A G E N E R A L D I F F E R E N T I A L E Q U A T I O N I N PSYCHO- PHYSICS: DERIVATION A N D E M P I R I C A L TEST

HANNES EISLER

Psychological Laboratory, University of Stockholm, Sweden

EISLER, H. A general differential equation in psychophysics: derivation and empirical test. Scand. J . Psychol., 1963, 4, 265172.-By a reasoning similar to Fechner’s a general psychophysical differential equation is developed. Its integration allows the calculation of intrasubjective relations, provided the Weber functions of the two pertinent variables are known. The method is empirically validated for the relation between category and magnitude scales and for the similarity function. Theoretical implications are discussed.

Intrasubjective relations in psychophysics, i.e. relations between subjective variables, have attracted some attention in recent years (Ekman, 1961; Ekman, Engen, Kiinnapas & Lind- man, 1963). Two examples, explored in more detail below, are (a) the relation between category and magnitude scales, and ( b ) the similarity function.

A common method of arriving at such relations is to measure each variable separately and then relate them by plotting one as a function of the other, in the hope of finding a mathematical description of the function depicted in the graph.

This paper presents another method. It will be shown that the notions by which Fechner arrived at his logarithmic ‘law’-which claims to relate intensity of sensation to stimulus magnitude-may be extended to intrasubjective relations. This application, however, entails both a generalization and a restriction. Fechner postulated the constancy of the uncertainty of the dependent variable (the ‘sensation scale’) and thereby defined this de- pendent variable. The development given below holds for any (reasonable) Weber function for both variables. (A Weber function is any function relating variability of a variable to its central tendency. Weber’s law is a special case of a Weber’function.) On the other hand, the dependent variable is no longer measured ‘by definition’; it can be measured directly and its Weber function determined empirically.

Fechner’s method-disregarding the problem to which it was applied-has met with some criticism. Stevens (1961a, b) pointed out that a measure of uncertainty such as the jnd cannot be regarded as a differential, so that the differential equation derived by Fechner is a pseudo differential equation. Luce & Edwards (1958) contend that Fechnerian integra- tion requires a linear Weber function of the independent variable when the variability of the dependent variable is to be constant. The latter criticism has been dealt with in some detail in Eisler (1963 b).

However, Fechnerian integration has made it possible to derive the relation between category and magnitude scale for a number of continua (Eisler, 1962a; 1963~2, b; Eisler & Ottander, 1963). The aims of the present paper are

I. T o give a rigorous mathematical derivation of a general psychophysical differentia1 equation (‘GPDE’), and

Scad. J . Psychol., Vol. 4,1963 265

266 HANNES EISLER

2. to investigate to what extent the numerical values of the parameters obtained experi-

(a) the relation between magnitude and category scales, and (b ) the similarity function.

mentally can be predicted by integration of the GPDE for

DERIVATION OF THE GENERAL PSYCHOPHYSICAL DIFFERENTIAL EQUATION

Let @ denote a stimulus variable and x and y two subjective variables, e.g. magnitude and category scales. A psychophysical scaling experiment usually aims at the two functions x = f l ( @ ) , y =gl(@). Assume that these functions are known. If we increase @ by a given increment A@, both x and y will increase. Let us denote the increments in x and y , Ax and Ay. Assuming that the functions f l and g , increase monotonically, let us consider the inverse functions = f ( x ) , @ =g(y). After adding the increments mentioned above we obtain

@+A@=f(x+Ax)=g(y+Ay). (1)

Let us now define the increment in x in the following way

Ax = haz(x). (2)

The auxiliary variable h may be regarded as a kind of scale factor for the Weber function a,(x) of the independent variable x. Furthermore, let us add the testable assumption

Ay = ha&) + o(h), rim - = 0, h4O h

where a,(y) is the Weber function of the dependent variable y . The fact that (2a) cannot be a definition, too, but must be an assumption is easily realized.

A given Ax uniquely determines a A@ which in its turn can be said to determine Ay. The test of assumption (2a) will be used to validate the mathematical model.

Assuming f ( x ) and g(y) to be analytic functions, a Taylor expansion of f[x+ha,(x)] and g ! j +ha&)] yields:

Subtracting CD =f(x) =g(y) and dividing by h yields

Scad. 1. Psycbol., Vol. 4,1963

A GENERAL DIFFERENTIAL EQUATION IN PSYCHOPHYSICS 267

Letting h tend towards zero we obtain, since limh,o o,(h)/h = 0,

a&) = - dg(y) a,(y). *) dx dY

Substituting @ for f ( x ) and g ( y ) we obtain

Equation (4) is a General Psychophysical Differential Equation (GPDE), integration of which gives the intrasubjective relation between the two variables whose Weber functions enter into the equation. Note that equation (4) is not a pseudo differential equation; the measures of uncertainty are nowhere in the derivation identified with differentials, though the result is equivalent to such a procedure. Furthermore, the GPDE is based on infini- tesimal increments; a finite h would not permit us to disregard higher order derivatives.

EXPERIMENTAL CHECK OF THE VALIDITY OF T H E GPDE (a) The relation between category and magnitude scales

For this problem y in equation (4) is identified as the category scale K and x as the magnitude scale cy. The Weber functions are obtained from intraindividual SDs of magnitude estimates and category ratings. A number of studies have shown that the category scale can be regarded as a Fechner integral with respect to subjective magnitude (cf. Eisler, 1962b; 1963b). The term 'Fechner integral' here implies that the Weber function of the dependent variable K is constant, i.e. u&K) - k,. Equation (4) for the case at hand may thus be written

Integration yields

The integral can be evaluated by numerical integration if u(y) is not known as an explicit function. In most cases, however, the following Weber function obtains:

q Y J ) ' h r y + 4 (7)

giving

When a magnitude estimation and a category rating experiment is carried out for the same set of stimuli, all parameters can be computed. The validity of the GPDE and its integration can be tested by computing the constant k , from equation (6) or (9) and comparing it with the directly computed intraindividual u,(K). This comparison has been made for all data available to me except the experiments on subjective angular velocity (Eisler 8z Ottander,

Scad. 1. PJychol., Voj. 4,1963

HANNES EISLER 268

1963). These data have been excluded since the category scale used only three categories, giving too crude a measure for aK(K). The experiments employed for the comparison are listed in Table I.

TABLE I . Experiments corresponding to Fig. I . __ ~

Exp. NO. Reference Continuum Remarks

I L Eisler, 1962 a Loudness I: log spacing, ambient of white noise noise not controlled

11: sone spacing, ambient noise not controlled

IS Softness 111: log spacing, sound- of white noise proofed chamber

2 Eisler, 1963 a Smell of 1-111: different standards in amyl acetate magnitude estimation

3 Eisler, 1963b Length of I: stimulus exposure 0. I 2 sec. 11: stimulus exposure 0. 50 sec. 111: stimulus exposure as long

lines

as 0 wished

For experiments Nos. I and 2, k, is computed from formula (9), for experiment No. 3 from formula (6). In Fig. I a&) is plotted as a function of K, and k, inserted as a straight line paral- lel to the K axis. In experiment No. 2 three series of magnitude estimation are carried out, yielding different kl’s; a variation in the calculation of a,(y) for one of the series gives a some- what deviating k,, so that, all in all, four values of k, enter into the comparison between aK(K) and kl for experiment No. 2.

Fig. I shows that, in general, a#) is roughly constant except at the extremes of the rangc. The only clear deviation from this principle is found for intensity of smell, experiment No. 2. I am not able to offer any explanation for this exception. The low SD values at the extremes found in all plots are probably due to 0s’ greater ability to identify these stimuli (cf. Gamer, 1953) resulting in spuriously low SDs; the same ‘end effect’ is noticed in the Weber functions of the magnitude scales, e.g. Eisler (1962~). The agreement between uK(K) and k1 can be considered fair (again with smell as an exception); usually the k, values are somewhat too low. Perhaps this is because of the difference in the computation methods used for the intra- individual SDs: the method used for the magnitude scales is described in Eisler (1962~); for the category scale the standard method has been used.

It can be concluded that the data on magnitude and category scales presented confirm the mathematical model developed above.

(b) The similarity function A number of studies of undimensional similarity (Eisler & Ekman, 1959; Eisler, 1960;

Ekman, Goude & Waern, 1961) have demonstrated that similarity s i f (on a scale between o and I) between stimuli i and j with the corresponding subjective magnitudes tp; and y, observes the relation

S c a d . 1. Psycbol., Vol. 4,1963

269 A GENERAL DIFFERENTIAL EQUATION IN PSYCHOPHYSICS

c 0 .-

4- m .- > (D

-0

-0

m V c

u)

m 3

L

m L.

- 2 .- Q C

an, , , , , , , .

0"

.-

06

1 2 3 4 5 1 3 5 7 9

Category

3 : u 0s 3 :m

m- scale

FIG. I. Intraindividual SDs as a function of the corresponding category scale for the experiments listed in Table I. (I L, I S: loudness and softness of white noise, 2: intensity of smell, 3: length of lines). The horizontal lines are K1 values computed by Fechnerian integration and can be

regarded as predicted SDs.

If we are to derive this relation by the GPDE, the Weber functions used must be comparable. Since a similarity judgment is given for a pair of stimuli, the judgments from which the sub- jective magnitudes are derived likewise have to refer to stimuli pairs. A suitable method is ratio estimation, in which the subjective ratio of pairs of stimuli is judged. Letting the ratio y f / y j = q f j , (0 < q l j < I), and leaving out indices, we can transform equation (10) into

2q S--

1+q'

Accordingly, we obtain from equation (4)

Data taken from Eisler (1960) demonstrate that both Weber functions are parabolas passing through the origin, described by the two equations

Fig. z gives the coefficients of variation as functions of q and s , respectively, for these data. Inserting (13) in (12) we obtain

Scad. 1. Prychol., VoZ. 4,1963

" N K S EISLER 270

and hence

Integrating yields I S I q 1

ak, U - s bk, b - q bk, - In - =- In -+- In C,

3 ? IV

y. 0 0.2 w-

0 0.2

U

.-

0 0 u Corrected estimated o Corrected estimated ratio (3') similarity (s')

FIG. z. Coefficient of variation as a function of ratio G') and similarity (s') estimations. The variables are corrected (see text). The straight lines are fitted by the method of least squares.

and thus

Solving for s yields

We introduce the empirically testable

Condition I: ak, /bk, = I .

Condition I changes (14) into

a% b - q

S=-

I +- cq *

b - 4

We have two boundary conditions, making it possible to evaluate C: ( I ) If q =o, then s "0 .

This condition is fulfilled for all finite C. ( 2 ) If q = I , then s = I. This condition gives

UC b - I .__

I =-

I + - C

b - I

and hence b - I C--. a-x

Scand. 1. Psycbol., Vol. 4,1963

A GENERAL DIFFERENTIAL EQUATION IN PSYCHOPHYSICS 271

Inserting (16) in equation (15) and rearranging we obtain

a(b - I) b(a - I) 4

S== b-a *

I +- q b(a- I)

Comparing the coefficients of q for equations (11) and (17) yields the conditions

a(b - I) b - a b(a- I) b(u- I)=’’

”2; -

both leading to a(b+ I )

b Condition 11: - = 2.

Now let us proceed to the data. As explained in some detail in Eisler (1960), the empirically obtained ratio matrix was not consistent and the ratio estimates q had to be transformed to the ‘correct’ ratios q’ by q‘ = .6581 q (q c .75) (A misprint in the paper quoted gives the trans- formation coefficient 4681.); q’ = 1.4777 q - .6098686 (q 2.75). The similarity estimates had to be transformed according to s’ - (I/I .og3) s. The intraindividual SDs were transformed ac- cordingly. Furthermore, the average ratios $ given in Table 3 of the quoted paper are used. From these data the pertinent constants are computed by the method of least squares (see Fig. 2) as k, = .400, a = 1.033; k, = .366, b = I .o91. Hence ak,/bks = 1.03 as compared to the I required by Condition I, and a(b + I ) /b = 1.98 as compared to the 2 required by Condition 11. Thus the empirical values are only a few per cent off what the model predicts. It ought perhaps to be mentioned that I have not calculated the coefficients of q in formula (17) directly because differences between numbers of same magnitude enter into the expressions, entailing a great decrease in accuracy. For instance, an error of I per cent in u leads to an error of about 30 per cent in (a - I). Therefore it is not to be expected that these coefficients can be obtained with good enough precision.

I t can be concluded that the data on ratio and similarity estimation presented confirm the mathematical model developed above.

CONCLUDING REMARKS

The findings presented above give rise to a few general reflections. I. The idea of using uncertainty of measure (variability, confusion, ‘noise’ etc.) in scaling

procedures in the vein of Fechner and Thurstone has sometimes met with disapproval (e.g. Stevens, 1961a, b). Variability is regarded as a nuisance and, to the extent to which it is inevitable, should be ignored as far as possible. To my mind, this attitude is caused by confusion as to the object of physical and psychological measurement. I quite agree with Stevens’s position when the measures obtained constitute the object of the investiga- tion, as for instance loudness measurements for acoustical purposes or the application of subjective scales in human engineering. In psychophysical theory, however, the object of investigation is not so much the continuum scaled but rather the measuring instrument employed-the organism. The scatter obtained is thus of genuine interest, it constitutes a characteristic of the organism, The lawfulness obtaining between variability and central tendency of subjective scales tightens the theoretical network of psychophysics.

2. The GPDE offers a method for expressing intrasubjective relations mathematically.

Scad. 1. P J Y C ~ O ~ . , Vol. 4,1963

HANNES EISLER 272

These relations are often more complicated than the Weber functions from which they can be derived. For instance, the relation between category and magnitude scales has been known empirically for a long time but defied mathematical description; only the applica- tion of Fechnerian integration has led to formula (9). Likewise the similarity function was found after a laborious trial and error process involving many tentative functions; the pro- cedure used in this paper may have led to a quicker and more direct solution of the problem.

3. One may ask whether the GPDE and its integration reflect some kind of subjective mechanism. This assumption seemed credible regarding the category scale because this fulfills the requirements of a discrimination scale; the mechanism would correspond to a transformation of the ‘perceptual’ scale (magnitude scale y) by a function rendering varia- bility of the transformed scale constant. Such a transformation is equivalent to the process of Fechnerian integration. However, Fechnerian integration is a special case of the applica- tion of the GPDE, the case in which the Weber function of the dependent variable is constant.

Since the differential equation can be applied also to other Weber functions (cf. similarity), the fact that the Weber function of the category scale is constant need not be primary; instead of regarding the category scale as a transformed magnitude scale it can just as well be a scale ‘of its own’ with the property of a constant Weber function. I n this case Fech- nerian integration would not correspond to a subjective mechanism despite its power to predict one set of data from another set. For the similarity function it seems highly probable that the subjective mechanism does not detour via uncertainty from ratios to similarities; for the relation between ‘discrimination’ and ‘perceptual’ scales this is still an open question.

This research has been supported by the Humanistic Faculty of the University of Stock- holm. The theoretical part was done during a visit to the Institute for Mathematical Studies in Social Sciences, Stanford University. I wish to express my thanks for the hospitality of the Institute and the generosity of the Social Sciences Research Council, which made this visit possible. I am indebted to Ralph Roskies and Dr. B. Ajne for valuable discussions on mathematical points.

REFERENCES

EISLER, H. (1960). Similarity in the continuum of heaviness with some methodological and theoretical considerations. Scund. J. Psychol., I, 69-81.

EISLER, H. (19620). Empirical test of a model relating magnitude and category scales. Scund.

EISLER, H. (1962b). On the problem of category scales in psychophysics. Scund. J . Psychol., 3, 81-87.

EISLER, H. (19634 . How prothetic is the conti- nuum of smell? Scund. J. Psychol., 4 , 29-32.

EISLER, H. (19636). Magnitude scales, category scales, and Fechnerian integration. Psychol.

EISLER, H. & EKMAN, G. (1959). A mechanism of subjective similarity. Actu Psychol., 16,

EISLER, H. & OTTANDER, C. (1963). On the problem of hysteresis in psychophysics. J. exp. Psychol., 65, 530-536.

J. Psychol., 3, 88-96.

Rev., 70, 243-253.

1-10.

EKMAN, G. (1961). Some aspects of psychophys- ical research. In W. A. ROSENBLITH, (Ed.), Sensory communication. New York Wiley.

EKMAN, G., ENGEN, T., K~NNAPAS, T. & LIND- MAN, R. (1963). A quantitative principle of qualitative similarity. [Rep. Psychol. Lab., Univ. Stockholm, No. 152.J

EKMAN, G., GOUDE, G. & WAERN, Y. (1961). Subjective similarity in two perceptual con- tinua. y. exp. Psychol., 61, 222-227.

GARNER, W. (1953). An informational analysis of absolute judgments of loudness. J. exp.

LUCE, R. D. & EDWARDS, W. (1958). The deriva- tion of subjective scales from just noticeable differences. Psychol. Rev., 65, 222-237.

STEVENS, S. S. ( 1 9 6 1 ~ ) . To honor Fechner and repeal his law. Science, 133, 80-86.

STEVENS, S. S. (1961 b). Toward a resolution of the Fechner-Thurstone legacy. Psychome-

Psychol., 46, 373-380.

trika, 26,35-47.

Scand. 1. Psycbol., Vol. 4, r963