O N T H E PROBLEM OF CATEGORY SCALES I N PSYCHOPHYS I C S

HANNES EISLER

Psychological Laboratory, University of Stockholm, Sweden

EISLER, H. On the problem of category scales in psychophysics. S c u d . J. Psychol., 1962, 3, 81-87.-In the first part of the paper it is shown that, if three assumptions are granted, the category ecale must be logarithmically related to the magnitude scale: K = alogy +/?. In the second part of the paper, the relation K=alog(krp + q ) +/I is derived, starting from the assumptions that the category scale is a pure function of discrimination and that discrimina- tion is appropriately described by the linear generalization of Weber’s law for prothetic continua. We can reconcile these two formulas by defining the zero-point of the magnitude scale as the point where variability vanishes.

Over the past few decades psychophysics has experienced a revival in a modern form. Direct methods have been used for the measurement of sensation, and subjective scales have been constructed for more than twenty perceptual continua. A comprehensive report on two kinds of direct scaling was made by Stevens 8z Galanter (1957), who presented subjective scales for a dozen continua. Since 1957 the number of subjective continua investigated has almost doubled (Stevens, 1961), a testimony to the persistent interest and the rapid development in the new psychophysics.

Stevens (1960~) has distinguished three kinds of subjective scales: (I) discrimination or confusion scales, which are obtained from measures of uncertainty or scatter among the observers’ judgments (for instance, the summing of jnds); (2) ratio scales, which are based on direct judgments of subjective magnitudes (sensations) or of ratios among subjective magnitudes; (3) partition scales, which are based on judgments of subjective differences or distances.

The present paper attempts an investigation of the theoretical relation between ratio or magnitude scales and partition scales. The argument will be confined to the subclass of partition scales called category scales, leaving open the question whether the argument is applicable to all types of partition scales,

In the construction of a category scale the observer is typically presented with the lowest and highest stimulus of the set to be scaled, together with the lowest and highest number to be used, e.g., I and 7 on a 7-point scale. The observer is instructed to assign to each stimulus an integral number within this range of numbers. In his instruction to the observer, the experimenter may or may not stress equating the subjective intervals between successive scale values, probably with little or no effect on the outcome.

Unfortunately, from the point of view of scaling theory, the magnitude scale and the category scale do not agree within a linear transformation, at least on the continua that Stevens defines as prothetic (loudness, brightness, etc.). In attempting to elucidate this disagreement, Stevens (1960~) writes: ‘Observers are so constituted that they are unable to partition a prothetic continuum without a systematic bias. . . . Why is this so? The answer seems to hinge on the fact that a person’s sensitivity to differences (measured in subjective units) is not uniform over the scale-a fact related to Weber’s law. A given difference that is

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large and obvious near the low end of the range is much less impressive in the upper part of the scale. This asymmetry in the observer’s appreciation of differences produces a syste- matic bias whenever he tries to effect partitions on a prothetic continuum’ (pp. 52-53).

Though this description is very helpful in clarifying the phenomenon, the problem of defining ‘sensitivity to differences’ independently of the fact to be explained remains unsolved.

The purpose of this paper is to demonstrate that for prothetic continua a linear relation is not to be expected between magnitude scales and category scales. If a few assumptions, whose validity can be subjected separately to experimental test, can be granted, the category scale ought to be a logarithmic rather than a linear function of the magnitude scale.

The idea of the mathematical treatment that follows was suggested by an ingenious experi- ment of Torgerson’s (1960a) in which he obtained magnitude and category scales for the lightness and darkness of Munsell neutral gray paper chips.

The argument is based on the following three assumptions: (i) A reciprocal (hyperbolic) relation holds between the magnitude scale of an attribute

(e.g., lightness) and that of the opposite attribute (darkness): if one of a pair of stimuli is experienced as twice as light as the other, it should also appear half as dark.

(ii) A complementary relation holds between the category scale of an attribute (e.g., light- ness) and that of the opposite attribute (darkness): a stimulus that is judged 2 on a 7-point category scale of lightness should be judged 6 on the corresponding scale of darkness.

(iii) The function relating the category scale to the magnitude scale is independent of the direction of the attribute: the same relation should hold between the category scale and the magnitude scale whether the observer tries to judge lightness or darkness.

From these assumptions it follows that the category scale should be a logarithmic function of the magnitude scale. Before proceeding to the mathematical proof, let me make the argu- ment more concrete.

The argument is based on two mathematical statements: (a ) if two variables are non- linearly related, they cannot both be linearly related to a third variable; and (b) if two variables are linearly related, a third variable that is linearly related to either one is also linearly related to the other.

Let us assume that the magnitude scale of, for example, loudness is a linear function of the category scale of loudness. It would then also be a linear function of the category scale of softness, since, according to assumption (ii), the two category scales are linearly related. The magnitude scale of softness, being nonlinearly related to the magnitude scale of loudness (assumption i), cannot then be a linear function of the category scale of softness. Thus as- sumption (iii) would be vitiated because the functions relating magnitude and category scales for the two opposite attributes would be linear for one and nonlinear for the other. It follows that the assumption of a linear relation between one of the magnitude scales and its corresponding category scale, made at the beginning of the argument, is incompatible with the three assumptions. This incompatibility is illustrated in Fig. I, page 86. If the relation between the category scale and the magnitude scale for one attribute is linear (shown by the straight line), assumptions (i) and (ii) give us the relation for the opposite attribute (uppermost curve). The two functions will converge into one, however, if we assume that each category scale is a logarithmic function of the correponding magnitude scale (middle curve).

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ON THE PROBLEM OF CATEGORY SCALES IN PSYCHOPHYSICS 83

THEOREM. Let the magnitude scales for the two opposite attributes be x and y and one of the

Proof. Assumption (iii) defines the other category scale as f (y ) . Assumption (ii) yields category scales f(x). Under assumptions (i) to (iii) it follows that f(x)=alogx+@.

f ( 4 +f(Y) = c and assumption (i) yields

This leads to the functional equation

y = k/x.

f ( 4 +fW4 = c. Taking the derivative with respect to x yields

and

Let f’ be g. Substituting in eq. (4) and rearranging yields

(3)

The solution of this functional equation is g(x) = a/x:

a k a x x k / x

x . - = - . -

which is an identity.

define a nonconstant w =g*/g. Then

To prove the uniqueness of the solution, assume two different solutions g and g* and

a p y x ) = w(x)g(x) = - w(x).

X

Inserting g* in eq. ( 5 ) yields

w(x) = w(R/x). (6 )

Since the units of the two magnitude scales are independent, it must be possible for k to assume any positive finite value. Equation (6) is therefore an identity with respect to both x and K; thus w(x) must be constant; and the solution g(x) = a/x is unique.

Integrating g(x) = f ’(x) = a/x yields

f(x)=alogx+p,

where a and /? are determined by the extreme values of the category scale, the base of the logarithm, and the subjective magnitudes of the two extreme stimuli.

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84 HANNES EISLER

The function derived above agrees with one of the two possible laws that can, according to Luce (1959), relate an interval scale to a ratio scale. The other law, a power function, does not satisfy eq. (3).

DISCUSSION

Several arguments have been offered to the effect that a logarithmic relation should obtain between category and magnitude scales. A few of them may be of interest here.

(I) Torgerson assumes that the observer is able to experience only one quantitative relation between two stimuli (1960a, 1960b). If I have understood Torgerson’s argument correctly, it runs as follows. If we pick a set of stimuli so that the ratio between successive stimuli is constant in a magnitude estimation experiment, the same set would give equal intervals in a category rating experiment. Thus the scale obtained in an experiment calling for difference judgments is logarithmically related to a scale obtained by an experiment calling for ratio judgments. Torgerson draws the conclusion that an observer is unable to distinguish between sensory ratios and sensory distances. He generalizes further that this one relation, which may result in a magnitude or a category scale, depending on the experiment, is the only relation an observer experiences.

At first glance, Torgerson’s contention seems to be further corroborated by some work on similarity carried out by Ekman, Goude & Waern (1961), for example. Again, the set of stimuli that gives equal ratios in a ratio estimation experiment and equal intervals in a category rating experiment will also give equal similarities between successive pairs of stimuli in a similarity estimation experiment. However, the relation that holds for similarity is not the simple ratio between two stimuli but a single-valued monotonic function of that ratio. If the observer is instructed to estimate in per cent the subjective ratio between a pair of stimuli, and the similarity of the same pair in per cent of identity, he will give us different figures. It seems idle to discuss whether the perceived relation is the same and only its interpretation differs. The same line of reasoning may be applied for the magnitude and category scale. Their loga- rithmic relation may have other sources than the observer’s apparent inability to distinguish between ratios and differences.

The concept of similarity probably also explains the apparent inconsistency Torgerson found when he let his observers estimate differences directly and compared the results with differences computed from magnitude estimates. As pointed out above, similarity is a single-valued func- tion of the subjective ratio of pair of magnitudes (Eisler & Ekman, 1959). Torgerson’s observers probably estimated difference in the sense of the reverse of similarity. If he had instructed them to estimate sensory distance instead, the outcome might well have been different. (2) Galanter & Messick (1961) scaled loudness by the method of magnitude estimation and

category estimation. By processing the category judgments by the method of successive inter- vals they obtained a logarithmic relation between the two scales. They give three possible explanations: ‘The result suggests that the processed category scale generated by analyzing the variability among category judgments is such that equal distances on the processed category scale correspond to equal ratios on the scale of subjective magnitude. The result could also suggest that processed category scaling on a prothetic continuum produces equated ratios rather than equated intervals, generating what Stevens [On the psychophysical law, Psychol. Rev., 1957, 64, 153-1811 has called a logarithmic interval scale. An alternative, and at present experimentally indistinguishable interpretation, is that category judgments reflect equated intervals on a log scale’ (Galanter & Messick, 1961, p. 369).

(3) Ekman, Goude & Waern (1961) hypothesize that it is similarity between successive pairs of categories that is equated in category scaling. Because of the similarity relation, equal similarities and equal ratios would result in the same scale. Their explanation is thus operationally indistinguishable from that of Galanter & Messick (1961).

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85 ON THE PROBLEM OF CATEGORY SCALES IN PSYCHOPHYSICS

(4) Junge (1960) relies on mathematics in deriving the logarithmic ‘category scale equation’. He defines difference as I-q, (st, =similarity between stimuli i and j ) and the corresponding interval on the category scale R , - R,, -ARC = k(I - s,,). Making use of the similarity relation (Eider & Ekman, rg59), substituting differentials for differences, and integrating yields

R, = kologS + C, where R, is the category value and S the corresponding subjective magnitude.

All these attempts at explanation have one thing in common: they assume that the category scale is an exact logarithmic function of the magnitude scale. As Stevens & Galanter (1957) have amply demonstrated, the category scale really lies somewhere between a linear and a logarithmic function; usually the deviation from the log function is small but not negligible. Galanter & Messick ( I 961) eliminated this discrepancy by substituting the scale of ‘successive intervals’ (a processing of the confusions between categories) for the direct category scale. I want to attack the problem from a different angle.

Let us regard the task in category rating as a kind of nominal scaling, in which we try to identify the stimuli and space the categories in such a way that successive categories are equally easy to discriminate. This way of conceiving the task is backed up by an experiment of Witte (1960). Witte’s subjects first judged a set of stimuli in two categories and then the same set in three categories. The stimulus value corresponding to the middle category in the second experiment coincided with the point of maximum uncertainty (equal number of judgments for the two categories) in the first experiment. This conception of the observer’s task says nothing about subjective distance. (In a bisection experiment, for instance, the middle stimulus would be equally easy or difficult to confuse with either of the outer stimuli. However, the subjective ratio between the two sensory distances 60 obtained need not be unity.) What we obtain is a pure discriminability scale.

According to Stevens (1957) variability (in terms of subjective jnds), is the inverse of discrimination, and it grows with magnitude on prothetic continua. Assuming the linear generalization of Weber’s law for subjective continua (see, for instance, Bjorkman, 1958), we have

o= ky + q,

where u is the standard deviation in subjective units and y subjective magnitude; k and q are constants. Substituting d y for a and carrying out a Fechnerian integration we obtain

dy - a log ( k y + q ) + P , K = S R y + 9 - where K denotes category scale values. Depending on the quantity q/k, we obtain functions lying between a linear (R=o) and a purely logarithmic (q=o) relation between the category and magnitude scales. This range is illustrated in Fig. 2. The development given above may also explain why the category and the magnitude scales coincide on metathetic continua: here the variability is constant over magnitude (Stevens, 1957), i.e., k = o , and the integral becomes a linear function of magnitude.

At this point I turn once more to the paper of Stevens & Galanter (1957) in which they give a detailed explanation of the category scale. They maintain that three factors are at work, intent, discrimination and expectation. Of discrimination, they write: ‘First let us

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86 HA"B EISLER

2 0 0 w I- !T 0

MAGNITUDE SCALE MAGNITUDE SCALE

Fig. I . Fig. 2. Fig. 3. FIG. I . The category scale as a function of the magnitude scale. If a linear relation is assumed for one continuum (straight line), conditions (i) and (ii) yield the function for the opposite con- tinuum (uppermost curve). For a logarithmic function, however, the two curve8 converge

(middle curve). The arrows indicate the direction of increasing stimulus values. FIG. 2. The category scale as a function of the magnitude scale for different values of the para-

meter q/k. FIG. 3. The linear generalization of Weber's law for subjective continua.

see how discrimination might affect the outcome. We have already noted that the width of . observer's categories tends to vary inversely with his ability to discriminate among the stimuli. If this were the only factor operating it would be possible to predict the form of the category scale in aprecise manner [italics mine]. The argument would proceed from the fact that on prothetic continua the increment required for a detectable difference is roughly proportional to the magnitude being judged.

If we were to assume that the category width is proportional to the discriminal uncertainty, and that the uncertainty is proportional to the magnitude being judged, it follows that the category scale would be logarithmic. In other words, in the extreme case in which relative discriminability is the controlling factor, the curve relating category judgment to magnitude scale should approximate a logarithmic form' (p. 379).

Since the curvature of most category scales when plotted against magnitudes is less than logarithmic, Stevens & Galanter conclude that discrimination is not the only factor deter- mining category judgments. But according to the argument presented above, the assumption of the linear generalization of Weber's law is sufficient to account for deviations from the 1 ogarithmic form.

The model proposed in the first part of this paper requires that the category scale be a logarithmic function of the magnitude scale, whereas the formula derived in the second part states that the relation between the two scales can lie anywhere between linear and logar- ithmic. How can this contradiction be resolved? If the equation K =crlog(ky+ q) +/I is accepted as the correct description of the relation between category and magnitude scales, then at least one of the assumptions made in deriving the pure logarithmic relation must be wrong. A scrutiny of Torgerson's data seems to indicate that it is assumption (i), the one which requires reciprocity between magnitudes of opposite continua, that does not quite hold. But let us look once more at the linear generalization of Weber's law, which is described in Fig. 3. If we regard the y intercept as the zero point of the magnitude scale (defining zero as the point where the variability vanishes), or expressed mathematically, if we make the following substitution: y' =y + q/k, y =p' - q/k then the integral becomes:

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ON THE PROBLEM OF CATEGORY SCALES IN PSYCHOPHYSICS 87

and we are back at the purely logarithmic relation. I t is thus to be expected that adding the constant q/k to the magnitude values will both straighten out the reciprocal relation between magnitudes for opposite continua and render the category scale a purely logarithmic function of the magnitude scale so ‘corrected’.

The conclusion seems to be that magnitude estimation and category rating measure different things, perhaps even ‘tap different sensory mechanisms’, to quote an expression of Luce & Edwards (1958). Therefore, both scales may be considered ‘true’ in their own right. The more interesting one is the magnitude scale, because ( I ) it is a ratio scale, whereas the category scale is only an interval scale and thus contains less information; (2) it seems possible to derive the category scale from the magnitude scale (and its standard deviation), but not vice versa; and (3) the magnitude scale seems to be an ‘extensive scale’ (Bergmann & Spence, 1944) since experiments on the addition of sensations carried out by Goude (see Ekman, 1961) yielded results predictable from the magnitude scale.

I should like to acknowledge my considerable debt to S. S. Stevens, whose tonic criticism and unrelenting disagreement with some of my ideas spurred me to repeated reformulations of my thinking on these topics. This research was supported by a grant from the National Science Foundation (Psycho-

Acoustic Laboratory Report PNR-261) and was done while the author was Research Fellow at the Psycho-Acoustic Laboratory, Harvard University, Cambridge, Mass., U.S.A.

REFERENCeS

BEXGMANN, G. & SPENCE, K. W. (1944). The logic of psychophysical measurement. Psy- chol. Rev., 51, 1-24.

BJORKMAN, M. (1958). Some relations between psychophysical parameters. [Rep. Psychol. Lab., Univ. Stockholm, No. 65.1

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

EKMAN, G. (1961). Some aspects of psychophys- ical research. In W. A. ROSENEILITH (Ed.), Sensory communication. New York: MIT Press & Wiley. Pp. 35-47.

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

GALANTER, E. & MESSICK, S. (1961). The relation between category and magnitude scales of loudness. Psychol. Rev., 6, 363-372.

JUNGE, K. (1960). The category scale equation. Scand. J . Psychol., I , I 12-1 14.

LUCE, R. D. (1959). On the possible psychophys- ical laws. Psychol. Rev., 66, 81-95.

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

1-10.

STEVENS, S. S. (1957). On the psychophysical law. Psychol. Rev., 64 , 153-181.

STBVENS, S. S. ( 1 9 6 0 ~ ) . Ratio scales, partition scales, and confusion scales. In H. GULLIKSBN & S. MESSICK (Eds.), Psychological scaling: Theory and applications. New York Wiley. 4 . 4 9 - 6 6 .

S m s , S. S. (1961, 1960b). Psychophysics of sensory function. In W. A. ROSENBLITH (Ed.), Sensory communication. New York: MIT Press & Wiley. Pp. 1-33. Also published in Amer. Scientist, 48, zz6--253.

STEVENS, S. S. & GALANTER, E. H. (1957). Ratio scales and category scales for a dozen per- ceptual continua. J. ucp. Psychol., 54, 377- 411.

TORCERSON, W. J. ( 1 9 6 0 ~ ) . Quantitative judg- ment scales. In H. GULLIKSEN & S. MESICK (Eds.), Psychological scaling: Theory and applications. New York Wiley. Pp. 21-31.

TORCERSON, W. J. (1960b). Distances and ratios in psychophysical scaling. [Rep. 58-G-00x4, Lincoln Lab., Mass. Inst. Technology.]

WITTB, W. (1960). Uber Phiinomenskalen. Psychol. Beitr., 4, 645-674.

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