in Experiment 2, seven different base durations ranging from 50 to 1,000 msec were employed. It was found that filled intervals (continuous tones) were discriminated more accurately than empty intervals (with onset and offset marked by clicks) at the 50-msec base duration, whereas no performance differences could be shown for longer ones. The findings are consistent with the notion of a unitary timing mechanism that governs the timing of both filled and empty auditory intervals, independent of base durations. A likely conceptual framework that could explain better performance with filled as compared with empty intervals represents an information-processing model of interval timing that evolved from scalar timing theory. According to this account, a performance decrement observed with empty intervals may be due to a misassignment of pulses generated by an internal pacemaker.     

Judging the relative duration of multimodal short empty time intervals

Three experiments address the cause of the different performance levels found in time discrimination of empty intervals with durations near 250 msec. Performance differed according to the kind of sensory modality that marked the intervals. With a procedure in which the type of marker was randomized from trial to trial, it was shown that variability of discrimination judgments could not be attributed entirely to the variability of the criterion on which a judgment was based. Such a randomization slightly affects discrimination but provokes a reorganization related to marker conditions of the probabilities of judging an interval to be short or long. Moreover, it was shown that within intramodal conditions, physical characteristics of markers influence the discrimination performances. To account for the results generated with different marker-type intervals at 250 msec, we propose that two types of processor may be involved in duration discrimination: one is specifically related to a given sensory modality, whereas the other is aspecific and responsible for discrimination of intermodal intervals.     

The effects of sex, age, and interval duration on the perception of time

The present experiment examined the interactive effects of sex, age, and interval duration on individual’s time perception accuracy. Participants engaged in the duration production task and subsequently completed questionnaires designed to elicit their temporal attitudes. The overall group of 100 individuals was divided evenly between the sexes. Five groups, each composed of 10 males and 10 females, were divided by decades of age ranging from 20 to 69 years old. The specific time estimation task was an empty interval production procedure composed of 50 trials on each of four different intervals of 1, 3, 7, and 20 s, respectively. The presentation orders of these intervals were randomized across participants but yoked across the sexes within each of the respective age groups. Analysis of the production results indicated significant influences for the sex of the participant while age did not appear to affect estimates of these short durations. Temporal attitudes, as reflected in responses to time questionnaire inquiries, did however exhibit significant differences across age. The contending theoretical accounts of such sex and age differences are considered and explanatory accounts that present a synthesis of endogenous and exogenous causal factors are discussed in light of the present pattern of findings.     

Contrasting task demands alter the perceived duration of brief time intervals

Four experiments were performed to assess the effects of task differences on duration judgments. Experiments 1 and 2 used the method of reproduction in prospective, within-subjects designs; their results supported previous research on the effects of task difficulty. Both experiments, using tasks that varied along somewhat different dimensions, found that subjects provided reproduction values that varied inversely with task difficulty. That is, while subjects tended to underreproduce across all tasks, the more difficult the task performed during the target interval, the greater the extent of the under-reproduction. Experiments 3 and 4 used a modification of the reproduction method by placing demands upon the subjects during both the target interval and the reproduction phase of each trial; they demonstrated that the greater the degree of contrast between demands made by the task performed during the target interval and those made during reproduction, the less accurate the duration reproduction. The results are discussed in terms of the contextual and resource allocation models of duration estimation.     

Speed contingencies, number of stimulus presentations, and the nodality effect in equivalence class formation.

Two experiments compared performances on conditional discrimination tasks using single-participant designs with and without speed contingencies. Experiment 1 was a systematic replication of Spencer and Chase (1996). Experiment 2 presented equal numbers of training and testing trials. In each experiment, 2 female undergraduates participated. Each formed three five-member and three seven-member equivalence classes in Experiments 1 and 2, respectively. Response speed was an inverse function of nodal number and relational type in Experiment 1, but only of relational type (i.e., baseline, symmetry, transitivity, and equivalence) in Experiment 2, with and without the speed contingency. Accuracy tended to peak without the speed contingency in both experiments. Adding the speed contingency increased response speeds in both experiments, more so in Experiment 2 with a lower limited hold for I participant. The results of Experiment 2 cast doubt on previous reports of the nodality effect and on the notion of "relatedness" among class members, and they support a reinforcement-contingency, rather than a structural, account of equivalence class formation.     

The power law for the perception of rotation by airline pilots

The purpose of this study was to determine the power laws for the perception of rotation about the three major body axes. Eighteen airline pilots made magnitude estimates of 5-sec pulses of nine angular accelerations having a range of acceleration x time of 10–150 deg/sec. The results showed that (1) the power law with an exponent of 1.4 describes the subjective motion of these pilots for all three major body axes, (2) the power law also describes the perception of motion for individual pilots with a substantial range of exponents, (3) there were significant correlations among the exponents for the three body axes, and (4) the data suggest that the power law over the wide range used may be more complex than implied by a formula with a single exponent.     

Remembered duration: working memory and the reproduction of intervals

On the basis of attention allocation models of time estimation, the role of working memory in prospective duration reproduction is explored. In four experiments, adult participants performed a counting task (duration, 400 sec) that allowed coordinative and sequential demands on working memory to be varied. After completing the counting task, the participants reproduced the time that they had worked on this task It emerged that (1) increased coordinative demands on working memory (but not increased sequential demands) reduced the accuracy of prospective duration reproduction (Experiments 1 and 2), (2) presenting context information during the reproduction phase enhanced the accuracy of the reproduced duration (Experiment 3), and (3) individual differences in coordinative working memory capacity affected duration reproduction in the same direction as the experimental manipulation of coordinative task demands (Experiment 4). The results suggest that attention allocation models of time estimation may benefit from taking a more differentiated view of the types of attentional demands that affect temporal cognition.     

Effects of temporal shapes of sound markers on the perception of interonset time intervals

This study investigated how the temporal characteristics, particularly durations, of sounds affect the perceived duration of very short interonset time intervals (120-360?ms), which is important for rhythm perception in speech and music. In four experiments, the subjective duration of single time intervals marked by two sounds was measured utilizing the method of adjustment, while the markers' durations, amplitude difference (which accompanied the duration change), and sound energy distribution in time were varied. Lengthening the duration of the second marker in the range of 20-100?ms increased the subjective duration of the time interval in a stable manner. Lengthening the first marker tended to increase the subjective duration, but unstably; an opposite effect sometimes appeared for the shortest time interval of 120?ms. The effects of varying the amplitude and the sound energy distribution in time of either marker were very small in the present experimental conditions, thus proving the effects of marker durations per se.     

Size judgment: The presence of a standard increases the exponent of the power law

Ss judged the size of circles by the methods of magnitude estimation and magnitude production, with or without a standard figure present. The presence of the standard raised the exponent of the power law. The effect was largely independent of the position of the standard in the range of stimuli. It is argued that the standard’s effect was not mediated by a difference in apparent size between perceived and remembered standards, a change in Ss’ response strategies, a decrease in sequential dependencies, or other previously identified bias effects.     

The effects of type of interval,sensory modality,base duration,and psychophysical task on the discrimination of brief time intervals

The present study was designed to investigate the influences of type of psychophysical task (two-alternative forced-choice [2AFC] and reminder tasks), type of interval (filled vs. empty), sensory modality (auditory vs. visual), and base duration (ranging from 100 through 1,000 ms) on performance on duration discrimination. All of these factors were systematically varied in an experiment comprising 192 participants. This approach allowed for obtaining information not only on the general (main) effect of each factor alone, but also on the functional interplay and mutual interactions of some or all of these factors combined. Temporal sensitivity was markedly higher for auditory than for visual intervals, as well as for the reminder relative to the 2AFC task. With regard to base duration, discrimination performance deteriorated with decreasing base durations for intervals below 400 ms, whereas longer intervals were not affected. No indication emerged that overall performance on duration discrimination was influenced by the type of interval, and only two significant interactions were apparent: Base Duration × Type of Interval and Base Duration × Sensory Modality. With filled intervals, the deteriorating effect of base duration was limited to very brief base durations, not exceeding 100 ms, whereas with empty intervals, temporal discriminability was also affected for the 200-ms base duration. Similarly, the performance decrement observed with visual relative to auditory intervals increased with decreasing base durations. These findings suggest that type of task, sensory modality, and base duration represent largely independent sources of variance for performance on duration discrimination that can be accounted for by distinct nontemporal mechanisms.     

The effect of embodying the elderly on time perception

The present study investigated the perception of stimulus durations represented by elderly faces or by young faces. In a temporal bisection task, participants classified intermediate durations as more similar to a short or a long reference duration. The results showed that the durations represented by elderly faces were less often classified as “long” than the durations represented by young faces. According to internal clock models of time perception, this shortening effect is due to a slowing down of the speed of the internal clock during the perception of elderly faces. Analyses also revealed an interaction between sex of face and sex of participant such that this shortening effect occurred only when the participants share the same sex than the stimulus faces. As discussed, this finding is quite consistent with embodied cognition approaches to information processing, but alternatives accounts are also considered.     

Ordinal properties of perceived average duration: simultaneous and sequential presentations

When a subject evaluates the duration of two simultaneous events, the perceived durations are combined in a nonlinear manner, whereas comparable evaluations of successive events yield linear combinations. Fifty-four subjects rated average durations of pairs of time intervals ranging from 0.5 s to 10.0 s. The intervals making up a pair were presented successively and simultaneously on separate occasions within the same presentation sequence. The order of data was consistent with models previously proposed for perceived average duration. A nonmetric analysis of the combined ordinal data from both conditions yielded a single set of measures of perceived duration with ratio-scale properties. The common scale for the two conditions is consistent with differences in results from simultaneous and successive monitoring of time intervals, stemming from differences in the way duration information is combined rather than how it is perceived. the derived scale is related to physical time by a power function with a 1.06 exponent.