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Scandinavian Journal of Psychology, 1992, 33, 339-358
Time perception: Effects of sex and sound intensity on scales of subjective duration
HANNES EISLER and ANNA D. EISLER Department of Psychology, Stockholm University, Stockholm, Sweden
Eider, H. & Eider, A. D. ( 1992). Time perception: Effects of sex and sound intensity on scales of subjective duration. Scandinavian Journal of Psychology, 33, 339-358.
Six women and six men reproduced ten time intervals varying in logarithmic steps between 1.3 and 20s. The durations were indicated by white noise of 10, 25, 40 and 55 d 3 SL. different sound intensities in different sessions. It was found that (i) greater sound intensity entails shorter reproductions, and (ii) reproductions by male observers are shorter than those by female, although for both (i) and (ii) there is an interaction with the standard durations. The data were treated in accordance with the “parallel-clock model” (Eider, 1975). whereby the parameters of the psychophysical power function are determined from duration reproduction data. As in previous experiments (Eider, 1975). the data showed a break in the function entailing two segments. The effect of sound intensity could be attributed to the exponent, which was lower for stronger noise, and the effect of sex to the weight coefficient of the upper relative to the lower segment of the psychophysical function, the coefficient being lower for men.
Key words: Psychophysics, time perception, mathematical modeling, sex, sound intensity.
H. Eider, Depurlment of Psychology, Stockholm University. S- 106 91 Stockholm, Sweden
With the parallel-clock model (Eider, 1975, 1976, 1977, 1981a, 1981b, 1984, 1990; Eisler & Eisler, 1991), subjective duration can be scaled without the use of numbers, thereby eliminating one of the possible sources of error in direct psychophysical scaling. The model builds on the reproduction of standard durations. According to Zakay (1990) reproduction is a “compatible design” and shows less variability than other methods.
The aim of the present experiment was to attempt to attribute possible effects of intensity and gender to parameters of the psychophysical power function for duration, using the parallel-clock model.
Both gender and intensity effects on time perception have been studied before, with inconsistent outcomes. A survey of Fraisse (1967, 1984), Doob (1971), Zelkind & Sprug ( 1974) and our own database of Psychological Time resulted in 36 references regarding the effect of intensity and 69 regarding the effect of sex on experienced time. They are considered in the section “Comparisons with other studies”.
The parallel-clock model is based on duration reproduction; the total subjective duration (from the onset of a standard duration to the offset of the reproduction) is assumed to accumulate in one sensory register, and the subjective reproduced duration in a second register. The observer terminates the ongoing reproduction when the difference between the :ontents of the two registers equals the content of the second register. (For details, see iection “The model”.) This corresponds, from the point of view of the researcher, to haloing the total subjective duration. Accordingly, the reproduction (equal-setting) of the observer- which is number-free-can be’used for scaling. The “1/2” derives from the model, not from the observer. This is an important difference from the common direct scaling methods, which rely on numbers emitted by or presented to the observers. Scales constructed from reproduc- tion data which avoid possible number biases in the observers can therefore hardly be
340
described as limited. as Allan (1983) did when she wrote: “. . . . if only reproduction exponents can be estimated, then the [parallelclock] model loses its appeal”.
Without claiming that the present experiment once and for all answers the problem of the effect of gender and sound intensity on time perception, we will first show how both these independent variables affect the length of the reproduced durations, a finding independent of any particular model for time perception, and then how these differences relate to the parameters of the particular version of the psychophysical power function which the parallelclock model requires (see Eisler, 1974).
H. Eider and A . D. Eisler &and J Psycho1 33 (1992)
METHOD In principle, the method is the same as described by Eisler (1975).
Obseruers Six women and six men, most of them students of psychology, participated in the experiment. Their ages ranged from 21 to 57 years, with a median of 31.5 years.
Apparatus A white-noise generator was connected via an amplifier and an attenuator (smallest step 0.1 dB) to a loudspeaker. Between the amplifier and the loudspeaker there was a gate, controlled by a timer (smallest step 0.01 s). The timer opened the gate for the standard duration to be presented. Immediately after the end of the standard duration the timer closed the gate for 300 ms, and then opened it again. The silent interval of 300 ms was experienced as an interruption of the ongoing sound. At the second opening of the gate an electric stop watch started, and stopped simultaneously with the sound when the observer pressed a microswitch.
A Briiel & Kjzr noise level meter measured the sound intensity at the observer’s position before the start of every experimental session.
Stimuli The noise stimuli varied in both duration and intensity. There were ten standard durations, ranging in logarithmic steps from 1.3 to 20s (1.3. 1.8,2.5,3.3,4.5,6.0, 8.1, 11.0, 14.8,20.0s), and four sound intensity levels, measured in dB above the individual observer’s hearing threshold (sensation level, SL), namely. 10, 25, 40 and 55 dB SL.
Procedure The observers participated in four sessions, one for each level of sound intensity. The order of the four intensities was randomized individually. The ten standard durations were all presented six times in an individual pseudo-random series in each session, except for the first session, when they were presented eight times. The first two presentations of each duration in this session were considered as preliminary trials to familiarize the observers with the experimental situation, and were excluded from the data treatment. A session lasted about 1 h 15 min. No feedback was given.
At the first session the hearing thresholds of the observers were measured. They were comparatively high, because the experimental room was not sound-insulated. They ranged from 16 to 20dB re 20 pN/m’. No observer had to be discarded because of hearing loss.
The observers were told that the experiment was to deal with time perception and were requested to remove their watches. They were to listen to two time intervals, separated by a short interruption, and indicated by a sound from the loudspeaker. The length of the first would vary, and the second was to be terminated by a button press when it had lasted as long as the first. It was stressed that the observer should react naively-as it feels-and not count or mark time in any other way.
Before the start of the session the noise generator was switched on. The sound intensity was adjusted by means of the attenuator to the level to be applied in the current session.
For every trial, the standard duration was set on the timer. the observer was forwarned by a “Now” from the experimenter, and the trial was started by pressing a button on the timer. After the trial, the reading on the stop watch (the reproduced duration) was recorded to the nearest 0.01 s. and the next setting of the standard duration was made.
Scand J Psycho1 33 (1992) Scales of subjective duration 341
TREATMENT OF T H E RAW DATA
Before applying the parallel-clock model to determine the parameters of the power functions, we want to ascertain whether the variables of intensity and sex had any effect, independently of the application of any model.
Arithmetic means for the six reproductions of each of the ten standard durations for each of the four sound intensities were computed, separately for each observer, see Table I . The distribution of the total sum of the reproductions did not differ significantly from normality (SAS program UNIVARIATE). With the data of Table 1, a three-way analysis of variance (mixed design, BMDP2V) was carried out with Sex(2) x Sound intensity(4) x Duration( 10). As Table 2 shows, the two main effects of interest (Sex and Sound level) were significant, as well as the interactions between Duration and Sex and between Duration and Sound intensity. The main effects are illustrated in Fig. I , together with the corresponding plots for two observers, the latter in order to show the large individual differences. It should be born in mind, however, that what is shown in Fig. 1 is the sums over all ten reproductions, which means that the values are dominated by the longer durations. The interactions (see Figs 2 and 3 ) demonstrate (i) that the slope, when reproductions are plotted against standard durations, is greater for female than for male observers, that is, that, for standard durations longer than about 4 s, the female observers give longer reproductions, and (ii) that for higher sound intensities, reproductions are shorter. The latter finding is clearest for the highest sound intensity.
Discussion
It can be concluded so far that, for the present experiment, there is a significant effect of sex on the length of the reproductions, dependent on the standard duration, as well as a significant effect of the different sound intensities.
The interactions of the effects of Sex and Sound intensity with Duration help to explain the conflicting results from other researchers because most studies make use of only one, and a few of only two or three durations. Thus, depending on the particular value(s) of the duration(s), effects may or may not be found.
COMPUTATION OF T H E PARAMETERS OF T H E POWER FUNCTION FOR DURATION
The parallel-clock model
The parallel-clock model was briefly described in the introduction and may be understood most easily by considering Fig. 4. The observer’s button press occurs, as indicated by the arrows, when the difference between the contents of the two registers equals the contents of the register containing the reproduction.
Fig. 4 shows, likewise, that when the observer reports equality of standard duration and reproduction by pressing the button, the total subjective duration (from the onset of the standard to the offset of the reproduction) is twice the subjective duration that corresponds to the reproduction. Accordingly, when plotting the raw data, reproductions are plotted against total durations (see, e.g., Fig. 5). Subjective duration may be regarded as an intervening variable between the physical standard duration and the physical reproduction (Eider, 1987~).
Rule and coworkers (Curtis & Rule, 1977; Rule & Curtis, 1985; Rule et al., 1983) found a nonlinear composition rule for duration when two simultaneous durations have to be combined by the observer, e.g., by requiring the estimation of the average of the two
342 H . Eider and A . D. Eider Scand J Psycho1 33 ( 1992)
Table I . Mean of six reproductions for 10 durations, 4 SLr and 12 obieruers in seconh
0. SL Standard duration No dB 1.3 1.8 2.5 3.3 4.5 6.0 8.1 11.0 14.8 20.0
FI
F2
F3
F4
F5
F6
MI
M2
M3
M4
M5
M6
10 25 40 55 10 25 40 55 10 25 40 55 10 25 40 55 10 25 40 55 10 25 40 55
10 25 40 55
10 25 40 55 10 25 40 55 10 25 40 55
10 25 40 55 10 25 40 55
1.30 1.54 1.35 1.23
1.06 1.29 1.47 1.61
1.07 0.94 1.33 0.76
1.49 1.60 1.21 1.10
1.06 0.99 1.01 0.86
1.46 1.58 1.68 1.48
1.16 1.76 1.55 1.72
1.41 1 S O 1.06 1.09
1.22 1.34 1.28 I .67
1.62 1.40 I .23 1.31
1.80 1.68 1.19 0.88
2.11 1.37 1.44 I .07
1.77 1.61 1.58 1.62
1.90 1.79 1.89 1.71
1.85 1.48 1.47 1.77
2.08 2.14 1.59 1.61
1.61 1.51 1.39 1.32
2.15 2.59 2.16 2.20
I .53 1.58 1.78 2.05
1.83 1.85 1.74 1.95
1.63 I .69 1.60 I .96
2.23 1.63 I .67 1.79
2.29
1.60 1.24
2.59 2.02 1.85 1.62
I .78
2.68 2.58 2.25 2.35
2.30 2.61 2.23 2.20
2.84 2.53 2.05 2.27
2.65 2.98
2.26
2.27 1.98 1.93 1.86
2.64 2.9 1 2.97 2.84
2.10 2.46 2.91 2.59
2.54 2.47 2.16 2.05
1.53 2.08 1.78 2.51
2.76 2.35 2.45 2.19
2.64 1.78 2.22 1.79
2.90 2.19 2.42 2.14
2.81
3.23 3.46 2.64 2.70
2.76 3.55 3.64 2.71
4.27 3.62 2.84 3.43
3.27 3.67 3.03 2.97
3.12 2.92 2.77 2.66
3.49 3.64 3.45 4.13
2.70 3.00 3.50 3.14
2.96 2.78 3.25 3.05
2.56 2.50 2.17 2.42
3.56 2.75 3.78 3.13
3.63 2.55 2.97 2.26
3.27 2.70 3.75 2.71
4.19 4.34 4.25 4.02
3.93 4.77 3.87 4.23
4.80 4.26 4.49 4.33
4.81 5.03 4.82 4.27
4.26 4.42 4.32 4.16
4.82 4.21 3.89 4.71
3.72 3.61 3.53 4.29
3.56 4.28 4.27 3.90
2.69 2.33 3.59 2.71
5.35 4.40 4.30 4.09
4.13 3.98 2.87 2.99
4.85 4.06 4.88 3.36
5.81 6.37 6.33 5.37
4.86 4.93 4.98 4.39
6.64 5.80 5.55 5.70
6.33 6.22 5.26 4.83
5.79 5.77 5.59
5.79 5.39 4.69 5.72
5.1 I 4.65 4.76 4.56
4.84 5.74 5.76 4.68
3.80 3.62 4.77 4.24
6.34
5.25 4.83
4.84 3.80 3.72 3.85
5.39 4.41 5.16 4.30
5.87
5.08
6.78 8.27 6.70 7.25
5.55 6.45 5.84 5.66
8.74 7.44 7.50 6.74
8.56 6.82
6.86
7.14 7.81 7.44 6.56
6.88 6.41 6.86 7.56
6.27 5.88 6.29 6.51
6.2 I 7.21 7.32 6.41
6.03 5.3 1 5.61 5.03
7.49 6.54 6.48 5.77
6.66 4.15 4.4 I 4.52
5.93 5.38 6.45 4.70
6.28
10.31 10.46 9.68 9.41
7.28 6.12 8.28 6.85
11 S O 10.75 10.32 8.68
9.86 9.40
7.9 I 9.06 9.66
10.68 8.0 I 9.09 9.02 7.64 8.69
8.69 7.74
9.33
8.49 8.37 8.29 7.85
8.03 7.38 9.08 7.64
9.53 7.77
8.27
8.96
8.70 6.82
9.12 8.54 5.6 I 4.89
6.89 6.75 5.94 6.04
12.87 12.56 12.41 13.71
9.43 9.01 8.66 8.59
14.59 14.18 14.21 10.73
12.47 10.17 10.55 9.65
13.99 13.74 14.20 11.63
9.47 1 I .39 10.91 11.61
10.76 10.92 10.37 11.43
10.33 1 I .09 10.85 10.62
10.48 10.97 1 1.27 11.95
11.77 10.44 9.74 9.57
9.49 7.15 6.74 6.11
10.17 8.06 8.63 6.42
17.23 18.64 16.84 14.90
11.26 12.44 13.95 I I .22
19.91 17.89 18.42 13.91
15.47 14.38 13.56 12.72
17.52 17.18 19.99 18.59
15.69 12.87 13.62 15.48
15.04 15.49 13.83 16.20
12.30 16.48 15.31 13.29
16.04 15.78 17.04 16.0 I 14.63 11.31 13.97 9.23
14.59 8.12 8.84 7.69
14.85 12.34 11.32 8.93
Note: F = Female. M = Male observers.
%and J Psycho1 33 (1992) Scales of subjective duration 343
320-
?a-
280
Table 2. Analysis of variance (BMDP VZ, mixed design) of duration reproductions
--\ /-- .\ --- *-
C- c-
Sum of Degrees of Tail Source squares freedom Mean square F probability
S 80.99168 Error 123.27296
L 3 1.83044 L X S 2.4227 1 Error 70.29023
D 7886.32038 D x S 72.10742 Error 303.48754
L x D 25.141 48 L x D x S 4.86660 Error 119.59877
1 10
3 3
30
9 9
90
27 27
270
80.99 168 6.57 0.0282 12.32730
10.61015 4.53 0.0098 0.80757 0.34 0.7932 2.3430 1
876.25782 259.86 <0.0001 8.01 194 2.38 0.0185 3.37208
0.93 1 17 2.10 0.0016 0.18024 0.41 0.9966 0.44296
Note: S =sex, L = sensation level, D = duration.
10 25 40 55 Sensation LeveL, db
Fig. I . Left panel: Mean for six observers of the sum of reproductions of ten durations as a function of sensation level. Solid line: female observers; Dashed line: male observers. Right panel: The correspond- ing sum for two individual observers. demonstrating individual differences.
durations. In a later study, Rule (1986) showed that the nonlinearity is not perceptual. because durations are judged in the same way, whether presented singly or simultaneously. The nonlinearity has to appear in a later stage of the processing, and depends on the particular task. For the parallel-clock model the conclusion to be drawn is that the processing does not reach this later stage for the task of reproduction. There are two differences between taking the average of two durations and reproducing a duration: (i) Taking an average requires a conscious cognitive process carried out on (the memory of) the subjective durations, which is not the case with reproduction, and (ii) this cognitive process is carried out only once. after the offset of the stimulation. whereas reproduction of durations, if conceived as a similar cognitive process, would require continuous “calcula- tions” of the difference between two subjective durations, whether according to the parallel- clock model or any other model. Such an achievement would be beyond the capacity of even
344 H . Eisler and A. D. Eider %and J Psycho1 33 (1992)
14 -
u) 12-
C -2
10 - C 0 -2 4 8 - 0 3 U 0 6- L 0- a lx 4 -
2 -
0
16
I t
Q) 12
10 C -2
C 0 -2 s , 8 0 Y
-0 0 6 L (1 a
a 4
2
I
? I I I t & I , , , 1 0 2 4 6 8 10 12 14 16 18 20
0 1 I I I I I i I I I I 0 2 4 6 8 10 12 14 16 18 20
Standard duration in s Fig. 2. Mean reproductions for female ( 0) and male (0) observers as a function of standard duration. Inset: The region near the origin.
16 1 D
Fig. 3. Mean reproductions as a function o f standard duration with sensation level as parameter. 0 = 10, 0 = 25, 0 = 40, D = 55 dB SL. Inset: The region near the origin.
the most prestigious arithmetical juggler. Thus the process has to be automatic. like the working of an electric circuit (which could easily be constructed as an analogue of the
As in previous investigations (Eisler, 1975. 1976; see also Eisler, 1990; Eisler & Eisler, 1991) the plots of the raw data consist of three (sometimes two; for one case out o f the 48 in the
. parallel-clock model).
Scand J Psycho1 33 (1992) Scales of subjectiae duration 345
standard and onset o f variabLe variobLe
PhysicaL t i m e Fig. 4. Duration reproduction according to the parallelclock model. Subjective versus total physical duration (left curve) and versus reproduction duration (right curve). When the difference berween these two subjective durations (upper arrow) equals the subjective reproduction duration (lower arrow). the observer reports equality between standard and reproduction by shutting OR the sound. (From Eider, 1989. Copyright 1989 by Springer. Reprinted by permission).
present experiment, one) straight lines, that is, they exhibit typically two discontinuities. These two breaks in the raw data correspond to one break in the psychophysical function:
Y = (@ - O0,)fl, @ < Ob, ( l a )
Y = a ( @ - (D0,)fl, @ > Qb, ( l b )
where Y is subjective duration, @ physical duration, the exponent, a a proportionality constant indicating a subjective weight applied to the upper segment (it is set equal to 1 for the lower), QOu and QOor the subjective zero for the upper and lower segment of the power function, respectively, and @b the duration value at which the break occurs. The two segments of the power function differ accordingly in the subjective zero @*, the weight a, or both.
According to the parallel-dock model the subjective reproduction Y r is half of the subjective total duration Y,, yielding (see Equation 1)
(Qr - @o,)B = ( a / 2 ~ @ ~ + @, - ( ~ ~ , , ) f l , (2b)
- @dB = (a /2) (@s + - @ o u ) f l , ( 2 4
where the total physical duration @, equals the sum of the standard duration and the reproduction 0,. When @, (and thus (0,) is <@,, both duration values lie on the lower segment (Equation 2a). When Qr (and thus @,) is >a,,, Equation 2c is applicable; in this equation a can be disregarded, since it cancels from both sides. If 0, is > @ b and mr < @ b ,
i.e., total duration and reproduction are on different segments, Equation 2b applies.
346 H. Eider and A . D. Eider %and J Psycho1 33 (1992)
a K
2-
14 -
12-
C 0 = 10- 0 L 1 0
> = 0 0
cn
a I -
- 9 I-
4-
2-
B" 04 I I I I I I
0 5 10 15 20 25 30 : 16
0 : I I I I I I
0 5 10 15 20 25 50 Duration In seconds
Fig. 5. Upper panel: reproduced vs. total duration; Lower panel: the psychophysical function (subjective vs. physical duration) for observer M6, 40 dB SL. 0 denote total durations and 0 reproductions in the lower panel. Note that the break at about 6 s (lower panel) corresponds to a break at the same duration at both the abscissa and the ordinate in the upper panel.
Taking the /?th root of Equation 2 and rearranging shows that @, is a liner function of Or :
( 3 4 or = ( I / 2 ) ' /Bar + [ 1 - ( 1/2) "q Qo,, 0, < @*
Scand J Psycho1 33 (1992) Scales of subjective duration 347
The three straight lines in the plots of reproductions against total durations are a consequence of the segmentation described above: When the reproduction and the total duration lie on the same segment (upper or lower), the two outer straight lines ensue (Equations 3a and 3c); for the middle one, called the transition line, the total duration lies on the upper and the reproduction on the lower segment (Equation 3b).
This makes four parameters: The exponent B, the proportionality constant GL, and the two subjective zeros mot and Oow. In addition we have the breakpoint @b. The latter is unfortunately not always distinct, as we shall see when dealing with curve fitting.
Fitting of the parameters of the power function
Consider Fig. 5 . The lower panel shows the power function for duration with the break at about 6 s. We see that for the three lowermost points both total durations (filled circles) and reproductions (empty circles) lie on the lower segment, and for the three uppermost points they lie on the upper segment. This correponds to the two outer lines in the upper panel, which shows the plot of the raw data. They have to be parallel because the exponent is the same for both segments, and the slope thus ( I/2)'lfl. For the middle four points the total durations lie in the upper and the reproductions in the lower segment, corresponding to the transition (middle) line in the raw data plot with a slope of (a/2>''8. When the break point @,, is unique and distinct, as in the data depicted in Fig. 5, it corresponds to the abscissa of the left break and the ordinate of the right break in the upper panel. For a detailed derivation, see Eisler (1975).
An interesting feature of the data shown in Fig. 5 is that the reproduction of 11 s is shorter than that of 8.1 s (8th and 7th point). This feature, also found in other time perception data (Eisler, 1981a, e.g.), is a strong indication of different psychophysical functions for different regions of the investigated continuum, psychophysical functions that differ in at least one parameter.
As indicated above, the position of the break is not always quite as clear as in Fig. 5 and as Equation 1 subsumes. The psychophysical function may exhibit gaps, as well as overlap- ping segments (Eisler, 1990). In order not to rely only on the subjective impression of the plots, the following procedure was adopted. None, one, and two breaks were placed at all possible positions. Accordingly, the ten experimental points were partitioned into one, two, and three consecutive series.
For one series, no break was assumed in the psychophysical function, implying that all ten points lie on the same straight line in the plot of reproductions against total durations. Fig. 6 gives an example, the only one among the 48 data sets.
The case of three series is exemplified in Fig. 5. Here, the two breaks are placed between points 3 and 4 (counting from the left), and between points 7 and 8. In general, however, any one of the three series could encompass between one and eight points. For other data, the left break could, e.g., lie between points 1 and 2, and the right between points 3 and 4.
When only two series are assumed, the experimental range is so small that there is space only for the transition part, with the total duration in the upper and the reproduction in the lower segment of the psychophysical function, in addition to points where both reproduction and total duration lie in either the upper or the lower segment, but not in both. Each of the two series can consist of one to nine points. An example is shown in Fig. 7.
When the @b value is distinct, a restriction corresponding to the one for three lines applies: I f the single break corresponds to the upper one, the transition line must not extend beyond the @b abscissa to the left; if it corresponds to the lower one, the transition line must not extend above an ordinate value of Q b . It should be pointed out that, from the curve fitting only, for the case of two straight lines one cannot determine which one is the transition line,
1
348 H. Eisfer and A . D. Eider %and J Psycho1 33 (1992)
18 - u 16-
0 '4-
d l2-
e
TI K 0
8 la
K 0 - 10- I TI 8 U E
w 0
-
B '- 4-
2-
O ! I I I I I I
0 5 10 15 20 25 30 i5
40 -
35 -
0' 30- - c e 4 2.5- 8 w
0 20- - n 15-
10 -
5-
Pi BY.
0 , lep" I I I I I I
0 5 10 Is 20 25 30 55 I
Duration in seconds
Fig. 6. Same plot as Fig. 5 for observer F5, 40dB SL. It exemplifies the case without a break in the psychophysical function.
except when the choice of one line as the transition line would obey the restrictions (i.e., that the ordinate of its upper end is not greater than the abscissa of its lower end), whereas the choice of the other would not.
There are altogether 64 combinations of the placements of the break point(s). For each of them the parameters were fitted by a nonlinear method of least squares, weighting the squared deviations by the variances u (for the six reproductions) o f the points, using STEPIT
%and J Psycho1 33 (1992) Scales of subjective duration 349
a
2- k p O f 1 I I 1 I I
0 5 10 15 20 25 30
16
14 -
12-
t 0 F 10- 0 L 3 0 0 I- > 0 0
f vl
- c a 6-
2 IN$, , I I I
0 0 I 10 15 20 ZS 30
Duration in seconds
Fig. 7. Same plot as Fig. 5 for observer F3, 55 dB SL. It exemplifies the case with only one break in the raw data plot.
(Chandler, 1969). The estimated value of the reproduction &r was computed from the pertinent part of Equation 3 (solved for (Dr after replacing (0, by (D,v + (0,) for all ten points. The sum of the squared differences between the empirical and estimated reproductions, weighted with the corresponding variances u, i.e., [ X ( & r -(0, .)2/v], was minimized, This was done for each of the four sound levels for each of the 12 observers, entailing 64 x 48 = 3072 fitted functions.
350 H. Eisler and A . D. Eisler
RESULTS For 26 out of the 48 data sets the smallest sum of the weighted least squares deviations was obtained when the restrictions as to the placement of the break(s) described in the previous section were fulfilled. (These restrictions necessarily entail a worse fit, decreasing the opportunity of capitalizing on chance.)
The discontinuities are determined by the abscissas of the two points on both sides of the left, and the ordinates of the corresponding two points of the right break. If there is an overlap of these two regions, the position of (Pb is within this overlap. However, this requirement was relaxed somewhat: A distance between the regions not exceeding the largest standard error of the mean of any of the four points +0.5 s was accepted. The same criterion was also used for the cases with two lines. This relaxation allows small overlaps between the two segments of the psychophysical function.
For all data sets (except those of F5, 40 dB SL, see below) a placement of the breaks in accordance with the model was chosen. For the data sets where this did not correspond to the best fit, the best one is given in parentheses in Table 3.
For the exception mentioned above (F5, 40 dB SL) a single straight line was chosen. As can be seen from the upper panel of Fig. 6, it did not seem reasonable to break up this line into three, in spite of the deterioration of the fit from 5.96 to 10.51 (see Table 3, F5, 40 dB SL). Fig. 8 shows data from Observer F6, 25 dB SL. To the left are plots with the accepted placement of the break, and to the right plots with the placement that was rejected in spite of exhibiting the best fit. Note the large overlap of the two segments of the psychophysical function for the rejected placement!
Two comparisons with simpler models were made: (i) The use of only two parameters (j3 and (Po, corresponding t o slope and intercept), i.e., no break, and (ii) a linear model, implying keeping the exponent constant a t unity (p = I). The corresponding sums of squared deviations are found in Table 3 (under “Comparisons”, “2-par.” and “lin.”, respectively).
It is seen that the two-parameter model yields a much worse fit, see, e.g., F4, 40 dB. The break-free model has to be rejected. Unfortunately, as soon as a break is assumed, the number of parameters increases by three. It is not meaningful, as in multiple regression, for instance, to increase the number of parameters one by one.
Also a linear model has to be rejected. Not only because the fit is much worse (except when the fitted /? value of the full model is close to one, of course) but also because the restrictions as to the placement of the break are met in only 13 of the 48 data sets, compared to 26 for the full model, when only the best fit is considered. The deviation values given in Table 3 refer to the smallest value and should thus be compared with possible values within parentheses. An interesting case is M4, 40 dB, where the same fit is obtained for = 0.7 and j3 = 1.0. Here we have only two straight lines, entailing the same fit whichever we take as the transition line. However, a j3 value of one implies that the upper line becomes the transition line, which violates the model restriction.
Table 3 gives also the values of the fitted parameters and of (D,,, the latter as the midpoint of the overlap (or of the gap between the two regions mentioned before).
It should be noted that the range of the exponent /? values agrees quite well with that found in Eisler (1975). Furthermore, as seen in Table 4 and Fig. 9. the exponents are close to 0.9, given in Eisler ( 1976) as an average value of about 500 exponents from 1 11 studies. An exponent of 0.9 also agrees with more recent studies, using ratio setting, including reproduction (Allan. 1978). and magnitude estimation (Allan, 1983); however, Allan con- siders the deviation from a linear function with an exponent of unity negligible or caused by bias.
&and 1 Psycho1 33 (1992)
Scand J Psycho1 33 (1992) Scales of subjective duration 351
Table 3. Partitioning of points, position of break, power function parameters, and goodness o f j t for four sensation levels and 12 observers
0. No.
Sum of squared deviations
Comparisons SL. Part of dB points fl a @o, @O" 2-par. lin.
FI
F2
F3
F4
F5
F6
MI
M2
M3
M4
M5
M6
10 21414 25 61212 40 41214 55 71211 10 01218 25 31413 40 31512 55 11316 10 11316 25 11316 40 4/1/5 55 01317 10 71310 25 5/4/1 40 0/2/8 55 01416 10 6/21? 25 21315 40 O/O/ lO 55 61311 10 51312 2s 012p 40 21216 55 01317
10 51312 25 21414 40 21216 55 51213 10 61311 25 71211 40 31413 55 0/2/8 10 01416 25 0/4/6 40 21216 55 4/41? 10 4/41? 25 4/5/1 40 01317 55 5/4/1 10 5/4/1 25 01317 40 01416 55 5/5/0 10 4/3/3 25 41412 40 31413 55 5/4/1
5.3 0.93 0.95 0.14
6.8 0.79 1.26 0.31 15.1 0.89 0.94 0.1 1
2.4 0.68 1.45 0.23 6.1 0.87 0.79 0.25 7.2 0.84 0.82 0.48 3.2 0.59 1.20 1.07 3.9 1.00 1.22 -0.20 3.5 0.91 1.24 -0.19 5.9 0.97 1.05 -0.28
12.5 1.05 0.83 -0.13
2.2 0.71 1.32 0.64 18.7 1.01 0.76 0.16 11.2 1.05 0.60 0.23 2.7 0.73 1.08 0.66 2.7 0.67 1.24 0.80
12.2 1.00 0.84 -0.22 4.4 0.89 1.16 -0.04 0.0 1.00 1.00 -0.38
13.0 1.01 0.84 -0.48 9.7 1.02 0.67 0.14 2.9 0.71 1.50 0.64 3.8 0.75 1.09 0.87 2.8 0.74 1.25 1.03
9.7 0.85 0.95 0.12 4.8 0.85 0.89 0.26 3.7 0.75 1.15 0.67 8.9 0.82 0.96 0.72
11.6 0.79 0.89 0.56 15.9 0.88 0.91 0.38 7.0 0.83 1.04 0.20 2.2 0.76 1.41 0.02 2.6 0.89 0.86 -0.20 2.4 0.87 0.83 -0.12 3.5 0.85 0.85 0.23 7.4 0.73 1.07 0.80 9.6 0.96 0.79 0.42 9.4 0.76 0.91 0.53 3.1 0.70 1.23 0.83 8.9 0.88 0.62 0.18
10.5 0.83 0.86 0.87 2.3 0.52 0.44 1.64 2.7 0.54 1.33 0.59 7.6 0.76 0.63 0.03 6.9 0.82 0.80 1.09 7.4 0.77 0.82 0.56 6.1 0.80 0.87 0.53 8.2 0.81 0.63 0.19
- 0.45 -3.07
1.66 - 1.94
1 .05 -3.01 - I .95
2.06 0.35 0.48
-0.22 I .80
-4.40 - 6.80
1.23 1.79
0.76 - 2.27
-0.38 - 1.63 -5.12
I .48 0.80 1.91
-0.84 -0.66
0.76 0.9 I
- 1.85 -0.15 -0.22
0.69 - I .06 - 1.38
0.00 2.74
-3.07 - 2.03
I .76 - 7.39 -0.54
1.61 1.22
- 7.44 -1.35 -2.11 -2.17 -6.26
1.78 9.34 6.1 1 7.27 (4.44) 7.22 (5.24) 4.95 5.96 (5.65) 2.30 (2.07) 3.80 (3.62) 8.69 (3.71) 3.30
12.42 ( 11.44) 1.18 1.83 4.41 (4.28) 2.83 1.22 ( 1.20) 1.61
10.51 (5.96) 2.81 2.99 9.14 (3.56) 5.89 2.93
0.65 5.19 (4.1 1) 4.07
3.81 (2.25) 7.30 2.83 6.05 (5.32) 8.59 0.68 (0.53) 8.40 4.61 2.15 (0.79) 2.79 2.17
1 1.32 (8.77) 4.79 (3.24)
10.74 (9.67) 1.84 1.38 2.54 (2.02) 3.48 1.90 4.76 (4.14)
2.11 (1.52)
10.64 24.42 26.13 11.81 17.88 54.88 20.89 23.73 24.47 27.52 14.86
103.69 15.78 25.18
125.12 28.55 . 5.86
8.17 10.51 30.47 22.89 60.70 27.18 34.62
2.70 10.66 2 1.67
5.07 10.47 20.24 19.22 21.37 28.10 28.16 23.49 62.01 16.48 26.28 42.60 71.88 7.09
17.48 63.53 72.25 6.57
13.90 37.1 I 36.85
3.29 9.84
19.88 6.67 8.47 6.53
10.19 22.01 3.64 7.9 I 5.13
23.75 1.32 2.42 5.39 3.01 1.16 3.35 5.99 2.90 3.15 9.15 6.77 3.84
1.78 7.26 9.64 3.17 5.09
11.71 3.80 6.12
17.19 5.71
12.05 6.10 0.99
11.93 2.17
16.18 3.24
14.13 6.20
12.65 3.66 8.9 I 6.18 6.53
Note: F = Female. M = Male observers. Regarding columns under the heading "Sum of squared deviations" see text.
352 H . Eider and A . D . Eider Scand J Psycho1 33 (1992)
U-.
0-
3
: - - 5 5 + e
2 '
O d 0 I rn u 20 ?A so 33
F6 25 db SL 0
I4
U-
C I 0 -
5 '- = 0 0
E :: '- a v)
4-
1-
0 7
/*
/*
I4
U-
= C I 0 - 5 '- 0 0
E ::' a v)
Fig. 8. Same plot as Fig. 5 for observer F6. 25 dB SL. The data points in the upper panels are the same. The psychophysical function shown in the left lower panel was chosen although the data interpretation according to the right panel gave a better fit. As can be Seen in the right. lower panel. the two segments of the psychophysical function have a large overlap. which is improbable. Furthermore. by having the outer lines in the upper right plot represented by single points, the requirement of their parallelism becomes void. thereby unduly improving the fit.
/*
f
a://, , ,
SEX, SOUND INTENSITY, AND POWER FUNCTION PARAMETERS
Five two-way analyses of variance (mixed design, BMDPZV) were carried out with sex and sound intensity as independent variables and the four parameters of the power function (/I, u, (Do,,,) and the position of the break ( O h ) as dependent variables. Two of these analyses showed significant effects, for Sound Intensity on /I (Table 4) and for Sex on u (Table 5). The distribution of these two dependent variables was not significantly different from normal, nor was the distribution of 01,, (SAS UNIVARIATE). Nonparametric tests of
Table 6 gives the means of the exponent /j and Fig. 9 shows them as a function of sensation level, separately for female and male observers. The tendency for /l to decrease with increasing sensation level is clear for both sexes, but mwe so for women. There is also a difference between the sexes for the three lower sensation levels, but the scatter renders this
and @,, did not, of course, show any significant effects.
Scand J Rychol 33 (1992) Scales of subjective duration 353
Table 4. Analysis of variance (BMDPV2, repeated measures) of f l
Sum of Source squares
S 0.06908 Error 0.21786
L 0.10216 LS 0.04529 Error 0.31338
Degrees of Tail freedom Mean square F probability
1 0.06908 3.17 0.1053 10 0.02179
3 0.03405 3.26 0.0351 3 0.01510 1.45 0.2492
30 0.01045
Note: S =sex, L = sensation level.
Table 5. Analysis of varinnce (BMDPVZ, repeated measures) of OL
~ ~ ~~~~ ~ ~ ~~~~~
Sum of Degrees of Tail Source squares freedom Mean square F probability
S 0.23171 1 0.2317 1 4.99 0.0495 Error 0.46404 10 0.04640
L 0.19447 3 0.06482 1.12 0.3555 LS 0.13552 3 0.0451 7 0.78 0.5131 Error 1.73213 30 0.05774
Note: S =sex, L = sensation level.
Table 6. Means of the exponent fl
dB SL
Sex 10 25 40 55 Mean ~ ~~~~~ ~ ~
F 0.942 0.915 0.849 0.769 0.869 M 0.857 0.775 0.745 0.794 0.793
Mean 0.900 0.845 0.797 0.782 0.831
Note: F = Female, M = Male observers.
difference nonsignificant. The effect of sensation level on the exponent is in agreement with the finding that reproductions are shorter for higher sound intensity (for longer durations).
Fig. 10 and Table 7 shows the corresponding a values. We see that a, the unit of the upper segment compared to the lower, is smaller for male than for female observers, with the exception of 40 dB SL. This is again in agreement with the findings for the raw data, see Fig. 2.
Comparison with other studies
In the introduction we mentioned that we found 36 references dealing with intensity differences and 69 concerning sex differences in experienced time. However, most of them do not allow a comparison with our results. More often than not the researchers studied less than four durations; mostly only one. In many papers the results were reported inappropriately, e.g., only the “error” without specifying its direction, or only reports of statistically sigxiificant results, likewise without indicating the direction. There were also a number of duration discrimination investigations irrelevant for the present research. Regarding intensity, the effect of empty intervals bounded by stimulation of different intensity was investigated, or empty
354 H. Eciler and A . D. Eider Sand J Rychol 33 (1992)
1.00
0.95 -
\ s \ \ \ \ \ 0.80 - \
0.75-
0.70- I I , , I , I I I , 10 15 aD 25 30 s Y) 45 50 55 80
Sound intensity, db Fig. 9. The exponent f l as a function of sound intensity. Solid line: female; Dashed line: male observers (vs. dB SL); Dotted line: Zelkinds (1968/1969) data from 24 female observers (vs. dB SPL).
intervals compared with filled intervals of different intensities. For sex differences, about one third dealing with time perspective or time concepts had to be excluded. What remains are four studies dealing with intensity and four with sex differences.
Studies dealing with the effect of intensity on timeperception. Allen & Hicks (1979) obtained exponents of 0.97, 0.98 and 0.89 for 75, 55 and 35 dB, based on duration estimation in seconds; no significant effect. Different groups were used for the three intensity levels, which tends to blur any small differences. Deal & Oyer (1975) had observers reproduce tones of 40, 50, 60, 70 and 80 phons and did not get a significant effect either. In their case the result is probably due to the long inter-stimulus interval (between offset of standard and onset of reproduction) of 15 s. Steiner (1968) used magnitude estimation with 20, 50 and 90 phons of white noise, and obtained exponent values of 0.80, 0.81 and 0.78. She used the same group, but had ten other treatments (empty intervals and tones of different frequencies and intensities) in addition in the same session, though the 13 treatments were presented separately. This procedure probably explains her result. For neither of these three studies did we have raw data, which would allow a closer scrutiny, available. More interesting is Zelkind's ( 1968/1969) dissertation. He obtained significant effects both for duration estima- tion (in seconds) and for reproduction of 1000 Hz tones of 20, 30, 40, 50 and 60 dB SPL from his 24 female observers. A linear regression on the logs of mean estimates vs. the logs of standards gives exponents of 0.82, 0.74, 0.73, 0.76 and 0.69, in order. Applying the parallel-clock model to his reproduction data yields corresponding exponents of 0.9 1, 0.88, 0.85, 0.83 and 0.78. These values are very close to our results, as an be seen in Fig. 9.
Studies dealing with the effect of gender on time perception. Adkins (1972) obtained a significant sex difference, using estimation in seconds of eight standards between 2 and 22 s. Reading off his data from Fig. 1, transforming them to seconds, and computing slope and intercept from the log-log values of estimates and standards yielded exponents of 0.93 for female and 0.87 for male observers. The scale unit (proportionality constant of the power function) was 1.20 for females and 1.05 for males, which is in the same direction as our
Scand J Psycho1 33 (1992) Scales of subjective duration 355
Sensation Level., db Fig. 10. The weight coefficient 01 as a function of sensation level. Solid line: female; Dashed line: male observers.
Table 7. Means of relative weight a
dB SL
Sex 10 25 40 55 Mean
F 0.982 1.020 1.050 1.131 1.046 M 0.859 0.802 1.078 0.889 0.907
Mean 0.921 0.911 1.064 1.010 0.977
Note: F = Female. M = Male observers.
finding of the effect of gender on a. (A linear fit of the raw data yields almost the same goodness of fit, with slopes of 0.94 and 0.69 for females and males, respectively.) The results of Carlson & Feinberg (1970) are difficult to interpret. They obtained no sex differences for estimation, but for production; for the task of reproduction the result depended on experimental conditions. However, two features of their experiment should be emphasized: The observers were not instructed not to count, and all three tasks were active, i.e., they were required to press a key also for the presentation of standards in estimation and reproduction. This procedure should tend to confuse the observers as.to the standards: are they delimited by the extinguishing of the light indicating the duration or by their releasing the key? Note that the authors found a difference between the sexes for production, where this ambiguity does not exist. Gilliland & Humphreys (1943), likewise using estimation, production, and reproduction, did not obtain any effect of gender. From their ANOVA table the F value for the effect of sex can be computed as 0.08; so low a value points to either a calculation error or some correlation among the data which we are unable to penetrate. Finally, Swift 8t McGeoch (1925) did not obtain any sex effect, using estimation in seconds for durations outside the range studied by us, namely between 30 s and 5 min.
356 H. Eisler and A . D. Eisler Scand J Psvchol 33 (1992)
3 0 L
G 0
C 0 2 4 2 m 0
n
Fig . .
9.0
8.0
7.0
8.0
5.0
4.0
I ! . Position of the break U+, as a function of sensation level. Solid line: female; Dashed line: male observers.
In going through these studies we have tried to point to differences in the experiments which may explain the lack of effects. It is always difficult to find out what lies behind divergent results. It should perhaps be kept in mind that the list given above does not include the great number of studies showing significant effects without giving their direction, whereas studies showing non-significant effects are included, since there is no direction.
The break Though no significant differences were obtained for the position of the break, a,, it may be of interest to study Fig. 11. [For the data set without break (F5, 40 dB SL), the break was assumed to occur at 0.0 s.] The close agreement between the sexes is surprising, as well as the minimum at 40 dB SL. Under the assumption that a break implies a switch from one neural loop to another (see Eisler, 19876,1990), the durations where the break occurs should not be distribated continuously. A closer scrutiny of the distribution of the (Pb values shows that their range is about the same for all sensation levels except 40 dB SL, namely from about 2 s to about 15 s. It is thus the relative frequency of the break point occumng at short or long durations that determines the average shown in Fig. 11. For 40 dB the highest value is 7.18 s; one could say that the category of breakpoints at long durations is empty. These observa- tions, as well as the agreement between female and male observers, may indicate a neurophysiological mechanism.
GENERAL DISCUSSION We can summarize our findings as (i) greater sound intensity entails shorter reproductions, and (ii) the reproductions by male observers are shorter than those by female, though both interact with the standard durations.
The effect of sound intensity could be attributed to a variation in the exponent of the psychophysical power function. One may speculate whether the lower exponent for louder
Scand J Psycho1 33 (1992) Scales of subjective duration 357
noise and the lower exponent for children (and rats) (Eisler, 1976, 1984) are related findings. “A low exponent implies that the subjective flow of time is fastest in the beginning of an interval . . . (compared with a uniform flow of physical time)” (Eisler, 1976, p. 1167). This concentration to the beginning of an interval also seems plausible for comparatively strong stimulation.
The effect of gender could be attributed to the weight of the upper relative to the lower segment of the psychophysical function. The weight (or unit) is smaller for men, who reproduce long durations (where it counts) shorter without violating the instruction. We refrain from any further explanations in terms of biological, personality or social psycholog- ical differences between women and men.
The break in the psychophysical function is an ubiquitous phenomenon in time perception and is not limited to duration reproduction experiments (Eisler, 1975, 1981a). It may be interpreted as an automatic attempt to maximize accuracy for different portions of the time continuum, analogously to the change in the range of a voltmeter for different voltages (Eisler, 1987b).
The exponent probably is a measure of the rate of internal clocks, the “sensory registers” mentioned in the Introduction. However, it should be emphasized that an exponent different from unity implies a rate continuously changing with the ongoing duration. The absolute rate cannot be determined from the present experiments. However, the weight a- of the upper segment is the ratio of its rate to the rate of the lower segment.
As a general conclusion we may state that there are two kinds of variations of the rate of the internal clock: (i) The continuous change of rate expressed by the parameter 8, and (ii) the sudden change of the absolute rate at the break point, expressed by the parameter a. The first is influenced by intensity, the second depends in part on gender.
This investigation was supported by the Swedish Council for Research in the Humanities and Social Sciences.
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Received 3 July 1991
