Weber's Law
DOI:10.1093/acprof:oso/9780199228768.003.0014
Abstract and Keywords
This chapter is all about changes and developments related to the Weber's Law during the past fifty years. Two ideas have transformed our understanding of sensory discrimination and of sensation since 1958. These are the ideas that sensory discrimination is differentially coupled to the physical world and that there is no absolute judgement. During the 1960s, many researchers, prompted by signal detection theory, proposed models for Weber's Law. Most of these models were based on particular sensory modality and aimed to explain the law without recourse to a logarithmic transform.
Keywords: Weber's Law, sensory discrimination, sensation, absolute judgement, signal detection theory, sensory modality
If X is a stimulus magnitude and X + ΔX is the next greater magnitude that can just be distinguished from X, then Weber's Law states that ΔX bears a constant 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, if ΔX bears a constant proportion to X, so also does X + ΔX, and
(1) 
This idea is due to Fechner (1860/1966), who envisaged a logarithmic transform 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
(2) 
a relation known as Fechner's Law. This was the consensus in 1958. It led electrophysiologists to look for logarithmic relationships in sensory pathways 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 superpose) sensation was correctly reflected in magnitude estimates and was related to stimulus magnitude by a power law,
(3) 
not a log law (eqn 2). And, of course, there was signal detection theory (Swets et al. 1961; Tanner and Swets 1954).
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.)
(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 (p.181) 
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.)
Two ideas have transformed our understanding of sensory discrimination and of sensation since 1958. The first says that sensory discrimination is differentially 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 models for Weber's Law, usually with respect to some particular sensory modality. (p.182) 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 primary 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 amplitude 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 ΔX. But Gaussian noise is a random function of time, each stimulus being randomly selected from a set of possible waveforms. A simple geometric argument 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 accomplished. 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 neurones 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 independent. Instead, they combine in square measure to provide a combined input that is a close approximation to Gaussian noise of power L. Weber's 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)?
Figure 13.3 (A) Two Poisson traces of equal density and opposite polarity, representing 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.)
Figure 13.4 The χ2 model 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.)
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 (p.185) logarithmic transform happens to work because the logarithm of a χ2 variable (with the number of degrees of freedom that are commonly needed to model sensory discriminations) happens to be approximated very closely by a normal variable (Johnson 1949). Fechner's Law derives solely from this mathematical 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 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 correct. 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
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.)
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 preceding 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 difference 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 variability would be observed if the exponent β in equation (3) or in
(5) 
were a random variable.
Let us throw Stevens' power law (eqn 3) away. Suppose, instead, that the comparisons between successive stimuli, Xn - X n-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, β takes 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 predecessor (a higher-level analogue to the differential coupling of sensory discrimination) 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 often 1-kHz 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 quantitative 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 (p.187) relative to the preceding stimulus, and the comparisons are no better than ordinal. So judgements of stimuli and of sensations (assuming those judgements 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 (eqn 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 magnitude 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
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.)
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 magnitudes 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 13.3), the background of Gaussian noise, half-wave rectification 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 experimental psychologists care to engage with. For example, many visual scientists (e.g. Klein 2001) use a Weibull function to approximate psychometric functions, 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 complexity, 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 χ2 density functions, but that is not the real point. What matters (p.189) most is that human participants are unable to identify single stimuli absolutely, in isolation. Analysis of magnitude estimation and absolute identification data shows, first, that each stimulus is judged relative to its predecessor in the experiment and, second, that that comparison is little better than ordinal (Laming 1984, 1997). There is no empirical distinction between judging 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 reference 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 discrimination between two 1° 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 diagram (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 symmetrical, whereas that for detection of the line (b) is asymmetrical. The superposition 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 increment 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, (p.190) 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 accuracy is clearly five categories, not three. How are the two extra categories distinguished? Stewart and co-workers (2005, p. 892) propose that participants have an internal standard for scaling the logarithm of the ratio between 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 identification is still to be resolved.
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