Features of Similarity

Amos Tversky Hebrew University Jerusalem, Israel

 

The metric and dimensional assumptions that underlie the geometric represen­tation of similarity are questioned on both theoretical and empirical grounds. A new set-theoretical approach to similarity is developed in which objects are represented as collections of features, and similarity is described as a feature-­matching process. Specifically, a set of qualitative assumptions is shown to imply the contrast model, which expresses the similarity between objects as a linear combination of the measures of their common and distinctive features. Several predictions of the contrast model are tested in studies of similarity with both semantic and perceptual stimuli. The model is used to uncover, analyze, and explain a variety of empirical phenomena such as the role of common and distinctive features, the relations between judgments of similarity and differ­ence, the presence of asymmetric similarities, and the effects of context on judgments of similarity. The contrast model generalizes standard representations of similarity data in terms of clusters and trees. It is also used to analyze the relations of prototypicality and family resemblance.

 

Similarity plays a fundamental role in theories of knowledge and behavior. It serves as an organizing principle by which individuals classify objects, form concepts, and make gen­eralizations. Indeed, the concept of similarity is ubiquitous in psychological theory. It under­lies the accounts of stimulus and response generalization in learning, it is employed to explain errors in memory and patter recogni­tion, and it is central to the analysis of con­notative meaning.

 

Similarity or dissimilarity data appear in different forms: ratings of pairs, sorting of objects, communality between associations,

 

This paper benefited from fruitful discussions with Y. Cohen, 1. Gati, D. Kahneman, L. Sjoberg, and S. Sattath. Requests for reprints should be sent to Amos Tversky, Department of Psychology, Hebrew University, Jerusalem, Israel.

errors of substitution, and correlation between occurrences. Analyses of these data attempt to explain the observed similarity relations and to capture the underlying structure of the ob­jects under study.

The theoretical analysis of similarity rela­tions has been dominated by geometric models. These models represent objects as points in some coordinate space such that the observed dissimilarities between objects cor­respond to the metric distances between the respective points. Practically all analyses of proximity data have been metric in nature, although some (e.g., hierarchical clustering) yield tree-like structures rather than dimen­sionally organized spaces. However, most theoretical and empirical analyses of similarity assume that objects can be adequately repre­sented as points in some coordinate space and that dissimilarity behaves like a metric dis­tance function. Both dimensional and metric assumptions are open to question.

 

It has been argued by many authors that dimensional representations are appropriate for certain stimuli (e.g., colors, tones) but not for others. It seems more appropriate to repre­sent faces, countries, or personalities in terms of many qualitative features than in terms of a few quantitative dimensions. The assessment of similarity between such stimuli, therefore, may be better described as a comparison of features rather than as the computation of metric distance between points.

 

A metric distance function, b, is a scale that assigns to every pair of points a nonnegative number, called their distance, in accord with the following three axioms:

 

Minimality:

Symmetry

The triangle inequality:

b(a,b) + d(b,c) >_ d(a,c).

 

To evaluate the adequacy of the geometric approach, let us examine the validity of the metric axioms when b is regarded as a measure of dissimilarity. The minimality axiom implies that the similarity between an object and itself is the same for all objects. This assumption, however, does not hold for some similarity measures. For example, the probability of judging two identical stimuli as "same" rather than "different" is not constant for all stimuli. Moreover, in recognition experiments the off­diagonal entries often exceed the diagonal entries; that is, an object is identified as an­other object more frequently than it is identi­fied as itself. H identification probability is interpreted as a measure of similarity, then these observations violate mjnimality and are, therefore, incompatible with the distance model.

 

Similarity has been viewed by both philoso­phers and psychologists as a prime example of a symmetric relation. Indeed, the assumption of symmetry underlies essentially all theo­retical treatments of similarity.

 

Contrary to this tradition, the present paper provides empirical evidence for asymmetric similarities

b(a,b) >_ b(a,a) = 0.

E(a,b) = b(b,a).

and argues that similarity should not be treated as a symmetric relation.

 

Similarity judgments can be regarded as extensions of similarity statements, that is, statements of the form "a is like b." Such a statement is directional; it has a subject, a, and a referent, b, and it is not equivalent in general to the converse similarity statement "b is like a." In fact, the choice of subject and referent depends, at least in part, on the relative salience of the objects. We tend to select the more salient stimulus, or the proto­type, as a referent, and the less salient stimu­lus, or the variant, as a subject. We say "the portrait resembles the person" rather than "the person resembles the portrait." We say "the son resembles the father" rather than "the father resembles the son." We say "an ellipse is like a circle," not "a circle is like an ellipse," and we say "North Korea is like Red China" rather than "Red China is like North Korea."

 

As will be demonstrated later, this asym­metry in the choice of similarity statements is associated with asymmetry in judgments of similarity. Thus, the judged similarity of North Korea to Red China exceeds the judged similarity of Red China to North Korea. Like­wise, an ellipse is more similar to a circle than a circle is to an ellipse. Apparently, the direc­tion of asymmetry is determined by the relative salience of the stimuli; the variant is more similar to the prototype than vice versa.

 

The directionality and asymmetry of simi­larity relations are particularly noticeable in similies and metaphors. We say "Turks fight like tigers" and not "tigers fight like Turks." Since the tiger is renowned for its fighting spirit, it is used as the referent rather than the subject of the simile. The poet writes "my love is as deep as the ocean," not "the ocean is as deep as my love," because the ocean epitomizes depth. Sometimes both directions are used but they carry different meanings. "A man is like a tree" implies that man has roots; "a tree is like a man" implies that the tree has a life history. "Life is like a play" says that people play roles. "A play is like life" says that a play can capture the essential elements of human life. The relations between the interpretation of metaphors and the assessment of similarity are briefly discussed in the final section.

 

The triangle inequality differs from minimal­ity and symmetry in that it cannot be formu­lated in ordinal terms. It asserts that one distance must be smaller than the sum of two others, and hence it cannot be readily refuted with ordinal or even interval data. However, the triangle inequality implies that if a is quite similar to b, and b is quite similar to c, then a and c cannot be very dissimilar from each other. Thus, it sets a lower limit to the simi­larity between a and c in terms of the similari­ties between a and b and between b and c. The following example (based on William James) casts some doubts on the psychological validity of this assumption. Consider the simi­larity between countries: Jamaica is similar to Cuba (because of geographical proximity); Cuba is similar to Russia (because of their political affinity) ; but Jamaica and Russia are not similar at all.

 

This example shows that similarity, as one might expect, is not transitive. In addition, it suggests that the perceived distance of Jamaica to Russia exceeds the perceived distance of Jamaica to Cuba, plus that of Cuba to Russia -Contrary to the triangle inequality. Although such examples do not necessarily refute the triangle inequality, they indicate that it should not be accepted as a cornerstone of similarity models.

It should be noted that the metric axioms, by themselves, are very weak. They are satis­fied, for example, by letting b (a,b) = 0 if a = b, and d(a,b) = 1 if a Pd b. To specify the dis­tance function, additional assumptions are made (e.g., intradimensional subtractivity and interdimensional additivity) relating the di­mensional structure of the objects to their metric distances. For an axiomatic analysis and a critical discussion of these assumptions, see Beals, Krantz, and Tversky (1968), Krantz and Tversky (1975), and Tversky and Krantz (1970).

 

In conclusion, it appears that despite many fruitful applications (see e.g., Carroll & Wish, 1974; Shepard, 1974), the geometric approach to the analysis of similarity faces several difficulties. The applicability of the dimen­sional assumption is limited, and the metric axioms are questionable. Specifically, minimal­

ity is somewhat problematic, symmetry is ap­parently false, and the triangle inequality is hardly compelling.

The next section develops an alternative theoretical approach to similarity, based on feature matching, which is neither dimensional nor metric in nature. In subsequent sections this approach is used to uncover, analyze, and explain several empirical phenomena, such as the role of common and distinctive features, the relations between judgments of similarity and difference, the presence of asymmetric similarities, and the effects of context on simi­larity. Extensions and implications of the present development are discussed in the final section.

 

Feature Matching

Let A = (a,b,c,.. .) be the domain of objects (or stimuli) under study. Assume that each object in A is represented by a set of features or attributes, and let A,B,C denote the sets of features associated with the objects a,b,c, re­spectively. The features may correspond to components such as eyes or mouth; they may represent concrete properties such as size or color; and they may reflect abstract attributes such as quality or complexity. The character­ization of stimuli as feature sets has been employed in the analysis of many cognitive processes such as speech perception (Jakobson, Fant, & Halle, 1961), pattern recognition (Neisser, 1967), perceptual leaming (Gibson, 1969), preferential choice (Tversky, 1972), and semantic judgment (Smith, Shoben, & Rips, 1974).

 

Two preliminary comments regarding fea­ture representations are in order. First, it is important to note that our total data base concerning a particular object (e.g., a person, a country, or a piece of furniture) is generally rich in content and complex in form. It in­cludes appearance, function, relation to other objects, and any other property of the object that can be deduced from our general knowl­edge of the world. When faced with a particular task (e.g., identification or similarity assess­ment) we extract and compile from our data base a limited list of relevant features on the basis of which we perform the required task. Thus, the representation of an object as a col­

 

 

Figure I. A graphical illustration of the relation between two feature sets.

section of features is viewed as a product of a prior process of extraction and compilation. Second, the term feature usually denotes the value of a binary variable (e.g., voiced vs. voiceless consonants) or the value of a nominal variable (e.g., eye color). Feature representa­tions, however, are not restricted to binary or nominal variables; they are also applicable to ordinal or cardinal variables (i.e., dimensions). A series of tones that differ only in loudness, for example, could be represented as a sequence of nested sets where the feature set associated with each tone is included in the feature sets associated with louder tones. Such a represen­tation is isomorphic to a directional unidimen­sional structure. A nondirectional unidimen­sional structure (e.g., a series of tones that differ only in pitch) could be represented by a chain of overlapping sets. The set-theoretical representation of qualitative and quantitative dimensions has been investigated by Restle (1959).

 

Let s(a,b) be a measure of the similarity of a to b defined for all distinct a, b in A. The scale s is treated as an ordinal measure of similarity. That is, s(a,b) > s(c,d) means that a is more similar to b than c is to d. The present theory is based on the following assumptions.

 

1. Matching:

s(a,b) = F (A f1 B, A - B, B - A).

The similarity of a to b is expressed as a function F of three arguments: A(1 B, the features that are common to both a and b; A - B, the features that belong to a but not to b; B - A, the features that belong to b but not to a. A schematic illustration of these components is presented in Figure 1.

 

2. Monotonicity:

s(a,b) >- s(a,c)

whenever

An BDA()C, A-BCA-C,

and

B-ACC-A.

 

Moreover, the inequality is strict whenever either inclusion is proper.

 

That is, similarity increases with addition of common features and/or deletion of distinc­tive features (i.e., features that belong to one object but not to the other). The monotonicity axiom can be readily illustrated with block letters if we identify their features with the component (straight) lines. Under this as­sumption, E should be more similar to F than to I because E and F have more common features than E and I.

 

 Furthermore, I should be more similar to F than to E because I and F have fewer distinctive features than I and E.

 

Any function F satisfying Assumptions 1 and 2 is called a matching function. It measures the degree to which two objects-viewed as sets of features-match each other. In the present theory, the assessment of similarity is described as a feature-matching process. It is formulated, therefore, in terms of the set­theoretical notion of a matching function rather than in terms of the geometric concept of distance.

In order to determine the functional form of the matching function, additional assump­tions about the similarity ordering are intro­duced. The major assumption of the theory (independence) is presented next; the remain­ing assumptions and the proof of the represen­tation theorem are presented in the Appendix. Readers who are less interested in formal theory can skim or skip the following para­graphs up to the discussion of the representa­tion theorem.

 

Let t denote the set of all features associated with the objects of A, and let X,Y,Z,...etc. denote collections of features (i.e., subsets of 4~). The expression F(X,Y,Z) is defined when­ever there exists a, b in A such that A (1 B = X,

t1D ca

A - B = Y, and B - A = Z, whence s(a,b) = F(A (1 B, A - B, B - A) = F(X,Y,Z). Next, define V.-, W if one or more of the following hold for some X,Y,Z: F(V,Y,Z) = F(W,Y,Z), F(X,Y,Z) = F(X,W,Z), F(X,Y,V) = F(XYW).

The pairs (a,b) and (c,d) are said to agree on one, two, or three components, respec­tively, whenever one, two, or three of the following hold: (A n B) - (C (1 D), (A - B) - (C - D), (B - A) = (D - C).

3. Independence: Suppose the pairs (a,b) and (c,d), as well as the pairs (a',b') and (c',d'), agree on the same two components, while the pairs (a,b) and (a',b'), as well as the pairs (c,d) and (c',d'), agree on the remaining (third) component. Then

s(a,b) >_ s(a',b') iff s(c,d) >- s(c',d').

To illustrate the force of the independence axiom consider the stimuli presented in Figure 2, where

A n B - C n D = round profile = X, A' (l B' = C' fl D' = sharp profile = X', A - B = C - D = smiling mouth = Y, A' - B' = C' - D' = frowning mouth = Y', B - A = B' - A' = straight eyebrow = Z, D-C=D'-C'=curved eyebrow        V.

By independence, therefore,

s(a,b) = F(A n B, A - B, B - A) = F(X,Y,Z) >- F(X',Y',Z) =F(A'nB',A'-B',B'-A') = s(a',b')

if and only if

s(c,d) = F(C fl D, C - D, D - C) = F(X,Y,Z') >_ F(X',Y',Z')

= F(C' n D', C' - D', D' - C~ = s(c',dI.

Thus, the ordering of the joint effect of any two components (e.g., X,Y vs. X',Y') is inde­pendent of the fixed level of the third factor (e.g., Z or Z').

d          c'

FiSwe 2. An illustration of independence.

 

It should be emphasized that any test of the axioms presupposes an interpretation of the features. The independence axiom, for example, may hold in one interpretation and fail in another. Experimental tests of the axioms, therefore, test jointly the adequacy of the in­terpretation of the features and the empirical validity of the assumptions. Furthermore, the above examples should not be taken to mean that stimuli (e.g., block letters, schematic faces) can be properly characterized in terms of their components. To achieve an adequate feature representation of visual forms, more global properties (e.g., symmetry, connected­ness) should also be introduced. For an inter­esting discussion of this problem, in the best tradition of Gestalt psychology, see Goldmeier (1972; originally published in 1936).

 

In addition to matching (1), monotonicity (2), and independence (3), we also assume solvability (4), and invariance (5). Solvability requires that the feature space under study be sufficiently rich that certain (similarity) equa­tions can be solved. Invariance ensures that the equivalence of intervals is preserved across factors. A rigorous formulation of these as­sumptions is given in the Appendix, along with a proof of the following result.

 

Representation theorem. Suppose Assump­tions 1, 2, 3, 4, and 5 hold. Then there exist a similarity scale S and a nonnegative scale f such that for all a,b,c,d in A,

(i). S(a,b) > S(c,d)          iff s(a,b) > s(c,d);

(u). S(a,b) = Of (A (1 B) - af(A - B) - tef (B - A), for some B,a,9 >- 0; (iii). f and S are interval scales.

The theorem shows that under Assumptions 1-5, there exists an interval similarity scale S that preserves the observed similarity order and expresses similarity as a linear combina­tion, or a contrast, of the measures of the common and the distinctive features. Hence, the representation is called the contrast model. In parts of the following development we also assume that f satisfies feature additivity. That is, f (X U Y) = f (X) + f (Y) whenever X and Y are disjoint, and all three terms are defined'.

Note that the contrast model does not define a single similarity scale, but rather a family of scales characterized by different values of the parameters 0, a, and 0. For example, if B = 1 and a and p vanish, then S (a,b) = f (A n B) ; that is, the similarity between objects is the measure of their common features. If, on the other hand, a = d = I and 0 vanishes then -S(a,b) = f(A - B) + f(B - A); that is, the dissimilarity between objects is the measure of the symmetric difference between the respec­tive feature sets. Restle (1961) has proposed these forms as models of similarity and psycho­logical distance, respectively. Note that in the former model (B = 1, a = p = 0), similarity between objects is determined only by their common features, whereas in the latter model (B = 0, a = d = 1), it is determined by their distinctive features only. The contrast model expresses similarity between objects as a weighted difference of the measures of their common and distinctive features, thereby al­lowing for a variety of similarity relations over the same domain.

 

The major constructs of the present theory are the contrast rule for the assessment of similarity, and the scale f, which reflects the salience or prominence of the various features. Thus, f measures the contribution of any par­ticular (common or distinctive) feature to the similarity between objects. The scale value f(A) associated with stimulus a is regarded, therefore, as a measure of the overall salience of that stimulus. The factors that contribute to the salience of a stimulus include intensity, frequency, familiarity, good form, and infor­mational content. The manner in which the scale f and the parameters (B,a.s) depend on Lhe context and the task are discussed in the following sections.

 

Let us recapitulate what is assumed and what is proven in the representation theorem. We begin with a set of objects, described as collections of features, and a similarity order­ing which is assumed to satisfy the axioms of the present theory. From these assumptions, we derive a measure f on the feature space and prove that the similarity ordering of object pairs coincides with the ordering of their con­trasts, defined as linear combinations of the respective common and distinctive features. Thus, the measure f and the contrast model are derived from qualitative axioms regarding the similarity of objects.

 

The nature of this result may be illuminated by an analogy to the classical theory of deci­sion under risk (von Neumann & Morgenstern, 1947). In that theory, one starts with a set of prospects, characterized as probability dis­tributions over some consequence space, and a preference order that is assumed to satisfy the axioms of the theory. From these assump­tions one derives a utility scale on the conse­quence space and proves that the preference order between prospects coincides with the order of their expected utilities. Thus, the utility scale and the expectation principle are derived from qualitative assumptions about preferences. The present theory of similarity differs from the expected-utility model in that the characterization of objects as feature sets is perhaps more problematic than the char­acterization of uncertain options as probability distributions. Furthermore, the axioms of util­ity theory are proposed as (normative) prin­ciples of rational behavior, whereas the axioms of the present theory are intended to be de­scriptive rather than prescriptive.

 

The contrast model is perhaps the simplest form of a matching function, yet it is not the only form worthy of investigation. Another

~ To derive feature additivity from qualitative as­sumptions, we must assume the axioms of an extensive stmctum and the compatibility of the extensive and the conjoint scales; see Kranta et al. (1971, Section 10.7).

n m

O N

matching function of interest is the ratio model,

f (A () B)

S(a~b) = f(Ar) B) +of(A- B) -} df(B - A)' a,# >> 0,

where similarity is normalized so that S lies between 0 and 1. The ratio model generalizes several set-theoretical models of similarity proposed in the literature. If a = 0 = 1, S(a,b) reduces to f (A (l B)/f(A U B) (see Gregson, 1975, and Sjoberg, 1972). If a - 0 - +}, S(a,b) equals 2f(An B)/(f(A) + f(B)) (see Eisler & Ekman, 1959). If a - 1 and 0 = 0, S(a,b) re­duces to f(AO B)/f(A) (see Bush & Mosteller, 1951). The present framework, therefore, en­compasses a wide variety of similarity models that differ in the form of the matching function F and in the weights assigned to its arguments.

In order to apply and test the present theory in any particular domain, some assumptions about the respective feature structure must be made. If the features associated with each object are explicitly specified, we can test the axioms of the theory directly and scale the features according to the contrast model. This approach, however, is generally limited to stimuli (e.g., schematic faces, letters, strings of symbols) that are constructed from a fixed feature set. If the features associated with the objects under study cannot be readily speci­fied, as is often the case with natural stimuli, we can still test several predictions of the contrast model which involve only general qualitative assumptions about the feature structure of the objects. Both approaches were employed in a series of experiments conducted by Itamar Gati and the present author. The following three sections review and discuss our main findings, focusing primarily on the test of qualitative predictions. A more detailed de­scription of the stimuli and the data are pre­sented in Tversky and Gati (in press).

Asymmetry and Focus

According to the present analysis, similarity is not necessarily a symmetric relation. Indeed, it follows readily (from either the contrast or the ratio model) that

s(a,b) - s(b,a)     iff of(A - B) + df(B - A) -af(B-A)+df(A-B) iff  (a - O)f(A - B) = (a - a)f(B - A).

Hence, s(a,b) - s(b,a) if either a - 10, or f(A - B) = f(B - A), which implies f(A) - f(B), provided feature additivity holds. Thus, symmetry holds whenever the objects are equal in measure (f(A) = f(B)) or the task is non­directional (a - d). To interpret the latter condition, compare the following two forms:

(i). Assess the degree to which a and b are similar to each other.

(u). Assess the degree to which a is similar to b.

In (i), the task is formulated in a nondirectional fashion; hence it is expected that a = B and s(a,b) = s(b,a). In (ii), on the other hand, the task is directional, and hence a and d may differ and symmetry need not hold.

If s(a,b) is interpreted as the degree to which a is similar to b, then a is the subject of the comparison and b is the referent. In such a task, one naturally focuses on the sub­ject of the comparison. Hence, the features of the subject are weighted more heavily than the features of the referent (i.e., a > 0). Con­sequently, similarity is reduced more by the distinctive features of the subject than by the distinctive features of the referent. It follows readily that whenever a > 0,

s(a,b) > s(b,a)    iff f(B) > f(A).

Thus, the focusing hypothesis (i.e., a > d) implies that the direction of asymmetry is determined by the relative salience of the stimuli so that the less salient stimulus is more similar to the salient stimulus than vice versa. In particular, the variant is more similar to the prototype than the prototype is to the variant, because the prototype is generally more salient than the variant.

Similarity of Countries

Twenty-one pairs of countries served as stimuli. The pairs were constructed so that one element was more prominent than the other (e.g., Red China-North Vietnam, USA-Mexico, Belgium-Luxemburg). To verify this relation, we asked a group of 69 subjects' to select in

The wbiects in all our experiments were Israeli eollege students, ages 18-29. The material was pre­sented in booklets and administered in a group setting.

each pair the country they regarded as more prominent. The proportion of subjects that agreed with the a priori ordering exceeded J for all pairs except one. A second group of 69 subjects was asked to choose which of two phrases they preferred to use: "country a is similar to country b," or "country b is similar to country a." In all 21 cases, most of the subjects chose the phrase in which the less prominent country served as the subject and the more prominent country as the referent. For example, 66 subjects selected the phrase "North Korea is similar to Red China" and only 3 selected the phrase "Red China is similar to North Korea." These results demon­strate the presence of marked asymmetries in the choice of similarity statements, whose direction coincides with the relative promi­nence of the stimuli.

To test for asymmetry in direct judgments of similarity, we presented two groups of 77 subjects each with the same list of 21 pairs of countries and asked subjects to rate their similarity on a 20-point scale. The only differ­ence between the two groups was the order of the countries within each pair. For example, one group was asked to assess "the degree to which the USSR is similar to Poland," whereas the second group was asked to assess "the degree to which Poland is similar to the USSR." The lists were constructed so that the more prominent country appeared about an equal number of times in the first and second positions.

For any pair (p,q) of stimuli, let p denote the more prominent element, and let q denote the less prominent element. The average s(q,p) was significantly higher than the aver­age s(p,q) across all subjects and pairs: 9 test for correlated samples yielded 1(20) = 2.92, p < .01. To obtain a statistical test baud on individual data, we computed for each subject a directional asymmetry score defined as the average similarity for comparisons with a prominent referent, that is, s(q,p), minus the average similarity for comparisons with a prominent subject, s(p,q). The average differ­ence was significantly positive: 1(153) = 2.99, p < .01.

The above study was repeated using judg­ments of difference instead of judgments of similarity. Two groups of 23 subjects each

participated in this study. They received the same list of 21 pairs except that one group was asked to judge the degree to which country a differed from country b, denoted d(a,b), whereas the second group was asked to judge the degree to which country b was different from country a, denoted d(b,a). If judgments of difference follow the contrast model, and a >,S, then we expect the promi­nent stimulus p to differ from the less promi­nent stimulus q more than q differs from p; that is, d(p,q) > d(q,p). This hypothesis was tested using the same set of 21 pairs of countries and the prominence ordering established earlier. The average d(p,q), across all subjects and pairs, was significantly higher than the average d(q,p): 9 test for correlated samples yielded 1(20) = 2.72, p < .01. Furthermore, the aver­age asymmetry score, computed as above for each subject, was significantly positive, 9(45) =2.24,p<.05.

Similarity of Figures

A major determinant of the salience of geo­metric figures is goodness of form. Thus, a "good figure" is likely to be more salient than a "bad figure," although the latter is generally more complex. However, when two figures are roughly equivalent with respect to goodness of form, the more complex figure is likely to be more salient. To investigate these hypotheses and to test the asymmetry prediction, two sets of eight pairs of geometric figures were con­structed. In the first set, one figure in each pair (denoted p) had better form than the other (denoted q). In the second set, the two figures in each pair were roughly matched in goodness of form, but one figure (denoted p) was richer or more complex than the other (denoted q). Examples of pairs of figures from each set are presented in Figure 3.

 

A group of 69 subjects was presented with the entire list of 16 pairs of figures, where the two elements of each pair were displayed side by side. For each pair, the subjects were asked to indicate which of the following two state­ments they preferred to use: "The left figure is similar to the right figure," or "The right figure is similar to the left figure." The positions of the stimuli were randomized so that p and q appeared an equal number of times on the

 

 

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