Multidimensional similarity: An experimental and theoretical comparison of vector, distance, and set theoretical models II. Multidimensional analyses: The subjective space

Acta Psychologica 41 (1977) 335-363 0 North-Holland Publishing Company

MULTIDIMENSIONAL SIMILARITY: AN EXPERIMENTAL

AND THEORETICAL COMPARISON OF VECTOR, DISTANCE,

AND SET THEORETICAL MODELS*

II. MULTIDIMENSIONAL ANALYSES: THE SUBJECTIVE

SPACE

Hannes EISLER** and Edward E. ROSKAM University of Stockholm, Sweden and University of Nijmegen, Holland

Received September 1976

Based on three kinds of judgments of the same stimulus pairs, viz. similarity ratings, multidimensional ratio ratings, and commonality ratio ratings, two mathematical models connecting these three data sets are compared and contrasted. One is in the spirit of Ekman’s content approach (E-model), the other in terms of a connection between set measures and distance metric (R-model). Whereas in Part I of this study the internal and mutual consistency of these data sets were investigated, the present paper deals with the multidimensional representation of the perceptual space.

The E-model uses component analysis of scalar products to arrive at the

subjective space; the R-model conceives of the negative logarithm of similarity as a city-block distance and applies multidimensional scaling algorithms. Surprisingly, two fundamentally different hut equally interpretable configurations were obtained. Both models provide a satisfactory fit to the data. hut the results are somewhat more favorable for the R-model. Also, whereas the E-model appears to require four dimensions, the R-model does not seem to need more than three, and two would appear to he sufficient in several

respects. Both multidimensional analyses were validated externally by satisfactory postdiction of independently collected data on the same stimuli (magnitude estimates of certain attributes of the stimuli), as well as of similarity judgments on another set of stimuli of the same kind. The results also sustained the overall coherence of the models. Taking Part I and Part II of this study together, it seems as if no definite conclusion can be drawn as to the overall superiority of either model.

* This work has been supported by the Swedish Council for Social Science Research.

** Requests for reprints should be sent to Hannes Eisler, Department of Psychology, University of Stockholm, Box 6706, S-l 13 85 Stockholm, Sweden.

0. Introduction

In part I of this study (Eisler and Roskam I9771 a series of experiments was described, in which three kinds of qLIalltitative judgments (ratings) were obtained in accordance with different instructions to the subjects for the same stimuli - patterns of pairs of ii~n~inous points in a plane - viz. similarity ratings (S-data), commonality ratio ratings (C-data), and InLlltidimensionaI ratio ratings (Q-data). Two different models, named E- and R-model, were presented, and a number of relations between these three data sets were studied. The data slightly favored the E-model.

The main assumption behind the attempt to relate these different data sets by modeIs is that the stimuli are represented in an invariant subjective perceptual space, and that different instructions give rise to different processings of the points (coordinates) of this space, leading to the different ratings obtained in the experiment. The present Part II of the study deals with this perceptual space. Each of the two models allows a multidimensional analysis. The E-model permits the construction of a scalar-product matrix, from which the coordinates of the space are obtained by component analysis. The R-model derives a city- block metric where the distances are the negative logarithms of the similarities. A more detailed description is given in sections 2 and 3, respectively. To confirm the results, two validation procedures are used. The first is based on a comparison between the outcome of the multidimensional analyses in terms of factors or dimensions, and the stimulus attributes as perceived - i.e. rated - directly. The pertinent experiment, mentioned briefly in Part I, is described in section 1. Section 4 deals with the other validation procedure. In another experiment. using stimuli of the same type as in the main experimental series, a study is made of how well similarity ratings for these new stimuli can be predicted from the configurations obtained in the main experiments. Finally, section 5 contains a general discussion of the results of both Part I and Part II.

1. Estimation of length and angle

Throughout the main experimental series 15 stimuli were used. Each stimulus consisted of a pair of luminous points shown on a vertical

H. Eisler. E. E. Roskamhfultidimensional similarity, II 331

A C G n a

B F J N

R E I n

0 H L

0 50 100

Fig. 1. The stimulus (A) and percept (B and C) spaces. The percept spaces are constructed from

length and angle ratings. A: Each stimulus was made up of two points. The lower left is not

indicated in the figure and has the x coordinate of A and the ,V coordinate of D, and can be

regarded as the origin. The second point of each stimulus is situated at the position of the

corresponding letter (A-O). B: The perceptual space when the length and angle ratings are

plotted as polar coordinates, implying that the subjective attributes are extensions in the x and y directions, i.e. the subjective x and ): coordinates. C: The perceptual space when length and

angle ratings are plotted as Cartesian coordinates, implying that the subjective attributes are

length and angle.

display in a dark room. The lower left point was always the same and could be conceived of as the origin in a coordinate system. The other point could be in any of 15 positions in the first quadrant, so that the 16 points were evenly distributed over a square. The stimuli are described in more detail in fig. 2 of Part I, together with their notation, the letters A-O. Fig. 1A in the present report likewise shows the positions of the stimuli; the fixed point (lower left) is omitted, and the letter shows the position of the variable second point. Since in all rating

338 fi. Eisler, E. E. Roskarn/Multidimcnsional similarity, II

tasks two stimuli were required, the display always showed two pairs of points side by side (cf. fig. 1 in Part I).

In order to study to what extent stimulus attributes inferred from the multidimensional analyses would agree with the corresponding attributes perceived directly, ratings were made of lengths and angles of the stimuli.

When length was to be estimated, stimulus 0 was used as the standard. The instruction ran: “The length that is indicated by the two bright points in one of the squares is always to be considered the standard. The two bright points in the other square constitute the variable. Your task is to estimate the length of the variable as a percentage of the length of the standard.”

In the angle-rating experiment, stimulus C was the standard. The instruction read: “The line that is indicated by the two bright points is to be regarded as one side of an angle, the other side being an imaginary horizontal line. The angle in one square is always to be regarded as the standard. Your task is to estimate the variable angle as a percentage of

the standard.” The same 15 subjects who had carried out the multidimensional ratio

ratings served as subjects in these tasks. Each subject made four ratings of each length and four ratings of each angle. The means of the 60 values thus obtained are given in table 1. These data were briefly mentioned as M-data in Part I.

Depending on whether extension in the horizontal and vertical directions, or length and angle are regarded as the subjective attributes, two different two-dimensional configurations can be obtained from these data. They are given in fig. 1B and lC, respectively. In fig. 1 B the estimates are plotted as polar coordinates (the angle ratings were multiplied by 0.9 to obtain degrees). In this plot the subjective attributes are conceived of as X- and ?:-coordinates, so to speak, thereby yielding a close replica of the stimulus arrangement. The difference in extension in the .Y- and y-directions, embedding the percepts in a rectangle rather than in a square, is probably due to the vertical-horizontal illusion (see, e.g., Kiinnapas 1957). In fig. 1 C the length and angle ratings are plotted as Cartesian coordinates rather than as the polar ones in fig. 1 B, which implies that length and angle (as rated) are conceived of

as the subjective attributes. Contrary to fig. lB, the ratio between the units of the two axes is arbitrary; we have chosen one in good agreement with the outcome of the multidimensional analysis according to the R-model.

H. Eisler, E. E. Roskam/Multidimensional similarity, II 339

C

340

2. The vector model

II. Eisler. 6 I?. Roskam/Muitidimensiorzal similarity, II

(E-model)

The E-model is a modification of Ekman’s content model (see, e.g., Ekman and Sjijberg 1965), and, as was shown in Part I, builds on the equivalence between a set theoretical and a vector representation. It agrees with Ekman’s model in the general feature that the perceptual

space is obtained by component analysis of a scalar-product matrix B with the elements bii:

(2.1)

(2.2)

where Vi is the vector length of percept 1 corresponding to stimulus i in the r-dimensional subjective space, ~ii is the angle between the two vectors representing percepts I and J, and xip is the coordinate of percept I in dimension 11.

In Ekman’s (1963) original model the vector lengths and cosines were obtained from the multidimensional ratio estimates, or, alter- natively, for the case of constant vector lengths (Vi = vi) from the similarities. However, as was shown in Part I, section 3, according to the E-model both the C- and the Q-data allow an estimation of the vector lengths or percept measures. These vector lengths agree within a power transformation, and are denoted by ~Zi when derived from Q-data, and by nz(O (or pi) when derived from C-data. Also the cosines can be derived from both data sets, yielding somewhat different values, denoted by wii and Gii when derived from the Q- and the C-data, respectively.

For clarification, let us summarize first the basic equations ex- pressing the structure of each kind of judgment, as they were derived in Part I. The lefthand sides of the following three model equations represent the theoretical Q-, S-, and C-values; m(I) is the measure of percept 1 (regarded as set).

(2.3)

sii = (2.4)


(3.5)

Note that Gii is defined as the larger commonality ratio, and that the relations between the three kinds of judgment were established by assuming’

(2.6)

hi - p,y. (2.7)

The B matrix can be constructed from bii = vivjcosrPq, see eq. 2.1. Before going into how the v and cos@ values are determined it should be noted that in Part I a distinction was made between observed (experimental) values, and values obtained by using internal consistency

requirements for the Q- and C-data applying a logarithmic least-squares procedure to derive estimates 0 and ?. Here only these log-least-squares fitted values are used throughout and therefore the ^ signs are omitted.

As mentioned before, the vector length can be derived from the Q-data according to (5 denoting estimation)

“, = ,2 & nj=n2 (1’2n) , I I ( 1 j=l qji

as well as from the C-data:

j=n 6-Q

i. 1 (I/n)

vi=/.+; n - ) j=l cji

where IZ is the number of stimuli. Likewise, the cosine can be computed from the Q-data as

coscjii = oii s k&g

(2.8)

(2.9)

(2.10)

’ In order to simplify expressions in the following, m(I) is denoted by pC As was shown in

Part 1 (e.g. table 21, pi, though numerically the same as m(I), is a denotation belonging to the R- rather than to the E-model.

342

and from the C-data as

COS$~~ = $ jj 2 max (cii, L)~).

H. Eider, E. E. Roskam jMultidimensional similarity, II

(3.1 1)

So there are altogether four possibilities:

I. bii = l~~l~~~~,. (both from Q-data);

II. bji = lzihj$ij (from Q-data and C-data);

III. bii = pipjwii (from C-data and Q-data);

IV. bij = pipjtiji (both from C-data).

Note that none of them involves estimation of the parameters (Y or y (eqs. 2.6 and 2.7). The constant h- in (2.10) is an under- or overestimation

parameter, as discussed in Part I, where its value was found to be approximated best by 0.93.

It is impossible to state a priori which of the four combinations for constructing the B matrix, and hence the perceptual space, is ‘correct’.

However, it was found in Part I that the similarity ratings (eq. 2.4) are best predicted when the vector lengths are taken from the Q-data and the cosines from the C-data; so we expect that this combination (case II above) will also yield the ‘best’ perceptual space.2

Component analyses were carried out with all four combinations, and two procedures were used to decide which B matrix, and which number of dimensions, is ‘best’.

The rationale of the first procedure is the following. The model equations 2.3 through 2.7 permit mutual prediction of one kind of data from another kind, e.g. one can predict ~ij (C-data) from ‘lij and qji (Q-data). But ~ji can be reproduced as well from the coordinates obtained from the component analysis of a B matrix for every dimensionality separately, for instance from the B matrix corresponding to

case IV, which is not constructed from Q-data but from C-data. We

* As can be seen from (2.4), it is possible to estimate $ii from the S-data and the Q-data by

$Q G Sij”i+h .)/ 2 min C$,h$. The calculations for cases II and IV wcrc also carried out with these values / or cos$p but since the results were almost identical to those with (2.10), they arc not dealt with !‘urther.

Note also that the assumption cos@u = $ij was incorporated in the presentation in Part I, e.g. in eqs. 3.10, 3.14, and 3.15, in anticipation of case II for construction of the scalar product matrix of the perceptual space.

II. Eisler, E. 15 Roskam/Multidimerxsional similarity, II 343

expect component analysis of this B matrix to give coordinates that reproduce the actual Gii values at least as well as the latter can be estimated from Q-data; a much closer reproduction may be taken as an indication that the component analysis is capitalizing on chance, that is that too many dimensions have been extracted. Therefore we should extract as many dimensions as are required for the two kinds of prediction to match as closely as possible.3

The second procedure seeks to predict the actual length ratings and angle ratings (as described in section 1) from the coordinates obtained by the component analyses of the B matrices. Again, the ‘best’ B matrix is the one which predicts these ratings best.

We shall now present these tests in detail.

2. 1. Compariso~z of mutual prediction UIKI reproduction from B

matrices

Let hi (i,j = 1, . . ., H) denote the reproduction of Dii as the scalar

product

based on the r first dimensions of the component analysis of a matrix B. To distinguish values predicted from another kind of data, the latter

are indexed with a double prime (“). Throughout this section we write

17, V, o, and $ (without ^) meaning the estimates as obtained by eqs. 2.8 through 2.1 1. The values used in the following for k, cy, and y are taken from Appendix B in Part I, which also gives the measures for goodness of fit for mutual prediction.

Cuse I. This is actually Ekman’s original model, except for the constant k We have, since

(2.12)

a The eigenvalues for the four B matrices were not of much help to decide the dimensionality

344 H. Eider, E. E. RoskamlMultidirnetlsional similarity, II

for the reproduction of (I;,. from the coordinates. (The summations go from 1 to r.)

According to the E-model, the y values can be obtained from the

C-data in two ways. The first one uses the c values directly:

where subscripts max and min indicate the pairs [(ii),($)] , respectively. The

measures p :

+; .

(2.13)

the maximum or minimum of other way uses the percept

(2.14)

Also the Q-data themselves (that is the values of 12, fitted to those data), can be used together with the S-data to estimate ~1;;:

. (2.15)

(Eqs. 2.13-2.15 above are given as eqs. 6.10, 6.11, 6.12 and 6.18 in Part I; the parameters cy, y and k were fitted by the method of least squares.)

The goodness of fit for eqs. 2.13-2.15 has been given in Part I; these measures4 were approximately 0.93, 0.92 and 0.99. The corresponding figure for eq. 2.12 is 0.92 when three dimensions are extracted, and 0.97 for four dimensions; for details, see table 2. This means that Case 1 and 4 dimensions cannot be rejected on the basis of these comparisons.

Case II. For this case we have, since

$Iii = hihj$ij “ihi’

(2.16)

4 The proportion of explained variance I/ = 1 - c(z - z’)‘/z(z - 5)’ (z given and Z’ estimated

values) was chosen as a measure of goodness of fit because the correlation coefficient would

incorrectly assume a linear function between the variables, see Part I, fn. 9.

H. Eider, E. b: Roskam/MuItidimensional similarity, II

and, for mutual prediction, from Q-data:

and from Q- and S-data combined:

q/j; = hi + hi

2 min (hi,hi) . sq .

34s

(2.17)

(2.18)

These predictions are to be tested against the values of the observed larger commonality ratio (cf. eqs. 6.3 and 6.21 in Part I).

The goodness of fit is 0.81 for eq. 2.17 and 0.91 for eq. 2.18; the corresponding value for the reproducibility from the component analysis (eq. 2.16) is 0.90 with four dimensions, and 0.95 with five. It appears that case 11 with four dimensions cannot be rejected.

Case 111. For this case eq. 3.12 is not appropriate, since the E-model assumes that h is a power function of p (with the exponent y = 1.45 taken from Part 1, Appendix B, eq. 6.12). Hence we have, since

0; 1 ~ __ = ~_ bj.ib~i k

(2.19)

For mutual prediction, the same equations as for case I apply here. The goodness of fit for reprodllcin, I,, 0 L ..‘s from the coordinates is not higher

than 0.91 for as many as five dimensions, indicating that case III should be rejected.

Cuse I I/. For this case the same equation, (2.16), as for case II applies for the reproduction of the $’ values from the coordinates, and its fit should be compared, again as for case II, by using eqs. 2.17 and 2.18 for

mutual prediction. The results indicate that case IV and four dimensions cannot be rejected.

In summary (see table 2), we conclude that Case III (COSbij from Q-data and 11~ from C-data) must be discarded, because the proportion

346 li. iz’isler, 15 E. Roskam/Multidimensional similarity, II

Table 2

Proportions of explained variance in + (left column) and in 4 (right column) from component

analysis in two to five dimensions (upper parts) and from mutual prediction (lower part). Equations used are given in parenthesis. Measure of goodness of fit described in footnote 4.

Dimensionality

2

3 vi ; hi

4

5

COSOij = ~ij COS~ij ’ 0.93 ~

Case II Case I

_b 0.264

(2.16) 1

0.720

0.902 (2.12)

1

0.923

0.975

0.948 0.984

(2.16)

YI”

0.762

0.907

0.955

(2.19)

Case III

0.629

0.870

0.909

0.912

Mutual

predictions”

(2.17)c 0.812 (2.131d 0.929

(2.181c 0.907 (2.141d 0.924

(2.151e 0.987

a Goodness-of-fit values taken from Appendix B in Part I, except for eq. 2.13, where the value

is taken from table 4 in Part I.

b Standard error of estimate exceeds variance.

’ Using Q-data only.

d Using C-data only.

e Using Q-data and S-data.

of explained variance for this combination is below the values for mutual prediction, even for as many as five dimensions. Each of the other three cases shows that four dimensions give the best agreement, in terms of explained variance between variables computed from the coordinates and from mutual prediction. The choice between them has to rely on the second procedure, to be described next.

2.2. Prediction ofletzgth and angle ratings from coordinates

This procedure consists in rotating the coordinate matrices for two to five dimensions so that the first two dimensions agree as well as possible with the estimated lengths and angles in the least-squares sense, The lengths were calculated as the vector lengths

I-I. Eisler, E. E. Roskam/Multidimensional similarity. II 341

Table 3

Proportion of explained variance in estimated length and angle according to eq. 2.20.

Dimensionality COS~ij = ~ ij cosoij 2 0.93 6

vi&hi

Case II Case I

0.911 0.791

0.931 0.893

0.977 0.906

0.978 0.911

Case IV Case III

0.923 0.828

Vi’ Pi 0.94 1 0.893

0.968 0.944

0.980 0.947

and the angles in degrees as Bi = arctg(.xi2/xil ), where the second subscript in the coordinates refers to the rotated dimensions. Denoting observed values, i.e. the ratings, by o, the following sum of squares was minimized:

Ci [(a ’ oli - b.Zi)’ + (0.9 00i - 0i)2 1. (2.20)

The constant a was determined so that the variance of a . J equals the variance of 0.9 . oB, which makes length and angle contribute equally to the variance. The constant b = Ci (a . oli) Ii/Xi (Zi)’ relates the units of the lengths as rated and as computed as vectors from the coordinates and constitutes the least-squares solution. A computer program including Chandler’s (1969) STEPIT was used to obtain the ratational transformation matrix that minimizes expression (2.20).

Table 3 gives the proportion of explained variance computed from (2.20) for all four B matrices and two to five dimensions. Also these data indicate that the correct dimensionality is four, because the fit improves up to the fourth dimension, but not appreciably from the fourth to the fifth. Furthermore, it is quite clear that the fit is rather bad when ~0s~~~ is taken from the Q-data; of the remaining two possibilities the one with the vector lengths deriving from the Q-data (case II) gives the best fit for the four dimensions required. These are the same variables that enter into the similarity equation 2.4, in agreement with our expectations.

Fig. 2 shows the subjective space for the best combination of vari-

348 H. Eider, E. ~5’. RosliamlMultidirnensional simi1arit.v. II

A K

3.0- c

c 0

2.5 -

N

2.0- 6

F J

1.5 -

E I H

l.O- &q

0.5 -

L H , 0,

0 0.5 1.0 I.5 2.0 I

I.5

N

1.C

0.5 I-J J

B 0

K

r, 0.0 -n

I E

F

-0.5 - DFI C

L

-1.0 B"

-1.0 -0.5 0.0 0.5 I.0 1.5

Fig. 2. The subjective space according to the E-model. The coordinates are obtained by

component analysis of the scalar product matrix with vector lengths 11 and cosines of angles $

and rotating so that the first two dimensions (panel A) agree as well as possible with the length

and angle ratings. Panel B shows dimensions 3 and 4 after rotation to a criterion which

attempted to keep the ‘frame’ of the space (points A, B, C, D, H, L) together.

ables (case II) after rotation to best fit. In fig. 2A the second dimension is plotted against the first, and we can see the close agreement with fig. 1 B. The vertical-horizontal illusion is still stronger. Note that even

such a feature as the slight upward slope of the row A-E-I-M is recovered in the subjective space. On the other hand, column A-B-C and row D-H-L are not quite orthogonal. Probably the cues for verticality and horizontality were stronger in the angle-rating task than in the other

rating tasks, or, to express the same idea differently, the subjects committed the stimulus error in the angle-rating task. and only then.

Fig. 2B shows the remaining two dimensions after rotation plotted VS. each other. The picture is not quite clear, but the points show a

tendency to form polytomous qualitative dimensions, similar to what Torgerson (196.5) found for certain data. The fourth dimension is dichotomous: the ‘ceiling’ of the pattern, comprising the points C, G, K, and 0, is separated from the other points. The third dimension is trichotomous and somewhat similar to the third dimension in the R-model: (1) the horizontal and the vertical ‘frame’, comprising the points A, B, C, D, H, L, (2) the ‘corner’ K, N, 0, and (3) the remaining points in between.

An interesting, and, perhaps, surprising feature of dimensions 3 and 4

H. Eisler, E. E. Roskam/Multidimensional SimilariT, II

COMPUTED LENGTH COMPUTE0 ANiLE

Fig. 3. Ratings of length and angle plotted against length and angle computed from the

coordinates of the first two dimensions of the E-model, see fig. 2A.

are the negative loadings. If this result holds it would mean that the introduction of a new dimension may also decrease the similarity of two percepts, which seems rather reasonable. The conception gets some support from the work of Lundberg and Devine (1975) on negative similarity.

Fig. 3 shows rated length and angle vs. length and angle computed from the coordinates of the first two dimensions. The fit for length is very good: so is the one for angle except for 0” and 90”. As we shall see in similar plots from the R-model (figs. 5 and 71, there too the largest deviations are found for these angles. This agrees with what one might expect from the previously mentioned stimulus error for these angles.

3. The distance model (R-model)

3.1. Multidimensional scaling of similarities

Whereas the vector model dealt with in the preceding section uses C- and Q-data to arrive at the subjective space, the distance model con-

sidered here employs S-data. The distances can be seen as some un- specified monotonic decreasing function of the similarities, requiring a ‘nonmetric’ multidimensional scaling algorithm, or a function can be specified, allowing a ‘metric’ method. Also, different values of the Minkowski parameter may be used. The stress values for the various

350 H. Eisler, 6 E RoskamlMultidirnensiorlal similarity, II

Table 4

Stress values for the multidimensional scaling analyses.

Distance

Metric Euclidean City block

(Minkowski 2) (Minkowski 1)

Dimensionality

2 3 I 2 3 I

Monotonic decreasing

with s.. 11

0.107

I

0.031 1 0.090 0.020

dii = c _ ~ii 0.136 0.059 0.141 0.073

dii = - In(csii) 0.129 0.077 1 0.116 0.055

Note. The nonmetric analyses (first row) were carried out with MINISSA, the metric with

MRSCAL, xe Roskam (1972).

analyses are given in table 4. A closer description of these analyses, together with the subjective spaces obtained, is found in Roskam (19721, who used MINISSA for the nonmetric and MRSCAL for the metric analyses.

The R-model derives a city-block metric for distances ulii defined as the negative logarithm of similarities sij:

p=r

dij = - In (c.s$) = 2 p = 1 I xip - sip I . (3.1)

where X is an arbitrary and thus dispensable scale parameter of the coordinate space, and Y is the dimensionality of the space: i and /’ refer to the two stimuli in a pair, and c is a normalizing constant of proportionality comparable to the additive constant in multidimensional scaling which transforms interval-scale data irlto ratio-scale distances (cf., e.g., Torgerson 1958). In this case, the introduction of such a normalizing constant requires some further justification. (First, it should be noted that this constant does not reflect the free power parameter in the models for the similarity data and for the commonality data, as discussed in section 4 of Part I). We have argued that the similarity scale used in the experiment (namely from zero for perfect lack of similarity to ten for perfect identity of a pair, sub- seqently converted into a zero-through-one scale) is not 174~ SC

If Eisler, E. E. Roskam JMultidimensional similarity, II 351

insensitive to response attitudes on the part of the subject: perfect identity is physically impossible, and even physically identical stimuli may be somewhat noisy perceptually, to the effect that stimuli which are identical by their definition in the experiment would not necessarily receive the maximal similarity rating. The normalizing constant, c, serves to shrink or stretch the similarity scale to compensate for this effect, and to permit zero distance between points of identical stimuli where the subject may have underrated the similarity, or may have felt a need to go beyond the upper limit of the similarity scale. In effect, a value of c = 0.83 was found to give the best fit for a two-dimensional solution, indicating a tendency on the part of the subjects to over- estimate the similarity, or, possibly, to ignore minor perceptual differences.

Table 4 confirms the distance model, and its particular formulation by the R-model, inasmuch as, for the metric analysis, application of eq. 3.1 gives the lowest stress, and the city-block metric is better than the Euclidean for the nonmetric analyses. (The fact that the stress is lower for the nonmetric than for the metric analyses is due to the lesser restraints.) In the following, only analyses based on eq. 3.1 will be dealt

with. Fig. 4 shows the two-dimensional MRSCAL solution. Clearly, the

two dimensions are length and angle, cf. fig. 1C. Like in the previous section, rated length and angle were fitted to the coordinates on the two dimensions. Since the zero points and the units of the coordinates are arbitrary, a best-fit linear tr ar.sformation of the coordinates was

Fig. 4. Second vs. first dimension of the two-dimensional MRSCAL

R-model.

analysis act ding to the

352 H. Eider, E. I:‘, RoskamlMultidimensional similarity, II

-0.5 0.0 0.5 I. OIWENSION 1 DIHENs~~N 2

Fig. 5. Ratings of length and angle plotted against the corrcspond:ng dimensions of the

two-dimensional MRSCAL analysis (R-model), set fig. 4.

Table 5 Proportion of explained variance for length and angle from the city-block analyses.

Nonmetric

Metric

Dimensionality

2

0.990

0.990

3

0.976

0.977

carried out, using the same principle as (2.20), but allowing three more parameters, viz. a scale unit for the angle dimension, and additive constants for both dimensions. The fit is shown in fig. 5. The corresponding calculations were also carried out with the three-dimensional solution.

The results are given in table 5. There is no difference between the nonmetric and the metric solutions. The two-dimensional solutions show a slightly better fit, which is surprising, since the ‘correct’ solution seems to be three-dimensional, see below. The explanation probably is that, with more dimensions, any error in the data (both random error and violations of the metric model) tends to show up in the configuration, thereby decreasing the stress partially by capitalizing on such errors, while the decrease of the stress cannot entirely be attributed to error. (In other words: a significant improvement of a configuration must not be taken as an improvement in every respect.)

The same data were also run with Eisler’s (1973) MCITYB algorithm, which allows for testing of dimensionality. The test indicated a three-

H. Eisler, E. E. Roskam/Multidimensional similarit)‘, II 353

A

A B C

F c K

E J 0 N

n

0 n L

6 I

M N E I

F J K 0

G

Ii * C

R L

0

Fig. 6. The three dimensions plotted pairwise against each other for the three-dimensional

MCITYB analysis according to the R-model.

0.0 0.2 0.4 0.6 0.8 I.0 DIMENSION 1

Fig. 7. Ratings of length and angle plotted against the first two dimensions of the three-

dimensional MCITYB analysis (R-model), see fig. 6.

dimensional solution (p < lo- 6 for improvement of fit in going from two to three dimensions). The stress for this solution was 0.053 and the proportion of explained variance in length + angle 0.987. Fig. 6 shows the three dimensions plotted pairwise against each other, and fig. 7 the

354 Il. Eider, E. E. RoskamlMultidimensional similarity, II

fit of the first two dimensions to length and angle. in the three- dimensional MCITYB solution, the scale unit of the second (angle) dimension is less compared with the first (length) dimension than it is in the two-dimensional MRSCAL solution. The third dimension is rather similar to the one described in section 3 (see fig. 2B). However. it seems to show only a dichotomy: the horizontal-vertical ‘frame versus the reJnaining points. Possibly the two points nearest the origin, A and D, can also be seen as separated from the rest.

Both these solutions as well as the solution from the vector model agree excellently with the length and angle ratings, with a proportion of explained variance between 0.98 and 0.99. Note that the coordinates of the vector model are ratio-scaled and those of the distance model interval- scaled, the latter thus requirin g three more parameters. On the other hand, the vector model required four dimensions, compared with three for the distance model, and five more parameters to achieve the best- fitting rotation.

3.2. Subjective space and set measwes

The assumed relation between similarity and distances in a metric perceptual space, which is obviously well sustained by the multi- dimeJlsioJlal analysis and by the close relation between two dimensions and magnitude judgments of the stimulus attributes (length and angle), provides only indirect corroboration of the validity of the R-model. As outlined in Part I, section 4, the distance model was derived by assuming a logarithmic relation between set measures, which can be

represented in a space with city-block metric, and similarity judgments. It follows that the set measure of each percept can be reproduced from the multidimensional analysis. In particular, the R-model assumed:

MZi = z: pXip 3

m 1nj = C, min (.yip,sjp ),

tniu i = C, max (.yip,,yi\;.p ),

and the following definition of similarity (writing p = exp(rn)):

(3.2)

(3.3)

(3.4)

(3.5)

Il. Eisler, 6 E. Rosh-amlMultidin~ensiona1 similarity, II 355

leading to the distance formula (3. I). In order to establish the relation with the commonality ratio ratings, the model for the latter was assumed to be

(3.6)

Now, as estimates of ln(pi) = mi were available from the analysis of the C-data, it is only natural that the multidimensional analysis should

permit of reproducing these estimates with a fair degree of approxima- tion, particularly so since the similarity judgments and the commonality ratio judgments fit each other very well, and both are highly consistent internally (the internal consistency of the similarity data can be concluded from the excellent fit of the multidimensional analysis, albeit by virtue of the assumption of the particular distance model).

Letting lyli stand for the estimate derived from the analysis of the

C-data, and taking into account the necessary fr, e parameters and free

choice of origin in the multidimensional analysis, we must reproduce /rzi

by, say

a C, (* sip + hp> = a C, (k xip) + B = rni, (3.7)

(allowing uniform dilatation, reflection, and translation of the metric space). Since the mi values were scaled such that their average equals zero, and since the space was centered at its origin (MRSCAL coordinates), we will find B trivially equal to zero when a least-squares fit of (3.7) is sought for. For some reflection of the coordinate space, we expect that the sum of the coordinates of a point is proportional to the comesponding m. The results of these analyses turned out to be mode-

rately satisfactory. The correlation between m and 171’ is 0.83 for the three-dimensional solution, and 0.77 for the two-dimensional solution (with the same orientation of the first two dimensions; with other reflections, these correlations were substantially lower), corresponding to a proportion of explained variance of 0.69 and 0.60, respectively. These correlations are poor compared to the very satisfactory fit between the distances and the logarithms of similarity data, and also compared to the high correlation between the coordinates of the scaling solutions and the length and angle judgments of the stimuli. On the other hand, if the model were false, we would not have expected that an increase in dimensionality would improve the correlation between 1y1 and m’. We

356 H. Eisler, 15 E. Roskam/Multidimensional similarity, II

may therefore look upon these results as corroborative evidence for the

assumed connections between set measures, multidimensional representation, similarity, and commonality as expressed in eqs. 3.1 through 3.6.

For the sake of completeness it should be mentioned that, following

Hays (1958) or Restle (1959), a city-block distance Dii can be defined in set theoretical terms as follows:

Dii = m(l) + 172(J) - 2m(mJ). (3.8)

The variables entering into eq. 3.8 were computed from the C-data.

The solutions after subjecting the Dii values to multidimensional scaling algorithms (both MRSCAL [city-block] and MCITYB were used) yielded a worse fit than both the E- and the R-models. The configura-

tion resembled most the E-model space. This outcome is perhaps not too surprising since the D-distances show a product-moment correlation of 0.88 with the Euclidean distances

derived from the variables used in the E-model. (The correlation between Dji and cEii according to eq. 3.1 amounts to 0.81.) Thus the interpretation of Dii as a city-block distance has to be rejected for our data.

4. Validation of the subjective spaces by another experiment

The experiment reported below was undertaken before the experiments described in Part I and in section 1 of Part II. It was designed to test a (faulty) hypothesis regarding similarity, using a similar type of stimuli, but rather than combining all with all to obtain a complete similarity matrix, the sample of points in the plane was denser.

4.1. Procedure

The stimuli were points in a plane, mimeographed as minute black spots (about half a millimeter in diameter) on white sheets (296 x 208 mm). The ‘origin’ of the left stimulus of a pair was 60 mm above the lower (and longer) edge of the sheet, and 60 mm to the right of the left (shorter) edge. The distance between the ‘origins’ of the left and right stimuli (point pairs), parallel with the lower edge, was 120 mm.

H. Eisler, E. E. RoskamjMultidimensional similarity, II

Table 6

351

Coordinates in cm of the right points (with the left points as origins) of the two stimuli constituting a pair together with empirical similarities.

Pair no. I

X

Stimulus

Y J

X Y

Rated similarity (average of 68 ratings)

I 5 0 0 4 0.469 2 0 3 10 1 0.127 3 4 8 1 0 0.087 4 1 6 9 0 0.298 5 0 1 3 9 0.121 6 10 2 1 3 0.197 I 8 1 2 9 0.252 8 1 2 1 6 0.305 9 2 5 1 1 0.390

10 10 10 2 4 0.399 11 6 9 1 0 0.198 12 6 2 3 I 0.373 13 8 4 3 8 0.502 14 8 3 4 I 0.533 15 5 4 9 I 0.583 16 9 5 5 1 0.623 17 2 6 5 5 0.691 18 6 3 8 2 0.698 19 6 8 4 9 0.832 20 0 10 4 10 0.685 21 9 9 1 1 0.326 22 8 8 2 2 0.438 23 1 I 3 3 0.497 24 6 6 10 10 0.641 25 5 5 4 4 0.809

There were 26 pairs of stimuli, each pair on a separate sheet. The coordinates of the right points with respect to their origins, the subsumed axes parallel to the edges of the sheet, are given in centimeters in table 6 for each pair. (The pair (7; 5), (7; 5) with the theoretical similarity of unity is not included. The mean rating was 0.947.)

The experiment was run as a group experiment and the 17 Ss were to write their similarity ratings on the sheets. Every S made four judgments of each pair, with reversed right-left position of the two stimuli constituting a pair for two of the judgments. Table 6 gives the arithmetic means of the 68 ratings per stimulus pair.

4.2. Data treatment

Given the coordinates of two stimuli in all dimensions of the subjective space, one can compute their theoretical similarity. For the E-model, the vector lengths and

358 H. Eisler, E. E. Roskam/Multidimensiorla[ similarity, If

the cosine of the angle between the vectors have to be calculated from the coordinates of the perceptual space, and entered into eq. 2.4. For the R-model the city-block

distances dii are computed as the sum of the intradimensional differences of the coordinates and the similarity si; is obtained from

- (a:tf..+h) Sij = e B (4.1)

where a and b are fitted parameters. To obtain the coordinates, a nomogram was constructed for each dimension

(and each subjective space), based on the fifteen points used in the main experiment. The physical scale was adjusted so that the squares from the main experiment

and the present experiments coincided. Fig. 8 gives an example. The desired coordinates were obtained by interpolating.

For tJ,e E-model, the computed values were almost throughout too large for all dimensic nalities. However, introducing a scale factor Cc~s/Cs*, where es is empirical and s computed similarity, improved the fit considerably. This scale factor was 0.76, 0.77, and 0.79 for two, three, and four dimensions, respectively, equivalent to multiplying the empirical similarity ratings by a scale factor of 1.3 (appr.). Table 7 gives the proportions of explained variance.

For the R-model, the parameters a and b were fitted by the method of least squares after taking the logarithm of eq. 4.1. Again, table 7 gives the fraction of explained variance of the similarity ratings (not of their logs). For three dimensions,

CONTtNT HOOEL ROTF(TEtl DItlEN510NS

OIHENSI~N 2

NR I J

x Y x Y

Fig. 8. An example of the nomograms used to obtain subjective coordinates for the points of

the validation experiment. The + , +, T, etc. denote the points from the main experiment, the coordinates of which were used to construct the nomogram. The X denote the points of the validation experiment, the coordinates of which were determined by means of the nomogram.

H. Eisler, 6: 6: Ros~amlMultidirnensional similarity, II 359

Table I Proportion of explained variance.

Model Algorithm Number of dimensions 2 3 4

E-model Component analysis 0.827 0.839 0.838

R-model MRSCAL

MCITY B 0.882

0.857 0.898

the best-fitting value for b was found to be - 0.038 (and for a 1.07). This implies a multiplicative scale factor of 0.96 for the similarity judgments.

4.3. Discussion

Bearing in mind the coarseness of the method used to determine the coordinates, and the differences in the experimental settings, the agreement is astonishingly good.’ The slight superiority of the R-model is probably at least partly attributable to the use of one more free parameter, so again it proved impossible to decide between the models. Note that increasing the dimensionality beyond three does not improve the fit of the E-model. This finding probably indicates that the fourth dimension is tied to the particular experiment; presumably points on a paper are experienced somewhat differently from luminous points in a dark room. The unexplained lo-10% of the variance need not he due to experimental or cal- culating errors or flaws in theory; the Ss may have experienced some other

unknown dimension(s) in this experiment.

5. Discussion and conclusion

This section is mostly devoted to a comparison between the two models, the E- and the R-model. Section 5.1 gives a comparison between the two subjective spaces with some more general remarks on vector- and distance models. Section 5.2 will attempt an overall comparison, including the findings in Part I of this report.

’ An attempt to weight the dimensions differently in the vein of Eisler and Knoppel (1970, fig.

4) orcarrolland Chang (1970), using Chandler’s (1969) routine STEPIT improved the fit only

negligibly.

360 H. Eisler, E. E. Roshzm/Multidimensional similarity, II

5.1. A conzparisorz between the nzultidilrlensiorzal analyses according to the E- ard R-r?lotlels

In comparing the two models, and the perceptual spaces to which they give rise, two aspects must be kept apart: (1) the correctness of the models, both in terms of meaningfulness (interpretability) of the subjective spaces and in terms of fit, and (2) the expediency of either model. Sjijberg (1975) gives a very illuminating overview of the content and distance models in general, listing their advantages and disadvan- tages, but he does not seem to keep the two aspects as clearly apart as one would like.

The usual finding is that when a vector model with its component analysis, and a distance model with any of the multidimensional scaling algorithms are applied to the same data set, some of the bipolar dimensions found in the distance model are represented as two orthogonal dimensions in the vector- (content-) model. An example is Ekman’s (1954) color study, where he obtained five factors by component analysis, whereas Shepard’s (1962) reanalysis using multidimensional scaling (Euclidean metric) yielded the familiar color circle, i.e. two bipolar dimensions. Even if this latter representation is more parsimonious, requiring only two dimensions, the two poles of a bipolar dimension are not always as clearly opposites as in the color case (cf. Ekman 1970). A necessary condition for meaningful replacement of

two orthogonal dimensions by a single bipolar one is that the space with two dimensions cannot be filled, so that ‘more’ of one dimension necessarily implies ‘less’ of the other. But even if this is the case, it may still be a moot question whether the more parsimonious representation also is the more valid. What we have called ‘external testing’ in Part I might be invoked to make the final choice.

One of the most exciting findings in the present analyses is that our data are at variance with the usual results. The first two dimensions we obtain from the two models cannot be transformed into each other by either rotation or replacing the orthogonal dimensions of the vector model by a single bipolar dimension. This result may be due to the particular stimuli used in the present experiment. Furthermore, these different two first dimensions seem to be equally meaningful and interpretable, since it seems equally reasonable that the subjects attend to degree of horizontal and vertical extension (the E-model) as to length and angle (the R-model). Accordingly, as to meaningfulness of


the first two dimensions, the superiority of either model cannot be established.

As to the remaining dimensions, the E-mode! requires four and the R-model only three dimensions altogether. Both third dimensions can be interpreted, though not as self-evidently as the first two. Also the fourth dimension of the E-model is interpretable, but still less clear-cut.

There are three ways in which the fit may be compared: (a) the goodness of fit when the startin g variables are reproduced from the coordinates, (b) the goodness of fit of rated length and angle when compared to length and angle computed from the coordinates, and (c) how well the similarities of the ‘validation experiment’ can be predicted from the coordinates of the subjective spaces. However, these comparisons are difficult to make because the number of parameters differs for the two models, due to the fact that the E-model yields ratio- scaled and the R-model interval-scaled coordinates, and that the E-model space requires rotation, whereas the city-block space of the

R-model is fixed. The E-model is consistent within itself inasmuch as mutual predic-

tion between variables yields the same proportion of explained variance as the prediction of the same variables from the coordinates for four dimensions, about 0.90. The goodness of fit for the R-model is measured in terms of stress, and for three dimensions 0.055 is obtained. The MCITYB algorithm computes the multiple correlation R2, yielding a proportion of explained variance of 0.98. As already pointed out it is difficult to make a just comparison. The E-model requires one more dimension, but has a fixed origin, and is not as free in the scale units as the R-model, permitting the latter more capitalization on chance. Although 0.98 does indicate a much better fit, also 0.90 seems to be acceptable.

Comparing the proportion of explained variance of length + angle ratings, the E-model yields 0.977 (four dimensions) and the R-model 0.977 for three and 0.990 for two dimensions. Again the result is difficult to judge. The R-model employs more parameters in the fitting of the ratings, but the space of the E-model has to be rotated to the best fit, which can also be seen as making use of a number of free parameters. The fit to both models may be described as excellent.

Also when the two models are compared for their ability to predict the similarity ratings from the validation experiment, no clear decision can be made. Taking three dimensions into account for both models,

362 II. Eisler, E. E. Roskam/Multidimensional similarity, II

we obtain a proportion of explained variance of 0.84 for the E-, and of 0.90 for the R-model. For these data, however, the R-model has a certain advantage: the E-model requires multiplication of the empirical similarity ratings with a factor of about 1.3, whereas the corresponding factor for the R-model is 0.96.

All in all, no firm decision can be made as to the superiority of either model in terms of ‘correctness’. The fit according to all criteria is more than satisfactory for both. It is, however, slightly better throughout for the R-model, so that the data for the perceptual space seem to give the R-model a slight edge.

As to expediency, the R-model is clearly to be preferred for the following two reasons: (1) The R-model requires only S-data, which are easily collected, whereas the E-model builds on C- and Q-data, the collection of which makes greater demands on the subjects; (2) the subjective space of the R-model is fixed (city-block), whereas the

E-model requires rotation. It should, however, be mentioned that the E-model yields ratio-scaled coordinates, which is preferable to Ihe

interval-scaled coordinates of the R-model, since the higher scale level contains more information.

5.2. A brief general comparisorz between the two models

In Part I of the present report it was shown that in terms of the relation between the three data sets, C-data, Q-data, and S-data, the E-model was slightly superior. For the relations where the two models differed, the fit was slightly better for the E-model, and this model also required one less parameter. The study of the perceptual space, on the other hand, given in the present Part II of the study, demonstrated a slight superiority of the R-model. Accordingly, it must be concluded that the present data do not establish the superiority of either model. In part, this outcome may be attributable to the particular stimuli chosen for the present series of experiments. Hopefully, a different type of stimuli might yield a decision.

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Documents

Multidimensional similarity: An experimental and theoretical comparison of vector, distance, and set theoretical models II. Multidimensional analyses: The subjective space