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One aim of psychophysics is to measure the subjective intensity of sensation. We easily hear that a tone of given intensity has a particular loudness, and that a tone of higher intensity sounds louder still. The measurement problem lies in the question ‘How much louder?’ Direct scaling is a particular way of answering that question: the observer is asked to assign numbers corresponding to the subjective magnitudes of given stimulus intensities, thus providing the capability of saying that, for example, one tone sounds twice or ten times as loud as another.

## 1. History

### 1.1 Fechner’s Solution

- Fechner ﬁrst proposed a solution, albeit an indirect one, in 1860 (Boring 1950). He accepted Weber’s law—that the diﬀerence threshold (the physical size of a diﬀerence in intensity needed to tell one signal from another) grows in proportion to signal intensity, determining a constant of proportionality, the Weber fraction, unique to each sensory continuum. Fechner assumed that all diﬀerence thresholds are subjectively constant and therefore can serve as the unit of measurement for scales of sensation. He concluded that subjective magnitude grows by constant diﬀerences as stimulus magnitude grows by constant ratios, a logarithmic relation now known as Fechner’s law.

### 1.2 Category Rating And Magnitude Scaling

In Fechner’s laboratory, there also evolved a direct approach to measuring sensation magnitude. This was the method of absolute judgment (later called category rating), in which the subject assigns stimuli to categories according to some subjective aspect. Originally designed to study esthetic judgment, it was adapted by 1920 to studying sensation magnitude; early ﬁndings using this method supported Fechner’s law.

In the 1950s, vs. S. Stevens reﬁned another and related direct scaling procedure in which the subject estimates the apparent magnitude of a signal by assigning to it a numerical value—hence, magnitude estimation. Data from this method were better ﬁtted by a power law.

## 2. Methods

All the direct scaling methods require an observer to judge a series of stimuli varying along (at least) one dimension. For instance, the experimenter might present a pure tone at several sound pressure levels covering a 1000:1 range and request judgments of loudness, or present a light disk at several luminanoes covering a 10:1 range and request judgments of brightness.

### 2.1 Category Rating

The observer is instructed to assign each stimulus to a category, which may be designated descriptively (very soft, soft, … very loud) or numerically (1, 2, … , 7); the assignments should create subjectively equal category intervals. The allowable response values, including the extreme values, are speciﬁed by the experimenter. Usually the number of categories ranges from 5 to 11, although smaller and larger values are sometimes used. Typically, the stimulus range does not exceed 10:1, although it may do so.

### 2.2 Magnitude Estimation And Production

In magnitude estimation, the observer is instructed to assign to each stimulus a (positive) number so that the number reﬂects its apparent intensity. There are no limits on allowable response values. Originally, one stimulus value was designated as standard and a numerical value assigned to it as modulus. Later, however, the use of designated standard and assigned modulus was abandoned and the observer instructed to choose any numbers appropriate to represent apparent magnitudes. Typically the stimulus range is at least 100:1 and is often greater.

In magnitude production, stimulus and judgmental continua are interchanged; the experimenter presents a set of numbers, and the observer produces values on some intensive continuum to eﬀect a subjective match for each presented number.

### 2.3 Cross-Modal Matching

The use of numbers, which has provoked many objections, can be avoided by instructing the observer to match intensities on one physical continuum directly to intensities on another. For example, the observer may produce luminances to match sound pressure levels so that brightness of a light disk is equivalent to loudness of a tone, or the observer may produce handgrip forces to match odor intensities so that perceived eﬀort is equivalent to strength of smell.

## 3. Achievements Of Direct Scaling

### 3.1 A Quantitative Phenomenology

These methods made possible a quantitative phenomenology of the various sensory systems. For example, in hearing, the importance of salient variables, such as frequency and intensity, to the ability of the observer to detect a sound or to tell the diﬀerence between two sounds, had long been known. Furthermore, by having the observer match tones of varying frequency and intensity to a ﬁxed reference tone, equal loudness contours were established. What remained unknown were the loudness relations among these contours. Magnitude estimation, by providing a numerical reference scale, permits the observation that one tone sounds twice, or ten times, or half as loud as another. Thus, magnitude estimation has been used to map the eﬀects of variables known to inﬂuence perception for loudness, brightness, tastes and smells, and tactile sensations, as well as internal states such as perceived eﬀort, fatigue, and satiety (Stevens 1975). It has also been used to study continua without a physical measure (e.g., seriousness of crime, severity of punishment, and their relationship) (Sellin and Wolfgang 1964). Category rating has been employed to study the cognitive integration of perceptual dimensions (Anderson 1981).

### 3.2 Intermodal Comparisons

Direct scaling also provides the capability to compare and contrast phenomena that occur in several modalities, such as temporal and spatial summation, sensory adaptation and recovery, and spatial inhibition (brightness contrast and auditory masking), at levels above absolute threshold (Marks 1974). For example, a persisting olfactory stimulus of constant intensity produces a smell intensity that diminishes in strength over time. That diminution can be assessed by asking an observer to assign numbers to the apparent magnitude of the smell at diﬀerent elapsed times, e.g., after 5 s, 10 s, 30 s, 60 s, … ; in this way, the course of sensory adaptation can be traced for this stimulus and for others of diﬀering initial intensities. When the procedure is repeated in a diﬀerent modality, the parameters of the adaptation curves can be compared.

### 3.3 Stevens’ Power Law

The third major achievement of direct scaling is the discovery that, to at least a ﬁrst approximation, equal stimulus ratios produce equal judgmental ratios. This nearly universal relation is called Steven’s power law, after vs. S. Stevens who established it: subjective magnitude is proportional to intensity raised to some power. Further, that exponent takes distinctive values for each stimulus continuum, ranging from 0.3 for luminance to 2.0 or more for electric shock.

Later experiments showed that the magnitude exponents predicted cross-modal matching exponents: if continua A and B had magnitude exponents a and b, and if a cross-modal match of B to A was obtained, the resulting exponent was a b. Indeed, the cross-modal exponents for a variety of continua are connected in a transitive network (Daning 1983), a ﬁnding with important theoretical implications.

## 4. Problems Of Direct Scaling

### 4.1 Use Of Numbers As Responses

Critics questioned treating the numbers that subjects assigned to stimulus magnitudes as if they were measurements. However, the practice was validated by the discovery that the results obtained with number as the matching continuum are in agreement with the results obtained with cross-modal matching (Stevens, 1975). Furthermore, in cross-modal matching, some other stimulus continuum can substitute for number, since the choice of a reference continuum is, for most purposes, arbitrary. Whether the numbers used by an observer constitute a direct measure of sensation is an unresolved, and perhaps unresolvable, question (Teghtsoonian 1974).

### 4.2 The Psychophysical Regression Eﬀect

Since magnitude estimation (assigning numbers to physical intensities) and magnitude production (assigning physical intensities to match numbers) are inverses of each other, they should produce the same exponent. However, this is not the case. The size of the diﬀerence in exponents may be quite large, with the result that a precise value of the exponent for a given stimulus continuum cannot be speciﬁed; no satisfactory combination rule has been agreed upon. The size, and indeed the direction, of the diﬀerence depends on the range of physical intensities (or numbers) presented; for practical purposes, the smallest eﬀect is exhibited when the range is large. Some portion, at least, of this regression eﬀect depends on sequential eﬀects, the inﬂuence exerted by previous stimuli and judgments on subsequent judgments.

### 4.3 Category vs. Ratio Scales

Although both category rating and magnitude estimation purport to give scales of sensation magnitude, they do not, in general, agree with each other: both scales can be characterized as power functions, but the exponents are not the same. Much discussion has centred on whether subjects can make two kinds of judgments or whether the experimenter transforms a single kind of magnitude judgment by treating it as ratio or interval. However, an experimental determination of the sources of variance in the type of scale produced shows that instructions to judge either ratios or intervals account for almost none of the variance; much more important are such methodological variables as whether judgmental end-points are assigned or free and whether the range of stimuli is small or large. With free end-points and a large range, category judgment and magnitude estimation produce the same results; the obtained power functions have stable exponents characteristic of magnitude estimation (Montgomery 1975).

### 4.4 Local vs. Global Psychophysics

A long-standing conundrum in psychophysics has been the relation between the local (thresholds, both absolute and diﬀerence) and the global (measurements of subjective magnitude at levels above absolute threshold) (Luce and Krurnhansl (1988). Fechner believed he had combined the two by using diﬀerence thresholds as subjective units to yield a logarithmic scale of sensation. Stevens (1961) proposed, while honoring Fechner, to repeal his law and to substitute a power scale of sensation; he believed that one could not derive measures of sensation magnitude from threshold determinations. An early attempt to reconcile these two positions was R. Teghtsoonian’s argument (1973) that both diﬀerence thresholds and power law exponents are indices of dynamic range. He proposed that there is a common scale of sensory magnitude for all perceptual continua and that the observer’s dynamic range for each continuum maps completely onto that common scale. For example, the least sound intensity experienced has the same subjective magnitude as the least luminance, and the greatest sound intensity to which the auditory system responds has the same subjective magnitude as the greatest luminance. Thus the mapping of widely divergent dynamic ranges for the several perceptual continua onto a single subjective magnitude range determines their power law exponents, and the mapping of widely divergent diﬀerence thresholds onto a single subjective diﬀerence threshold determines their Weber fractions.

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