ON PSYCHOPHYSICS IN GENERAL AND THE GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION IN PARTICULAR

O N P S Y C H O P H Y S I C S I N G E N E R A L A N D T H E G E N E R A L P S Y C H O P H Y S I C A L D I F F E R E N T I A L

E Q U A T I O N I N P A R T I C U L A R

HANNES EISLER

Psychological Laboratories, University of Stockholm, Sweden

EISLER, H. On psychophysics in general and the general psychophysical differential equation in particular. Scand. r. Psychol., 1965, 6 , 85-1oz.-The first part of the paper discusses theoretical issues regarding the problem of measurement in psychophysics and the suitable choice of a subjective scale, with special reference to the views of Ekman, Ross, Suppes & Zinnes, and Treisman. T h e theory of the G P D E is discussed more fully than before. In the second part, further empirical confirmation of the GPDE is given for the following combinations of Weber functions: constant-constant, linear-linear, linear-parabolic, parabolic-parabolic.

As will be shown in the second, empirical, part of this paper, the general psychophysical differential equation (GPDE) developed and to some extent tested in a previous paper (Eisler, 1963~) seems to be valid in a variety of instances. Since its derivation together with its application to empirical data implies an isomorphism between the empirical set of objects and their relations on the one hand, and the set (or possibly a subset) of all real numbers with corresponding relations on the other (i.e. presupposes that numbers can be employed meaningfully in the subject area), a few comments pertinent to direct scaling in psychophysics may be in order, They are given in the first part of this paper, together with a few more specific comments on the GPDE.

THEORETICAL COMMENTS ON DIRECT SCALING IN PSYCHOPHYSICS AND T H E GPDE

The measurement problem First, let me say that I certainly do not wish to deny that differences exist between phy-

sical and psychophysical measurement. One such difference would be that the scatter- suitably measured as coefficient of variation-encountered in psychophysics usually is greater than the scatter found in physics. However, when a new phenomenon shows up with similarities and differences to one that is familiar and already christened, we have two choices. Depending on the importance we attach to the differences, we may define the scope of the new phenomenon and give it a new name or we may extend (possibly relax) the definitions of the familiar phenomenon to include the new one. This seems mostly a question of terminology that I cannot feel strongly about. If somebody prefers the term scaling to measurement in connection with psychophysics, I have no objections.

However, whether we talk about psychophysical scaling or measurement, for developing a quantitative theory, it seems clear that measurement requires isomorphism between the subject matter measured and the number system. This problem has been dealt with recently

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86 H A " E S EISLER

by Suppes & Zinnes (1963) and Ross (1964) in a theoretical way and by Goude (1962) both theoretically and empirically. The customary approach, and the one employed by these authors, is the deductive one. However, the use of measures (numbers) as input for the mathematical mill and the finding that the number output of this mill permits correct empirical predictions, leads us to believe that the above-mentioned isomorphism does exist. Of course, an inductive argument like this does not constitute a proof. But another conclusion may be drawn. As long as we have neither proved nor disproved deductively that the assignment of numbers to objects or events in the case in question constitutes measurement, we cannot know whether our procedure fulfils the necessary requirements. Let me exemplify by Stevens' method of magnitude estimation. It seemed to qualify well for what Suppes & Zinnes term 'pseudopointer measurement'-until Goude defined an empirical operation corresponding to addition in the number system, so that magnitude estimation can now be regarded as genuine pointer measurement since it has been shown to yield numerical values that correspond to those of a (fundamental) measurement.

The argument above may throw some light on ROSS' criticism of Stevens' claim that Stevens' method of assigning numbers to objects (or sensations) does not constitute measurement. Since Stevens has not presented formal representation theorems and the corresponding uniqueness theorems (these terms employed in the meaning of Suppes & Zinnes) and tested them out empirically, nothing was known about the necessary isomorphism. However, Ross' counter-claim-that Stevens' procedures are no measurement-has not been substan- tiated either. Goude's work-apparently unknown to Ross-seems to have decided the question in favor of Stevens.

Somewhat surprisingly, Ross is inclined to regard psychophysical scaling as vicarious derived physical measurement. I t is a well known fact that physical and psychophysical scales do not agree in most cases. Ross, however, claims that if the same elements are measured in accordance with different measurement theories on the ordinal level and the two scales agree, then an extension to different measurement theories of the additive type will lead to the same additive scale. This statement is based on a proof given in his monograph (1964, pp. I 10-1 I I). Unfortunately, the proof is incorrect.

Ross defines two empirical operations of combination, A and A* respectively, corresponding to addition in the number system:

N(z ) =N(x) +N(y)=zP(xAy),

N"(z) =N"(x) +N"(y)=zP(xA*y).

N(x) denotes the number assigned to x according to the measurement theory and P the equivalence operation corresponding to ' =' in the numerical system. Ross now picks two equal elements so and s t , i.e. soPsg, and applies the first operation of combination, finding an element o4 in his set such that (s0As~)Po4. From the above he derives the following two equations:

N(S0) +N(s:)=N(o,),

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PSYCHOPHYSICS AND GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION

However, the second of these equations implies (soA*s~)Po,, about which we know nothing. In fact, if we have two measurement theories which differ as to A and A*, i.e. as to the empirical operations of combination employed, it is highly improbable that two elements combined in different ways will be equivalent to the same third element.

Thus different scales can be expected from different operations of combination, in agreement with empirical results for, e.g. physical and psychophysical scaling with weights. These empirical operations of combination-here I am following Ross-define the pertinent branch of science and the measurement theory determines the properties or attributes of the objects investigated. If we place two weights on the same pan of a scale and find a third weight to balance them, we are dealing with physical weight in physics; if we, for instance, let a subject lift two weights in succession and then find a third weight which feels as heavy as the first two together (Goude, 1962), we are dealing with subjective weight in psychology.

Furthermore, Ross claims that Stevens in his direct scaling methods develops a subjective psychology. In a sense Ross is right, of course, since the subjects communicate their-ne- cessarily subjective-impressions. However, I would prefer to call psychophysical scaling ‘objectivized introspection’ and one needs to be a very tough-minded behaviorist to refrain from the information obtainable through introspection for psychophysical theory. (Ross admits the possible value of psychophysical scales for practical purposes.) I would like to reserve the term ‘objective science’ for cases where different scientists agree on the outcome of their investigations, whatever the source of their data.

87

Choice of scale In contrast to Ross (1964), Treisman (19644 1964b) accepts Stevens’ psychophysical

scaling procedures as measurement. However, he criticizes Stevens’ choice of scale and advo- cates a logarithmic rather than a power law for the psychophysical function. Ekman (1964a, 1964b) is undecided on the issue.

Treisman contends that the relation between two percepts, standard (ys) and comparison ( y J , resulting from a fractionation experiment in which the subject was instructed e.g. to ‘pick out a weight half as heavy as the standard from among the comparison weights’, may be defined equally well by yc -ys = log 1/2 as by yJyS = 1j2. The former ‘rule of assignment of numbers to objects’ results in a logarithmic scale, whereas the latter yields the power function for the same set of data when the subjective scale is related to the physical one.

As Phillips (1964) points out, these two rules constitute a choice from an infinite number of possible rules. In Suppes & Zinnes’ (1963) terminology, we have different representation theorems. These authors demonstrate nicely, e.g. by proving six representation theorems for the B.T.L. system (which is based on relations between relative frequencies of paired comparisons), that many such theorems are possible. Therefore the choice from among representation systems must be made on other grounds and I want here to discuss three criteria: (a) Simplicity of theory of measurement, (6 ) simplicity of substantive theory, and (c) communicability. (Regarding the terms ‘theory of measurement’ and ‘substantive theory’, see Luce, 1959.)

Simplicity of theory of measurement. Given a representation theorem, one can prove a uniqueness theorem defining the scale type. The weaker the scale, the more free parameters,

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and, in this sense, the more complicated the measurement theory. Suppes & Zinnes, commenting on the number of free parameters between the sixth representation theorem for the B.T.L. system and the suggestion of a seventh, write: ‘It may be observed, however, that, although there is no strict mathematical argument for choosing a representation, say R,, that yields a ratio scale rather than one, say R,, that yields a log interval scale, it does not seem scientifically sensible to introduce an additional parameter such as k that is not needed and has no obvious psychological interpretation.’ It may be added that the information value of a scale decreases with increasing number of free parameters.

Although no strict representation theorems exist for the two relations dealt with by Treisman, and thus no uniqueness theorems either, it seems more than plausible that the relation resulting in a power function yields a ratio scale-with one free parameter-and the relation resulting in a log function an interval scale-with two free parameters. This point also lies at the root of the mistake made by Treisman (1964a), pointed out by Stevens (1964) and disputed by Treisman (1964b) regarding the value of cross-modality matching for supporting the power function. A substantive theory containing the log function y =

ulog(rjb/b) (y subjective and C$ physical magnitude) must be invariant under a linear transformation-the log function being an interval scale-and the factor a thus cannot carry any information-as Treisman would have it do when he wants to predict the result from crossmodality matching from, say, fractionation experiments. Luce & Galanter (1963) say the same thing by pointing out that a is a free parameter, the arbitrary scale unit.

Thus, from the point of view of simplicity of the measurement theory, the power function seems to be preferable to the log function.

Simplicity of substantive theory. Here, too, Stevens’ ratio scale seems superior. Let us look at the similarity function, demonstrated for the first time for the continuum of pitch (Eisler & Ekman, 1959). Similarity judgments sit for a pair of percepts corresponding to stimuli i a n d j were found to be related to the corresponding scale values yi and y j obtained by the fractionation method according to the following equation:

If the fractionation experiments had been interpreted in the vein of Treisman and the corresponding scale values are denoted by R, the similarity function would take one of the following forms, depending on whether Treisman considers an exponentiation appropriate for the similarity judgments, too:

Taking the simplest possible form, we obtain

which requires an exponentiation and thus is more involved than Equation ( I ) .

The second substantive theory I want to discuss is the well known non-linear relation

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PSYCHOPHYSICS AND GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION 89

between category (partition) and magnitude scales for prothetic continua. According to Treisman (1964~) ‘category scaling is similar to magnitude estimation save that the set of numbers which can be used is restricted, and they are supposed to be equally spaced.’ In essence, however, there are two differences. In category rating the subjects estimate the stimuli only in terms of whole categories, usually as integers, and furthermore there is an upper and lower limit to the categories whose use is permitted. That it is the second restriction which is important and which yields the typical category scale is apparent from two experiments considered in conjunction. Rubin (see Stevens & Galanter, 1957, Fig. 8) scaled loudness of white noise using an ‘unlimited point scale’ (which essentially means a lower but no upper limit) and obtained a scale that exhibited the category effect for low values but agreed with the magnitude scale for the rest. A complement to this experiment is one carried out by Ekman, Lindman & William-Olsson (1961) in which subjects estimated volumes of cartographic symbols on a scale with fixed extreme values (I and 3500 for the smallest and biggest stimulus, respectively, in accordance with the volumes the symbols were intended to represent) and full freedom to choose any number between the extremes they considered appropriate. The result was a scale that was concave downward when plotted against the physical volume scale in log-log coordinates, i.e. the category effect was still more pronounced than in a typical category rating experiment. (The corresponding magnitude scale followed the power law.)

In my opinion (see Eisler, 1962a, b, 1963a, b, c; Eisler & Ottander, 1963) we have two different scales, methodologically characterized as open-ended (magnitude scale) and closed (category scale). The relation between these scales and their corresponding Weber functions constitutes a special case of the GPDE and can be investigated quantitatively. The closure of the category scale results for the ideal case in a constant Weber function (real data show deviations at the end points, cf. Eisler, 1963a) and the category scale may thus be computed by Fechnerian integration with respect to the magnitude scale. It is of course impossible to say, where in the neurological stimulus-response chain the difference between the two scales lies. In view of the recent discoveries of efferent nerve fibers to receptors (see e.g Livingstone, 1959), I would not be surprised if the difference exists already at peripheral level, in contrast to Treisman’s view ‘. . . that the central effect of the stimulus . , . is related to the stimulus intensity in the same way, whatever the psychophysical task . . .’

The arguments above are based on Stevens’ rules of number assignment. Treisman offers another explanation, based on his log function, for the observed discrepancies between the two scales. He regards the numbers obtained by e.g. magnitude estimation as exponen- tials of the ‘correct’ sensations and considers exponentiation a difficult task for the subjects. (As to Treisman’s second point, I agree whole-heartedly.) Treisman further notes that it would be still more difficult to select equal intervals on this exponential scale, as required by e.g. category estimation, and the subjects would not succeed completely. Since complete failure would result in the log function and complete success in the magnitude scale, we should obtain something in between, as indeed we do empirically. This model seems rather artificial, complicated and not very amenable to quantitative treatment (and thus also difficult to disprove). The same goes for Junge’s (1962) model for the relation between magnitude and category scales, which is based on some combination between a ‘ratio factor’ and an ‘interval factor’ in the subjects.

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Treisman also seeks support for his argument from an experiment by Galanter & Messick (1961) in which a successive intervals scale proved to be a pure log function of the corresponding magnitude scale, while the category scale lay in between. As Suppes & Zinnes (1963) have shown for the general case and the author (Eider, 1965) in some detail for the case at hand, different representation theorems for the Thurstonian or successive intervals scale result in functions which may be linearly or logarithmically related to the magnitude scale. The experimenter’s choice of representation theorem is more or less arbitrary and has nothing whatever to do with the behavior of the subjects and their possible proficiency in arithmetic.

Thus, as far as the simplicity of substantive theory is concerned, the power function seems preferable to the log function.

Communicability. In a way, the problem of communicability is merely a pragmatic question. In my opinion, rather weighty reasons must be given before refraining from the face validity of the direct scales and applying another rule of measurement than the one im- plied by the instruction to the subject. I can imagine the experimenter laughing as he listens to two subjects in a scaling experiment discussing which weights they found half as heavy as the standard, because he knows better: surely they chose a weight that felt log z lighter than the standard.

But more seriously, is not this perhaps a matter of terminology? Probablywe have a series of neural transformations between stimulus and response. There may be a log function, somewhere in the system, but why stop there and call just this transformation of the stimulus ‘sensation’? I find Treisman’s own analogue-about a subject watching a display that seemed continuously visible to him in spite of saccadic eye movements and shifting threshold-a good example. The level of our discourse will determine whether we regard his seeing as continuous or not. The two levels co-exist and neither one seems preferable to the other. Similarly, in psychophysics it seems most profitable to accept the direct scales as they come, while in neurophysiology the log function may prove more useful.

Ekman (1964~) considers another possibility according to which the ‘true’ psychophysical function between percept and stimulus might be a log function. If subjective numbers, too, were a log function of mathematical numbers, a magnitude estimation experiment could be regarded as a matching experiment of sensations to numbers. Denoting the stimulus variables by S (sensations) and N (numbers) and the corresponding subjective variables by R, and RN, matching, i.e. equating Rs and RN, yields

a + p log N = y + 61og S,

logN = a + nlog S, or

which can be rewritten as the power function

N = bS”.

By this approach Ekman circumvents the difficulty which Treisman does not regard as such: that the experimenter’s rule according to which numbers are assigned does not agree with the instruction given to the subject. However, the criticism ventured against the log

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PSYCHOPHYSICS AND GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION 9’

function as to simplicity of theory of measurement and substantive theory holds for Ekman’s log function as well. Also from the point of view of communicability it is difficult to see what can be gained by introducing a subjective number function, particularly when it is realized that numbers constitute a very suitable general standard. Stevens (1964), in order to evade the number problem, suggests another standard for psychophysics, viz. length of lines. This continuum has an exponent of one and thus the acceptance of this standard would not change the exponents obtained for different continua by the use of numbers. Judgment of length of lines is a natural task-in contrast perhaps to many other continua-and agreement between physical and subjective magnitudes for length of lines as expressed by an exponent of one seems a simpler model than a logarithmic relation.

As to Ekman’s question whether the subject’s quantitative estimates represent subjective states or merely stimulus values chosen to match unknown subjective states, a comparison with MacKay’s (1963) model may be illuminating. MacKay, too, proposes a matching of two log functions and his matching equation is formally identical with Ekman’s. However, the symbol N , denoting mathematical numbers according to Ekman, signifies sensations for MacKay. MacKay’s model is a neurophysiological one; according to him the matching takes place between an impulse frequency generated by the receptors and a centrally evoked impulse frequency that is supposed to ‘counterbalance’ the first one. The transformation functions of both stimulus intensity and sensation to impulse frequency are assumed logarithmic. If Ekman wants to identify ‘subjective state’ with nervous impulse frequency, his and MacKay’s models merge. On the other hand, we find ourselves on the neurophysiological level, whereas the discussion concerns psychophysics.

Thus, from the point of view of communicability, the power function seems to be preferable to the log function.

The GPDE

The following theorem expresses the general psychophysical differential equation (GPDE): Between two monotonically related subjective variables x and y which are functions of the sanze physical variable 4, and their corresponding Weber functions az(x) and all(y), the latter based on intraindividual standard deviations, the following relation obtains:

A proof will be given again (cf. Eisler, 1 9 6 3 ~ ) because I shall be returning to it in the discussion. Here I will use the somewhat more streamlined proof given by Mashhour ( 1964).

Proof. Starting from x=fi(CD), y =g,(CD) we consider the inverse functions CD = f ( x ) , CD =g(y). If we add an increment Ax to x, we have the corresponding increment in CD, A@, and thus also in y, Ay. Ax is defined as Ax(h)=ha,(x) and Ay is assumed to be Ay(h)= ha,(y) +o(h), lim,,[o(h)/h] =o. Dividing and letting h tend towards zero yields

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1.e.

There are two minor differences from the GPDE given in Eider ( 1 9 6 3 ~ ) . The first is the weakening of the restriction that the two subjective variables must have a monotonically increasing relation. In fact, a monotonical relation suffices; as it happens, in the empirical part of this paper, the GPDE is shown to hold also for variables with a monotonically decreasing relation. (In this case, of course, A y is negative if AX is positive, and Emh4[ - Ay(h) /Ax(h)] =

The other change is the emphasis on intraindividual SDs as the base for the Weber functions. This of course concerns the empirical content of the GPDE and not the mathematical model.

Formally the GPDE is identical with the formula of the theory of errors used in physics by which the uncertainty in the dependent variable is computed when the uncertainty of the independent variable is known (cf. Hald, 1960) . As a matter of fact, it was from there (together with the Fechnerian argumentation) that I got the idea of the GPDE, as is reflected in my original derivation (Eider, 1963a). In the theory of error, this is an approximative formula, obtained by a Taylor expansion. Since in physics variability, e.g. the SD, is usually small, the higher terms of the Taylor series can be neglected without appreciable error. This is not the case in psychophysics: on the contrary, the SDs obtained are rather large. The crucial difference between GPDE and the theory of error lies in the introduction of the auxiliary variable h and its tending towards zero. This limes transition is supposed to describe a psychological mechanism in single observers. GPDE in physics, for instance, would be either meaningless or false. Meaningless, if the two related variables are measured separately; in this case the power of the different measuring instruments would determine the errors and no connection between errors would exist. False, if the error in one variable is computed from the error in the other; the Taylor theorem must then be used and the possible neglect of higher terms in the series depends on the precision intended and the size of the known error.

- dy /dx . )

The GPDE is concerned with the uncertainty of psychophysical observations. I have chosen the intraindividual SD as a measure, ( I ) because the interest centers around the behavior of single individuals and its lawfulness (which may become obscured in group data), ( 2 ) because it is a comparatively stable measure from the statistical point of view (picking, for instance, the quartile deviation would not change anything, since it is roughly proportional to the SD), and (3) because the derivation of the GPDE requires a measure of the increments of the same sort (unit) as the variables to which the increments are added, excluding for instance, the variance.

This does not mean that I am completely happy with the intraindividual SD as a measure of uncertainty, nor with any of the methods used for computing averages; in certain cases I have found somewhat unusual computing methods necessary (Eider, 1962 u). One reason is purely statistical: the distributions found for magnitude estimation data are usually not normal (cf. Eider, 1965), which sometimes may distort the meaning of common statistics, which are based on normally distributed data. The second reason is connected with the first: the problem is to get hold of the psychologically relevant variables; the uncertainty of psychophysical judgments is probably not the result of a stochastic process in the same sense as is the variation in the diameter of screws produced in a factory.

The view that a relation between two subjective variables is connected with their variability is contested in a paper by Schneider & Lane (1963), describing an experiment on magnitude and category production of loudness and softness of white noise. Their-in my

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opinion-most striking finding is that the downward concavity of the category scale of softness, when plotted against the corresponding magnitude scale, is greater than logarithmic, in contrast to an experiment using the estimation methods (Eisler 1962a), in which this concavity was less than logarithmic not only for loudness (in agreement with Schneider & Lane, 1963) but also for softness. This seems to be a general finding for ‘inverse’ continua, as e.g. shortness of lines and smallness of squares (Stevens & Guirao, 1963)~ when a production method is used. (Regarding the difference between estimation and production methods, see also Ottander & Eisler, 1965.)

Schneider & Lane (1963) find the same Weber functions for both loudness and softness for each type of scale in physical units and conclude that variability in physical units cannot be at the root of the difference between the category scales of loudness and softness. But what about variability in psychological units? The authors compute it by calculating, for each single production, the corresponding psychological scale value by means of the power function derived from the magnitude production experiments. For these subjective scale values a mean and an SD is computed for all five psychological values (stimulus numbers) to which loudness and softness productions were made. If this method worked as it should, the means calculated in the way described had to agree with these stimulus numbers. However, the points in Fig. I I

of Schneider & Lane’s paper reveal large differences between these two sets of subjective scale values, the one given by the experimenter and the other computed from the productions and the power function. Thus the calculation procedure employed seems doubtful. These calculations by Schneider & Lane show close agreement between the Weber functions for loudness and softness for the magnitude scales in psychological units and they conclude that neither can psychological variability account for the difference between the two category scales. It is remarkable that unlike almost all Weber functions for magnitude scales I have seen, both these pass through the origin. (The same holds for the Weber functions in physical units.) The curvature of a category scale when plotted against the corresponding magnitude scale is largely determined, according to the GPDE model, not by the slope of the Weber function of the magnitude scale (in terms of which Schneider & Lane argue) but by the intercept of the Weber function on the magnitude scale axis.

I am inclined to conclude from Schneider & Lane’s experiment that at present it is impossible to calculate subjective variability in a meaningful way from production data. Neither does the GPDE work, since the subjective Weber functions computed by this method from the Weber functions in physical units and the power function, pass through the origin too. The GPDE is thus not valid for psychophysical but only for intrasubjective relations.

FURTHER EMPIRICAL CONFIRMATION OF THE GPDE

All of the data presented below stem from earlier experiments, carried out before the idea of the GPDE arose, and most are taken from previous publications. The general treatment is as follows: the parameters of the Weber functions a,(x) and oy(y) of the two subjective variables x and y are evaluated. The GPDE dy/dx =o,(y)/a,(x) is integrated and the numerical value of the integration constant determined by fitting y = f ( x ) . Since the derivative seems to be more sensitive to deviations between empirical and theoretical values, I have followed a suggestion by Mashhour (1964) and compared the ratios of the SDs (u,,/a,) to the derivativefl(x). Where this course is not followed, a precise description of the data treatment is given. I t should be noted that the GPDE is tested by fitting only one

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HANNES EISLER 94

free parameter, the integration constant. I t would certainly have been possible to obtain still better fits than those presented below, by allowing the parameters obtained from the Weber functions to vary when fitting the integrals of the GPDE.

I have tried to make an inventory of available data and the following combinations of Weber functions will be tested: ( I ) constant-constant, (2 ) linear-linear, (3) linear-parabolic, (4) parabolic-parabolic.

Combination of Weber functions: constant-constant As a matter of fact, I have yet to meet a constant Weber function, so the heading of this sec-

tion is somewhat misleading. This is also why no constant-linear or constant-parabolic combinations are investigated, though the relation between magnitude and category scales may be regarded as an approximation of the combination constant-linear. However, since it is the ratio between two Weber functions that enters into the GPDE, a constant ratio may justify its inclusion under the heading constant-constant.

TABLE I. Ratio of intraindividual SDs of category ratings of softness to loudness of white noise. Data from Eisler (19620.).

Stimulus No. I 2 3 4 5 6 7 8 9 10 Meana

Series 1 (1.414) 1.092 .863 .861 1.122 1.078 .985 1.325 1.265 (2.000) 1.07 Series I1 - 1.118 .813 .757 .970 1.103 1.462 .922 1.128 (1.225) 1.03 Series 111 - 1.051 .714 .990 1.021 .983 1.145 1.755 1.166 (0.802) I . I O

a In computing the means, the extreme values (in parentheses) are excluded.

The data presented derive from three experiments on category rating of loudness and softness of white noise (Eisler, 1962~). As Table I shows, the mean ratio between corresponding Weber functions is 1.07, 1.03, and 1.10 for the three experimental series, respectively, and variability is small for different stimuli. From the GPDE we have (since the category scale of softness y is a monotonically decreasing function of the category scale of loudness x)

- dy/dx = k .

Integration yields y = -kx + C. Fig. z in the paper from which the data are taken gives the data points of the functions y =f(x) and the lines y = - x + C. The fit is almost perfect. The fit for the derivative is seen from Table I . (The integration constant disappears in the derivative.)

Combination of Weber functions : linear-linear The data presented derive from three experiments of magnitude estimation of loudness and

softness of white noise (Eisler, 1962~). Denoting softness magnitudes by y and loudness magnitudes by x, we have the two Weber functions

ag = b ( y -yo), 02 = k r ( x -xo),

and, keeping in mind that softness magnitudes decrease with increasing loudness magnitudes, the GPDE yields:


PSY CHOPHYSICS AND GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION 95

Integrations yields

k log (y -yo) = - $! log (x - xo) + log c.

50.00

20 L O 6 0 2 0 4 0 60 20 L O 6 0

L o u d n e s s ( m a g n i t u d e s c a l e )

FIG. I. The derivative of the magnitude scale of softness of white noise with respect to the magnitude scale of loudness of white noise as a function of the magnitude scale of loudness for 3 experimental series (Eider, I 962 a). Circles: the ratio between intraindividual SDs of softness and loudness for corresponding stimulus values.

I t was not considered worthwhile determining C by a least squares fit; instead l o g e was computed for every stimulus and the mean over the ten stimuli taken. For the three experimental series logC was 2.6025, 2.4672, 2.5367. Since

the derivative becomes

k - Cky/’kr dx (x - x O ) h / k ~ + l *

The derivative and the corresponding values of the SD ratios for the three experimental series are given in Fig. I. Because of the large range the ratios are plotted in logarithmic coordinates . The fit seems good for all three series.

Combination of Weber functions: linear-parabolic

The data used here derive from an experiment in which the unpleasantness of electrical stimulation was scaled with magnitude estimation and category rating (Ekman, Franken- haeuser, Levander & Mellis, 1964). (These data have also been treated by Mashhour, 1964.) This experiment has attracted my attention for three reasons in particular: (I) the authors claim that ‘the relation between the category and the magnitude scale in the present experiment, as well as in several other investigations, will have to be described in terms of a different quantitative principle’ after having referred to Fechnerian integration (Eider, 1962~2, b, 1963 c), (2) the category scale in this case exhibits an irregularity for the lowermost points and I was interested to see how the GPDE could cope with this, and (3) as will be shown presently, the Weber function for the category ratings is here-unlike other experiments, clearly parabolic. This may be because it was not a category rating experiment proper, since the subjects were allowed to give ratings in terms of decimal fractions of categories. Before proceeding to the data treatment, I should like to thank the authors for making their raw data available.

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Only the data for the 14 subjects taking part in both the magnitude estimation and the category rating experiment are used. The means and the corresponding intraindividual SDs for the category ratings were computed in the usual way. For the magnitude estimations, the scale values were obtained by computing the arithmetic mean for each subject (of his four estimates) and the geometric mean of these 14 arithmetic means. The intraindividual SDs were computed as described in Eisler ( 1 9 6 2 ~ ) . (The standard 20 was presented for each judgment and thus no SD could be computed.) These latter calculations, together with the fitting of the

TABLE 2. Empirical and computed data from an experiment on unpleasantness of electrical stimulation. (See text.)

2

3 4 5 6 7 8 9

I 0

2.239 5.48 I 8.800 13.914 20

26.440 35.051 44.147 5 I .698

I .065 2.440 2.746 3.499

5.610 7.187 7.363 5.881

-

I .04

2.18 2.63 3.42 4.13 5.01

5.94

1.75

6.44

0.21 5

0.390 0.420 0.483 0.504 0.6 I 8 0.632 0.573 0.425

0.202

0.160

0.138 0.153

- 0.110

0.088 0.078 0.072

0.90 0.80 1.04 1.49 1.55 1.42 2.01 2.18 1.86 2.75 2.97 2.57 3.55 3.72 3.42 4.30 4.37 4.23 5 . r s 5.09 5.14

6.47 6.15 6.36 5.91 5.71 5.88

2608 1927 2063 2529 2613 2748 2795 2543 2494

0.109 0.155 0.157 0.142

0. 134 0.118 0.094 0.07 I 0.056

a Category scale computed by Method 2. See text. Category scale computed by Method I . See text. Category scale computed by Method 3. See text.

straight line describing the Weber function according to a method likewise described in Eisler (rg62u), was carried out with a high-speed computer (Ottander, Backstrijm & Pettersson, 1963). The stimulus values (in arbitrary units), scale values and SDs are given in Table 2. The Weber function for the magnitude scale is given in Fig. zA. In fitting the straight line, the two extreme points have been excluded (cf. Eisler, 1 9 6 2 ~ ) . Except for these points, the Weber function is well approximated by a straight line that does not pass through the origin, in agreement with the usual findings. Its equation is ow = o . r q p (y + 11.08). In the subsequent treatment, the GPDE has been used in three different methods.

Method I . As in previous work, the Weber function of the category scale is considered constant. Denoting the category scale by K and the magnitude scale by y, Fechnerian integration yields K =alog(w - y o ) +B, where vo is the magnitude intercept of -11.08. The two parameters a and are considered free and fitted by the method of least squares to the function mentioned above. The category values KCb' thus computed are given in Table z as well as in Fig. z C .

The fit appears acceptable. From a and the, slope of the Weber function of the magnitude scale, the value of the Weber function-assumed constant- for the category scale is calculated as 0.495 and introduced as a straight line parallel to the category scale axis in Fig. 2B. A comparison with the correct-parabolic-Weber function shows how good-or bad-an approximation a constant Weber function can be regarded as being for these data. Most Weber functions for category scales (proper) exhibit a constant middle section and decreasing sections towards the extremes (Eisler, 1963 a). For these cases, a constant Weber function constitutes a better approximation, of course.

Scand. I . Psychol., Vol. 6, 1965


Method 2. The ratios of the SDs a&, are integrated numerically by the trapezoid method with respect to the magnitude scale. The integration constant is determined so as to give the least sum of deviations from the empirical values. T h e ratios and the values of the integral are given in Table 2 (Kla’). Fig. 2C demonstrates that the fit to the data points is better for this integral than for the curve computed according to Method I despite the fact that only one free parameter is used.

U n D l e a s a n t n e s s

0 70 20 30 40 50

M a g n i t u d e s c a l e

6

5 Y

n - : L

? 3

>r L 0

Y n U

2

1

0 10 20 30 LO 50

M a g n i t u d e s c a l e

I ’

0.00 - 1 3 5 7

C a t e g o r y s c a l e

0.20

la

UI

r 0

0 0 3 5

.I? 0.10 1 n LL

0.05

. , D

0 10 20 30 LO 5 0

M a g n i t u d e scale

FIG. 2. (A) The Weber function of the magnitude scale of unpleasantness of electrical stimulation (Ekman et al., 1964). The straight line is fitted by a modification of the method of least squares (excluding the points marked by triangles). (B) The Weber function of the corresponding category scale. The parabola is fitted by the method of least squares (excluding the point marked by a triangle, which corresponds to the standard in the magnitude estimation experiment; this always preceded the category rating experiment). The horizontal line is the Weber function predicted from Method I-see text. (C) The category scale as a function of the magnitude scale (circles). The category scale computed according to Method I (chain-dotted), Method z (dashed) and Method 3 (continuous)-see text. (D) The derivative of the category scale as a function of the magnitude scale. Concerning the dashed part of the curve-see text. Circles: the ratio between intraindividual SDs of the category and the magnitude estimations. Triangles: Corresponding ratios of the two fitted Weber functions.

Method 3. A parabola of the form

ug =K,(K-a) ( b -K)

is fitted to the Weber function of the category scale by means of the method of least squares with the parameters k, = 0.037, a =0.296, b =8.590, see Fig. 2B. T h e GPDE becomes

7-651941 Scand. J. Psychol., Vol. 6,1965

98 HANNES EISLER

Integration yields

I K - a 1nC I ~- In----+--- - KK(b - a) b - K kK(b - a ) k, In (Y -Yo),

or, solving for K

Since the constancy of the integration constant C testifies to the correctness of the equation above, C has been computed for all stimulus values and is given in Table 2. It is seen that variability is small; however, the second and third points, corresponding to the stimulus values 3 and 4, deviate from the other 7 values (italics in Table 2). Since the category scale irregularity mentioned in the beginning of this section seems to concern these points and the two Weber functions do not show any corresponding anomaly, it may be concluded that the irregularity reflects a shift in the integration constant. The psychological meaning of this shift is unclear, as is the importance of the integration constant in general.

Instead of using the method of least squares, C was determined as the mean, excluding the two anomalous points. The category scale values can now be computed and are given in Table z (K")); the curve is plotted in Fig. 2C. The fit (with one free parameter) is very good. The next step is the computation of the derivative dK/dy. This is done using the mean of C for all points except the two anomalous ones, for which their own C values are used. The values of the derivative are given in Table 2. Fig. 2 D shows the derivative (continuous line) as a function of the magnitude scale. The dashed line in the region of the two deviating points shows the derivative if the mean value of C had been used there, too. The empirical ratios of the SDs u&, are indicated with circles and the corresponding theoretical ratios computed from the two Weber functions, the straight line and the parabola, with triangles. Note the difference between empirical and theoretical ratios for the end points, reflecting the fact that the two extreme points of the Weber function of the magnitude scale do not lie on the straight line. Con- sidering that this curve is based on only one free parameter, which has not even been determined by the method of least squares, the fit appears very good.

Codinat ion of Weber functions: parabolic-parabolic

The data used here stem from an experiment on similarity and ratio estimation of pairs of weights (Eider, 1960). It was shown that the similarity function

(where s i j denotes the similarity on a scale between o and I, between the two percepts yi and y j corresponding to stimuli i and j ; q,, is the ratio y i / y j of these percepts) is valid if the ratio estimations are corrected so that the empirically obtained ratio matrix becomes internally consistent. The corrected data have been treated previously (Eider, 19634 and it was shown that the parameters of the similarity function can be computed by means of the GPDE. Here, the course outlined in the beginning of the second part of this paper will be followed both for the uncorrected and the corrected data.

Scand. J. Psychol., Vol. 6, 1965


For all four kinds of data-uncorrected and corrected similarity and ratio values-the Weber functions can be fitted by parabolas passing through the origin, of the form ot = kx(u -x); the GPDE thus becomes

FIG. 3. T h e derivative of the function relating similarity to ratio judgments of heaviness (Eider, 1960) as a function of the ratios for uncorrected ( A ) and corrected (B) data. Circles: uncorrected (A) and corrected (B) ratios of intraindividual SDs of Similarity and rntio estimates.

Integrating and solving for y yields

2.5

2.0

1 5

1 0

5

.O .? .L .6 8 1 0 .O 2 ,L .6 .0 10

U n c o r r e c t e d r a t i o e s t i m a t e s C o r r e c t e d r a t i o e s t i m a t e s

see Eider (1963a). Let us first investigate the uncorrected data. The method of least squares yields k, = 0.392,

b = 1.093, k, =0.365, a = 1.130. Setting e g l b k z =A, computing A for all 21 data points and taking the mean (instead of using the method of least squares) yields A = I .065. The derivative of the above equation with these numerical values is given as a function of the ratio estimates in Fig. 3A together with the corresponding SD ratios u,/u,. Taking the big scatter into account, the fit can be considered fair.

The computations are much easier for the corrected data, Denoting the corrected ratios by q and the corrected similarities by s we have (Eider, 1960, 1 9 6 3 ~ ) q =0.6581x, ( X < .75), 4 =

1 . 4 7 7 7 ~ -0.6098686, ( x > .75) and s = ( ~ / r . o g g ) y . Carrying out the corresponding corrections on the SDs yields os/ua = 1.39 uJu,, ( x < .75) and a,/., =0.619 uu/uz, ( x 2 .75). The derivative of the similarity function

ds 2

dq- ( I + d2 together with the corrected SD ratios is plotted as a function of the corrected ratio estimation in Fig. 3B. The fit appears very good. [It should not surprise anyone that the agreement is not very good when the points of the left graph are compared with the curve of the right graph in Fig. 3 (Mashhour, 1964).]

Magnitude and category scales for evaluative judgments

Judgments of unpleasantness (of electrical stimulation) have been dealt with above. I t i s somewhat doubtful whether these judgments should be denoted as ‘evaluative’ in the same sense as the data presented below.


I00 H A ” E S EISLER

The first set of data stems from an unpublished experiment carried out by Mrs. Birgit Johansson at the Swedish Institute for Food Preservation Research and I wish to thank her for making them available. Eleven subjects judged the goodness of 5 mixtures of lemon juice and sugar solution four times each on a category scale between I (worst mixture imaginable)

TABLE 3. Magnitude estimations ( y ) and category ratings ( K ) with their intraindividual SDs (av, a=) for the goodness of 5 mixtures of lemon juice and sugar solution.

~

Stimulus No. %sugar %juice Y CtP K CK

I 3.5 3.5 2.89 1.83 2.71 0.79 2 28 3.5 3.94 1.23 3.30 1.20 3 I 0 I 0 10.44 3.36 5.18 1.18 4 3.5 28 2.76 I .04 2.68 0.86 5 28 28 4.40 2.03 3.55 1.03

and 7 (best mixture imaginable). Seven of these subjects judged the same mixtures four times each on a magnitude scale with 10 as an imagined standard of average goodness. The data are given in Table 3. A least squares fit to the Weber function of the magnitude scale gave an intercept yo of - 1.36. The scatter among the 5 points was large and the yo value is probably not much more than an educated guess (cf. Eider, 1963b). Method I was used; combination of Weber functions: linear-parabolic. Fig. 4A gives the category scale as a function of log (y - yo) with a and j3 of the equation K = alog(y - yo) + determined by the method of least squares. The fit seems good but the computed value of the Weber function-assumed constant-for the category scale is 0.67 and thus lies below all uK values. However, as an approximative description the equation does fairly well.

The other ‘evahativ‘e’ experiment presented here was carried out by Dr. Trygg Engen at Brown University and I wish to thank him for placing these data at my disposal. A group of z I subjects magnitude-estimated 48 Lightfoot Pictures as to pleasantness-unpleasantness of facial expression, with a neutral expression as the standard with the value 10. Nine of these 48 pictures were repeated 4 times (just mixed in with the rest) and constitute the magnitude estimation data used here. Another group of 96 subjects rated these nine pictures once each on the corresponding category scale between I and 9. The data are given in Table 4. The same method as above was applied and the computed yo was - 1.9.

Fig. 4B gives the category scale as a function of log (y - yo) together with the straight line fitted by least squares. The fit seems good except for picture no. 4, which was shown at the beginning of the category rating experiment and has obtained too low a category value. This point is marked by a triangle in the figure. Since there is only one category judgment from each subject, no intraindividual SD could be computed.

TABLE 4. Magnitude estimations (geometric mean, y), the corresponding intraindividual SDs (cry) and category ratings (arithmetic mean, K ) for pleasantness-unpleasantness of g Lighlfoot

Pictures.

Picture No. 4 I 1 I 3 I4 18 2 1 32 33 49

w 24.5 1.8 14. I 18.7 4’1.4 7.3 23.9 8.1 6.8 Cv 6.51 0.74 2.35 3.24 7.48 1.86 5.56 1.34 1.92 K 6.4 2.2 5.9 6.9 8.5 3.7 7.4 4.5 4.0


PSYCHOPHYSIC S AND GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION I01

"/ 1 FIG. 4. Category scale as a function of the logarithm of the corrected magnitude scale pleasantness for (A) goodness of facial of expression lemon juice, (Light- (B) 2 ?i 1 /,/ , I !I/, 1 foot Pictures). The straight lines are fitted by the method of least squares. z The triangle in B marks the stimulus shown first to all subjects in the category a5 l o 1 5 0 5 1 0 1 5

,X

rating experiment.

CONCLUSION

The GPDE thus seems to be valid in quite a number of instances, supporting the view that psychophysical measurement-in the sense of isomorphism between the subjects' judgments and the number system-can be carried out. It is also of interest to note that different but lawfully related judgments for the same set of stimuli can be obtained in accordance with the instructions given. As an instance of as yet unsolved problems in connection with the GPDE, I should like to mention the psychological meaning of the parameters used, both the parameters appearing in the Weber functions and the unavoidable integration constant.

I am indebted to Dr. Gunnar Goude and Dr. Lennart SjBberg for many valuable discus- sions and suggestions. This investigation was supported by the Swedish Council for Social Science Research.

REFERENCES

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EISLER, H. (1963~) . A general differential equation in psychophysics: derivation and empirical test. Scand. J . Psychol., 4 , 265-272.

EISLER, H. (1963b) How prothetic is the continuum of smell? Scand. J . Psychol., 4, 29-32.

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EKMAN, G. (1964~). A note on alternative psychophysical laws. [Rep. Psychol. Lab., Univ.

EKMAN, G . (1964b). Is the power law a special case of Fechner's law? Percept. mot. Skills, 19,

EKMAN, G., FRANKENHAEUSER, M., LEVANDER, S. & MELLIS, I . (1964). Scales of unpleasantness of electrical stimulation. Scand. J. Psychol., 5 , 257-261.

EKMAN, G., LINDMAN, R. & WILLIAM-OLSSON, W. (1961). A psychophysical study of cartographic symbols. Percept. mot. Skills, 13, 3 5 5-368.

GALANTER, E. & MESSICK, S. (1961). The relation between category and magnitude scales of loudness. Psychol. Rev., 68 , 363-372.

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MASHHOUR, M. (1964). On Eider’s general psychophysical differential equation and his Fechnerian integration. Scand. J. Psychol., 5 , 225-233.

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Scand. J . Psychol., Vol. 6, 1965

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