CHAPTER 1 INTRODUCTION Psychophysics is commonly defined as the quantitative branch of the study of perception, examining the relations between observed stimuli and re- sponses and the reasons for those relations. This is, however, a very narrow view of the influence it has had on much of psychology. Since its inception, psychophysics has been based on the assumption that the human perceptual system is a measuring instrument yielding results (experiences, judgments, responses) that may be systematically analyzed. Because of its long history (over 100 years), its experimental methods, data analyses, and models of underlying perceptual and cognitive processes have reached a high level of refinement. For this reason, many techniques originally developed in psycho- physics have been used to unravel problems in learning, memory, attitude measurement, and social psychology. In addition, scaling and measurement theory have adapted these methods and models to analyze decision making in contexts entirely divorced from perception. In this book we concentrate on the traditional link between scaling and psychophysics, but with a mind toward elucidating the broader role of scaling in psychology. There are many possible ways to study the human organism as a measur- ing instrument. Some of these merely measure sensitivity to change in the environment, whereas others attempt to sketch a person's internal organiza- tion of the world. It is our intent to describe each within its own philosophi- ----------------------------------------------------------- 2 INTRODUCTION cal and historical framework to further an appreciation of its full beauty and complexity. Several major themes are recurrent throughout all these methods. We will explore these themes and show how they lead to a shared set of beliefs (paradigm in the sense of Kuhn, 1962) that transcends individ- ual approaches. The theme of this paradigm is that human beings perform as measuring instruments interpreting the myriad of physical stimuli impinging upon sensory systems from the surrounding environment. These stimuli come in many forms, but all can be analyzed as values along component dimensions, such as light intensity, sound pressure, and size; and each of these dimen- sions is open to physical measurement, at least in theory. The key question is "How does the human being use sensory and cognitive mechanisms to perceive the type and amount of stimulus energy?" In brief, psychophysics investigates the correspondence between the magnitude of stimulus proper- ties as assessed by the instruments of physics and as assessed by the perceptual systems of people. PSYCHOPHYSICAL THEORY Many psychologists unfamiliar with the subtleties of psychophysics are apt to treat it more as technology than as science. This is because their exposure to the field is often limited to the blind application of specific methods without consideration for the historical and theoretical rationale behind their devel- opment. The scientist working in psychophysics sees matters in quite a different light. To this person, psychophysical theory shapes the interpreta- tion of results as well as the experimental procedures. Whatever the subject population, the theorist searches for invariant relations among stimuli and responses and the constraints these relations place on models of the underly- ing mechanisms of information processing. The technical machinery re- quired to reach this goal is important and interesting in its own right, but for most scientists this aspect is clearly subservient to substantive issues in psychology. In brief, a psychophysical theory is a set of statements (assumptions) that describe how an organism processes stimulus information under carefully specified conditions. The assumptions usually concern hypothetical processes that are difficult or impossible to observe directly. Once these assumptions are made explicit, however, formal models can be devised. The validity of the theory can be tested by comparing observations against the predictions of the model. In other words, a theory represents a set of "reasonable" guesses about exactly how a person behaves as a measuring instrument when asked to judge properties of stimuli. ----------------------------------------------------------- PSYCHOPHYSICAL THEORY 3 Detailed predictions of what a person will actually do in an experiment are based on models especially designed to test one or more theories. Although in recent years the terms "model" and "theory" have often been used interchangeably, a model is thought to be a concrete synthesis of the assumptions of a theory. This synthesis specifies the interrelationships among the postulated primitives of the theory. Often these statements are in the form of mathematical formulas, computer programs, or logical truisms. In this way they are both more specific and yet more general than the theory giving rise to them--more specific in that the theory, through its models, is now amenable to laboratory test, and more general in that an abstract model may be used to quantify theories in many areas of study. In some cases, a model combines only a few explicit assumptions and therefore can be readily accepted or rejected by an examination of relevant data. This is particularly true when mathematical functions are fit to results in hopes of producing a more economical representation. For example, the mathematical functiony = ax+ b might be employed as a model, wherey is the magnitude of a subject's estimate, x is the stimulus intensity giving rise to that estimate, and a and b are determined by examining pairs of x andy values. By appropriate statistics we can quickly decide whether or not the model is adequate to describe the data. This is not to say that no assumptions are involved in its use. Critical assumptions may simply be well hidden or left unstated; for instance, it may be assumed that the relation between stimulus (intensity) and response (magnitude) may be modeled by a smooth curve without jumps or discontinuities. One important example of such a descriptive model is the power function, which has played a dominant role in modern psychophysics, mainly because it handles an impressive array of experimental results. For example, a person's perception of the intensity of electric shock can be studied by comparing numerical estimates (verbally stated) with actual physical inten- sities. One can often approximate the relationship between stimulus intensity (S) and mean response magnitude (R) by the equation R =XS  (1.1) where X and n are constants. The X is merely a scaling constant that depends on the particular experimental conditions. By contrast, the attribute under study is characterized by a unique value of the exponent n. For electric shock, n is approximately 3. This general relation, often called Stevens's law, holds for many stimulus continua, including visual line length, sound intensity, luminance, and force of handgrip (Chapter 5). The interesting invariant here is the power function, because it transcends the particular nature of the stimulus and suggests an underlying order in the perceptual transformation of the environment. ----------------------------------------------------------- 4 INTRODUCTION Some investigators argue that this is all we can ever expect from a psychophysical function and that consequently there is no need to delve further into its underlying assumptions. On the other hand, one can argue that a theory is still required to explain why the power function works so well, and this theory will come equipped with assumptions and rules about why it holds under some conditions and not under others. We offer such a theoretical structure in Chapters 4 and 5. SCALING MODELS The level of understanding in various scientific disciplines is closely linked with the extent to which quantitative methods are employed. This is true in psychology, where the concept of a dimension or a scale extends beyond the field of perception and psychophysics. A scaling model is employed in a variety of situations to represent empirical relations among a set of observa- tions by the mathematical relations among a set of numbers (Krantz et al., 1971). Hence, we can "scale" not only the perceived intensity of electric shock but also such attributes as the excellence of handwriting or the desirability of automobiles. Therefore, the use of scaling models in psycho- physics is a subset of their total use in psychology. While emphasizing psychophysical applications throughout this work, we also hope to provide some insights into some of the broader issues raised by the use of scaling theory. MEASUREMENT AND SCALE TYPES One of the key ideas throughout psychophysics and scaling is the concept of measurement invariance. That is, across a variety of conditions, the data from experiments should be consistent in predictable ways. The theory need not say that the response to a given stimulus will be absolutely invariant over many replications, but only that the pattern of response magnitudes will stay the same. Measurement theory is the study of these patterns; numbers are assigned to objects or attributes so that the relations among the attributes are represented by the relations among the numbers. Such a pattern is called a scale type. Although a fair variety can be formulated (Coombs, Raiffa, and Thrall, 1954; Krantz et al., 1971), we will concentrate on the four most commonly distinguished: ratio, interval, ordinal, and nominal (Stevens, 1946, 1951). The thing to note here is that the type of scale employed in any instance depends on theoretical expectations and restricts the kinds of ----------------------------------------------------------- MEASUREMENT AND SCALE TYPES 5 statements one can make about empirical data represented by points along the scale. Less information about the empirical world is conveyed by a nominal scale (essentially a set of labels) than by a ratio scale (a ruler, for example). The rules of numerical assignment are logical or mathematical, and the objects or attributes are phenomena whose existence is agreed to on either empirical or theoretical grounds. In psychophysics the stimulus attributes are light intensity, sound level, electrical current, and weight; the measuring instruments include the photometer, sound level meter, voltmeter, and balance; and the rules of interpretation of the system are taken from mathematics (for example, 1 q-1 = 2, 3 > 2, 5 -5, and 4- 7). An unusual side of this enterprise is that numerals (emitted by subjects) are treated as numbers satisfying the rules of mathematics. In this way, the subject is treated both as the originator of an attribute (numerals) and as the measur- ing instrument for that attribute. Only in a few special cases is the assignment of numbers absolute. One example is the counting of distinct objects. In most situations, however, numbers are assigned relative to some standard. For example, the values assigned to lengths depend on the unit of measure (the length of an object could be given the value 100 if the units were centimeters and the value 1 if the units were meters). This does not mean that the values assigned to lengths can be anything. In this case the numerical assignments are con- strained by what will later be more formally described as a ratio scale. For instance, a pair of lengths assigned 12 and 24 cm could not be assigned the numbers 1 and 50 in another unit of measure. This would not satisfy the rules of a ratio scale, since 24/12-50/1. (The values 20 and 40 would satisfy the requirement, since 24/12 = 40/20.) Often the requirements of a ratio scale are too stringent, for reasons either theoretical or experimental (the data do not satisfy the conditions). But luckily we can substitute other criteria for the equal ratio requirement and still "do science." Different sets of criteria determine different types of scales, each implying a particular uniqueness for the relations among the data represented. As the uniqueness requirements of the ratio scale are relaxed, we obtain the interval, ordinal, and nominal scales. It should be emphasized that there is nothing sacred about these four scale types. This classification scheme is due primarily to the relative simplicity of specifying and testing their requirements. In addition, it helps us to decide which mathematical models are appropriate for analyzing the data. A related point of view is that these four types specify the degree of invariance expected when scaling data. This means that certain relations among scale values will remain true when another scale is constructed from a second data set obtained under the same, or similar, conditions. ----------------------------------------------------------- 6 INTRODUCTION Table 1.1 Hypothetical results from light measurement Black box 1, Ratio scale Lights Scale Trials A B C D E transformations 1 2 5 8 10 15 x 2 6 15 24 30 45 x' --= 3x 3 1 2.5 4 5 7.5 x" =0.5x Black box 2, Interval scale Lights Scale Trials A B C D E transformations 1 2 5 8 10 12 x 2 5 8 11 13 15 x'=x+3 3 0 1.5 3 4 5 x" -= 0.5x - 1 Black box 3, Ordinal scale Lights Scale Trials A B C D E transformations 1 2 5 8 10 12 x 2 2 6 6.5 9 200 x' = monf(x) 3 1 75 90 92 99 x" --- monf(x) Black box 4, Nominal scale Lights Scale Trials A B C D E transformations 1 2 5 8 10 90 x 2 3 1 6 2 0 x'=f(x) 3 8.5 60 1.5 90 10 x" --f(x) f(x) is a relabeling of each light such that all the new labels are different from one another. To illustrate the connection between theoretically expected invariances and data, consider a hypothetical experiment. Let us say we have a generator that produces lights of five different intensities. (We will denote the lights as A, B, C, D, and E.) To avoid some philosophical problems, let us assume that light A is always of a constant intensity and that the same is true for the other four lights. Suppose also that we have four "black boxes" ----------------------------------------------------------- MEASUREMENT AND SCALE TYPES 7 (could be people, photometers, or Martians), each producing scale values consistent with one of the four scale types (a ratio-scale black box, an interval-scale black box, etc.) Let us suppose, further, that through some misfortune we have mixed up the labels and now wish to accurately relabel the boxes. To do this, we run an experiment and collect responses from all four boxes. This is done on three occasions (replications) for each of the five lights. The hypothetical results are summarized in Table 1.1. Ratio Scale Black box 1 produces responses that maintain their relative sizes across trials. For instance, the ratio of responses for lights A and C is always 1/4=6/24= 2/8. This could be interpreted as meaning that once a value for the standard, A, is established, the value for C is fixed. Alternatively, this means that values obtained from one trial can be transformed into those from another by a single multiplier. That is, each repetition (trial) produces a scale, and to go from a measure x from one scale to a measure x' on another, multiply by a positive constant a: x>x', x'=ax, a>O (1.2) Other qualities make this pattern of numbers unique among the four scale types to be mentioned later. The ratio scale is the only one in which the concepts of "twice" or "ten times as bright" have any meaning. One of the distinctive characteristics of a ratio scale is the existence of a unique zero, so a stimulus attribute maintains the same zero point on all trials regardless of the multiplicative constant a. In addition, a ratio scale satisfies all the criteria for the other three scale types. So, even though black box 1 could be labeled as any one of the four types, it is probably best to say it produces numbers on a ratio scale, because this gives us the most information about the invariance of the scale values. The empirical data (measures) are best modeled by the numerical rules of a ratio scale. Interval Scale Next, let us consider the data collected from black box 2 in Table 1.1. Here the ratios among the measures of the four lights can vary over trials (for example, comparing measures of A and B on trials 1 and 2, 2/5  5/8), but the relative size of the intervals is retained, as well as their intensity order. To ----------------------------------------------------------- 8 INTRODUCTION illustrate this, consider the interval AB and the interval CD on trial 1. IA-/l=12 - 51=3 Ic-/l=l-01=2 Rather than look at the ratios of the scale values, as we did for ratio scales, we look at the ratio of the two intervals IC-Ol _2 IA -al 3 This ratio is the same for each of the other trials. For instance, [C-D__.[= 14-3__] 1 _ 2 la-BI 1.5-ol 3 on trial 3, Hence, transformation from a measure x on one trial to x' on another is given by the linear equation known in mathematics as an affine transforma- tion: ' ' (3) xx, x =ax+b a>O 1. If each pair of trials produces a set of measures that constitute a scale, corresponding measures may be related by a wider range of transformations than that permitted for ratio scales. That is, adding a constant is now a permissible transformation. This also eliminates the significance of zero (zero always being mapped to zero in a ratio scale) since it can now be trans- formed into numbers other than itself [i.e., (x = 0)+ b = b]. Measures satisfy- ing the conditions for an interval scale also satisfy all conditions for ordinal and nominal scales, but we shall label black box 2 as producing interval scale results. As in the case of box 1, this gives us the most information about the invariance of the scale values. Ordinal Scale Among the measures from black box 3, only the order of scale values is constant over trials. Light A is always less intense than light B, and so on, but the ratios and the ratios of intervals among scale values may change from trial to trial. The integrity of the ordinal scale is maintained by any ----------------------------------------------------------- MEASUREMENT AND SCALE TYPES 9 monotonic transformation: x-->x', x'--monf(x) (1.4) where x' is a monotonic function of x. Therefore, if); > v, then);' > F, since f(x) is a transformation that maintains the order of); and v. Nominal Scale The measures from black box 4 seem to have very little consistency. All we can say is that the measures within a single trial distinguish the lights from one another. In addition to a loss of ratio and interval information, stimulus order may change from trial to trial. The simple substitution of unique labels within each trial is indicated by the expression x' =f(x). The only information available is whether two stimulus attributes are the same or different. Labeling can be said to produce a scale of sorts, but since numbers need not be involved (we could use letters, or dogs' names), it is debatable whether this scale should be included in the list (Torgerson, 1958). In any event, the nominal scale adds a final note to the hierarchy of types most often discussed in the literature. CONCLUDING COMMENT We began the previous section by claiming that numbers could be attached to stimuli according to different rules. In the measurement example, the meaning of numerical reports depends upon theoretical considerations and the invariant properties of the measures over trials. Therefore, the critical question is: What scale properties are left invariant? The answer determines the scale type and hence the particular significance given to the measures. In order for scaling to be a scientifically useful tool, one must know the invariances and permitted scale transformations in the data produced. This allows measurement in, and sometimes generalizations to, situations beyond those in which the scale was originally established. It also restricts the various analyses that may be performed on the data. The major point, then, is this. If a person is viewed as a measuring instrument in the psychophysical paradigm, the pattern of responses will contain certain invariances over different experimental conditions. Theories capitalize on these invariances by organizing responses into a scale type, which restricts the kinds of meaningful statements we can make about the ----------------------------------------------------------- 10 INTRODUCTION data being modeled. Only four scales have been discussed here since they are the ones most relevant to modern psychological research. Occasionally in psychophysics, often on the physics side, it is helpful to deal with other types of scales (see Krantz et al., 1971). For example, scale values may be invariant up to a power transformation. x-->x', x'=cx , c>O, ¾>0 (1.5) Assuming x and x' are positive, this formula may be rewritten as follows: log x' = ¾1og x + log c (1.6) We see that with suitable transformations of the initial scale values, the formula is identical to that prescribed by an interval scale (Eq. 1.3). In both cases two values are needed to specify the scale: the multiplicative constant a or , and the additive constant b or log c. For this reason, scales that are invariant up to a power transformation are called log-interval scales. In actuality, the early history of psychophysics contains little in the way of references to scale types, whose importance only became apparent during the course of applying classical methods to new problem areas. However, the psychophysical paradigm of the human-as-a-measuring-instrument was laid down from the start in the writings of Gustav Fechner who also provided the field with its first theory and accompanying mathematical model (the logarithmic function). This historical development is the subject of Chapter 2. REFERENCES Coombs, C. H., Raiffa, H., and Thrall, R. M. "Some views on mathematical models and measurement theory." Pychological Review, 1954, 61, 132-144. Krantz, D. H., Luce, R. D., SupDes , P., and Tversky, A. Foundations of Measurement. I. Additive and Po.lynomial Representations. New York: Academic, 1971. Kuhn, T. S. The Structure of Sdentific Revolutions. Chicago: University of Chicago Press, 1962. Stevens, S.S. "On the theory of scales of measurement." Sdence, 1946, 103, 677-680. Stevens, S.S. (Ed.) "Mathematics, measurement, and psychophysics." Handbook of Experimental Pychology. New York: Wiley, 1951, 1-49. Torgerson, W. S. The0y and Methods of Scaling. New York: Wiley, 1958. ----------------------------------------------------------- CHAPTER 2 CLASSICAL PSYCHOPHYSICS: FECHNER'S LAW Our study of psychophysics begins most naturally with i'ts founder, Gustav Fechner, a German philosopher and physicist who combined his philosophi- cal inclinations and scientific talents into the production of the now classic Elemente der Psychophysik, published originally in 1860. This work contains a strange mixture of philosophy, mathematics, and experimental method, making it difficult to treat individual aspects of his approach in isolation. In this and the succeeding chapter, we will outline the kinds of problems Fechner posed, his suggested solutions, and the relative success of these efforts compared with more recent formulations. Fechner's major concern was with philosophical matters. His guiding mission throughout a productive intellectual life was to undercut material- ism as a dominant style of thought. In place of this view, he championed a more mystical approach to philosophy and science, giving full attention to the conscious experience he believed accompanied all physical processes. Most remarkably, in anticipation of later thinkers, Fechner attributed consciousness to organisms at all levels of the animal hierarchy, as well as to 11 ----------------------------------------------------------- 12 CLASSICAL PSYCHOPHYSICS: FECHNER'S LAW other living matter (such as plants; Fechner, 1848; Goude, 1962). To him the physical environment was a womb containing the life spirit, which could find complete unity with the universe only after death of the corporeal form. A true theoretician, he clearly states in one of his books (1905) that, after death, the human spirit is able to comprehend all patterns, relationships, and meanings of the universe simultaneously. To Fechner, this was tantamount to comprehension of God, whom he equated with the spirit of the universe. PHILOSOPHY OF PSYCHOPHYSICS Elemente der Psychophysik is limited to one manifestation of this philosophical position: the conscious experience (sensation) accompanying activity of the brain induced by external stimuli--in short, the mind-body problem. For Fechner, this area of inquiry served only as an example, which hopefully would generalize later to broader epistemological issues. When it came to psychophysics, Fechner dealt primarily with three concepts and their interrelations: (1) the external physical environment, (2) the brain (called bodily) activity accompanying environmental stimulation, and (3) the conscious perception (sensation) accompanying these external and internal physical processes. Of the three possible pairings, the mapping between brain activity and conscious sensation was the most critical. Fechner termed the theoretical study of this relationship "inner psycho- physics." Unfortunately, as so often happens in science, this most crucial relationship proved to be inaccessible to direct observation. Therefore, one could only speculate about the exact properties of inner psychophysics, and this created a seemingly insurmountable barrier. To avoid this barrier, Fechner hypothesized that measured brain activity and subjective percep- tion were simply alternative ways of viewing the same phenomena. One realm of the universe did not depend on the other in a cause-and-effect fashion; rather, they accompanied each other and were complementary in the information conveyed about the universe (Fechner, 1907, 1966; Arguelies, 1972). In this regard, his world view was similar to that introduced some years later by the physicist Neils Bohr (1961) to explain the apparently contradic- tory behavior ascribed to light when defined in alternative ways (waves or particles). That is, a single phenomenon may be modeled in several ways with each model revealing only part of the overall picture. The complemen- tarity principle in psychophysics involved a distinction between what Fechner called the "night view" (materialism, body physiology) and the "day view" (mysticism, mind). ----------------------------------------------------------- PHILOSOPHY OF PSYCHOPHYSICS 13 (environment) Figure 2.1. DAY VIEW (wisdom) OUTER PSYCHOPHYSICS I I I [ NIGHT VIEW ] (moteriolism) Schematic diagram of Fechnerian psychophysics. This conception is illustrated in Fig. 2.1, where the environment N is shown impinging on the person and producing an internal event. When viewed from below the page, one sees the M of "materialism," but from above one sees a IV, which might stand for conscious "wisdom." The geometric pattern on the page is identical in either case, but its interpreta- tion varies depending upon one's location vis-a-vis the page. Acceptance of this dual but functionally interdependent view of the same phenomena represents Fechner's basic epistemology. The scientific problem then be- comes one of finding the correct transformation between these two standpoints and expressing this transformation mathematically. Now consider the following transformations. The change from perception of M to W in Fig. 2.1 is accomplished by rotating the page 180 ø. The transformation required to change from the night (material) to the day (psychological) view of perceptual events is not so obvious. Before focusing on that primary issue, it is helpful to state the possible relationships among N, W, and M and Fechner's ground rules concerning them (Fig. 2.1). The hypothetical relation between bodily activity (M) and conscious sensation (W) is inner psychophysics. The relation between M and N involves principles of physics, chemistry, and physiology. This relation was of interest ----------------------------------------------------------- 14 CLASSICAL PSYCHOPHYSICS: FECHNERS LAW to Fechner, but the measurement of M lay outside the power of existing scientific technology. Instead he assumed that environment (N) and sensa- tion (W) were the subject matter of outer psychophysics. This field was the basis of Fechner's admirable experimental work and is the precursor of modern psychophysics. In this chapter, however, we are most interested in the proposed transformation between M and W (inner psychophysics). In the next chapter the experimental approaches to outer psychophysics are discussed. INNER PSYCHOPHYSICS Fechner was primarily looking for a quantitative means to express the relation between brain activity (M) and sensation (W) following the pre- sentation of environmental stimuli. Since he assumed a one-to-one corre- spondence between elements in N (environment) and M, he indirectly measured the latter using a quantitative scale of the N intensity. (He actually spoke of the amount of internal kinetic energy.) That is, environ- mental stimuli (say, line lengths, or lights presented to the eye) have their M analogs in the central nervous system. Theoretically, the length of each line or the intensity of each light could be measured and ordered by the magnitude of this internal physiological effect. Second, in his search for a mathematical relationship between M and W, Fechner assumed that sensation could be measured (theoretically) and properly located along an internal scale. The function relating values of physical and psychic magnitude then represents the psychophysical transfor- mation. The general form of this transformation was thought to be the same for all physical attributes (lights, sounds, lengths, smells). In effect, the idea was that conscious experience is unitasv, can be ordered along an intensity scale, and bears a constant relation (except for changes in stimulus parame- ters) to both internal brain processes and external conditions giving rise to those processes. Hence, a pluralistic, materialistic environment is mapped into a unitary scale of conscious sensation. Although some mechanism must also be provided in this scheme for recognizing qualitative differences among attributes, this difficulty did not receive much attention from Fechner and consequently played no role in his further development of a quantitative mapping between M and W. We see that Fechner had little hope of measuring M and W directly, and therefore the mapping between them was similarly out of scientific reach. How, then, did he decide upon a particular mapping? We can, of course, never know the actual conditions leading to the proposed mapping. We do know, however, that he was familiar with the work of the mathematicians ----------------------------------------------------------- INNER PSYCHOPHYSICS 15 Bernoulli, Laplace, and Poisson, who proposed that the subjective utility of an increase in personal wealth was a function of the amount one already had. Different increments in monetary wealth are required to produce equally noticeable subjective changes for the rich and poor. Not a highly debatable hypothesis! For example, in order to obtain the same significant increment in subjective utility, both the rich and the poor might have to double their existing wealth (see, for example, Bernoulli, translated by Sommer, 1954). In expressing this rule quantitatively, the subjective utility of money is logarithmically related to its monetary value. This in turn was the rule Fechner used to describe the relation between M and IV. He tells us that the solution came in a flash of intuition, a hunch, as he lay in bed on the morning of October 22, 1850 (Boring, 1950; Fechner, in translation, 1966). This date, "Fechner Day" is still celebrated in scattered laboratories and homes throughout the world. Since we can be sure a man of his caliber was familiar with numerous functional relations, it would be a mistake to think he simply extended the earlier mathematical formulations on utility of money to apply to psychophysics. The hunch was undoubtedly the outcome of many hours of sifting through alternatives, casting about for new formula- tions, and checking ideas against reality (as he knew it from the experimen- tal literature). On the other hand, the early mathematical work probably influenced the quantitative expression of this important hunch. Whatever the actual antecedents, the field of psychophysics was founded on the following premise: A geometric increase in brain activity ( M) is accompan- ied by an arithmetic increase in conscious sensation (W). In other words, a constant increment in IV is paralleled not by a constant increment in M, but by an increment in M that is proportional to the existing amount. This relation is shown in Fig. 2.2, where equal arithmetic steps in W are accompanied by geometrically increasing steps in M. That is, the units on the IV axis are stepped off in equal intervals of size c. On the other hand, these equal steps of W are accompanied by geometric changes (equal ratios) in M of the form M 1  kM o M = k'kM o (2.1) Mr_ 1: k'- 1Mo M, = k'Mo where M 0 is a threshold or initial value. Here, i is the number of steps along ----------------------------------------------------------- 16 CLASSICAL PSYCHOPHYSICS: FECHNER'S LAW o w o =o Mo-l c=l k=l.2 M..Z = k i,ø 5 M 2 M t M4 0 I0 20 30 40 50 60 M, meterielism (arbitrary units) Figure 2.2. Fechner's conception of inner psychophysics. Equal steps on the sensation scale (W) are related to equal ratios on the materialist scale (M). the W axis and k is a constant ratio obtained in the following way: M, -- =k (2.2) Mi_ 1 ki-lMo It is convenient to graph the relation so that a constant interval on the W axis accompanies a constant interval (instead of ratio) along the M axis. This can be approached by taking logarithms 1 of both sides of Eq. 2.2: lnMi-lnM _ --Ink 1 1Logarithms are the inverse of the exponential function, so if the following equality is true for constants c and b then the following equality is also true: lnc= b. This is read: "log of c to the base e is equal to b." In this chapter, all logarithms are to base e (a constant approximately equal to 271828), which is written "In" (natural log) instead of the more verbose "1oge." ----------------------------------------------------------- OUTER PSYCHOPHYSICS 17 Since k is a constant, Ink is a constant, so if we assume that W is a linear function of the logarithm of M, each constant interval c corresponds to a constant interval ln k. Typically, a linear function is written y--ax, where x and y are the two variables and a is a constant slope. In our case, x corresponds to lnM, y corresponds to W, and a is the ratio c/Ink. Therefore, the equation may be rewritten as follows: c W-- 1-lnM (2.4) Although the general form of Eq. 2.4 is considered to be invariant, the constants c and k may depend on the stimulus attribute and the experimen- tal conditions. Fechner himself did not give much attention to these vari- ables, but modern psychophysics does, as we will see in later chapters. With some modifications (outlined later), this logarithmic relation is the one proposed by Fechner to describe the transformation between the materialis- tic and mentalistic views as reflected in the smaller, but most crucial, arena of inner psychophysics. OUTER PSYCHOPHYSICS As noted earlier, the original idea for the psychophysical transformation was based on a hunch, and Fechner was surely not satisfied to let it rest at that. Gonsequently, he proceeded to develop a theoretical rationale for the logarithmic function that depended more on logic than on personal whims and historical precedent. Since he knew what to look for--that is, the final outcome had to agree with his intuitions--the task was much easier, although success was not inevitable. His eventual solution was truly creative and still stands as one of the most interesting theoretical arguments in the history of psychology. We will now trace a line of thought that probably captures at least some of the factors considered in arriving at a rational argument for the logarithmic function. Although the relation between the internal events M and W was the object of attention, it was clear that this relation could not be verified by direct measurement. Therefore, an alternative approach was necessary. It was possible, of course, to obtain a direct measure of N, the environmental stimulus. This, then, was a starting point, although there was no hope of measuring the brain processes triggered by N. That left IV. If some measure of W were possible, however indirect, the relation between N and IV could probably be determined. Fechner called the experimental quest for this relation "outer psychophysics." With the assumption of a one-to-one connec- ----------------------------------------------------------- 18 CLASSICAL PSYCHOPHYSICS' FECHNERS LAW tion between N and M, one could infer the all-important mind-body function within the realm of inner psychophysics. This situation may be clarified by reexamining Fig. 2.1. The problem, then, is to find a relation between N and W, where the former can be measured directly and the latter cannot. The initial condi- tions for this problem are diagrammed in Fig. 2.3. Suppose the N-meter in Fig. 2.3 is an instrument recording the magnitude of an external stimulus impinging on a person along some dimension of interest (such as light, sound, or length). Treat the W-meter as purely hypothetical for the moment. It would be nice if the W-meter could measure internal sensation so changes in it could be correlated with changes in the N-meter. Fechner thought that reading the W-meter was impossible or, at best, would produce unreliable results. In other words, a person would be incapable of reporting the absolute magnitude of his or her sensations by saying something like: "This light is 17 units on my sensation scale," or "This sensation is three times as intense as the previous one." Presumably a person could report whether one sensation were less than, equal to, or greater than the intensity of another. The cornerstone of all Fechner's measurement procedures was this assump- tion that an internal scale of sensation does exist. Differences between sensations can be detected but their absolute magnitudes cannot. Two types of changes in scale value are relevant here. The first is the change from zero sensation to unit 1; that is, the threshold or starting point of the W-meter is triggered by the stimulus intensity (magnitude) necessary for any sensation at all to be experienced. This is called the absolute threshold (or absolute lower threshold to distinguish it from the upper threshold encountered at the other extreme of the intensity scale). The second type of change proceeding from one unit to the next on the W scale is the smallest (environment)  (sensation) i I N- METER W-METER Figure 2.3. A schematic view of outer psychophysics. The external environment (N) is observable, the internal sensation (W) is not. ----------------------------------------------------------- OUTER PSYCHOPHYSICS 19 difference in intensity required for one stimulus to be perceived as different from a second. This is called the just-noticeable-difference (jnd) in sensation. Corresponding stimulus changes are known as &fference thresholds, difference limens (DL's), or stimulus jnd's. In what follows we will try to maintain the distinction between these two sorts of jnd's (subjective and physical). At this juncture we arrive at a critical postulate: All jnd's measured hypothetically along the scale of sensation are assumed to be of identical size. Another way of saying this is to state that the W-meter operates only in whole number steps of a size equal to one jnd. A jnd in sensation is a constant in this scheme. Given this postulate, we can proceed to construct a quantitative transformation between stimulus intensity measured in absolute (physical) units, and subjective magnitude measured in jnd (sensation) units. Here is how this would work in an ideal experiment. Initially, both the N-meter and the W-meter are set at zero. This is probably impossible in practice (we might approach it temporarily by extreme measures, such as suspending a person, blindfolded, in a tub of water kept at body tempera- ture, etc.), but for our purposes it is not really that important. In any case, we slowly crank up the intensity along some physical dimension, say, sound, and continuously monitor the absolute output with the N-meter. For a while, the person hears nothing, but eventually a sensation is aroused, the subject reports this fact, and simultaneously the W-meter jumps to unit 1 on the sensation scale. The current reading on the N-meter is then said to represent the absolute stimulus threshold for that particular dimension under those particular conditions. The stimulus intensity is then increased from the current energy level registered on the N-meter, and once again we wait until the person says the subjective magnitude has increased (changed one unit). The W-meter now reads "two" jnd units, and the corresponding stimulus value can be recorded from the N-meter. The increment on the N-meter over the previous value is called the stimulus ind. The experimenter maps the relation between succes- sive units of W and corresponding intensities of N until the upper threshold is reached. This hypothetical procedure is shown schematically in Fig. 2.4. Each jnd on the W-meter can be linked to a corresponding intensity from the N-meter, which gives some hypothetical results from our experiment. If the W values are plotted against the N values, the transformation between the two scales is apparent. This procedure is done in Fig. 2.2, where M may be replaced by N. The function is, of course, logarithmic and illustrates the results Fechner both expected and required to verify his hunch about the transformation between N and W and consequently between M and W. Appropriately enough this function has since been known as Fechner's law. ----------------------------------------------------------- 20 CLASSICAL PSYCHOPHYSICS; FECHNER'S LAW MEASUREMENT N-METER W-METER SEQUENCE (observoble) (unobservoble) NULL POSITION 2 ABSOLUTE THRESHOLDI.7 JND I =-- 1.8 JND 2  2.3 Figure 2.4. Hypothetical results of a method to obtain absolute threshold and stimulus jnd's. The successive thresholds on the sensation scale (subjective jnd's) measured by the W-meter are of equal size (1 unit). The corresponding stimulus values measured by the N-meter vary as indicated (1.7, 1.8, 2.3). The magnitude of a sensation is defined as the sum of the jnd's beginning at a threshold value. The corresponding stimulus intensity can be taken directly from the N-meter and equals the sum of the stimulus jnd's. Presum- ably, a person experiencing an intensity would be unaware of this break- down into a series of jnd's on either a subjective or physical level. In other words, it is important to understand that the summation discussed is a theoretical transformation between environmental and sensation values. We need not believe that people are actually aware of some little adding process involving jnd units. One must also realize that a logarithmic function is not the only possible transformation that can arise from such an adding proce- ----------------------------------------------------------- MATHEMATICAL DERIVATION OF FECHNERS LAW 21 dure. The nature of the final relation depends on the assumed size of all units on the W scale as well as on the experimentally determined sizes of the stimulus jnd's. More will be said about this in Chapter 4. Let us summarize our progress so far. Fechner's major interests lay in philosophy, where he attacked a purely materialistic view of the universe by claiming equal status for conscious awareness. He pared down this general concern to the study of the transformation between brain states and sensa- tions. This discipline was formally called inner psychophysics, but since it was not amenable to direct study, outer psychophysics was invented. This enterprise was designed to explore the quantitative relation between external intensities and internal sensations. Because the former could be measured directly, but the latter could not, a theoretical trick was introduced. The key idea was that a series of experimentally determined stimulus jnd's was linked theoretically with a series of sensation jnd's. Each increment in sensation was triggered by one stimulus ind. The latter is a measurable quantity, such as 1 cm or 3 foot-lamberts (a measure of luminance). The total sensation magni- tude is the sum of sensation units associated with the appropriate stimulus level. The outcome of this procedure could subsequently be compared with Fechner's hunch that the true relation between M and W is logarithmic. Having satisfied himself that such an approach was theoretically sound, Fechner then set out to formalize the equations needed to describe the results of this hypothetical procedure. MATHEMATICAL DERIVATION OF FECHNER'S LAW The major derivation is illustrated by the following line of argument. Suppose we keep the assumption that steps are equal along the W scale. Therefore, for each step, A W= c. Suppose also that the stimulus jnd (AN) is a function of N; that is, AN=f (N). Now we can combine the two equalities and set up the following equation2: aw = c (2.5) AN f(N) At this point a decision is necessary with respect to f(N). Fechner was confident that AN was proportional to N: AN=J(N)=kN (2.6) 2The symbol delta (A) may be loosely interpreted as "change in," so A W is "the change along the W scale," and AN is "the change along the N scale." ----------------------------------------------------------- 22 CLASSICAL PSYCHOPHYSICS: FECHNER'S LAW This confidence came from two sources. First, there was evidence that the experiment just described with the N and W meters would yield Eq. 2.6. These data were obtained by Weber (1846) and will be discussed more fully in the next chapter. Second, Eq. 2.6 was consistent with Fechner's hunch that the psychophysical function was logarithmic. In other words, arithmetic increases in W were accompanied by geometric increases in N. Let us see why Eq. 2.6 was consistent with Fechner's views. We noted previously in the discussion of Fig. 2.2 that a geometric progression is equivalent to a constant ratio (k') between adjacent N values. N+ AN = k N 1 + -- --  (2.7) Substituting k for k'- 1, we have AN= kN and equating N with M in Fig. 2.2 completes the connection. Now returning to Eq. 2.5, we can substitute for f(N) to obtain AW c = kN (2.8) Rearranging, AW= cAN  (2.9) The terms A W and AN are small values, but they are still large enough to be measured and counted. But let us assume that what is true for values as small as jnd's is also true for even smaller values. This would imply that the relation (Eq. 2.9) is also true when A W and AN become infinitesimal. In other words, by substituting dw and dn for A W and AN, respectively, Eq. 2.9 may be rewritten as a differential equation, dw = C dn (2.10) where C is equal to c/k times a constant. This is used to make a smoother transition from discrete to infinitesimal values. Fechner justified this transi- tion by recourse to his "mathematical auxiliary principle" (more on this in Chapter 4). ----------------------------------------------------------- REFERENCES 23 Suppose further that the zero point on the W scale ( W 0 = 0) corresponds to a threshold, N 0. Then, integrating 3 both sides of Eq. 2.10 fwo%w=c f N dn (2.11) From calculus we know that the integral of dw is W and the integral of dn/n is lnN. Therefore, the result between N O and N is W= ClnN- ClnN 0 (2.12) W= CIn( N/ N o ) A linear function exists between W and In(N/No) with slope C. Perhaps the most important advantage of Fechner's approach is that only a single stimulus attribute must be measured. Most modern methods of sensory scaling require a person to match the intensity of one attribute to that of another (Marks, 1974). In order to make sense of the results, one should understand the psychophysical mapping of two attributes indepen- dently, but this is usually very difficult to do solely on the basis of the matching data. Consequently, one ends up grappling with Fechner's old problem of making realistic assumptions about internal jnd's except now one is stuck with two attributes instead of one. We will have more to say about this problem in later chapters. Thus far, we have described a hunch supported by a theoretical explana- tion. Fechner's next step was to search for empirical data to help bolster the viability of this "law" between sensation and stimulation (outer psycho- physics). Luckily, someone (E. H. Weber) at his own university had col- lected data to suggest that the relation between AN and N was indeed linear (Eq. 2.6). The details of this story are the subject of the next chapter. REFERENCES Arguelies, J. A. Charles Henry and the Formation of a Pychophysical Aesthetic. Chicago: University of Chicago Press, 1972. Bernoulli, J. "Exposition of a new theory on the measurement of risk." Translated by L. Sommer. Econometrica, 1954, 22, 23-36. Bohr, N. Atomic Physics and Human Knowledge. New York: Science Editions Inc., 1961. Boring, E.G. A Histoo of Experimental Pychology. New York: Appleton-Century-Crofts, 1950. 3Integration is equivalent to finding the area under a curve. In the case of Eq. 2.11, we could also derive a value by drawing the curve y= 1/n and finding the area contained under the curve for n, taking all values between N O and N. ----------------------------------------------------------- 24 CLASSICAL PSYCHOPHYSICS: FECHNER'S LAW Fechner, G. T. Elements of Pychophysics, Vol. 1, Translated by H. E. Adler. New York: Holt, Rinehart and Winston, 1966. Fechner, G. T. Elemente der Pychophysik, Vol. 2. Leipzig: Breitkopf and Hartel, 1907. Fechner, G. T. The Little Book of Life After Death. Translated by M. C. Wadsworth. Boston: Little, Brown, 1905. Fechner, G. T. Nanna oder ilber das Seelenleben der Pflartzen. Hamburg: Verlag von Leopold Voss, 1848. Goude, G. On Fundamental Measurement in Pychology. Stockholm: Almqvist and Wiksell, 1962. Marks, L. Senso Processes: The New Pychophysics. New York: Academic, 1974. Weber, E. H. "Der Tastsinn und das Gemeinhl." In R. Wagner (Ed.), Handw'rterbuch der Physiologie, Vol. 3. Vieweg, Braunschweig, 1846, pp. 481-588. ----------------------------------------------------------- CHAPTER 3 CLASSICAL PSYCHOPHYSICS: WEB ER'S LAW The theoretical study of psychophysics was initiated by Gustav Fechner, but relevant empirical work began at least a hundred years earlier (Bouguer, 1760). The data reported by Fechner's contemporary, E. H. Weber (1846), however, have been the most influential in terms of both stimulating early developments and maintaining interests up to the present day. Weber's chief contribution was to show that sensitivity to changes in stimulus intensity is measurable and predictable. Since the measurement of sensitivity in terms of a just-noticeable-difference (jnd) is carried out with specific stimuli such as lights, sounds, odorants, and pressures, S and AS are used in place of the symbols N and AN that stand for general environmental inputs. In addition, we often will refer to stimulus continua or dimensions to emphasize the notion of a quantitative scale in place of the more general term attribute. Weber measured the stimulus jnd by experimental means similar to the procedure outlined in Chapter 2. The intensity of the stimulus was changed along some dimension until the subject noted a difference, and the physical increment required to accomplish this was the jnd. Its size (AS) was not 25 ----------------------------------------------------------- 26 CLASSICAL PSYCHOPHYSICS: WEBERS LAW constant but increased as a function of S. In fact, the AS needed to obtain a unit step in subjective magnitude (A W) was a linear function of the initial intensity S. The generality of this function led to its designation as Weber's lazJd: as = ks (3.1) There are several reasons why this law has had a lasting impact on psychophysics. The first was Fechner's persuasive influence on the scene. To reiterate, he thought that mental activity was logarithmically related to bodily activity (inner psychophysics), but this relation could not be directly observed. Because of this difficulty, he was willing to settle for an objective determina- tion of the function mapping external stimuli into sensation (outer psycho- physics). One side of this function could not be approached experimentally: internal sensation (W). Therefore, he simply assumed that all steps along the W scale were equal. Fortunately, the stimulus side of outer psychophysics could be measured experimentally and was described by Weber's law. This was ideal for Fechner, since he could then propose on theoretical grounds that A w = a (3.2) AS kS which could be rewritten as a differential equation and rearranged in a form called the fundamental formula: dw = C ds (3.3) $ Integrating both sides of Eq. 3.3 yields the logarithmic law: W= Cln (S/S o ) (3.4) where S O is the lower or so-called absolute threshold. It is important to realize that not just any relation between AS and $ leads to the logarithmic function. It is rather special in this regard. Fechner definitely required the empirical evidence of Weber's law to support his philosophical views about the mind-body connection, and so he naturally took every opportunity to inform the scientific public of its generality and importance. The second reason for the prominence of Weber's law was less tied to the personal status of those involved in its promulgation. Namely, the very ----------------------------------------------------------- CLASSICAL PSYCHOPHYSICS: WEBER'S LAW 27 existence of such a law made it clear that human perception could be studied reliably with scientific methods. A person could be considered a sort of measuring instrument, coding external stimuli in terms of an internal scale. The techniques used by Weber and expanded upon by Fechner were available to answer a number of interesting experimental questions about this coding. (1) Over what range is the stimulus effectively coded? (2) What is the resolving power of the coding for different intensities? (3) Does resolving power depend upon the stimulus continuum (for example, sound, light)? (4) How stable is the coding under changes in stimulus conditions (contextual variables)? We will consider questions 1, 2, and 3 in this chapter, reserving discussion of 4 for Chapter 6. Yet a third reason is given most often for the influence of Weber's law: It may provide a valid index of the relative sensitivity of different sensory channels. The classical approach to psychophysics postulates a fixed sensa- tion scale onto which all external stimuli are mapped. Therefore, the argument goes, differences in sensitivity to different continua are due to differences in the underlying sensory mechanisms mediating the transforma- tion onto the sensation scale. This situation is described by the scheme in Fig. 3.1. Each of the three stimulus continua is transformed by its own sensory system (magic box), but the output from all three converges on the same internal W scale. It is at this level that conscious perception arises and leads to a subjective report. Stimulus Figure 3.1. Schematic diagram of information flow from the environment to a subjective report. ----------------------------------------------------------- 28 CLASSICAL PSYCHOPHYSICS: WEBER'S LAW GENERAL THRESHOLD THEORY Having brought Weber's law to the fore, we can now discuss the methods used to measure the jnd (AS). Before proceeding, however, we ask the reader to put aside Fechner's philosophical inclinations and consider the methodol- ogy for observing the relation between AS and S. It should be recalled, for example, that S and AS are physical quantities. For instance, in a study of perceptual sensitivity to line lengths, S might be 10 cm and AS 1 cm. The Weber constant (k in Eq. 3.1) would then be .1. Basic to the ensuing argument is a theory in which the subject samples the stimulus at a particular moment and compares its magnitude with that of a threshold on the sensation scale. If the stimulus value is greater than the threshold, the subject reports an increment of the stimulus intensity (a jnd). Otherwise, no increment is reported. We will refer to this as the "classical theory." To see how this theory might be converted into a working model, first consider a situation where the stimulus sampling is infallible. The subject knows the exact subjective intensity of each stimulus. If it is also assumed that the internal threshold level is fixed, the subject would be infinitely sensitive to stimulus differences. From the standpoint of this theory let us first consider the detection of the presence or absence of a stimulus (the absolute threshold). In this case, a direct pipeline is assumed to link stimuli and sensations. What one reads on the stimulus scale is what one has on the sensation scale. Suppose now there is a fixed sensation threshold below which a stimulus is never detected (sensed) and above which it is always detected. Then, if a series of stimuli are presented in the general vicinity of the threshold, there will be a sharp separation between those that are noticed and those. that are not. Therefore, a step function is the proper model to describe the probability of detection as a function of stimulus intensity, with the rise occurring at the absolute threshold (as shown in Fig. 3.2, top). Next, suppose a standard stimulus is selected that is well above threshold. A subject is given the task of reporting whether a comparison stimulus is greater than or less than the standard. On the sensation scale there are three locations of interest: the standard, the threshold separating the standard from less intense stimuli, and the threshold separating the standard from more intense stimuli. Call the stimuli producing these locations S, S 2, and S 3. Then, as depicted in Fig. 3.2 (bottom), the probability of reporting that the comparison is greater than the standard can be plotted against the intensity of the comparison. Below S the probability is 0. From S to S 3 assume the subject guesses randomly over many trials. So the probability is .5. Beyond S 3 the probability jumps to 1 because comparison intensities are always reported to be greater than S 2. The appropriate model to describe ----------------------------------------------------------- GENERAL THRESHOLD THEORY 0.75 - 0.50 - 0.25 - 0 0 c 1.00- z  0.75 -- z 0 n  0.50 -- 0 I-- _O.25-- O Fixed Absolute /---Threshold STIMULUS TNTENSITY IU = S3-S  r I I I I I Standard S S 2 S 3 STIMULUS :INTENSITY Figure 3.2 (top) Theoretical step function predicted when the absolute threshold is fixed and the subject is infinitely sensitive to stimulus presence and absence. (bottom) Theoretical step function for determination of a jnd with fixed thresholds symmetrically located above and below the standard. this behavior consists of two steps, with the tread width of the lower one representing the distance (stimulus) between thresholds S and S 3. The jnd (AS) can now be defined as either the distance from S to S 2 or the distance from S 2 to $3. If the thresholds are roughly symmetrical around the stand- ard, it does not matter which measure is chosen, so an average is computed as AS = (S 3 - S)/2. The foregoing theoretical argument and the resulting means for obtaining a jnd are implicitly in effect throughout Chapter 2. In the hypothetical experiment mentioned there, stimulus and sensation meters were read to ----------------------------------------------------------- 30 CLASSICAL PSYCHOPHYSICS: WEBER'S LAW determine the size of the stimulus jnd at different intensities. Although a useful starting point for discussion, the empirical facts do not support this version of the classical theory. Rather, the experimental evidence indicates that subjects are not infinitely sensitive and are not even consistent in their evaluations of the same stimulus and the same threshold. This could be because of the inherent noisiness of sensory systems and the variability introduced through imperfect stimulus production and measurement. Therefore, it is probably best to accept the variability as inevitable and work with statistical measures for determining AS. That is, we accept the occurrence of uncontrolled factors such as quirks in the stimulus generator, residue from prior stimulation, noisy neurons, and so on, and incorporate their effects into the theory. In such a situation, it is often assumed that a bell-shaped (normal) distribution gives a reasonable picture of the sum of all uncontrolled factors. This mathematical model is a useful representation frequently encountered in the theoretical arguments of both classical and modern psychophysics. Returning then to the general theory, there are two primary ways to describe the location of an internal threshold in respect to an external stimulus, as demonstrated in Fig. 3.3. In each we have a stimulus generator, a subject receiving the stimulus, and the possibility for sensation awareness on the part of the subject. The appropriate intensity at each level is measured by the three scales (S, /, and W), where the I scale is at the interface between the person and the environment. Suppose, for our first example, that the stimulus generator A and the receptors are infallible, in that constant intensity readings are obtained on all three scales (Fig. 3.3, left) whenever generator A is cranked up to deliver a fixed intensity. The internal threshold, however, is jiggling around and thus produces a normal distribution of thresholds on the W scale. The fixed stimulus intensity is compared with a single value randomly picked from this internal distribu- tion. Over a large number of trials the threshold will fall above and below the stimulus in accord with the probabilities given by the normal distribu- tion. Therefore, we can visualize the stimulus as a probe dividing the distribution into two parts. On those occasions when the sensation threshold is below the stimulus, the latter will be noticed; and on those occasions when the sensation threshold exceeds the stimulus it will not be noticed. This is one possible notion of what happens when a stimulus is presented to a subject. Alternatively (Fig. 3.3, right), the stimulus generator B is fallible, and hence does not always produce the true intensity indicated by the S scale. This variability can theoretically be recorded by the I and W scales and describes a normal distribution. If the sensation threshold is considered fixed, however, it slices the distribution into two areas, leading once again to the ----------------------------------------------------------- GENERAL THRESHOLD THEORY 31  Fixed Stimulus / Variable Tntensity IW-SCALE Sensory ] I /I I I [ I Triterface IT-SCALE Stimulus I  11  I I I Tntensity S-SCALE STIMULUS GENERATOR A - Fixed Threshold // Variable W- SCALE T-SCALE S-SCALE STIMULUS I GENERATOR 1 B Figure 3.3. Schematic diagram of two classical threshold models. (left) A variable internal threshold is probed by a fixed stimulus intensity. (right) A variable stimulus is imposed on a fixed threshold. The scale types ($, I, and IV) are described in the text. situation depicted on the left at the level of the W scale. Since both conceptions of the problem lead to the same reliance upon the normal curve, the classical theory does not differentiate between the two. One realizes that in both cases the W scale does not yield to direct measurement, since it is a purely hypothetical construct. Only the S scale is observable; hence, it is assumed that both threshold and stimulus variability can be revealed by proper observation of the S scale. So we must proceed from there. Suppose now we have a series of stimulus intensities and the percent of occasions that each was judged more intense than the standard. This is still not equivalent to computing AS for a given S. Some decision must be made as to a cutoff percentage for the stimulus at one jnd above the standard (that is, what is AS?). The customary approach is to assume that a standard stimulus leads to variable effects described by the normal curve. Empirical percentages, corresponding to stimulus intensities, are plotted as points on ----------------------------------------------------------- 32 CLASSICAL PSYCHOPHYSICS: WEBERS LAW the distribution (or a curve derived from the distribution). The dispersion of this empirically determined curve is then used as a measure of AS. To gain an understanding of how this all works, one must know a little more about the normal curve. THE NORMAL CURVE AS A MODEL Given the central value (/) and the dispersion (o), the probability distribu- tion for the normal curve may be defined by the equation: f(x) 1 _(,/2)[(x_)/o] 2 (3.5) A particular value x, be it line length, brightness, or a value along some other continuum, is plugged into the equation, resulting in a value f(x), which is the probability density that x will occur. In other words, we may compute the probability that a stimulus of value x will occur if we assume a normal distribution and provide the central value (mean =/) and dispersion (standard deviation: o). Since Eq. 3.5 is a probability density function, the total area under the curve is 1. Area and probability are therefore synony- mous in what follows. By integrating Eq. 3.5, the area under any section of the curve, from x a to %, is determined.  In the methodology to be described shortly, x will usually be - c and x b will represent a stimulus cutoff on the x axis. That is, (3.6) Another convenient way to compute cumulative probability (area) is to dispense with the units of measure for x in favor of a standard z score. Note that the exponent in Eq. 3.6 can be rewritten as (3.7) The bracketed expression is the standard score used throughout statistics. It is a measure of the distance separating a raw score from the mean in units of standard deviation. One of the features of this measure is that the area under the normal curve between two particular z values is always a constant IThere is no analytic solution to the integral of the normal distribution, but approximations are available in mot statistics books. ----------------------------------------------------------- THE NORMAL CURVE AS A MODEL 33 independent of the original units of measure on x. You can obtain z values and their associated areas from tables in many of your favorite statistics books. By converting to z scores in Eq. 3.6, we obtain a standardized normal curve with a mean of 0 and a standard deviation of 1. All other raw scores can be represented as positive or negative z scores depending on whether they lie above or below the mean, respectively. These tabled scores are commonly used as indices of location along the x axis of a normal curve; more extensive treatment of the normal curve can be found in most statistics books (see, for example, McGee, 1971). To recap, the classical theory assumes that a stimulus intensity is com- pared to an internal threshold. There is noise somewhere in the system that may be represented as points with a normal distribution on one of the scales in Fig. 3.3. As a cutoff is moved along the S scale, the area to its left is the integral of the normal curve, as shown in Fig. 3.4. This cumulative probabil- ity (area) function is called the normal ogive. Since the mean and standard deviation uniquely specify the normal curve, they also specify the normal ogive, which is just another way of describing matters. Increasing the mean cumulative area from the left I / I I 1.0 0.75 0.50 0.25 S-SCALE ( arbitrary units) cumulative oreo fr3m the left 1.0 I-- 0.75 m  0 n o. 0.50 ,,, I-- 0.?-5 J S-SCALE (arbitrary units) Figure 3.4. Relationship between the normal curve and its integral. Two standard deviations are illustrated as well as a graphical procedure to obtain the mean from the cumulative probability function (after McGee, 1971). ----------------------------------------------------------- 34 CLASSICAL PSYCHOPHYSICS: WEBER's LAW simply shifts the function along the S scale; increasing the standard devia- tion decreases the steepness, as shown in Fig. 3.4. When applied to psychophysics, the normal ogive serves as a model of the relation between psychophysical judgments and stimulus intensity. For ex- ample, the probability of judging a comparison as more intense than a standard can be plotted against stimulus intensity after the manner of Figs. 3.2 and 3.4 (bottom). When certain mathematical constraints are met (Luce and Galanter, 1963), as they usually are for average data, this relation is called a psychometric function. Its relevance is this: If we can collect data on such a function, the mean and standard deviation (or some related measures) can be computed on the S values and used to describe the underlying distribution. In the case of the absolute threshold, the mean of the distribution is critical. On the other hand, the standard deviation is important in studies of the difference threshold. Standard deviations (o) are illustrated in Fig. 3.4, as well as a graphical procedure to obtain the mean from the cumulative probability function (after McGee, 1971). To briefly ummarize, the foregoing version of the classical theory assumes either that the stimulus is variable and the threshold fixed or, conversely, that the threshold is variable and the stimulus fixed. From either perspective it is concluded that the normal curve is a satisfactory model of the composite distribution. The psychophysical methods to be discussed next use the cumulative area (integral) of the normal curve to fit data points. The pattern of such points is often referred to as a psychometric function. PSYCHOPHYSICAL METHODS To many people the classical psychophysical methods are important because of their extensive range of application in experimental psychology. Learning the ins and outs of these methods, however, can also be the most boring exercise in psychology because of the many possible variations in application and analysis. However, several excellent treatments are available for those wishing to pursue this matter in more depth (for example, see Engen, 1972; Gescheider, 1976; Guilford, 1954). We will gratefully leave most of the variations for laboratory study and concentrate upon the three chief methods: adjustment, limits, and constant stimuli. All three methods depend on some technique whereby the presumed threshold is bracketed by comparison stimuli. The relevant statistic is either the mean of a group of measures or an index of variability such as the standard deviation. For the following examples, the model of a variable threshold and a fixed stimulus will be emphasized. The absolute and difference thresholds may be determined using any of the three methods. ----------------------------------------------------------- PSYCHOPHYSICAL METHODS 35 Method of Adjustment In this method the subject controls the intensity of the stimulus and attempts to match the threshold. To determine the absolute threshold, the stimulus intensity is preset to random points above and below threshold (the general region is found by preliminary research), and the subject adjusts intensity from this point until he or she either just detects the stimulus (ascending trials) or just ceases to detect it (descending trials). The adjusted intensity is then measured by the experimenter, and the mean value over a number of trials is the absolute threshold. The difference threshold (or limen) (AS) is obtained in a similar way. In this case, a standard stimulus is presented, and the subject matches its perceived magnitude by adjusting a comparison stimulus. Over a series of trials the mean adjustment is usually quite close to the value of the standard and is called the point of subjective equality (PSE). The PSE, however, tells us nothing whatsoever about the subject's sensitivity to stimulus change. Rather, this is indicated by the size of the difference limen, which is often defined as half the range (interval) of uncertainty around the PSE. There is no objective way to decide how this range should be determined. A common criterion is plus or minus one standard deviation, which encloses an area (probability) under the normal curve representing approximately 68% of the cases; one standard deviation (o) equals AS. Tradition dictates that this is the region of indifference, where comparison targets are "indistinguishable" from the standard. We could choose some other cutoff for our boundary, for example, 2 or .5 standard deviations, and these choices would not alter theoretical conclusions surrounding the cumulation of jnd's, Weber's law, and the like. Method of Limits In practice this method is essentially the same as the method of adjustment, except that the experimenter adjusts the stimulus intensity in a systematic way and the subject merely renders a yes-no (binary) judgment at each step as to whether the comparison is equal to or different from the standard. The calculation of the threshold value is also somewhat different. To measure the absolute threshold, the stimulus is randomly set to a value above or below the threshold (whose approximate location is found by preliminary exploration). The subject indicates the relative location of the stimulus by saying something like "I don't see the light" or "yes, I see it." The experimenter then increases or decreases intensity in equal steps, noting the subject's judgment each time. At some level the response will switch from "no, I don't see it" to "yes, I do see it" or vice versa. Generally, the ----------------------------------------------------------- 36 CLASSICAL PSYCHOPHYSICS: WEBER'S LAW procedure is continued a few more steps to ensure that the change in response is not a fluke. The mean of the values where the response category changes is calculated separately for ascending and descending trials, and the average of these two values is the absolute threshold. The difference limen is found in a similar fashion. The subject makes a judgment with respect to a fixed standard, as the comparison intensity is stepped in either an ascending or descending direction. The standard is bracketed, delimiting the region on the stimulus scale where the response category switches from "less than" to "not less than" on ascending trials or "greater than" to "not greater than" on descending trials. The width of this region is defined as the interval of uncertainty (IU). It is analogous to the region from - 1 o to + 1 o, as derived from the method of adjustment. Since the IU is assumed to be roughly centered on the mean, one-half the IU is ap- propriate as a measure of the difference limen in either a positive or negative direction from the standard intensity. As before, all measures are based on average data collected over many trials. Alternatively, the difference limen is measured by analyzing the distribu- tion of category switches on ascending and descending trials as if they were method-of-adjustment data. First, compute the standard deviation of the stopping points on the ascending trials only. Then compute the same measure on the stopping points of the descending trials only and average the two values. In this way we measure the jnd using the method-of-adjustment model while removing the effects of response anticipation or habituation that are often reflected in the difference in means of the ascending and descend- ing trials. The method of limits is quite time consuming. A common variation is the staircase or up-and-down method (Cornsweet, 1962; Engen, 1972, p. 20). Here, multiple measures of the threshold are secured on the same trial, since a trial consists of a sequence of judgments concerning the relative locations of the standard and the adjustable comparison. When the response category changes, the direction of the stimulus is reversed until the response switches back, and so on. That is, on an ascending trial the comparison will initially be judged less intense than the standard. But after a number of step increments the comparison will eventually be judged as more intense. At this point the experimenter initiates a descending series, etc. A plateau is usually quickly reached and the stimulus hovers around the threshold until a predetermined number of reversals occurs. In this sense, staircase can be viewed as an experimenter-controlled method of adjustment. Considerable information is obtained in a short time period, bringing relief to experi- menter and subject alike. The method of limits appears to provide a more intuitive measure of AS than the method of adjustment because it is more directly tied to changes in ----------------------------------------------------------- PSYGHOPHYSICAL METHODS 37 the response category. It is also interesting to note the similarity between this method and Weber's original procedure, where the subject indicated when the comparison was different from the standard as they were separated in a series of steps away from initial equality. This procedure is the origin of the term "just-noticeable-difference." In contrast, the comparison stimulus for the method of limits is initially set well above or below the standard and the subject must indicate when it is "just-noticeably-not-different" from the standard. This modification of Weber's method was introduced by Fechner. Since the abbreviation jnnd is longer and more confusing than jnd, the latter has been retained despite the misinformation conveyed about the experi- mental operations involved. As mentioned previously, the best way to think about the jnd (AS) is as a physical measure, which in turn reflects the dispersion of a hypothetical normal distribution. This stimulus jnd should not be confused with the internal jnd (A W), which the classical approach assumes to be of constant size, subjective, etc. Method of Constant Stimuli Once the inherent variability of perceptual processing is granted, applica- tions of the methods of adjustment and limits follow rather directly. One has the feeling that any intelligent person faced with similar requirements could have invented these methods. It always looks easy after the fact! The method of constant stimuli, on the other hand, provides a more imaginative ap- proach to the measurement of thresholds. Suppose there is a variable absolute threshold, and a comparison stimulus is introduced at different locations within the range of the thresholds. On any given trial the subject samples a single threshold from his or her distribution (which is assumed to be normal) and compares it against the stimulus. Over a number of trials we calculate the relative frequency with which the comparison stimulus is detected (that is, stated to be greater than the threshold). We can then plot the relative frequencies for various com- parison stimuli to obtain the psychometric function (see bottom half of Fig. 3.4). The mean of the distribution is the stimulus value corresponding to p: .5 on they axis. Even though this exact stimulus may not have been included in the experiment, it can be found quite easily by the graphical procedure illustrated in Fig. 3.4. The stimulus value so determined is the absolute threshold. The difference threshold is similarly obtained; a standard is fixed, and a series of comparisons symmetrically located around the standard are com- pared with it. In this case, the relative frequency of calling the comparison ----------------------------------------------------------- 38 CLASSICAL PSYCHOPHYSICS: WEBER'S LAW greater than the standard is plotted on the y axis against the comparison values plotted on the x axis. Sample data from an experiment using the method of constant stimuli are given in Fig. 3.5, based on a study by Weissmann, Hollingsworth, and Baird (1975). This experiment was conducted at a cathode ray terminal linked to a computer time-sharing system. The stimulus continuum was the number scale, so the particular stimuli were numbers presented on the scope of the terminal. Random samples were drawn from a "standard" distribution and a "comparison" distribution and presented in two separate rows. The subject judged whether the numbers in the parent distribution of the comparison set were larger or smaller than the numbers in the parent distribution of the standard set. The mean of the standard was fixed at 150 with a standard deviation of 5. The standard deviation of the comparison distribution was also 5, but the mean varied in 15 steps between 129 and 171 according to the usual procedure with the method of constant stimuli. The sample numbers were randomly chosen from these distributions, which were speci- fied beforehand by the experimenter. Numbers are not commonly used for this type of investigation, but they have the advantage of allowing us to exactly simulate the stimulus conditions presumed by the classical theory. The data analysis proceeds in the standard way. The PSE (mean) can be found graphically as previously illustrated for the absolute threshold. It is approximately 152. For the difference threshold a decision must be made concerning the appropriate statistic. One candidate is the standard deviation of the normal distribution assumed to underlie the psychometric function. I.OO 0.75 0.50 0.25 0 129 135 141 147 153 COMPARISON AS3 159 165 171 Figure 3.5. A psychometric function fit to data obtained by the method of constant stimuli. The PSE and AS (jnd) are obtained as illustrated (after Weissmann, Hollingsworth and Baird, 1975). ----------------------------------------------------------- PSYCHOPHYSICAL METHODS 39 But the measure commonly used with the method of constant stimuli is the distance along the stimulus axis corresponding to the difference between p = .5 and p = .75. The graphical determination of AS (--3) obtained with this index is shown in Fig. 3.5. By referring back to Fig. 3.4 it can be seen that large AS values imply large variability of the underlying distribution of stimulus or threshold values. If this method is considered too crude, more accurate curve-fitting procedures can be used (Bock and Jones, 1968; Guilford, 1954). Similarly, AS below the mean corresponds to the difference between p = .5 and p = .25. Either index is acceptable, as long as we are consistent in our choice of statistic. The need for consistency arises when we make assump- tions about the equality of the corresponding internal jnd's and sum them to obtain scale values. A more formal description of this requirement is expressed by the psycho- physicist's litany, "Equally often noticed differences are equal, unless always or never noticed." To illustrate how this places some constraints on the definition of AS, first define p(y,x) as the percentage of time that stimulusy is judged greater than stimulus x. So if p(y,x)=.75 and p(v,x)=.75, then y= v in terms of being equally discriminable from x. But if y is always discriminated as larger than x: p(y,x)=l, so any v with v>y also has p(v,x) = 1. However, we would not want to conclude that they were equally discriminable from x. A subject may clearly distinguish y and v from x but just as clearly distinguishy from v. Similarly, if x,y, and v are stimuli such that p(x,v)--O and p(y,v)=O, we again should not consider x and y as equally discriminable from v. Theoretically, then, we can use any probability cutoff above the absolute threshold to define AS as long as 0----------------------------- 40 CLASSICAL PSYCHOPHYSICS: WEBER'S LAW Surprisingly little research has been published on this problem. Early work by Fechner, together with experiments by Crozier and his associates (1940) indicate that jnd's obtained by different methods can be related by a multiplicative constant (that is, that jnd's are measured on a ratio scale). Specifically, if AS a is the jnd from the method of adjustment and AS c is the comparable value obtained by the method of constant stimuli, then ASa -- 4?ASc (3.8) In a recent study by Wier, Jesteadt, and Green (1976), these two methods were compared as a means for determining the jnd in sound frequency with a fixed 1000-Hz standard. Their results agreed with earlier research showing that the jnd obtained by the method of adjustment is approximately one-half the size of the jnd obtained by the method of constant stimuli. If this proportion were the same for all frequencies, the value of 4? in Eq. 3.8 would 1 be set at 7' The validity of Eq. 3.8 has two important consequences. First, the relationships between the jnd and other stimulus measures (for example, the stimulus intensity at which it is assessed) varies by a multiplicative constant as one changes psychophysical methods. And second, the relative sizes of Weber fractions among stimulus continua stay the same. That is, sensitivity to visual length of line would always be greater than sensitivity to sound intensity, etc., independent of the method (as long as the same one was used for both continua). This invariance is important for establishing sensitivity indices. It is obvious that different methods would yield different size jnd's since each has its own arbitrary criterion for determining a probabilistic jnd cutoff point. Two other considerations could also affect the relative sizes of the jnd's measured using different methods. First, psychophysical methods may differentially alter the variance of the underlying normal distribution. Hence, the size of the jnd would change accordingly, even if the same statistical criterion were employed in defining the jnd (for example, standard deviation or one-half the interquartile interval). A second possibility is that the variance of the distribution is constant, but the methods induce the subject to use different judgment criteria. The modern theory of signal detectability allows one to distinguish between these two possibilities, but full consideration of this topic is deferred until Chapter 8. One thing is clear. When we wish to compare jnd's for different stimulus conditions, it is best to rely on the same psychophysical method. This is particularly crucial when the goal is to develop functions describing the reliance of AS on stimulus intensity and attribute. In both of these cases, discussion eventually centers on the most famous empirical relation in psychophysics: Weber's law. ----------------------------------------------------------- WEBER'S LAW 41 WEBER'S LAW If AS is measured at a number of intensities S, the relation between the two is expressed by Weber's law, as given by Eq. 3.1. Since AS is a linear function of S, the steepness of the psychometric function (Fig. 3.4) must decrease as S increases. In other words, the variance of the underlying normal curve increases with intensity, as is illustrated in Fig. 3.4. The rate at which AS increases is k, the Weber fraction. This fraction is a dimensionless number indicating the sensitivity of a subject to a particular continuum. A__ =k (3.9) One often sees AS/S plotted directly against S, in which case the function is a straight line horizontal to the stimulus axis and with ay intercept equal to k. The smaller k is, the greater is the perceptual sensitivity. Figure 3.6 gives examples of these two graphs for three hypothetical continua (A, B, and C). Two empirical examples of the relation between AS and S are shown in Fig. 3.7 for the intensity of a 1000-Hz tone. The open circles on the graph represent data collected by Riesz (1928), who employed the method of limits in a somewhat unusual way. Two tones of similar frequency (1000 and 1003 Hz) were presented simultaneously. When the intensity of one was altered by the experimenter, the subject experienced a fluctuation in loudness (beats) of an apparently single tone. The difference threshold (AS) was defined in terms of the intensity change needed to produce beats? Riesz reported his results in energy units with respect to absolute threshold, and as such, the intensity range was considerable (1 to 108). Since sound can also be expressed in pressure units, and since pressure is proportional to the square root of energy, the stimulus range in those units is compressed to 104 , which is still too large for our purposes. (We will want to talk about some details in the region of 1 to 10.) Consequently, we show data only over the range of 1 to 100, where the points are based on some corrections of Riesz's calculations suggested recently by Burbeck (in press). 3 The filled circles on the graph were determined 49 years later by Jesteadt, Wier, and Green (1977). They employed a variation of the staircase method 2For further discussion of this method see Green, 1976 or Gulick, 1971. 3Our analysis to secure measures in pressure units proceeded as follows. From Burbeck's table of intensity I in logarithmic units (decibels, dB) we first computed I in energy units as 10 aB/ø for each level. Next, to convert to pressure we let S  . Values for AS then were obtained by determining AI in energy units and then transforming into pressure: AS /I +AI -S, where AS and S are in pressure units proportional to some standard intensity (e.g., .0002 dyne/cm2). This derivation was based on equations given by Green (1976, p. 255), who also advised us on their application in the present context. ----------------------------------------------------------- slope S (orbitrory units) 1.0 .75 .50 .25 œow Sensff/vity A B C H/h Sensit/vity S(arbitrary units) Figure 3.6. Two renditions of Weber's law for three hypothetical continua (A, B, and (left) The jnd (dS) as a function of stimulus intensity ($). (right) The Weber fraction (dS/S) as a function of $. moo0 Hz TONE / o o RIESZ (1928) 0 20 40 60 80 I00 STIMULUS ]'NTENSITY, S (pressure units) Figure 3.7. The jnd size dS as a function of &' for a 1000-Hz tone that varied in sound intensity. The open circles are transformed values from Reisz (1928). The filled circles are transformed values from Jesteadt et al. (1977). Representation of intensity is in pressure units (proportional to a reference value; e.g., .0002 dyne/cm'). The solid lines were drawn by visual inspection. For more details, see the text. 42 ----------------------------------------------------------- WEBER'S LAW 43 noted earlier in the chapter. A 1000-Hz tone was fixed in intensity while the comparison intensity was stepped toward or away from the standard depend- ing upon the subject's response. On a given trial the two tones were presented successively and the subject had to indicate which was louder. An error in identification led to an increase in separation between the two intensities until the subject correctly identified the tones, at which point the separation was decreased, and so on. The eventual measure of AS was the average separation of standard and comparison for approximately the last 20 turnaround points. Returning now to Fig. 3.7, we see that both sets of data can be fit reasonably well by straight lines, indicating that Weber's law holds over most of the range. The solid lines on the graph have been drawn in freehand to capture the major trends. The slope of the linear portions are k  .08 for Riesz and .11 for Jesteadt et al. Hence, greater sensitivity was evident in Riesz's experiment. At higher stimulus intensities (not shown), the Weber fraction slowly decreased to approximately .05 in both experiments (.10 in energy units). Apart from the general trends, important deviations from Weber's law occur near the lower absolute threshold. In the first place, linear extension of each function to the y axis does not intersect the origin. Therefore, a better description would involve the addition of a small constant on the right-hand side of Eq. 3.1. The second deviation is the slight curvature near threshold, suggesting that a linear function, with or without the additive constant, does not offer a completely satisfactory account of the results. Both of these deviations from Weber's law will be treated more thoroughly in Chapter 4. Empirically, k ranges from 0.02 for finger span to 0.24 for some chemicals used in studies of odor discrimination (Baird, 1970; Teghtsoonian, 1971). A representative list of Weber fractions for different stimulus continua is given in Table 3.1. One of the advantages of the Weber fraction as an indicator of sensitivity is its independence from physical units of measure on a ratio scale. For example, although distance may be measured in centimeters or meters, both AS and S are measured along the same physical scale, so k is simply a dimensionless ratio. Because of this property, one can compare sensitivities for different continua. If we further assume that the internal scale (W) is fixed for all continua, the relative sensitivity of different sensory systems can be inferred from k (or more accurately, from I/k). Since the jnd probably depends on the method used to obtain it, comparison of Weber fractions should ideally be made for data collected in the same way. If Eq. 3.8 is correct, the relative size of Weber fractions for different continua will be the same regardless of the method (as long as it is the same for each continuum). ----------------------------------------------------------- 44 CLASSICAL PSYCHOPHYSICS: WEBERS LAW lable 3/I Representative Weber fractions for common continua Weber fraction Continuum AS / S Finger span .02 Saturation (red) .02 Electrical (skin) .03 Position of point (visual) .03 Length of lines (visual) .04 Area (visual) .06 Heaviness .07 Brighiness (naive observers) .08 Loudness (1000 Hz, energy units) .10 Loudness (white noise, energy units) .10 Taste (salt) .14 Taste (sweet) .17 Skin vibration (100-1100 Hz) .20 Smell (several substances) .24 This is because the jnd size, and hence the Weber fraction, will increase proportionately (p) for all continua. As will become apparent in Chapter 4, all is not well with Weber's law as originally formulated. Alt.hough ap- proximately correct for many continua, failures do occur, especially for very weak or strong stimuli. The Upper Threshold It is difficult to use the classical methods to determine the upper threshold for a stimulus continuum. Experimental results for very intense stimuli are scarce, since there is always the risk of physiological tissue damage. No complete table of dynamic ranges (differences between lower and upper thresholds) of continua has been published, although it appears from some studies (Valter, 1970) that there are substantial differences in this regard. It has been suggested by Teghtsoonian (1971) that dynamic range is directly related to the Weber fraction (a logical conclusion from the classical notion of a fixed internal scale of constant range). So far the relevant data analyses do not lend much weight to this hypothesis (Valter, 1970). For instance, the smell continuum has large jnd's but a very short dynamic stimulus range ----------------------------------------------------------- LESSONS OF CLASSICAL PSYCHOPHYSICS 45 over which the sensory system reacts. This is clearly contrary to the hypothe- sis, although it still may apply to other continua; more analyses are required to find out for sure. Fechnerian Integration If Weber's law is correct and we grant Fechner his assumption about the equality of internal jnd's (A W), the logarithmic function (Eq. 3.4) follows. So Fechner was right! Well, maybe. In the next chapter we will see that Fechner was correct only to a first order of approximation and only when his special assumptions about internal jnd's are granted. In fact, Fechner was aware of some of these problems, but time constraints and his quest for a psychophysical law did not allow him to pursue the solution to these problems in greater detail. LESSONS OF CLASSICAL PSYCHOPHYSICS A human being is not a perfect measuring instrument, infinitely sensitive to changes in external stimulation. The variability of the threshold and the statistical nature of AS attest to this fact. In addition, the threshold variabil- ity is not constant; it depends on the intensity of stimulation in a linear way according to Weber's law. Moreover, sensitivity is variant across stimulus continua. The Weber fractions for different continua, such as light, sound, and weight, range at least from 0.02 to 0.24. And finally, Fechner's logarith- mic law states that humans are systematically nonlinear compared to physical instruments. According to this law, geometric increases in physical intensity are required to produce arithmetic increases in sensation (for instance, to cause an increment of 1 on the sensation scale, the physical intensity may need to be doubled or tripled), suggesting the following mind bender: As measuring instruments, humans are not calibrated the same as the instruments used to measure human responses. A second lesson is of more direct theoretical importance. The key assump- tion in classical psychophysics is that the sensation scale is unitary and is not distorted by various laboratory conditions. Without evidence to the contrary, such faith is necessary if one is to make any sense at all out of psychophysical data. Just as an organism must first attribute a fixed structure to its own nervous system before it is able to distinguish between external and internal changes, so too must psychophysical theorists assume constancy somewhere in the system if they are to infer something about sensory transformations. ----------------------------------------------------------- 46 CLASSICAL PSYCHOPHYSICS: WEBER'S LAW Several investigators have sought to verify this assumption by recourse to physiological data (see, for example, Stevens, 1972). But we have reason to pause here, because this time-honored theoretical pastime can easily lead us astray. Unless we exercise real caution, the search for relevant data in the physiological realm can turn into a self-fulfilling prophecy, where the investigator uncovers only those facts that agree with his or her preconcep- tions. This is one important lesson of classical psychophysics we would do well to recall as new techniques and data continue to proliferate in both the behavioral and physiological fields. REFERENCES Baird, J. C. "A cognitive theory of psychophysics: I. Information transmission, partitioning, and Weber's law." Scandinavian Journal of Pqychology, 1970, 11, 35-46. Bock, R. D., and Jones, L. V. The Measurement and Prediction of Judgment and Choice. San Francisco: Holden-Day, 1968. Bougner, Trait$ d'optique sur la gradation de la lumidre. Paris: 1760. Burbeck, S. L. "A reexaminafion of Riesz's intensity discrimination data." Journal of the Acoustical Society of America (in press). Cornsweet, T. N. "The staircase-method in psychophysics." American Journal of Pqychology, 1962, 75, 485-491. Crozier, W. J. "On the law for minimal discrimination of intensities. IV. AI as a function of intensity." Proceedings of the National Academy of Science, 1940, 26, 382-389. Engen, T. "Psychophysics. I. Discrimination and detection." Woodworth and ScMosberg5 Experi- mental Pqychology. Vol. 1: Sensation and Perception, J. W. Kling and L. A. Riggs (Ed.). New York: Holt, Rinehart and Winston, 1972, pp. 11-46. Gescheider, G. A. Pqychophysics.' Method and Theory. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1976. Guilford, J.P. Pqychometric Methods. New York: McGraw-Hill, 1954. Green, D. M. An Introduction to Hearing. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1976. Gulick, W. L. Hearing.' Physiology and Pqychophysics. New York: Oxford, 1971. Jesteadt, W., Wier, C. C., and Green, D. M. "Intensity discrimination as a function of frequency and sensation level." Journal of the Acoustical Society of America, 1977, 61, 169-177. Luce, R. D., and Galanter, E. "Discrimination." In R. D. Luce, R. R. Bush, and E. Galanter, (Eds.), Handbook of Mathematical Psychology, Vol. I. New York: Wiley, 1963, pp. 191-243. McGee, V. E. Principles of Statistics: Traditional and Bayesian. New York: Appleton-Century-Crofts, 1971. Riesz, R. R. "Differential intensity sensitivity of the ear for pure tones." Physical Review, 1928, 31, 867-875. Stevens, S.S. "A neural quantum in sensory discrimination." Science, 1972, 177, 749-762. Teghtsoonian, R. "On the exponents in Stevens' law and the constant in Ekman's law." Pqychological Review, 1971, 78, 71-80. ----------------------------------------------------------- REFERENCES 47 Valter, V. "Deduction and verification of a quantum psychophysical equation." Reports from the Institute of Applied Pychology, Universify of Stockholm, no. 13, 1970. Weber, E. H. "Der Tastsinn und das Gemeinffihl." In R. Wagner (Ed.), Handw'rterbuch der Physiologie, Vol. 3. Braunschweig: Vieweg, 1846, pp. 481-588. Weissmann, S., Hollingsworth, S., and Baird, J. C. "Psychophysical study of numbers: III. Methodological applications." Pychological Research, 1975, 38, 97-115. Wier, C. C., Jesteadt, W., and Green, D. M. "A comparison of method-of-adjustment and forced-choice procedures in frequency discrimination." Perception & Pychophysics, 1976, 19, 75-79. ----------------------------------------------------------- CHAPTER 4 DERIVATIO,N OF PSYCHOPHYSICAL LAWS It is universally acknowledged that Weber's law in its unmodified form is valid for the middle range of stimulus intensities but not for the very extremes near the lower and upper thresholds. This has been clear from the start, and Fechner considered Weber's law only as a first-order approxima- tion to the law operating in the realm of inner psychophysics. If other functions later superseded Weber's law, it was argued, these new functions would be open to the same theoretical approach: Set up a differential equation and integrate to determine the psychophysical law. In this chapter we see that Fechner was not entirely justified in taking this stance. In fact, most alternatives to Weber's law produce Fechner scales that violate the very assumptions used in their construction! Although realization of this difficulty would have surely distressed Fechner, it probably would not have altered his philosophical view concerning the relation between sensation and physiological processes. It is clear from his writings that he was convinced of the truth of Weber's law as applied to tuner psychophysics, regardless of its overall validity for outer psychophysics. Remember, the original logarithmic 48 ----------------------------------------------------------- ALTERNATIVE WEBER FUNCTIONS 49 law was built on an intuitive hunch, not on an inductive argument from empirical data. Nonetheless, deviations from Weber's law exist and deserve description. There are three primary types of distortions. 1. As intensity (S) approaches zero, the jnd (AS) cannot continue to shrink proportionally because of the absolute threshold (according to the classical theory). 2. A similar distortion occurs as intensity approaches the upper threshold. At some point near the upper end of the scale, the jnd must approach infinity--it becomes fruitless to coax a response from a burnt-out sense organ. Empirical results indicate that this deviation from linearity (if it could be measured completely) is a gradual one as intensity approaches either the lower or upper threshold. 3. More significant departures from Weber's law occur for certain con- tinua in all regions of their dynamic range. For instance, in the estimation of time duration, the jnd is not a constant proportion of length of time passed for any significant region of the continuum (Michon, 1964). A second counterexample is pitch perception. If sound frequency is thought of as a continuum, classical studies showed that the size of the jnd at any point is independent of frequency over a fairly extensive range (from 200 to 2000 Hz). For these values the jnd is approximately constant (Gulick, 1971). just recently, an experiment by Wier, Jesteadt, and Green (1977) resulted in higher sensitivity measures over this region and a systematic change in the difference threshold as a function of the standard frequency (in the direction of Weber's law). However, the rate of increase of AF (frequency) as a function of F was much greater at higher frequencies than it was at lower ones, in accord with the classical findings of Shower and Biddulph (1931). In addition, the more recent investigation found that the jnd also depended on the intensity (amplitude) of the tone, implying once again that perception of differences in tonal frequency is a complex matter (see their paper for discussion of the implications of these results for various psychophysical models of auditory perception). ALTERNATIVE WEBER FUNCTIONS In order to more accurately mirror observed phenomena, several alternative equations have been suggested to replace Weber's law (see especially Guil- ford, 1954). Each, however, is designed to handle special experimental conditions, so none has gained universal favor. To facilitate discussion, let us ----------------------------------------------------------- 50 DERIVATION OF PSYCHOPHYSICAL LAWS first call the relation between AS and S a Weber function regardless of its compliance with Weber's law. A shrinking jnd near the lower threshold can be represented by adding a constant to Weber's original formulation--a possibility noted by Fechner and Helmholtz in the last century (Miller, 1947). Thus we have AS--kS+a (4.1) In this equation a may be considered proportional to the absolute threshold. When S is zero, AS = a, which can be considered a correlate of the first step on the sensation scale. Since stimulus continua have diverse thresholds that vary with experimental conditions, Weber's law will be strictly true only for intensities above these effective thresholds. In any event, Eq. 4.1 must do at least as well as Weber's law, because a can take on the value of zero. Equation 4.1 has also provided better fits to data in several studies. Miller (1947) demonstrated its adequacy for auditory noise, as did Ono (1967) for judgment of line length. Equation 4.1 also is a more realistic model of the sound intensity results discussed in the previous chapter (Fig. 3.7). Although the jnd for sound grows linearly with intensity over a wide range, the function has a positivey intercept (a) that varies with tonal frequency. That is, the perceptual system seems to require a little boost before it begins to operate in accord with the unmodified form of Weber's law. By replotting the data from Fig. 3.7 we can obtain a more detailed view of what is happening around the absolute threshold and why an additive constant is required to faithfully describe matters. Before embarking on this venture, however, let us deal with some hypothetical jnd's to help lay the Table 4.1 Hypothetical results for jnd experiment s as as/s 1 .85 .85 2 .90 .45 5 1.05 .21 10 1.30 .13 100 5.80 .058 1,000 50.80 .0508 10,000 500.80 .05008 100,000 5000.80 .050008 ----------------------------------------------------------- ALTERNATIVE WEBER FUNCTIONS 51 basis for understanding the empirical results. Table 4.1 gives jnd size (AS) for a series of intensities, evaluated by the following equation: AS = .05S +.8 Imagine that an experimenter knows $, measures AS, and believes Weber's law, but doesn't know k and a. Then he might well try to summarize his findings by computing the Weber fractions listed in the right column of the table. True to psychophysical practice, he then might graph the Weber fraction as a function of $. Before doing this, however, he notes the huge values of $ and realizes that a compression of that scale is desirable to save paper and still maintain the essentials of the relationship. So, he computes the logarithms of $ and plots them along the x axis, as shown in Fig. 4.1. There are three points of interest about this graph. 1. As S approaches 1, log0 S approaches 0 and AS/S approaches .85. 2. The function flattens out very neatly as S increases beyond 100 (log0(100) = 2). Weber's law holds in this region. 3. The logarithmic scale has the effect of magnifying activity at the low end and smoothing activity at the high end. 1.00 00.50 AS = 0.05S + 0.8 S = I  I00,000 2 4 STIMULUS INTENSITY (IOglo S) Figure 4.1. Hypothetical function between the Weber fraction and the logarithm (base 10) of stimulus intensity. The function summarizes data from Table 4.1 obtained by an evaluation of Weber's law with an additive constant (Eq. 4.1). ----------------------------------------------------------- 52 DERIVATION OF PSYCHOPHYSICAL LAWS Note that the logarithmic scale is used here for convenience only. The scale has no deeper theoretical meaning. It is a good idea to be aware of the implications (or lack thereof) of such transformations, since they influence interpretations of the graphed results. One must be ever alert to the diverse ways in which psychophysicists push numbers around! So Fig. 4.1 is a picture of what occurs when data modeled by Eq. 4.1 are plotted as though one expected to obtain a straight horizontal line across the entire stimulus range (as in Fig. 3.6). Returning now to the findings on sound intensity, we can plot the points in the same fashion. First, Weber fractions were calculated from the results given in Fig. 3.7 for a 1000-Hz tone (Jesteadt, Wier, and Green, 1977; Riesz, 1928). Second, the hypotheti- cal Weber fractions from Table 4.1 were plotted as the dashed function in Fig. 4.2 together with the empirical findings. A logarithmic transformation of S was not necessary to secure a good description of the general trend. It is clear that the hypothetical function is quite similar to the Riesz data. Therefore, an additive constant would greatly improve the fit of Weber's law to these results. The set of results determined more recently by Jesteadt et al. are not as readily accommodated. The curvature near threshold is less pronounced than for either the Riesz data or the hypothetical function. Nonetheless, some improvement of the fit would also be gained here by adding a constant to the basic Weber function. 0'50I moo0 Hz TONE o o RIESZ (192'8) 0.40-I -' : JESTEADT, WIER  GREEN (1977) / As--o.05s + 0.8 o.3olt 0.10  0.05  ...... 0 20 40 60 80 I00 STIMULUS TNTENSITY, S (pressure units) Figure 4.2. The Weber fraction as a function of sound intensity in pressure units (proportional to a reference value; e.g., .0002 dyne/cm2). The empirical data are from Fig. 3.7. The dashed function is an evaluation of Eq. 4.1 with the parameter values given on the graph, but with the resulting Weber fraction (AS/S) plotted against S. ----------------------------------------------------------- ALTERNATIVE WEBER FUNCTIONS 53 Figure 4.2 appears to signal a resounding defeat for the original formula- tion of Weber's law. Caution is advised, however. In actuality, the Weber fraction is reasonably constant once the stimuli are clearly above the absolute threshold. The function begins to flatten out at fairly weak intensi- ties. To take a rural example, the intensities at which the curve begins to level off might be comparable to the ambient sound in an empty Vermont farmhouse. It is rather quiet. Although the additive constant definitely improves the fit in the neighbor- hood of the absolute threshold, there also appears to be a little curvature in the data when AS is plotted as a function of $ (Fig. 3.7). An alternative equation of general use that deals explicitly with curvature is the power function suggested by Guilford (1932, 1954): As = (4.2) When fitting data, the exponent g may vary between 1 and .5 and might even exceed 1 for some conditions. When g is 1, Eq. 4.2 reduces to Weber's law, and when g is .5, AS is a square root function of $ in agreement with an early proposal by Fullerton and Cattell (1892). Guilford and others have demonstrated the usefulness of the power function for studies of line length, sound intensity, and judgments of muscular tension (Guilford, 1932; Hov- land, 1938; Fitts, 1947). One of the potential drawbacks of this function is its nonlinearity in the middle range. This is unfortunate because most writers recognize the legitimacy of a constant Weber fraction in the middle range of intensities. However, the problem is not severe when the exponent is near 1. The generality of the power function should be explored more fully. To stay with our current examples in the auditory realm, the data of Riesz, Jesteadt et al., and others (Harris, 1963; McGill and Goldberg, 1968; Luce and Green, 1974) on the jnd for sound intensity are fit quite well by a power function with an exponent of approximately .9. That is, AS = kS'9 In order to see what this means when AS/S is plotted as a function of S (as in Fig. 4.2), we divide both sides of the above equation by S and obtain AS = kS .9-, S or AS = kS -'1 S If Weber's law held exactly, the exponent would be 0. So we are really ----------------------------------------------------------- DERIVATION OF PSYCHOPHYSICAL LAWS talking about a discrepancy of -.1. This small difference has led McGill and Goldberg (1968) to talk about the "near miss to Weber's law." Green (1976) gives a thorough review of the auditory literature on this point together with a discussion of the importance of this result for the formulation of neural and psychophysical models of intensity discrimination. None of these theories has much to say about the upper threshold. Indeed, it is possible to write equations describing both the upper and lower thresholds as well as the middle region of the scale. However, these equations quickly become messy and invariably contain four or five parameters to be estimated from the data. This is especially true of Valter's (1970) work, which is the most ambitious recent attempt to describe the relationship between intensity and the size of the jnd. Because of the number of free parameters in his model, it is even possible to fit nonmonotonic functions with special added twists at the lower and upper thresholds. By nonmono- tonic we mean that the size of the jnd may go up and down as stimulus intensity increases. Other models with the same number of parameters might do as well (for example, the combination of two hyperbolas), and it is probably necessary to introduce such complications if the entire Weber function is to be accurately described. Nonetheless, for a considerable range of intensities, Weber's law (with an additive threshold constant) is usually adequate. Of course, it is possible to construct a wide array of theoretical Weber functions in order to observe the consequences for psychophysical laws based on Fechnerian integration. One thing seems apparent in this regard. Since Weber's law holds tolerably well in the middle range, the integral of any type of Weber function that purports to fit empirical results will be ap- proximately logarithmic with sufficiently large values of $. Hence, it is unlikely that any significant deviations from Fechner's law can be expected if one grants the assumption of equal steps along the sensation scale (IV). HOW lEGITIMATE IS FECHNERIAN INTEGRATION? Recent mathematical developments have questioned the legitimacy of Fechnerian integration, because it does not seem appropriate for all types of Weber functions. The most extensive work on this topic was done by Luce and Edwards (1958), Krantz (1971), and Falmagne (1971, 1974). We will focus on the paper by Luce and Edwards. We begin by noting that Fechner's idea for measuring sensation is only one of many possible mathematical models relating stimulus intensity to an internal dimension. But all assume three things: ----------------------------------------------------------- HOW LEGITIMATE IS FECHNERIAN INTEGRATION? 55 1. There exists a function relating the magnitude of sensation to a corresponding physical stimulus. 2. The function is monotonic. An increase in the stimulus intensity will never produce a decrease along the sensation scale. 3. The function is everywhere differentiable. Changes in the stimulus produce smooth transitions from one sensation level to the next. Unfortunately, there is no direct way to monitor the sensation scale. There- fore, one may arbitrarily define values on this scale with respect to values on the stimulus scale. The oldest sensation scale, Fechner's, defines equal jnd's on the sensation side. With this view it is clear that the ultimate validity of the sensation scale may be neither proved nor disproved by empirical evidence. As discussed previously, the jnd's on the stimulus dimension need not be equal. In fact, the Weber function defining the size of each jnd at any point on the stimulus continuum is increasing with S. In a similar vein, we define a Fechner function as the relation between cumulative jnd's on the stimulus dimension and cumulative jnd's on the sensation dimension. This is not to be equated with Fechner's law, which is a particular Fechner function. To reiterate, the Fechner function can be derived from any Weber function, assuming constant sensation jnd's. For example, if we assume Weber's law (k = AS/S) and equal sensation jnd's (A W--1), we may write AS AW_ 1 k A W=  or A kS Next, rewrite it as a differential equation: dw__. a ds ks The correction constant a is a scaling factor needed to convert discrete values (AS and A W) to continuous values (ds and dw). Then we rearrange and integrate to obtain W= Cln S (4.3) So where C= a/k and S O is a threshold. The crucial step in this formulation is the conversion to a differential equation. This was accomplished by Fechner's "mathematical auxiliary principle," which states that whatever is true for differences as small as jnd's is also true for all smaller differences. In other words, jnd intervals may be divided into infinitesimal differences and integrated without distorting the ----------------------------------------------------------- 56 DERIVATION OF PSYGHOPHYSIGAL LAWS results. Unfortunately, the "principle" is generally erroneous and may be rigorously justified in only a limited number of cases, such as Weber's law. Even if Fechner's "principle" were true, a contradiction arises in the formulation of Fechner functions. He assumed that any Weber function could be integrated to produce a Fechner function. Luce and Edwards show that this is not true. It does, however, work correctly for Weber's law and the generalized version including the additive threshold constant. Let us ex- amine this case first. Its validity can be checked by the following argument. Fechner defined the sensation scale with equal intervals between adjacent scale values (A W---constant). This was done before integration secured the psychophysical function. The latter must also satisfy the original definition; that is, A W values must be equal for all points on the W scale after integration. Otherwise, we have a flat contradiction of the original premises. At this point it is convenient to use functional notation to emphasize that W varies as a function of $. Therefore, W(S+AS)= W(S)+AW (4.4) states that the sensation (W) produced by the stimulus plus stimulus increment (S+AS) equals that produced by the stimulus [ W(S)] plus one sensation increment (A W). Stated another way, a single stimulus increment (AS) results in a single sensation increment (A W). Now consider two stimuli separated by one jnd according to Fechner's law. For the first stimulus, W(S)-- Cln0 ø (4.5) and for the second, W(S+AS) = Cln (S+AS) (4.6) So Using the equality in Eq. 4.4 and subtracting 4.5 from 4.6 yields one sensation jnd (the larger stimulus minus the smaller). AW=Cln S+AS Cln0 (4.7) So A W= Cln S+AS (4.8) S A W-- Cln(1 + --) (4.9) Since both C and AS/S are constants, A W is a constant in compliance with ----------------------------------------------------------- HOW LEGITIMATE IS FECHNERIAN INTEGRATION? 57 the original definition. So Weber's law and Fechner's law satisfy this equal-A W requirement. One slight problem remains. We initially defined A W--- 1, but Eq. 4.9 will not satisfy this for all values of AS/S= k. The source of the problem seems to arise in going from discrete to continuous values via the constant a. This smoothing factor must change somewhat to accommodate different k values if Eq. 4.9 is to satisfy the initial definition that A W= 1. Substituting a/k for C and k for AS/S, and rewriting Eq. 4.9, we obtain AW---- 1----  ln(l+k) k ln(1 + k) Substituting the values of k into this equation reveals that a changes about 12% between k--0.02 and k--0.24 (see Table 3.1). This effect is relatively small but may prove important as more accurate measures become available for determining psychophysical functions (especially with regard to the power law discussed in the succeeding chapter). For other types of Weber functions, we get into more serious difficulty, as pointed out by Luce and Edwards. For example, in the case where AS increases by the square of S: AS (4.10) AS---- kS2; k -- S - If we define A W= 1, we can use the first of these equations and state AW 1 AS kS 2 Applying the "mathematical auxiliary principle" and rearranging: ads ks 2 where a is the correction constant mentioned previously. Integrating and using the functional notation to define sensation magnitude as a function of stimulus intensity: W(S)----+b (4.11) where b is a constant. Next, remember that each additional stimulus jnd adds a constant (A W = 1) to the count of sensation jnd's: W(S+AS)- W(S)--AW=I ----------------------------------------------------------- .8 DERIVATION OF PSYGHOPHYSIGAL LAWS In words: the sensation aroused by a stimulus of intensity (S+ AS) minus the sensation aroused by a stimulus one jnd less intense (S) is equal to a single sensation step (A W). Substituting for AS (Eq. 4.10), w(s+ks2) - Next we substitute from Eq. 4.11 for the two stimuli S+ kS 2 and S to obtain: - w(s)=[ which reduces to k(S+kS2) --+b =AW--1 AW=I '---- a 1 + kS (4.12) The question mark in the equation represents our skepticism about the relation expressed. Since a is a constant, k must equal zero for the equality in Eq. 4.12 to hold for all values of S. But Weber fractions of zero are not very interesting, so the "principle" may not be applied to define a Fechner function with constant-sensation jnd's. It may also be shown that very few Weber functions, aside from Weber's law, produce Fechner functions satisfy- ing the constant-A W requirement. Functional Equation Solution Luce and Edwards then show that accumulating jnd's one at a time is an alternative to using the "mathematical auxiliary principle." In fact, this can always be accomplished by graphical techniques once the Weber function is known. Starting from an arbitrary stimulus level So, the AS 0 at that intensity is plotted in two dimensions (stimulus intensity, sensation intensity) at the coordinates (AS0, 1). Next, stimulus intensity S O +AS0----S is plotted at the point (AS 0 + AS, 2) and so on. Once the points are known, the equation for the curve of best fit may be used as the Fechner function. The solution of the analogous mathematical accumulation of jnd's, called a functional equation (see Aczl, 1966), may be used to derive the Fechner function from the Weber function. Often it is computationally difficult to obtain a solution, and in some instances it is impossible to derive a Fechner function. On the other hand, when such a function does exist it is automatically consistent with the definition of constant-sensation jnd's. The complete Luce-Edwards argument is involved and will not be presented here. ----------------------------------------------------------- HOW LEGITIMATE IS FECHNERIAN INTEGRATION? 59 The difficulties encountered by these authors has resulted in a reformula- tion of Fechnerian integration by Krantz (1971). He states that integration produces meaningless results unless single steps along the stimulus dimension produce constant increments on the sensation scale. Existence of a solution may be determined by integrating the Weber function and checking the results in the manner of Luce and Edwards. In addition, Krantz explores the effects of an error term in the sensation dimension: W (S + AS ) - W(S ) --- constant + error In some cases, the difference between the functional equation and integra- tion solutions including the error term are small enough to justify either approach. In conclusion, Luce and Edwards demonstrate the inadequacies in the initial formulation of Fechnerian integration and the incompatibility of an equal-interval sensation scale with most Weber functions (that is, Weber's law is a special case). Fechnerian integration and the mathematical aux- iliary principle are discarded and replaced by functional equation solutions. Unfortunately, it may be excessively difficult to derive a functional equation solution from experimental data. If one is after a Fechner function without all the associated mathematical niceties, the best policy is simply to sum jnd's by a graphical procedure and fit a function to the data. This will be adequate for empirical formulations as well as for all but the most rigorous theoretical approaches. For most experimentalists this is the natural way to proceed anyway, with or without the blessing of mathematical consistency. Consequently, the discovery of a weakness in Fechner's argument, although theoretically important, has had little impact on laboratory practice. Alternative Definitions of AW If one entertains different assumptions about the nature of the sensation scale (W), different psychophysical laws will be derived. Until now we have assumed that A W is a constant. We have also noted that under certain conditions AS is sometimes constant, independent of S, whereas Weber's law states that AS is a linear function of S over the middle range of most continua. It is natural to propose a parallel Weber-type law for the sensation scale, where A W is a linear function of W (Ekman, 1959; Treisman, 1964). Other assumptions and empirically determined Weber functions are also possible, but the four alternatives just mentioned cover the basic conditions we expect to encounter in classical psychophysics. ----------------------------------------------------------- 60 DERIVATION OF PSYCHOPHYSICAL LAWS THE FOURFOLD WAY  Ignoring the objections to the mathematical auxiliary principle, a psycho- physical function is derived on theoretical grounds by treating A W/AS as a derivative and integrating. If we concentrate only on the two major possibili- ties for Weber functions (constant and linear), and for assumed sensation func