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Weber’s Law
Donald Laming
If Xis a stimulus magnitude and X+∆Xis the next greater magnitude that
can just be distinguished from X, then Weber’s Law states that Xbears a con-
stant proportion to X. As of 1958 (looking at my undergraduate notes),
Weber’s Law was attributed to a logarithmic transform somewhere in the
brain.For,ifXbears a constant proportion to X, so also does X+∆X, and
ln(X + DX) – lnX = constant (1)
This idea is due to Fechner (1860/1966), who envisaged a logarithmic trans-
form as the interface between outer psychophysics (the domain of stimuli)
and inner psychophysics (the domain of sensations). Fechner proposed that
sensation should be measured in units of just noticeable differences, so that
S = lnX + constant (2)
a relation known as Fechner’s Law. This was the consensus in 1958. It led elec-
trophysiologists to look for logarithmic relationships in sensory path-
ways and to place a quite disproportionate emphasis on a finding by Hartline
and Graham (1932) recording from a single ommatidium in the king crab,
Limulus. They found that the maximal frequency of discharge at onset
increased as the logarithm of luminance over about three log units (though
the sustained rate of discharge, measured after 3.5 s, followed a power law
instead).
At about the same time Stevens (1957) asserted that on prothetic continua
(continua for which stimulus magnitudes superimpose) sensation was cor-
rectly reflected in magnitude estimation and was related to stimulus magni-
tude by a power law,
S =aX
β
(3)
not a log law (eqn 2). And, of course, there was signal detection theory
(Swets et al. 1961; Tanner and Swets 1954).
14-Rabbitt-Chap13 7/5/08 2:24 PM Page 177
In combination with the logarithmic transform (eqn 1), the normal, equal
variance, model of signal detection theory gives a superbly accurate account of
the properties of discriminations between two separate stimulus magnitudes.
The manner in which this is achieved is illustrated in Figure 13.1. The presen-
tation of a stimulus of magnitude Xis represented by a random sample from a
normal distribution, mean lnX, normal with respect to ln(Stimulus magni-
tude). A discrimination between two magnitudes Xand X+∆Xcan then be
modelled with two normal distributions, and the standard deviation, σ, is a
free parameter at our disposal to adjust the discriminatory power of the
model to the discriminability actually observed. The two continuous density
functions in Figure 13.1 (means 0 and d) model the 52-ms data in Figure
13.2(a); dhas been set to 1.4 to generate the continuous operating character-
istic in that figure, and the vertical broken lines in Figure 13.1 are the decision
criteria that generate the data points. This model can be adapted to any stimu-
lus difference X:
d=ln(1 +∆X/X)/σ (4)
and the proportion of correct responses in a two-alternative forced-choice
task increases as a normal integral with respect to d. The stimulus difference
DONALD LAMING
178
Ln (Stimulus magnitude)
Probability density
0d
P("Brighter"| L)
P("Brighter"| L+∆L)
Figure 13.1 The normal model for discriminations between two separate stimuli. The
continuous density functions and the five criteria (dashed lines) model the 52-ms
data in Figure 13.2(a). Additional density functions (dotted curves) can be added as
required to generate a model for the entire continuum. The pale grey curves are
density functions transposed from Figure 13.4. (Adapted from Laming, D. ‘Fechner’s
Law: Where does the log transform come from?’ © 2001, Pabst Science Publishers.
Reproduced by permission.)
14-Rabbitt-Chap13 7/5/08 2:24 PM Page 178
Xenters into the calculations only via the ratio X/X, so that Weber’s Law
obtains everywhere. The dotted density functions in Figure 13.1 represent
other stimuli on the same continuum. This theory accounts for all the proper-
ties of discriminations between two separate stimulus magnitudes with
a numerical precision rarely encountered in experimental psychology—but,
I emphasize, only for discriminations between two separate stimulus
magnitudes.
Two ideas have transformed our understanding of sensory discrimination
and of sensation since 1958. The first says that sensory discrimination is dif-
ferentially coupled to the physical world,so that only changes in sensory input
are available as a basis for perception. The second idea says that there is no
absolute judgement. Instead, judgements of stimuli presented one at a time
depend on the preceding stimulus and the response assigned to it as a point of
reference; in addition, comparison with that preceding stimulus is little better
than ordinal.
Weber’s Law
Signal detection theory prompted many authors in the 1960s to propose mod-
els for Weber’s Law, usually with respect to some particular sensory modality.
WEBER’S LAW
179
P(“Yes”|Background alone)
P(“Yes”|Increment)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
52 ms duration
230 ms duration
P(”Brighter”| L)
P(“Brighter”| L+∆L)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Normal model
x2 model
52 ms duration
230 ms duration
(a) (b)
Figure 13.2 (a) Signal detection data for discrimination between two brief flashes of
light differing in luminance by 26% (Nachmias and Steinman 1965, experiment III).
The two sets of operating characteristics correspond to two different models: the
normal (Figure 13.1) and χ2(Figure 13.4). (b) Corresponding data for detection of a
bright line, 1.9-min arc in width and 18% greater than the background (Nachmias
and Steinman 1965, experiment II). (Adapted from Laming, D. Sensory analysis,
pp. 26 & 94. © 1986, Academic Press. Reproduced by permission.)
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For the most part, those models sought to explain the law without recourse to
a logarithmic transform. An unpublished manuscript from 1975 reviews 27
such essays, of which one suggested the first of the ideas.
By the early 1960s, microelectrode technique enabled recording from pri-
mary fibres in the auditory nerve (Kiang 1965; Tasaki and Davis 1955). Such
recordings revealed that primary discharges were synchronized with a specific
phase of the stimulus tone, and comprised a Poisson-like stream of impulses.
This suggested to McGill (1967) that auditory discrimination might be based
on a counting of these impulses. After a deal of complicated mathematics,
McGill derived Weber’s Law for the discrimination of the intensity of Gaussian
noise, but a square root law only for pure tones. Why the difference?
The difference results from the physical structure of the stimuli.A pure tone
is a mathematical function of time, and discrimination between one ampli-
tude of tone and another is limited only by the sensitivity of the discriminator.
McGill’s counting mechanism substituted a square root law for a constant X.
But Gaussian noise is a random function of time, each stimulus being ran-
domly selected from a set of possible waveforms. A simple geometric argu-
ment shows that, to distinguish one level of noise from another, one must
scale the set of waveforms by a constant multiplicative factor; that is, Weber’s
Law is a natural property of Gaussian noise.
Suppose, now, that the initial stages of transmission convert sensory input
into a sample of Gaussian noise. Figure 13.3 shows how this is accom-
plished. Light is transmitted as a Poisson stream of energy. The topmost trace
in Figure 13.3 is a sample from a Poisson process of density 1/2L. Sensory neu-
rones take both positive (excitatory) and negative (inhibitory) inputs, and this
is the positive input. The negative input is represented by the second trace in
panel A, another Poisson sample of density 1/2L, but now inverted. Panel B
shows the combination of these two inputs. The means, ±1/2L,cancel, and the
sensory process is thereby differentially coupled to the physical process. But
the quantal fluctuations do not cancel, because they are mutually independ-
ent. Instead, they combine in square measure to provide a combined input
that is a close approximation to Gaussian noise of power L.Webers Law
results. McGill’s (1967) study provided the source of the first idea, that sensory
discrimination is differentially coupled to the physical world.
Sensory neurones, of course, emit action potentials of one polarity only, so a
half-wave rectification follows. The positive-going excursions of the Gaussian
noise are output as a maintained discharge. Half-wave rectification loses
half the information in the original noise sample, but preserves the Weber
Law property. So how does this explanation compare with the logarithmic
transform (eqn 1)?
DONALD LAMING
180
14-Rabbitt-Chap13 7/5/08 2:24 PM Page 180
The answer is set out in Figure 13.4. The energy in a sample of Gaussian
noise has a χ2distribution, and the χ2densities in Figure 13.4 parallel the nor-
mal densities in Figure 13.1. They each have 72 degrees of freedom, chosen, as
before, to match the 52-ms data in Figure 13.2(a). The respective operating
characteristics are so similar that no experiment will discriminate between
them. This equivalence extends to all the properties of discriminations
between two separate stimulus magnitudes. This is demonstrated in Figure
13.4 by reproducing the normal distributions from Figure 13.1, but now plot-
ted as pale grey curves with respect to a linear (not the previous logarithmic)
abscissa. They underlie the corresponding χ2densities. (Likewise the χ2distri-
butions of Figure 13.4 are reproduced as the pale grey curves in Figure 13.1.)
There are, therefore, two quite distinct theories that model the properties of
discriminations between two separate stimulus magnitudes, each with superb
numerical precision. How to choose between the two?
WEBER’S LAW
181
A
B
1
2
1
2
–L
–L
0
0
Figure 13.3 (A)Two Poisson traces of equal density and opposite polarity, represent-
ing the inputs respectively to the excitatory and inhibitory components of a receptive
field. (B) Their sum, a Gaussian noise process centred on zero mean. (From Laming,
D. Sensory analysis, p. 80. © 1986, Academic Press. Reproduced by permission.)
14-Rabbitt-Chap13 7/5/08 2:24 PM Page 181
The difference needed between two separate stimulus magnitudes, Xand X
+∆X, before they can be distinguished is typically 25% (see Laming 1986,
Table 5.1, pp. 76–77). But if Xis added as an increment to a background
magnitude X, a difference of 2% can be detected (Steinhardt 1936), and a
sinusoidal grating can sometimes be detected with a contrast as small as 0.2%
(van Nes 1968). Sensory systems are therefore peculiarly sensitive to bound-
aries and discontinuities in the stimulus field. Such sensitivity is achieved by
differential coupling. Differential coupling is essential to the transition in
Figure 13.3 from a Poisson input to Gaussian noise. It accommodates a wide
range of phenomena for the detection of increments and sinusoidal gratings
(Laming 1986, 1988) and generates an asymmetrical operating characteristic,
skewed in the direction of the data in Figure 13.2(b).
The logarithmic theory in Figure 13.1 admits no such development, because
it does not incorporate any differential relationship to the physical stimulus.
So, although there are two theories that can each provide a superlative
account of the properties of discriminations between separate stimulus
magnitudes, only one of them can also accommodate the related properties of
the detection of increments and sinusoidal gratings. The normal model with
DONALD LAMING
182
Stimulus magnitude
LL
+∆L
Probability density
P(“Brighter”| L)
P(“Brighter”| L+∆L)
Figure 13.4 The χ2model for discrimination between two separate stimuli,
analogous to the normal model of Figure 13.1. The continuous density functions
and the five criteria (broken lines) again model the 52-ms data in Figure 13.2(a).
The dotted curves indicate some of the additional density functions that can be
added to model the entire continuum. The pale grey curves are the density functions
transposed from Figure 13.1. (Adapted from Laming, S. ‘Fechner’s Law: Where
does the log transform come from?’. © 2001, Pabst Science publishers.
Reproduced by permission.)
14-Rabbitt-Chap13 7/5/08 2:24 PM Page 182
logarithmic transform happens to work because the logarithm of a χ2variable
(with the number of degrees of freedom that are commonly needed to model
sensory discriminations) happens to be approximated very closely by a nor-
mal variable (Johnson 1949). Fechner’s Law derives solely from this mathe-
matical relationship (Laming 2001).
The psychophysical law
A psychophysical law is a relation between physical stimulus magnitude and
sensation. Equations (2) and (3) are psychophysical laws. Stevens (1957)
argued that only direct methods (e.g. magnitude estimation) gave unbiased
estimates of sensation. Subsequently he (Stevens 1966) showed that estimates
of the exponent in (3) were approximately consistent as between magnitude
estimation, magnitude production, and cross-modality matching. In view of
the purely mathematical origin of Fechner’s Law, this might appear to be cor-
rect. But that would be too simple.
Sometime in 1982 Christopher Poulton passed me a reprint (Baird et al. 1980)
that contained Figure 13.5. This led to the second idea. After Stevens’ death in
1972, magnitude estimation continued at Harvard University in the hands of
WEBER’S LAW
183
dB difference between successive stimuli
Correlation coefficient
1.0
0.8
0.6
0.4
0.2
0
40 30 20 10 0 10 20 30 40
Figure 13.5 Correlations between successive log numerical estimates in the
experiment by Baird et al. (1980). (Adapted with permission from Baird, J. C.,
Green,, D. M., and Luce, R. D. Variability and sequential effects in cross-modality
matching of area and loudness. Journal of Experimental Psychology: Human
Perception and Performance 6: 286. © 1980, American Psychological Association,
and reproduced with the permission of Oxford University Press from Laming, D.
The measurement of sensation, p. 129. © 1997, Donald Laming.)
14-Rabbitt-Chap13 7/5/08 2:24 PM Page 183
Duncan Luce, Dave Green, and their associates. Each participant now
contributed many more trials than in Stevens’ day, and autoregressive
analysis revealed that successive log estimates were positively correlated.
Figure 13.5 displays an example from the magnitude estimation of the
loudness of 1-kHz tones.
When the stimulus value is repeated to within ±5 dB, the correlation is
about +0.8; that is to say, the second log numerical assignment inherits about
two-thirds of its variability from its predecessor. So the stimulus on the pre-
ceding trial and the number assigned to it must serve as a point of reference
for the present judgement. That must still be so even when the difference
between successive stimuli is large, because participants cannot know until
they have judged the stimulus how it relates to its predecessor—large differ-
ence or small. The much smaller correlations when the difference between
successive stimuli is large must therefore reflect a greatly increased variability
of the comparison over large stimulus differences; such an increased variabil-
ity would be observed if the exponent βin equation (3) or in
log (Nn- Nn-1) log (Xn- Xn-1) (5)
were a random variable.
Let us throw Stevens’ power law (3) away. Suppose, instead, that the com-
parisons between successive stimuli, Xn-Xn-1, are no better than ordinal.
Equation (5) is then the mean resultant of a large number of ordinal
comparisons and, with respect to individual comparisons, βtakes on the
properties of a random variable. The model curve in Figure 13.5 results. The
data in Figure 13.5 suggest that (a) each stimulus is judged relative to its pred-
ecessor (a higher-level analogue to the differential coupling of sensory dis-
crimination) and (b) those comparisons are little better than ordinal.
The ordinal character of sensory comparisons is confirmed by Braida and
Durlach (1972, experiment 4). Participants were asked to identify stimuli
from sets of ten 1000-Hz tones, presented in different sessions at 0.25,0.5, 1, 2,
3, 4, 5, and 6 dB spacing. Identification did not become more accurate as the
spacing increased; instead, except for a purely sensory confusion at the
closest spacings, errors of identification increased in proportion to the spacing
of the stimuli (Laming 1997, pp. 150–3). This is what one would expect if the
comparison of each stimulus with its predecessor were no better than
< 'greater than', 'about the same as', 'less than' > This idea supports quantita-
tive models for a diversity of results from magnitude estimation and absolute
identification experiments (Laming 1984, 1977).
It follows from these experiments that there is no empirical distinction
between judging the stimulus and judging the sensation. Judgements are all
DONALD LAMING
184
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relative to the preceding stimulus, and the comparisons are no better than
ordinal. So judgements of stimuli and of sensations (assuming those judge-
ments to be distinct) can always be mapped on to each other, and sensation
does not admit measurement on a ratio or interval scale. In short, Stevens
power law (3) does not relate to internal sensation at all. How then does that
relation arise?
Stevens’ experiments nearly always used a geometric ladder of stimulus values—
equally spaced on a logarithmic scale. His participants had received a Western
scientific education and were well accustomed to ratios of numbers—
approximately equally distributed on a logarithmic scale (see Baird et al.
1970). Purely ordinal comparisons between one stimulus and its predecessor
lead to great variability in magnitude estimates, about 100 times the variability
of threshold discriminations (Laming 1997, pp. 120–2). Figure 13.6 presents
one set of data. The only meaningful relation between stimulus magni-
tude and numerical estimate is linear regression with respect to logarithmic
scales, and this equates to a power law. Poulton (1967) and Teghtsoonian
(1971) showed that Stevens’ exponents bore an uncannily precise relationship
to the log range of the stimulus variable—that is, Stevens’ participants
WEBER’S LAW
185
Duration of light(s)
1
2
10
20
100
0.5
5
50
Magnitude estimate/production
1 2 10 200.5 5
Magnitude estimation
Magnitude production
Figure 13.6 Matching of number to the duration of a red light. The open circles are
geometric mean magnitude estimates and the filled circles magnitude productions.
The vertical and horizontal lines through the data points extend to ±1
standard deviation of the distributions of log matches. (Data from Stevens and
Greenbaum 1966, p. 444, Table 2.)
14-Rabbitt-Chap13 7/5/08 2:24 PM Page 185
were fitting much the same range of numbers to whatever range of stimuli was
presented for judgement.
Where are we now?
Present-day understanding of Weber’s Law and of psychophysics is simpler
now than it was 50 years ago.
Weber’s Law
The key development has been the realization that the properties of a sensory
discrimination—signal-detection operating characteristic, psychometric
function, Weber fraction—depend on the configuration in which the two
magnitudes to be distinguished are presented. Formerly it was argued
(e.g. Holway and Pratt 1936) that Weber’s Law represented no more than the
minimum of a Weber fraction that increased at both low magnitudes and
high. It is now clear that for discriminations between two separate magni-
tudes Weber’s Law holds down to about absolute threshold, but that for the
detection of an increment it tends to a square root relation at low magnitudes
(e.g. Leshowitz et al. 1968).
The properties of sensory discriminations can now (Laming 1986, 1988) be
related to a small number of basic principles—the differentiation of sensory
input (see Figure 13.2), the background of Gaussian noise, half-wave rectifica-
tion of cellular output, and a local smoothing/summation of the sensory
process. These principles mean that discrimination is critically dependent on
the spatial and temporal configuration in which two stimulus magnitudes
are presented for comparison. The downside is that the mathematics needed
to relate principles to predictions are more complicated than most experimen-
tal psychologists care to engage with. For example, many visual scientists
(e.g. Klein 2001) use a Weibull function to approximate psychometric func-
tions, notwithstanding that the basis (‘probability summation’) on which that
function is derived has long been known to be contrary to experimental
observation (Graham 1989, pp. 158–9). In view of the mathematical complex-
ity, it is not surprising that the study of sensory discrimination is now out of
fashion.
The psychophysical law
Fechner’s (1860/1966) psychophysics was founded on an attempt to measure
internal sensations. That can now be seen to have been misconceived. Fechner’s
Law can be identified with a purely mathematical relationship between the
normal and log χ2density functions, but that is not the real point. What mat-
DONALD LAMING
186
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ters most is that human participants are unable to identify single stimuli
absolutely, in isolation. Analysis of magnitude estimation and absolute identi-
fication data shows, first, that each stimulus is judged relative to its predeces-
sor in the experiment and, second, that that comparison is little better than
ordinal (Laming 1984, 1997). There is no empirical distinction between judg-
ing the stimulus and judging the sensation. It is true that magnitude estimates
are still sometimes interpreted as measures of sensation (e.g. West et al. 2000),
a practice that I have dubbed the sensation error’ (Laming 1997, p. 25), in ref-
erence to Boring (1921). But it is now clear that sensation cannot be measured
in the sense that either Fechner or Stevens envisaged.
Two residual problems
Lest this survey should give the impression that in the matters of Weber’s Law
and the psychophysical law everything is now buttoned up, I finish with two
fundamental problems that still require resolution.
Asymmetrical operating characteristics in signal detection
Figure 13.2 presents two sets of signal detection data from Nachmias and
Steinman (1965). The left-hand diagram (a) shows the data from a discrimi-
nation between two primary circular fields in Maxwellian view, differing in
luminance by 26%. The right-hand diagram (b) relates to the detection of a
fine vertical line, 1.9-min arc in width, and 18% greater in luminance,
superimposed on the lesser background of part (a). All other details of the
experimental procedure, including the observer, were the same. The operating
characteristic for discrimination between two separate luminances (a) is sym-
metrical, whereas that for detection of the line (b) is asymmetrical. The super-
position of the line is but a small perturbation of the input, yet it leads to an
extreme asymmetry. Calculation based on existing theory says that, while the
characteristic in (b) should, indeed, be asymmetrical, the degree of asymmetry
should be so small as to be indistinguishable from symmetry (Laming 1986,
pp. 256–62). So, where does the extreme asymmetry in detection of an incre-
ment come from?
The limit to absolute identification
Experiments on the absolute identification of single stimuli routinely give a
limiting accuracy equivalent to the identification of five distinct magnitudes
without error (2.3 bits of information, except for colour and orientation
where there are, arguably, internal anchors; Garner 1962, Chapter 3). Pollack
(1952) provides a particularly compelling example. Comparisons with
the preceding stimulus are clearly not evaluated on an interval scale; indeed,
WEBER’S LAW
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if there is only that one point of reference, comparisons cannot be better than
ordinal: <'greater than', 'about the same as', 'less than'> But the limit to accu-
racy is clearly five categories, not three. How are the two extra categories
distinguished? Stewart and co-workers (2005, p. 892) propose that partici-
pants have an internal standard for scaling the logarithm of the ratio bet-
ween successive stimulus magnitudes—an absolute judgement of log ratios,
though not of magnitudes. But this proposal conflicts with the experiment
by Braida and Durlach (1972), in which wider stimulus spacing produces a
negligible increase in accuracy. The source of the limit to absolute identifica-
tion is still to be resolved.
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ResearchGate has not been able to resolve any citations for this publication.
Article
Suppose that, with rather few exceptions, the assignment of a number to a stimulus on one trial of a magnitude estimation or category judgement experiment serves as the point of reference for choosing an appropriate assignment on the next trial. This principle of relative judgement—relative to the immediate context—is developed to generate models for both magnitude estimation and category judgement experiments. It is applied, in particular, to the explanation of three, hitherto unrelated, phenomena: these are (a) the limited transmission of information in category judgements; (b) sequential constraints on the resolving power of category judgements; and (c) the autocorrelation of successive numerical magnitude estimates. Finally, some comparison is made between the principle of relative judgement introduced here and other contemporary ideas which have been addressed to one or more of these three critical phenomena.
Book
1. The origins of a controversy 2. Can sensation be measured? 3. Fehcner's law - the normal model 4. A reinterpretation of sensory discrimination - the chi-squared model 5. Stevens' power law 6. The physiological basis of sensation 7. Scaling sensation 8. Matching just noticeable differences 9. Judging relations between sensations 10. The psychophysical primitive 11. Why Stevens' law is a power law 12. The stimulus range 13. How then can sensation be measured?
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This paper reports the results of a series of experiments on tone pulses designed to test certain predictions of the preliminary theory of intensity resolution (Durlach and Braida, 1969) relevant to one‐interval paradigms. Resolution was measured in identification and scaling experiments as a function of the range, number, and distribution of intensities, and the availability of feedback. Some of the results, such as those on the dependence of resolution on range and number of stimuli in absolute identification, support the theory. Other results, however, such as those comparing resolution in identification with resolution in magnitude estimation for a small common range, indicate that the theory is inadequate and needs to be revised.
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From the light that falls on the retina, the visual system must extract meaningful information about what is where in our environment. At an early stage in this process, it analyzes the incoming sensory data along many dimensions of pattern vision. This book describes the current knowledge about this stage of visual processing, focusing both on psychophysical experiments measuring the detection and identification of near-threshold patterns and on the mathematical models used to draw inferences from such experimental results. Neurophysiological evidence is presented and compared critically to the psychophysical evidence. Orientation, spatial frequency, direction of motion, and eye of origin are among the many dimensions of spatiotemporal pattern vision for which experimental results and mathematical models are reviewed. Introductory material on psychophysical methods, signal detection theory, and the mathematics of Fourier analysis is also given. The preface gives a guide to the organization of the book and to what parts of the book can be read independently of one another. The last two chapters contain lists of references organized by dimensions of pattern vision. An appendix at the end of the book lists the assumptions used in the models both in order of appearance and in groups according to function.