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Does Stevens's power law for brightness extend to perceptual brightness averaging?

The relationship between the physical magnitude of a stimulus (e.g., luminance, weight, sound pressure) and the corresponding experienced magnitude (brightness, heaviness, loudness) has been the focus of intense scientific study since the founding of experimental psychology in the mid 19th century. One particularly fruitful method of investigating this psychophysical relationship was developed in the mid 20th century by S. Stevens (e.g., S. Stevens, 1957, 1959, 1965). The method is quite simple and is called magnitude estimation because observers simply rate the magnitude of the stimulus property of interest with respect to a standard magnitude. For example, a sound of a certain measured intensity would be presented to an observer, who is told to think of it as intensity 100 (the standard). For each subsequent stimulus, the observer rates (estimates) the perceived intensity with a number relative to the standard: A stimulus perceived to be twice as loud should be rated as 200; a stimulus that sounds half as loud as the standard should be given a rating of 50, and so forth.

Using this and related techniques, the relationship between physical magnitudes ([PHI]) and psychological magnitudes ([PSI]) has been determined for scores of continua (see, e.g., Poulton, 1967; J. Stevens & Shickman, 1959; J. Stevens, Mack, & Stevens, 1960; S. Stevens, 1965; S. Stevens & Galanter, 1957; S. Stevens & Guirao, 1963). Three broad classes of continua have emerged. In the simplest case, perception grows as a direct function of the stimulus intensity ([PSI] = k[PHI]). This is true of (inter alia) visual line length; intensity of vibration; and repetition rate of light flashes, tactual pulses, and auditory clicks (J. Stevens & Shickman, 1959; S. Stevens & Galanter, 1957; M. Teghtsoonian, 1965; R. Teghtsoonian & Teghtsoonian, 1970). Thus, for this class of stimuli, [PSI] grows as [PHI] across the continuum. (Note that there is an implicit exponent of 1.0 on [PHI] in this case. Also, the constant k, but not [beta], will vary with the physical units of measurement used, e.g., inches vs. centimeters.)

In the second class, psychological magnitude grows slowly at the extreme low end of physical intensity but thereafter grows very rapidly with increased physical intensity (e.g., tactual roughness, electric shock; see S. Stevens, 1959; S. Stevens & Harris, 1962; S. Stevens, Carton, & Shickman, 1958). This relation is characterized by Stevens's Power Law [[PSI][various][[PHI].sup.[beta]]] with [beta] > 1 and represents an expansion function, because experience grows more rapidly than physical intensity for medium and high values of the stimulus.

The third class includes stimuli for which psychological magnitude grows more quickly than physical at the low end of the scale, but then follows a compression function for medium and large stimulus intensities; large increases in physical magnitude produce little gain in perceived magnitude. This relation is characterized by Stevens's Power Law [[PSI][various][[PHI].sup.[beta]]] with [beta] < 1. Perceived loudness, intensity of coffee odor, and brightness follow compression functions (Reese & Stevens, 1960; Reynolds & Stevens, 1960; J. Stevens & S. Stevens, 1963). Stevens's Power Law, the obtained exponents, and the magnitude estimation technique have generally stood the test of time. Furthermore, though the discussion above is framed with reference to magnitude estimation, the validity of Stevens's Law does not require the use of that methodology. In fact, the known exponents for various perceptual continua have been validated with other methods. S. Stevens (1959, 1965), S. Stevens and Guirao (1963), and, J. Stevens et al. (1960) have shown that the exponents found using magnitude estimation methods replicate using cross-modality matching (adjusting intensity on one continuum to indicate the experience of intensity on another). For example, an observer might adjust the loudness of a tone to match the experienced length of lines, heaviness of hefted weights, or pressures applied to the palm (see, also, M. Teghtsoonian, 1980). Similarly, Stanley (1966) obtained [beta] [approximately or equal to] 1.0 for length using blindfolded observers who indicated perceived haptic length of wooden rods by adjusting the size of the gap between their (unseen) indexfingers. Thus, observers do not need to assign numerals to their sensations to produce the well-documented exponents. Recently, other psychophysical methods, including the method of constant stimuli and two alternative forced-choice methods, have been used in perceptual scaling experiments (Ariely, 2001; Chong & Treisman, 2003).

Given that this law characterizes the psychophysical relationship for stimuli judged one at a time, it seems natural to ask whether the law extends to cases where stimuli are judged as a group. Consider, for example, a small group of lines of several different lengths. The observer is asked to judge the average of the lengths shown in the ensemble, and this judged average length is measured using one of the many psychophysical techniques. Now, the Stevens's Exponent ([beta]) for visual length of lines judged one at a time is very close to unity, as verified many times with various techniques (e.g., Baird, Kreindler, & Jones, 1971; Baird, Romer, & Stein, 1970; Hartley, 1981; Hubbard, 1994; Kerst & Howard, 1978; Markley, Ayers, & Rule, 1969; Moyer, Bradley, Sorenson, Whitting, & Mansfield, 1978; Rule, 1969; S. Stevens, 1969; S. Stevens & Galanter, 1957; M. Teghtsoonian, 1965; M. Teghtsoonian, & Teghtsoonian, 1971; R. Teghtsoonian & Teghtsoonian, 1970; Walker, 2002; Zwislocki, 1983; Zwislocki & Goodman, 1980). If this exponent is valid for the experienced ensemble averages as well, then observers ought to produce an experienced average that is quite close to the average of the physical lengths. This is true because all lines (large or small) contribute to the average according to their actual size. Numerically, this means that the experienced average should be the same as the arithmetic average of the line lengths. In fact, Miller and Sheldon (1969) and Miller, Pedersen, and Sheldon (1970) performed this experiment and found evidence that the perceived ensemble mean was a linear function of the physical mean length with an exponent [beta] [approximately or equal to] 1.0, as did Weiss and Anderson (1969). Miller and Sheldon (1969) and Stanley (1974) demonstrated this for averaging of angle (tilt) of lines within ensembles (again, [beta] [approximately or equal to] 1.0). From this, Miller and Sheldon concluded that "from these considerations the 'power law' might be expected to apply to the SA [subjective average] when it holds for the unitary continuum" (p. 21). Weiss (1972), in an article titled "Averaging: An Empirical Validity Criterion for Magnitude Estimation," also expressed the utility of further testing the generalization of Stevens's Law to averaging, yet, interestingly, the last 40 years have seen little additional evidence for this assertion.

The success of Stevens's Law with ensembles of lines only begins to address the generalizability to averaging. Evidence from other sensory continua with [[beta] [not equal to] 1.0] is required. The present experiments seek to verify the applicability of Stevens's Power Law (as indicated by the [beta] value) for perceived brightness of ensembles of filled circles (spots) that vary in luminance.

The Stevens Exponent for perceived brightness as a function of physical luminance is typically between 0.25 and 0.35 (Barlow & Verrillo, 1976; Curtis, 1970; Drum, 1976; Fagot & Stewart, 1969; Hopkinson, 1956, 1960; Mansfield, 1973; Marks, 1968a, 1968b; Marks & Stevens, 1965; Osaka, 1980; Raab, 1962; Rudd, 2007; J. Stevens & Hall, 1966; J. Stevens & S. Stevens, 1963; J. Stevens et al., 1960; S. Stevens, 1966, 1970; Van Orden, Sturr, & Taub, 1987) for steady-state stimuli (no perceived flicker) presented foveally and for at least 100 ms (see the Appendix for some representative [beta] values from these studies). Adrian and Matthews (1927) and many others have found a cube root relation (i.e., exponent of 0.33) between the intensity of light falling on the retina and the frequency of neuron firing in the optic nerve.

The prediction is straightforward. For ensembles of items of various luminance, the perceived average of the brightnesses should follow a compression function with a Stevens Exponent [beta] [approximately or equal to] 1/3. Many methods could be used to test this hypothesis. Given that several recent studies of perceptual averaging have used the method of constant stimuli (e.g., Ariely, 2001; Chong & Treisman, 2003), I chose this method for the present experiments. Observers are shown an ensemble containing spots of different luminances followed by a single probe spot. The observer decides whether the probe spot (which varies in luminance from trial to trial) is brighter or darker than the mental average of the ensemble brightness. The probe brightness, to which an observer responds equally often "brighter" or "darker" over many trials, is called the "point of subjective equality" (PSE) and is considered to be equivalent to the mental average because the observer is unable to distinguish the probe brightness from the mental average of the ensemble brightnesses.

EXPERIMENT 1

Method

Observers

Twenty-two observers were recruited from an undergraduate Sensation and Perception course. All were naive to the purposes of the experiment.

Apparatus

Timing, stimulus presentation, and response collection were performed on a Macintosh 4400/200 computer with a LaCie Electron Blue IV 19-in. CRT monitor set to 1024 x 768 at 75Hz. The controlling software was written using the VideoToolbox routines (Pelli, 1997). All screen changes were synchronized to the vertical refresh. Responses were collected via an ADB keyboard. Viewing distance was approximately 50 cm (unrestrained), resulting in approximately 0.03[degrees] of visual angle per pixel.

Stimuli

Trial events. Each trial began with a 300-ms presentation of a small "+" in the center of the screen, followed by 300 ms of blank screen. Next, the ensemble of 16 filled circles (spots) was presented for 507 ms, followed by a mask of 44 lines for 212 ms. Upon termination of the mask, the probe spot appeared and was available until a response was made. The response triggered the erasure of the screen and the appearance of the fixation "+" for the next trial.

Ensembles and Probes. Ensembles contained 16 filled circles. Within an ensemble, there were four luminances represented in four spots each. The assignment of the four luminances to the 16 spot locations was randomized. There were three different ensembles and seven probe luminances per ensemble. Luminance (Yxy) in [cd/[m.sup.2]] of the ensemble and probe spots as measured with a Minolta CS100A chromameter are presented Table 1. Black level of the monitor was 0.10 [cd/[m.sup.2]]. All spots were 48 pixels in diameter or approximately 1.44 degrees across. With three ensembles and seven probes per ensemble, the minimum crossing is 21 trials. This minimum set was replicated for a total of 210 trials per observer (10 replications per trial). The ensembles were presented roughly centered on the midpoint of the screen. The individual spots were positioned in the cells of an imaginary regular 4 x 4 grid with approximately 110 pixels between cell centers vertically and horizontally. Randomly selected positional offsets of -6, 0, and +6 pixels in the vertical and horizontal directions were applied to each item to break up the regularity of the array. This jitter was introduced to reduce the probability of runs of identical items lining up in the ensemble and perhaps being processed groupwise, or the probability that observers would adopt a subsampling strategy rather than averaging display-wide (see Myczek & Simons, 2008). The probe spot appeared below the imaginary grid and thus was never drawn in a location previously occupied by ensemble members. The probe position was also jittered laterally 0[+ or -]12 pixels in 6-pixel increments.
Table 1

Luminace ([Y.sub.xy]) in [cd/[m.sup.2]] for the Ensemble and Probe Spots
in Experiment 1.

Ens: [Y.sub.xy] probe [Y.sub.xy] a-mean PSE
 [Y.sub.xy]

 1 8.9, 13.1, 18.4, 24.6 10.8, 13.1, 15.5, 18.4, 16.2 21.2
 21.4, 24.6, 28.3

 2 15.5, 21.4, 28.3, 36.4 15.5, 21.4, 24.6, 28.3, 25.4 23.5
 32.1, 36.4, 41.1

 3 24.6, 32.1, 41.1, 50.9 21.4, 24.6, 28.3, 36.4, 37.2 26.9
 45.5, 50.9, 56.0

Note. Ens = ensemble.


The mask was different for each trial and extended beyond the screen area previously occupied by the ensemble. The 44 lines in the mask were constructed according to the following algorithm: Pick a random starting point within the allowed mask rectangle (no closer than 100 pixels to either screen edge in the horizontal direction and no closer than 60 pixels to the edges in the vertical direction), then draw a line of thickness 5 to 11 pixels (selected at random) from the starting point to another random point within the allowed rectangle. This terminus became the start point for the next line, and so on, to 44 lines. The brightness of each line was selected randomly (with replacement) from within the range of spot luminances used in the entire experiment. The mask was composed off screen and presented as a single unit. A schematic version of trial events is presented in Figure 1.

[FIGURE 1 OMITTED]

Procedure

Observers were tested individually in a black-walled room after approximately 5 minutes spent adapting to the dark. The only sources of light in the room were a small desk lamp and the light from the monitor. The desk lamp was positioned to the left and slightly behind the front of the monitor, and the accessory hood supplied with the monitor was installed to prevent the lamp light from falling on the face of the CRT. Before data collection, the following instructions appeared on the monitor:
 You will see a series of displays containing filled dots of various
 shades of gray. The displays are presented briefly then replaced by a
 single gray dot. Decide if this gray dot is BRIGHTER than or DARKER
 than the average of the grays shown in the previous display of
 dots.


The response keys were indicated (">" for "brighter than" and "<" for "darker than"), and five unrecorded demonstration trials were shown. Once instructions were understood, the trial runner was started and the experimenter left the room. An observer-terminated rest break was provided after trials 80 and 160. No feedback regarding accuracy was given. Observers completed the experiment in one sitting of approximately 15 minutes.

Results

Data Reduction

For each of the 21 ensemble/probe combinations and for each observer, the number of "greater than" responses was tallied. This allows the creation of three p("gt") curves as a function of probe luminance for each observer. The PSE is the probe luminance that corresponds to the interpolated point on the curve where the p("gt") = 0.5. PSEs were computed using the probit analysis routine in psignifit version 2.5.6 (Wichmann & Hill, 2001), with a cumulative Gaussian fit to the p("gt") functions. Noisy functions due to inconsistent responding [non-monotonic increase in p("gt") as a function of probe luminance] can produce erratic PSE estimates. Data from five observers were rejected because the PSEs computed were outside the range of luminances in a given set of probes. Inspection of the p("gt") for these five generally revealed nonsystematic or flat responding as a function of probe luminance. Each of the remaining 17 observers contributed a PSE for each of the 3 ensembles.

An ANOVA with PSE as the dependent measure indicated that the effect of Ensemble (3 levels, within-observer) was significant, F(2, 32) = 33.2, p < .001, MSE = 4.1, as were all pairwise comparisons. In accordance with Stevens's Law, the log (PSEs) were regressed on the logarithm of the arithmetic mean of the luminances in the corresponding ensemble, and the slope of that fit is the exponent ([beta]) to the psychophysical power function. The mean PSEs are given in Table 1. The Stevens Exponent [beta] is 0.283, which is within the range of slopes found by J. Stevens and S. Stevens (1963) for dark-adapted observers (0.27 to 0.33; sec the Appendix). The data were then re-analyzed using a group approach: For each ensemble and probe, all p("gt") were accumulated across all observers. These functions (shown in Figure 2) were then used to re-compute PSEs. The resulting PSEs were within 0.1 of the mean of the individual PSEs, and the exponent was 0.279. Furthermore, including the data from the five rejected observers produced an exponent of 0.25. This demonstrates that the perceptual averaging of ensemble brightness produced a [beta] value consistent with that obtained in dozens of papers investigating brightness judgments of single items. The prediction made by Miller and Sheldon (1969) is therefore supported. However, to solidify this conclusion, a replication with a different set of luminances is in order.

[FIGURE 2 OMITTED]

EXPERIMENT 2

Experiment 2 is a replication of the first experiment with a new group of observers and a different range of luminances. The luminances for the ensemble members and probes are given in Table 2. This set of stimuli is brighter, overall, than those in Experiment 1, which should lead to a slightly elevated exponent due to a higher adaptation level. Furthermore, the black level of the monitor used in Experiment 2 was slightly higher (0.20 cd/ [m.sup.2]), which may also raise the adaptation level and increase the exponent. There is a larger range of luminances within the ensembles compared with Experiment 1.
Table 2

Luminance ([Y.sub.xy]) in [cd/[m.sup.2]] for the Ensemble and
Probe Spots in Experiment 2.

Ens: [Y.sub.xy] probe [Y.sub.xy] a-mean PSE
 [Y.sub.xy]

 1 9.8, 19.0, 32.1, 49.2 11.6, 19.0, 25.0, 32.1, 27.5 27.7
 36.1, 40.8, 54.2

 2 13.8, 25.0, 40.8, 59.5 9.8, 16.3, 25.0, 28.5, 34.7 29.5
 36.1, 44.4, 54.2

 3 19.0, 32.1, 49.2, 71.0 19.0, 21.8, 28.5, 40.8, 42.8 32.0
 54.2, 65.5, 71.0

Note. Ens = ensemble.


Method

Observers

Thirty-two observers were recruited from an undergraduate Sensation and Perception course. All were naive to the purposes of the experiment.

Apparatus, Stimuli, and Procedure

The experimental setup, stimuli, and procedure (type of computer, trial events, timings) were as in Experiment 1 except for the following changes: The luminances of the ensemble members and probes were different and are shown in Table 2; and a different LaCie Electron Blue IV 19-in. CRT monitor was used, and the testing room was slightly larger.

Results and Discussion

Data reduction and PSE computations were as in Experiment 1. Data from two observers were rejected for essentially random responding. Six observers were rejected because the PSEs computed were outside the range of luminances in a given set of probes. An ANOVA with PSE as the dependent measure indicated that the effect of Ensemble (3 levels, within-observer) was significant, F(2, 46) = 13.3, p < .001, MSE = 8.0, as were all pairwise comparisons. The Stevens Exponent [beta] based on the means of individual PSEs (see Table 2) is 0.319, which is in the expected range. The data were then reanalyzed using a group approach: For each ensemble and probe, all p("gt") were accumulated across all observers. These functions were then used to recompute PSEs and are shown in Figure 3. The resulting PSEs were within 0.7 of the means of the individual PSEs, and the exponent was 0.312.

[FIGURE 3 OMITTED]

It is not possible to determine whether the exponents in Experiment 2 were (at least numerically) slightly larger than in Experiment 1 due to slight differences in adaptation level or just natural variation. This is not a concern for the present article. Given that the exponents are slopes from log-log fits of PSEs estimated from proportion p ("greater than") and that these data are from relatively inexperienced observers, it is impressive that they converge near the cube root of luminance as expected. What is more important is that the averaging process under the present conditions appears to follow Stevens's Law.

General Discussion

The prediction made by Miller and Sheldon (1969) was supported across two perceptual brightness averaging experiments. It appears that to a first approximation, the perceived average of a property across many objects follows the same power function as for the same property perceived from a single object. The present experiments provide necessary, but not sufficient, data to support Miller and Sheldon's prediction. Further tests including continua with [beta] values greater than 1.0 (e.g., warmth, electric shock; see Poulton, 1967) would supplement the present results. It will also be important to determine whether ensemble luminances (or other properties) are scaled (according to [beta]) and subsequently averaged, or whether they are averaged then scaled, for comparison against the probe. The following two equations represent these two different processes. In the first procedure, Equation 1, all members of an ensemble would be scaled according to Stevens's Power Law and then the arithmetic mean taken:

[bar.[PSI]] = [1/16][16.summation over (i = 1)](k[[PHI].sub.i.sup.[beta]]). (1)

In the second procedure, Equation 2, the average of the ensemble members is taken, then scaled:

[bar.[PSI]] = k x [([1/16][16.summation over (i = 1)][[PHI].sub.i]).sup.[beta]],(2)

where [beta] is the Stevens Exponent and k is the intercept from the log-log fit. The range of stimuli used here does not advise on this theoretically important question because the numerical difference between the two is minute. One way to address this question might be to present the ensemble members sequentially rather than all at once. In this way, the stimuli are more likely to be averaged according to Equation 1. Weiss and Anderson (1969) used a serial presentation procedure for averaging of line length but found unfortunate recency effects. In essence, the items seen just prior to responding (the observer adjusted a variable line to indicate their mental mean) were overweighted in the mean produced. This would likely contaminate the results and render invalid any comparison with a simultaneous presentation condition. That said, the present experiments do suggest that to a first approximation, the Stevens Exponent for brightness [beta] [approximately equal to] 0.33 is valid for stimuli judged singly, and for the average brightness extracted from an ensemble of items, as was suggested by Miller and Sheldon (1969).

Early in the history of experimental psychology, forms of averaging are mentioned (e.g., James's [1890] concept of "coalescence" or Messenger's [1903] concept of "constructive combination"). Helson's Adaptation Level Theory (Helson, 1947; Helson, Michels, & Sturgeon, 1954) proposed that the experience of a current stimulus is influenced by running averages of recent stimuli. Demonstrations of visual averaging of orientations (Parkes, Lund, Angelucci, Solomon, & Morgan, 2001), of motion signals (Watamaniuk & Duchon, 1992), of spatial positions (e.g., Melcher & Kowler, 1999; Vishwanath & Kowler, 2003, 2004), and of faces structurally, emotionally, and aesthetically (e.g., Haberman & Whitney, 2007; Rhodes, Maloney, Turner, & Ewing, 2007), as well as auditory averaging of pitch (Holt, 2006), suggest that the ability to average stimulus properties may be widespread in perceptual processing. One conception of the averaging (ensemble statistics) is that it contributes to a slowly changing, temporally or spatially coarse normalized signal (global), whereas individual stimuli present transient, finer grained information (local) as input to perceptual or cognitive judgment processes (see Nishida, Ledgeway, & Edwards, 1997; Ulanovsky, Las, Farkas, & Nelken, 2004). Recent physiological evidence suggests that information about members (high-spatial/temporal-frequency information) and information about the ensemble (low-spatial/temporal-frequency information; e.g., the envelope) are registered and transmitted in parallel (Mareschal & Baker, 1998; Middleton, Longtin, Benda, & Maler, 2006). Working together in this way allows efficient coding and transmission of information (see Graham & Field, 2007; Schwartz & Simoncelli, 2001; Wainwright, 1999) and representation of the gist of a scene (Chong & Treisman, 2005a, 2005b; Rensink, 2000). If the generalization of Stevens's Law from single item judgments to group averaging is confirmed, research into scaling issues in ensemble processing will be greatly simplified. Given the renewed interest and vigorous debate concerning ensemble averaging (see, e.g., Ariely, 2001; Chong, Joo, Emmanouil, & Treisman, 2008; Chong & Treisman, 2003, 2005a, 2005b; De Fockert & Marchant, 2008; Emmanouil & Treisman, 2008; Myczek & Simons, 2008), further work determining the limits of the power law generalization is warranted.

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Appendix

Steven's Exponents, for a Sample of Studies With Foveal Viewing,
Luminance Ranges, and Target Sizes Similar to the Present Experiments

 Source Target Size [beta]
 (degrees visual angle)

Barlow & Verrillo, 1976 2.0 .24-.30
Fagot & Stewart, 1969 0.2 .33
Hopkinson, 1960 0.8-2.4 .30
Mansfield, 1973 0.7-1.5 .33-.35
Marks, 1968a 1.0 .25-.31
Marks & Stevens, 1965 2.5-4.0 .29-.32
Osaka, 1980 2.0 .33
J.C. Stevens & Marks 1965 4.0 .33
J.C. Stevens & Stevens, 1963 5.7 .26-.35
J.C. Stevens, Mack, & Stevens, 1960 5.0 .33
Van Orden, Sturr, & Taub, 1987 2.0-4.0 .26-.36

Note. Target sizes may be rounded from original. Other factors, such as
duration, target color, adaptation level, task (Magnitude Estimation,
Magnitude Production), and so forth, may have been different across
these experiments.


Ben Bauer

Trent University

Thanks to the observers for their patience and perseverance. Thanks also to two anonymous reviewers who helped improve this paper. Some data from these experiments were presented at the 17th annual meeting of the Canadian Society for Brain, Behaviour, and Cognitive Science.

Address correspondence concerning this article to Ben Bauer, Dept. of Psychology, Trent University Oshawa, 2000 Simcoc N., Oshawa, ON, Canada L1H 7L7. E-mail: benbauer@trentu.ca
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