SIMILARITY IN THE CONTINUUM OF HEAVINESS WITH SOME METHODOLOGICAL AND THEORETICAL CONSIDERATIONS

S I M I L A R I T Y I N T H E C O N T I N U U M O F HEAVINESS W I T H S O M E M E T H O D O L O G I C A L AND

THEORETICAL C O N S 1 D E R A T I O N S HANNES EISLER

Psychological Laboratory, University of Stockholm, Sweden

EISLER, H. Similarity in the continuum of heaviness with some methodological and theoretical considerations. Scand. J . Psychol., 1960, I , 69-81.-An experiment in which the subjects estimated (a) the ratio and (b) the similarity of pairs of stimuli, consisting of weights, is reported. The report reveals that the relation obtained in earlier investigations between estimated similarity and the two subjective magnitudes of the pair, holds good for the continuum of heaviness too. In connection with this the intraindividual standard deviation of ratio estimates and the problem of internal consistency of a ratio matrix are treated. In a speculative discussion the concept of similarity is scrutinized, the similarity function is tentatively extended to multi-dimensional cases and an attempt is made to connect similarity and discrimination learning.

The concept of similarity pervades many psychological theories. It is seldom unambiguously defined, however, and the investigations carried out on the subject are rather confusing. An attempt to clarify the phsychological mechanism lying behind the experience of similarity is being made through a series of experiments now being carried out at the laboratory and designed to investigate similarity at the simplest level possible, in presumably subjectively one-dimensional continua. The series was started on the metathetic continuum (in Stevens’ conception of the term, Stevens, 1957) of pitch (Eisler & Ekman, I959), followed by the continua of greyness and visual area (Ekman, Goude & Waern, 1960). All three investigations revealed the same relationship between the experience of similarity (expressed as a fraction of identity) si, and the two subjective magnitudes Ri and R, which correspond to the two stimuli S, and S, (S, < SJ, the similarity of which was to be estimated. This relation is expressed by the formula

The main purpose of the present investigation was to find out whether the same relationship holds good for the typically prothetic continuum of heaviness. A similar investigation is planned for the continuum of brightness.

The methods employed (ratio estimation and similarity estimation of pairs of stimuli) make possible the elucidation of several other questions in connection with the evaluation of the experimental data. These methodological problems will be treated in detail in the following sections, but are introduced in short below:

I . The possible influence of (a) the order in which ratio estimation and similarity estimation is performed, and (b) the method of estimating similarity (scale rating vs. percentage rating).

2. The standard deviation in ratio estimation. 3. The internal consistency of a ratio matrix and a sketch of the methods used to date for

4. A transformation rendering the obtained ratio matrix internally consistent followed by evaluating ratio matrices.

computation of scale values of heaviness.

The last section of the paper contains a more general discussion of theoretical problems in connection with similarity.

Scand. J. Psychol., I , 1960 69

70 HANNES EISLER

A. EXPERIMENTAL PROCEDURE

The investigation reported in this paper is based on two experiments. In both experiments the stimuli consisted of 7 identical cylindrical cardboard containers, numbered I to 7, weigh- ing 50, 125, 200, 275, 350, 425 and 500 g. The subjects were presented with the 21 possible pairs of stimuli in random order, every pair 4 times in each experiment. The weights were lifted in succession by the same hand, as many times as the subject desired, but at least three times in order to minimize the time order error. Preliminary trials preceded the experiments proper.

The task of the subjects consisted in the one experiment in ratio estimation (the lighter weight as a percentage of the heavier one) and in the other experiment in similarity estimation. The two experiments were separated by at least one week.

In both experiments the subjects were 12 undergraduate students of psychology; six of them carried out the ratio estimation before the similarity estimation, whereas the order was reversed for the other six. Half of the subjects estimated similarity on a scale between o and 10 (to one place, if desired), where o denoted no similarity at all and 10 identity, and the other half estimated similarity as a percentage of identity. All data obtained were converted into decimal fractions (of identity in the case of similarity

rating) before carrying out any other calculations.

B. METHODOLOGICAL PROBLEMS

I . Efect of the order of ratio estimation and similarity estimation and of the method of estimating simikzrity

In the first experiment of this series of similarity investigations (pitch, Eisler & Ekman, 1959)~ the similarity estimation preceded the ratio production in order to keep the subjects naive regarding the possible experience of subjective magnitude of the stimuli. One question the author tried to answer in the present experiments was whether this is a necessary precaution. Again, some criticism has been advanced, pointing out that estimation in scale values might be a sort of category scaling (Stevens, 1957), and thus possibly yield spurious results. This problem was investigated under the assumption that percentage rating, being a sort of magnitude estimation, yields correct results.

Both problems are suitably investigated at the same time by t-testing the interaction and the main effects of order between ratio and similarity estimation and method of similarity estimation, disregarding the different stimulus combination treatments as they are of no consequence in this connection. The means of a11 stimulus combinations are added together for every subject separately and the sums are treated according to Lindquist (1953, pp. 21-22>.

None of the 6 t-values reached the 5 %-level of significance and therefore it is concluded that neither the order of estimation nor the method of similarity estimation is of any real consequence. All values can be considered to derive from the same population and are treated accordingly below.

The data upon which the tests are based will be found in Tables I and 2 of Appendix I.

2. The standard dmiation in ratio estimation

In Fig. I a, the intra-individual standard deviations u,, (computed by averaging the individual variances) are plotted against the ratio estimations q (the means for all trials and subjects).

Scand. J. Psychol., Vol. I, 1960

SIMILARITY IN THE CONTINUUM OF HEAVINESS 71

0.3

02

0.1

b

02 0.4 0.6 0s 1.0 1.2 q 0.2 0.1 0.6 0.0 1.0 1.2 q

FIG. I a. Intra-individual standard deviation plotted against estimated ratios of heaviness. FIG. I b. Coefficient of variation plotted against estimated ratios of heaviness.

The data for this graph, together with the inter-individua1 standard deviations ue are shown in Table 3 of Appendix I. It is evident that a, is greatest around q = 0.5 and decreases towards zero and one. A parabola is fitted by the method of least squares (assuming its passing through the origin) and entered into the plot. The fit is rather close as is seen still better in Fig. I b, where the coefficient of variation, a,/q, is plotted against q. The equation of the parabola is

a, = 0.428 q - 0.392 q2 or a, = 0.392 q (1.093 - 4).

It is remarkable that the standard deviation seems to be independent of either stimulus of the pair and to depend solely on the ratio estimated. A rationale of the formula above might be as follows: The subject is working with two anchorage points (cp. Helson, 1947). His uncertainty (expressed as the standard deviation) is proportional to the distances of the estimated ratio from both anchorage points. The one anchor is clear: if he has only one stimulus, q estimated will equal zero without any doubt, i.e. with a, =o. The other anchor ought to be where both stimuli are identical. But at this point some uncertainty remains in the subject (a, > o for q = I);

only when q > r is the subject really ‘sure’ that the smaller stimulus ‘at least’ equals the larger one. We may regard the amount by which this anchorage point exceeds one as a kind of subjective Weber fraction. If we call it w, we get the general formula for the standard deviation of estimated ratios:

(2) a, = CQ(I+ w - q),

where q and ( I + w - q) are the distances from the anchorage points and C is a constant of proportionality.

A n interesting inference appears to be that by estimating the ratio of the greater to the smaller stimulus, all q > I , there would be only one anchorage point and thus O, should increase linearly with q. This is in agreement with an experiment carried out by Beloff (personal communication).

3. The internal consistency of a ratio matrix and methods of evaluating ratio matrues The procedure of estimating subjective ratios among stimuli in order to construct a

psychological scale has been employed in several forms. Simple painvise estimation provides a convenient form of the procedure and is the one used here. Different ways have been suggested for evaluating the ratios obtained.

According to Comrey (1950)’ the subjective scale is constructed by means of the subjective ratio of pairs of adjacent stimuli, obtained as a mean of the direct estimate of this ratio and a number of reproductions of it by estimates of other suitable pairs of stimuli. The procedure is

Scund. J. Psychol., Vol. I, 1960

HANNES EISLER 72

described in some detail by Guilford (1954). It suffers, however, from two weaknesses. One is that by successively multiplying scale values in order to obtain higher ones, every error of one scale value not only remains, but also increases in the next one. The other weakness is based on the reproduced ratios. If there are n stimuli, n - 2 reproductions are computed, each made out of two direct ratio estimates. But as each experimentally obtained ratio estimate is not used equally often, they are automatically assigned arbitrary weights in the final computation.

Both these weaknesses are remedied by the method for treating ratio estimation data suggested by Ekman, Bergstrom & Kiinnapas (1956). In the final evaluation of the ratio matrix, each scale value is computationally independent of all the others. Likewise, if one of the methods for averaging row and column values is used, each experimentally obtained ratio estimate appears four times, twice as its reciprocal.

Torgerson (1958, pp. 104-1 16) reports Comrey’s method critically before suggestinghis own. This latter is based on a least squares solution of the logarithms of the ratio estimates. His method is similar to Ekman’s et al. (1956), but by taking logarithms he avoids the possibly considerable error introduced by using the reciprocals of small fallible values. On the other hand, Comrey’s method, as distinguished from the other two, permits a check of the consistency of the subject’s estimates, as the values to be averaged for each ratio ought to show only random variation. The obvious way of reproducing the ratios by the scale values obtained and judging the size and distribution of the residuals cannot be considered satisfactory. One can heartily agree with Torgerson’s statement: A rigorous statistical test for goodness of fit has yet to be devised.

It is however possible to judge the internal consistency of a ratio matrix on surer grounds, as will be shown below.

As in all psycho-physical work, the basic assumption is that the subject is in fact doing what he is instructed to do. In the weight-lifting example reported by Guilford this is obviously not the case, and Guilford (1954, p. 220) writes: In the two-stimuli arrangement, there seems to be a constant error, . . . This is for the estimates of the same ratio (A/B, B/C, etc.) to increase as one goes farther away from the direct estimate in each column.

In the present paper, a more general method for investigating the internal consistency of a ratio matrix is proposed. The main point is that not only the estimates of the ratios of uc@cent stimuli, but all estimated ratios can be reproduced by pairs of other estimates. If there are n stimuli, n(n - I ) / Z estimates are obtained. The subjective value R, of stimulus S, enters into n - I qik-estimates (k =k i, k +j ) . If the estimate qir = Ri/R, is to be reproduced, there remain n - 2 of the above n - I estimates which can be paired off with n - 2 estimates containing R,:

Note that qri = I / qi,. Accordingly, each estimate can be reproduced in n - 2 ways, and in all n - I values of each ratio are obtained.

If there is satisfactory agreement, the n - 2 values for every pair are averaged and any of the methods mentioned above may be applied; moreover, a simpler and probably almost as exact a method is to make the subjective value of the greatest stimulus = I, and use the averaged ratio estimates of all the other stimuli with this one directly as scale values. 2(n - 2) + I

direct estimations enter every such scale value, and this ought to be enough to smooth out random errors.

The procedure is demonstrated in the next section by the use of data obtained in the present experiment.



4. Transformation of the ratio matrix obtained. Still another scale of heaviness

Table 3 of Appendix I shows the 5 reproductions of each of the 21 ratio estimates in the column ‘Reproduction 1’. For the sake of simplicity zeros and decimal points are left out. The values in bold type were obtained by multiplying the experimental values according to Equation (3), the others by dividing, i.e. multiplication with reciprocals. The outcome is quite unambiguous. The ‘ratio matrix’ is far from being internally consistent and can yield no meaningful values in its present form. A closer examination reveals that all values obtained by division are about equal and smaller than the original ones, and those obtained by multiplication are larger. Thus some kind of systematic ‘error’ appears to exist. A correcting factor, which cancels in the divisions and is squared in the multiplications, seems to afford a suitable transformation. A plot of the original q-values against the averaged reproductions of divisions, however, turns out similar to the one in Fig. 2.

Thus an appropriate transformation seems to be:

Q’ = aq, (4 < 0.75) (4)

Q’ = bq + c, (42 0.75), (5)

where q’ are the transformed ‘correct’ ratios.

The 105 reproduced values would give 105 equations from which a, b and c could be computed by the method of least squares. Of these, however, only 35 are different. T h e following possibilities exist. As qij = q i k / q ; k , we get:

Equations containing two or three expressions as in Formula ( 5 ) had to be excluded be- cause of computational difficulties. All equations containing the ratio 417 were excluded, too, as the values obtained by them deviated markedly from those obtained by the others. There remain, then, 8 equations of type (6) to compute a and 17 equations of type (7) to compute b and c. The result was:

a=o.5681 b = 1.4777 c = -0.60986860.

q‘ was computed according to Equations (4) and ( 5 ) and the values recorded in Table 3, Appendix I. Reproductions were calculated as before and are to be found in the columns headed ‘Reproduction 11’ of this table. I t is quite clear that the new ratio matrix is internally consistent. The empty places in the table are due t o the inapplicability of 41,. Finally, q’, was computed as the mean of q‘ and the pertinent reproductions. (In more precise work, every value ought to be weighted with the reciprocal of its variance.)

Fig. z shows a plot of q against q’. The reason for exchanging the dependent and independent variables is that it is better to start a possible explanation or even description of the transformation mechanism from the ‘correct’ ratios. It ought to be pointed out in advance that any explanation must necessarily be ad hoc and of a rather speculative nature. It must,

Scand. J . Psychol.. Vol. I, 1960

HANNES EISLER 74

R

P

Fig. 2. Fig. 3.

FIG. 2. Estimated ratios of heaviness plotted against the corresponding corrected ratios. FIG. 3. Subjective heaviness as a function of weight.

however, be remembered that the only alternative to the transformation of the otherwise useless ratio estimates would have been to throw the results of the experiments into the waste-paper basket, an alternative disliked by the author.

The graph illustrates the following facts. One straight line passes through the origin and the other through a point having the ordinate q = 1.089 for the abscissa I’ = I . These two points seem to agree with the anchorage points mentioned in section B.2. Furthermore, the two lines intersect at 4’=0.5. I t appears as if the subjects rated the distance from the nearest anchor. In the case of the origin, they add a certain fraction of this distance when giving their estimates. In the case when ‘identity’, i.e. q = I .089 is the nearest anchor, they diminish the distance by a certain fraction and subtract the remainder from ‘identity’. The dashed line indicates fictitious ratings, asuming the existing anchors without further ‘ameliorations’ from the subject. The thin lines, parallelled by arrows, illustrate the supposed mechanism.

The problem of the anchors has been touched upon in section B.2. The other part of the mechanism is harder to understand. A tentative explanation might be the size-weight effect. The lack of internal consistency of the ‘ratio matrix’ in the present experiment appears also in the experiment described by Guilford (1954) but seems to be absent in a similar experiment reported by Baker & Dudek (1955). Their subjects pulled a ring without seeing the weights used. Further on, in Flournoy’s experiments (I 894), the size-weight illusion disappeared when the subjects closed their eyes. In the present experiment, the weights being of the same sue, a tendency of q towards I seems reasonable.

In order to obtain scale values of heaviness, the subjective magnitude of the heaviest stimulus R7 is set = 100. The other values are obtained directly from Table 3 after multiplication by 100:

S in grams 50 125 200 275 350 425 500

R 2.9 9.5 19.1 32.7 52.1 73.6 100.0

The corresponding pIot is found in Fig. 3. A Iog-log-plot reveals some curvilinearity and therefore the simple power function is not adequate. The subjective scale agrees rather



well with the one found by Harper & Stevens (1948), which has the steepest slope (biggest exponent) of all heaviness scales reported by Baker & Dudek (1955) and Stevens & Galanter

1’957).

C. THE SIMILARITY FUNCTION

As mentioned in the introduction, Formula ( I ) has proved fairly correct in the few investigations made to date. However, in that connection the question arose as to whether the instruc- tion to rate similarity perhaps results instead in another subjective magnitude scale, so that the

FIG. 4. Experimental estimates of similarity plotted 0.2v , , , , , , against theoretical values of similarity computed from corrected ratio estimates of heaviness according to Equa- tion (IU). 0 0.2 O L 0.6 0.8 1.a stheor

similarity estimates were in reality ratio estimates of this other scale. If this were so these two scales would be connected through the relationship expressed in Formula (I). It can now be mathematically proved that, given two proper ratio matrices, their respective scale values cannot be connected according to the following:

where L denotes scale values derived from the ‘similarity ratio matrix’. The interested reader will fmd the proof in Appendix 11.

I n the present experiment, 9‘’ was transformed into Stheor according to Formula (I) written in the form

The observed similarity estimates semD are to be found in Table 4 of Appendix I (together with their intra- and inter-individual standard deviations 0, and ue and the computed stheor). Fig. 4 shows the corresponding plot with the straight line calculated by the method of least squares assuming its passing through the origin. Its slope turns out to be I .093 instead of I. This deviation is explained by referring to the anchorage point mentioned twice before, yielding Semp = I .093, when Stheor = I.

It can thus be concluded that the relationship found in other investigations holds good for the prothetic continuum of heaviness too.


76 H A W S EISLER

D. DISCUSSION OF THE EXPERIMENTAL RESULTS

One of the most interesting results of this investigation is the concept of anchorage points as starting points for the estimations to be made. Zero has always been implied as one of them, but the other has usually been supposed to be one. In the present experiments it turned out that the upper limit of the range in fact exceeds I by a kind of Weber fraction, denoted by w. eo has been determined by three independent methods as .093 (section B.2), .089 (section B.4) and .093 (section C). The agreement can be considered remarkably good. Work has yet to be done to investigate whether an anchor of this kind is generally found or whether it pertains only to the experiments reported here.

The second point of interest is the demonstrated advisability of examining the internal consistency of a ratio matrix before processing the data in any of the customary ways. The frequency of internally inconsistent ratio matrices must be examined, as must the more general applicability of the transformation described in section B.4. Similarly, the tentative explanation by the size-weight effect has yet to be confirmed.

It might be argued that the similarity function, Formula (I), cannot be considered valid in the investigation reported. Operationally speaking it is not, since both the ratio estimates and the similarity estimates necessitate a correction before they are included in the formula. Against this it can be said that the relation holds good for correct values of ratio and similarity estimates and is thus correct in principle. The author holds the latter point of view.

E. THEORETICAL DISCUSSION

It was mentioned in the introduction that the concept of similarity is used in many psychological contexts (see e.g. Attneave, 1950). In most of the investigations dealing with the topic, similarity is defined in one way or another and the results are rather obscure. The most common mistake met with is the mixing up of ‘objective’ and ‘subjective’ similarity. Similarity is an attribute of a pair of subjective magnitudes (corresponding to a pair of stimuli). The term ‘attribute’ is used as referring to psychological dimensions, to be distinguished from ‘properties’, referring to physical dimensions (Bergman & Spence, 1944). The scientist qua scientist can only define similarity arbitrarily. If he tries to give a definition he puts himself in the place of the observer, which is as incongruous as if he tried to define subjective loudness before making any psychophysical measurements. Similarity belongs intrinsically to the object language and can only by experimentation be transferred to the the metalanguage (Bergman & Spence, 1944), at least as long as we know so little about nervous processes.

It has been stated that similarity is intrinsically unscalable (Noble, 1957). As the relation ‘similarity’ is nontransitive, symmetric, irreflexive and nonconnected, it is not a serial one and thus it cannot be scaled. Now, it is never the relation but the elements of the pertinent univer- s u m that may possibly be scaled; that single stimuli (or, to be more exact, the corresponding subjective magnitudes) cannot be scaled regarding similarity is a rather trivial statement. If, instead, a universe made up of pain of stimuli is considered, with the relation ‘more similar than‘, then this relation is asymmetrical, transitive and connected and thus the attribute ‘similarity’ is scalable in principle.

Another question of interest is the relationship between (subjective) distance and similarity,

Scad. J. Psychol., Vol. I, 1960

SIMILARITY I N THE CONTINUUM OF HEAVINESS 77

Usually, it is taken for granted that similarity is a one-to-one decreasing function of distance. Now, a geometry (possibly non-Euclidian) can always be conceived of such that the statement above becomes true. At the same time, however, it becomes devoid of meaning. The matter is an entirely empirical one. In order to investigate whether the statement holds good for Eucli- dian geometry, we transform relation ( I ) in the following way:

I

where d is the subjective distance Rj-Rt. It is evident that similarity is not a one-to-one monotonic decreasing function of distance. This means that the same distance in subjective units between two percepts yields different similarity, depending on where in the continuum it is situated. I t is the ratio between two percepts which determines their similarity. Under the assumption of the validity of the n-th power law, which relates the subjective magnitudes to the corresponding physical measures of the stimuli, the interesting conclusion can be drawn that the ratio of the physical measures of the pair of stimuli determines their similarity as well.

A third point of interest concerns dimensionality and similarity. If, for the sake of simplicity, we restrict ourselves to perceptions, we can classify perceptual dimensions (or continua) in the following way. Let us divide perceptions into two categories, simple perceptions and derived perceptions. Simple perceptions have the aspect of quality (e.g. pitch) and quantity (e.g. loudness). The aspect of quantity is an energetic one and the corresponding continua are called prothetic (Stevens, 1957). Physiologically, they may correspond to nervous impulse frequency, i.e. a time pattern of nervous impulses in a certain cortical area. The qualitative aspect of perception may refer to loci in the brain, or, crudely, to the ‘senses’. For every perceptual quality there is a corresponding body of dimensions which may comprise a differen- tiable continuum (called metathetic, Stevens, 1957, e.g. pitch), a classifiable continuum (cg. fundamental colors) or only one category (e.g. heaviness). The above classification does not apply to derived perceptions. They are in a certain way independent of the ‘sense’ which elicits them. Examples would be subjective size, subjective velocity.

With this background in mind it seems reasonable to assume that the concept ‘similarity’ can be meaningfully applied to two percepts belonging to the same dimension or to the same body of dimensions, but not to two percepts belonging to different bodies of dimension. An observer instructed to estimate the similarity of e.g., two differently colored weights, is supposed to ask: In what respect? In other words, whereas the similarity between, say, pure red and pure green, is supposed to be zero, the relation ‘similarity’ does not exist between, say, a weight and a tone.

The author could not refrain from speculating about the similarity function if the percepts were multidimensional (but belonged to the same body of dimensions). The following condi- tions must then be fulfilled:

(i) If the two percepts have no dimension in common, their similarity must be zero. (ii) If the two percepts are identical, their similarity must be one. (iii) o <sU < I. (iv) Symmetry.

(v) Formula ( I ) must be obtained as a special case, when the number of dimensions is reduced to one. (Which would mean zero-coordinates for all dimensions but one in a space model.)

Let the two similar percepts Ri and R j be points in an n-dimensional space with the coordinates R, (ril, rie, . . . ri,) and Rj(rjl, yja . . . rjn). If the vectors from the origin to the points


HANNES EISLER 78

are denoted by Ri and R,, the following relationship for similarity si, satisfies the above condi- tions:

Ri Rj = Xi, * Mi, *

Xi,(xl, x2, . . . x,,) denotes a vector, the corresponding point of which has in every dimension the larger co-ordinate of the two pertinent percepts and Mu(? = (rt l + rj1)/2, m2 = (ri2 + rj2)/2, . . . . m, = (Itn + rjn)/2) has in every dimension the mean co-ordinate.

Finally, a problem concerning discrimination learning and similarity deserves mentioning. I n connection with the theory of learning, a similarity index is defined as the probability for the appearance of a reaction previously conditioned to a certain stimulus when another similar stimulus is presented (Bush & Mosteller, 1951). Let us, for the sake of argument, assume that there is a close monotonic relationship between this index of similarity and similarity sit according to Formula (I) . Now, the index of similarity is said to decrease during discrimination learning, and as it is defined in stimulus terms, whereas similarity si, is defined in subjective terms, the conclusion would be that during discrimination learning the R - S function which relates subjective magnitudes to physical measures, changes. Estimated ratios of pairs of stimuli would decrease. I n other words, after discrimination learning, the same stimulus would elicit another percept, but the same pair of percepts would yield the same similarity. This problem is being investigated at present at the Psychological Laboratory of the University of Stockholm.

SUMMARY

Two experiments were carried out yielding ratio estimates and similarity estimates of pairs of weights. It was found that:

I . (a) The order in which ratio and similarity rating are performed does not influence the results. (b) Whether the similarity ratings are carried out in scale values (between o and 10)

or as a percentage of identity is of no consequence. 2. The intra-individual standard deviations of the estimated ratios are independent of

either stimulus forming the pair. They are proportional to the distances from two anchorage points, so that a plot of them against the corresponding ratios yields a parabola. The two anchorage points are q - o and a ratio value exceeding I by a sort of Weber fraction.

3. Before processing data from ratio estimation, the internal consistency of a ratio matrix ought to be investigated. If it proves to be satisfactory, a method of obtaining scale values suggests itself. A method for examining the internal consistency is described.

4. This method is put to use with the experimental data. A transformation is carried out in order to obtain ‘correct’ ratios. An attempt has been made to give a rationale for the transformation in question. A magnitude function of heaviness has been constructed and compared with those of other authors.

5 . It has been shown that the similarity function found in previous investigations

is aIso valid in the continuum of heaviness.

Scad. J. Psychol., Vol. I, 1960


In Section E, theoretical discussion, the following problems were dealt with:

I. It was shown that 'similarity' is a scalable attribute. 2. It was shown that, in Euclidian geometry, similarity is not a one-to-one monotonic de-

creasing function of subjective distance. 3. The problem of similarity in multi-dimensional cases was briefly discussed and a simi-

larity function for this case suggested. 4. Based on some statements about discrimination learning and similarity it was concluded

that the R - S-function, relating stimuli and percepts, would change during discrimination learning.

APPENDIX I TABLE I. Sum of the ratio estimations (qtJ

for all stimulus combinations for each TABLE 2. Sum of the similarity estimations (s ir ) for all stimulus combinations for each

subject. subject.

si, rated si j rated si, rated sij rated in scale values as percentages in scale values as percentages

qn before sij 10.486 9.647 qi, before sij 10.367 I 1.605 13.171 13.472 8.955 12.042 11.457 9.338 9.860 9.330

q i j after sij 12.316 12.536 q i j after sij 9.568 9.666 14.357 12.763 11.588 9.469 12.098 11.088 10.81 I 7.036

TABLE 3. Intra- and inter-individual standard deviations, estimated ratios and the corresponding corrected and averaged ratios of heaviness for all stimulus combinations. For columns headed

Reproduction see text in section B. 4.

iii 0, 0, 4 Reproduction I 4' Reproduction I1 B'

.0784

.0763 *0519 .0285 -0235 .0225

.I117

.I369

.0762 ,0746 .oj68 .0976 .1182 .1161 .0886 .0673 .I311 .1131

.0915 -0747 .0470

.1z98

.0689

.0640 *o419 *0325 .0360 .rzo4 .0867 -0499 J3.571 .o514 .083 I .I 067 .0726 .0664 .0862 .083 I .075 I

.0522

.0735

.041z

290 290 317 314 371 162 152 161 181 337 98 90 114 205 178 69 72 120 IZI 114 60 89 84 91 75 66 65 63 64 56 557 480 511 489 484 310 287 309 295 615

162 138 289 313 238 IZI 226 218 203 179 646 561 631 609 608 431 395 388 355 704 332 285 263 563 496

768 626 637 584 677 526 467 432 470 768 412 323 367 654 623 840 733 740 694 799 647 553 542 556 811

221 193 188 416 392

2.51 196 391 423 350

287 290 I45 152

84 90 57

38 29 28 499 480 266 287 I82 188 '33 138

99 554 561 357 384 271 284

194 636 609 430 467

323 707 733

552

316 I 60

52

27 511

309 I88 126 94 63 I

388 260 189 636

36

43 2 329 739 542

314

89 52 39 29 489 295 I82

136

609 91

352 272 189 5 84 422

340 694 476

I47 86 57 39 30 484

198 300

12.5

95

397 260 185 607 469 330 685 519

603

.301 -149 .087 .055 .03 9 .029 .494 .293 .186 .131

.095

.593

.374

.267

.19r

.622 4 4 5 .327 .712 .521

747 781 695 726 .736

Scand. J. Psychol., VoZ. r. 1960

80 HANNES EISLER

TABLE 4. Intra- and inter-individual standard deviations, experimental similarity estimations and theoretical similarity of heaviness computed according to Equation ( I a ) .

,1277 .0867 .0670 4542 .oz81 -294 .1181 .I152

.0819

.0733

.0469

-1137

so593 e504

-044.5 -13.59 .I356 .o718 .0826 .0671

.0989

.0467

-5317 .3029 .I731 . I 148 .0813 .0669 .7860 -4750 .32I3 .2198 ,1565

.463

.259

. I 60

. I 04 -075 .056 .661 -453 .314 -232

*I74

.I 246

. I 027

.0885

.0800

.o91 I

.1086

.I I32 ,0801 .0995 .0678

.0987

.1421

.lo71

.0677

.1223

.I313

.0676

.0982

.06&

* 1033

.8360

.6002 -4052 .32I9 3340 .6948 .5494 .go12

.8988

.7498

.745

.544

.421

.321

.767

.616

.493

.832

.685 -848

APPENDIX I1

Proof that a similarity matrix cannot be a ratio matrix as well In section C the problem was put forward as to whether a similarity matrix, conforming to

Equation (I), can at the same time be a ratio matrix, so that, depending on the instructions to the subjects, two different magnitude scales for the same continuum could be obtained. I t will be proved below that two ratio matrices with the elements qij and s i j are incompatible, if

sij=Li/Lj

qfj = &I% where R and L denote scale values of the two magnitude scales.

Formula (I) can be transformed in the following way:

2 sij = ~

I + I / q i j J

sij

2 - sij yielding Qii = -*

We have further

From the last three formlas, we obtain

Scad. J. PsychoL, VoZ. I. 1960

SIMILARITY IN THE CONTINUUM OF HEAVINESS

and

Substituting (I I) for (12) and rearranging, we obtain the equation

If both si, and qU constituted elements in ratio matrices, the left side of Equation (13) ought to be identically zero. As this obviously is not the case, the two ratio matrices are incompatible.

T h i s investigation was supported by a grant from the Swedish State Institute of Psycho- logy and Education.

REFERENCES

ATTNEAVE, F. (1950). Dimensions of similarity. Amer. J. Psychol., 63, 516-556.

BAKER, K. E., and DUDEK, F. J. (1955). Weight scales from ratio judgements and comparisons of existing weight scales. 3 . exp. Psychol., 50,

BERGMAN, G., and SPENCE, K. W. (1944). The logic of psychophysical measurement. Psychol. Rev., 51, 1-24.

BUSH, R. R., and MOSTELLER, F. (1951). A model for stimulus generalization and discrimination.

COMREY, A. L. (1950). A proposed method for absolute ratio scaling. Psychometrika, 15, 3 17- 325.

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

EKMAN, G. (1958). Two generalized ratio scaling methods. J . Psychol., 45, 287-295.

EKMAN, G., EERGSTROM, B., and KONNAPAS, T. M. ( I 956). A comparison between two psychophysical scaling methods. [Rep. Psychol. Lab. Univ. Stockholm. No 37.1

EKMAN, G., GOUDE, G., and WAERN, Y. (1960).

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PSYC~ZO~. Rev., 58, 413-423.

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Subjective similarity in two perceptual continua. J . exp. Psychol. (To be published.)

FLOURNOY, TH. (1894). De l'influence de la perception visuelle des corps sur leur poids apparent. Annie psychol., X, 198-208.

GUILFORD, J. P. (1954). Psychometric Methods. 2nd ed. New York: McGraw-Hill.

HARPER, R. S., and STEVENS, S. S. (1948). A psychological scale of weight and a formula foritsderivation. Amer.3. Psychol, 61,343-351.

HELSON, H. (1947). Adaptation-level as a frame of reference for prediction of psychological data. Amer. J . Psychol., 60, 1-29.

LINDQUIST, E. F. (1953). Design and AnaZysis of Experiment3 in Psychology and Education. New York Mifflin.

NOBLE, C. E. (1957). Psychology and the logic of similarity. J. gen. Psychol., 57, 23-43..

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

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

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SIMILARITY IN THE CONTINUUM OF HEAVINESS WITH SOME METHODOLOGICAL AND THEORETICAL CONSIDERATIONS