J'OURNAL OF VERBAL LEARNING AND VERBAL BEHAVIOR 2, 446-452 (1963) Goodness of Pattern and Pattern Uncertainty 1 A visual configuration, or pattern, has qualities over and above those which can be specified by designating the physical properties of each element of the pattern. The Gestalt psychologists have particularly emphasized the fact that a pattern also has organization, that it exists as an entity. This Gestalt property does not exist in the same amount for all patterns, so we speak of the "goodness" of a pattern; and some patterns have more goodness than others. While many of the Gestalt principles of perception, such as proximity, similarity, continuation, symmetry, etc., have helped in understanding the nature of pattern goodness, there has not been a good unifying concept for the phenomenon. More recently, the concept of redundancy, as developed in information theory, has appealed to many psychologists as a possible unifying idea. Attneave (1954) developed a technique for estimating the redundancy of a single pattern, and Attneave and Arnoult (1956) described ways of generating visual forms with different amounts of randomness. Later Attneave ( 1957) showed that judged complexity of form was related to such things as number of turns in the figure, variability of angular change, and symmetry. Hochberg and McAlister (1953) and also Hochberg and Brooks (1960) have used a similar approach in studying the factors 1 This research was done under contract Nonr- 248(55) between the Office of Naval Research and Johns Hopkins University. This is report No. 25 under this contract. Reproduction in whole or in part is permitted for any purpose of the United States Government. W. R. GARNER AND DAVID E. CLEMENT Johns Hopkins University, Baltimore, Maryland 446 which lead a figure to be perceived as twodimensional rather than as three-dimensional. In these studies, an attempt was made to understand the perceived properties of figures by specifying some objective property or properties of the single stimulus which is perceived. As Garner (1962) has pointed out, there is a certain inconsistency in using the concept of redundancy with respect to single figures or patterns, since redundancy is a property of sets of patterns, not of single patterns. And yet there is a way in which the concept of redundancy, or its inverse, uncertainty, can be used in understanding the nature of pattern goodness. Suppose that each single pattern which a person sees is perceived not just as that one pattern, but as one of a set of alternative or equivalent patterns. In this case, we could speak of the relative size of the set of alternative patterns, and the greater the size of this psychologically inferred set of patterns, the greater would be the uncertainty of all patterns in that inferred set. These considerations led Garner (1962) to suggest the specific hypothesis that pattern goodness is inversely related to the size of such psychologically inferred sets. The specific purpose of the present experiment is to test this hypothesis. What is required is a specified total set of possible patterns, so that the number of objectively possible patterns is known. Then we obtain judgment of the goodness of each pattern, and also judgments of the size of the set of equivalent patterns from the total set that exist for each particular pattern. The hypoth-
PATTERN GOODNESS AND UNCERTAINTY 447 esis states that these two measures should be correlated The Stimuli METHOD Two kinds of stimulus patterns were used, patterns of dots and patterns of X's and O's. Dot Patterns. A total of 90 patterns was produced by placing dots in the centers of the cells of an imaginary 3-by-3 square matrix. The patterns were all those which can be generated with exactly five dots, and with the restriction that each row and each column contain at least one dot. This latter requirement was used so that each dot pattern would suggest the dimensions of the matrix, since no lines defined the location of the nine possible cells of the imaginary matrix. Each pattern was placed in the center of a white card 3 inches square, and the possible dot positions were a ~ inch apart. The dots themselves were typed on the cards by using the period. The card was otherwise blank except for a small red dot in the upper right corner which wa.s used to keep the cards correctly oriented. Several of these dot patterns are shown in Fig. 1, along with a code number for each. While our primary hypothesis is concerned with the number of psychologically inferred equi.valent patterns, it is possible to state the number of equivalents for each of these patterns by using an arbitrary rule concerning equivalents. The rule which has been used in designating each of the patterns in Fig. 1 as one from a given equivalence set is that of reflection and rotation, and the first number of the code indicates the number of patterns in each equivalence set by this criterion. For example, pattern 11 has only one equivalent (itself), since any reflection (mirroring) or 90-degree rotation simply reproduces the same pattern. So also is pattern 12 unique by this criterion. On the other hand, there are four 41 patterns by this criterion, since there are three other patterns which will produce the actual pattern shown in Fig. 1 if they are rotated or reflected. The two operations taken together constitute the equivalence criterion we are using, since in some cases both operations must occur together to change one pattern into another. Both operations are required, for example, to obtain the four equivalents of pattern 43. All told there are eight equivalence sets having four patterns each, so that 32 of the stimulus patterns were of this kind. In addition, there are seven equivalence sets having eight patterns each, making a total of 56 patterns of this kind (see Prokhovnik, 1959). X-O Patterns. In order to provide some check on the generality of our results, we also used 90 patterns in which each of the cells of the imaginary nine-cell matrix contained either an X or an O. The 90 patterns used were all those containing exactly five X's, and in which at least one X appeared in each row and each column of the matrix. Thus each of these patterns was exactly analagous to one of the dot patterns, and we can specify which of the X-O patterns should be the same as which dot pattern. These 90 patterns were typed on the same size of white card by using the capital letters in the same locations as the dots. Each card also had the red dot in the upper righthand corner to maintain correct orientation. The Tasks and Subjects Ratings. For the rating task, S was required to rate each of the 90 patterns for pattern goodness on a seven-point scale, with "1" for the best patterns and "7" for the poorest. The 90 patterns were presented to S in random order, different for each S, and S wrote his numerical rating on a sheet. He was allowed to look through the first several patterns before beginning his ratings in order to establish a general frame of reference, but then made his rating of each pattern before going on to the next.one. For an entire set of 90 patterns, this task required approximately 30 min. Altogether, 19 Ss (male undergraduates) made ratings of each of the two sets of 90 patterns, but they were divided into four groups of Ss. One group of five Ss rated X-O patterns on a first session, then the dot patterns on a second session, and then the X-O patterns again on a third session. A second group of four Ss (one S did not complete the ratings) rated dot patterns, then X-O patterns, and then X-O patterns again. A third group of five Ss rated X-O patterns, then dot patterns, and then dot patterns again. A fourth group of five Ss rated dot patterns, then X-O patterns, and then dot patterns again. These various schedules were used to allow us to obtain estimates of the reliabilities of the ratings, while counterbalancing first presentations of each of the sets. Groupings. A second group of 20 Ss (also male undergraduates) performed the second task. Half of them used dot patterns on a first session, and X-O patterns on a second session; the other.half used the patterns in the reverse order. Each session took approximately 1 hour. The task itself consisted of S's taking all 90 patterns and arranging them into groups according to a similarity criterion. It was explained to S that none of the patterns were identical, but that some were more alike than others, and he was to arrange them into groups so that each group had similar
448 GARNER AND CLEMENT patterns in it. lie was asked to use approximately eight groups, but he was allowed to use more (or fewer) groups if necessary. It was also explained to him that the groups did not have to be of equal size, and that he could use groups of any size that seemed most appropriate to him. Ratings RESULTS Each S rated either the dot patterns or the X-O patterns twice. The correlation between the first and second rating was determined by pooling data for all Ss; this correlation for pooled ratings of the 90 dot patterns was 0.73, and for the X-O patterns, the pooled correlation was 0.63. Each S also rated both the dot patterns and the X-O patterns, and it will be recalled that each dot pattern had an exactly equivalent X-O pattern. The correlation between ratings of dot patterns and X-O patterns was determined by pooling data of all Ss. This pooled correlation was 0.56. These pooled correlations concern primarily individual reliability, and are quite reasonable for a task of this sort. The main score for our purposes is the average rating for each stimulus pattern for each of the two sets of patterns. When these average ratings are compared for the two sets of patterns, the correlation is 0.93, indicating a high degree of consistency of ratings of stimulus patterns regardless of whether the actual pattern was formed with dots or with X's and O's. Thus we can conclude that the particular way in which the pattern is presented has little effect on the average rating of goodness of pattern. What slight differences existed between ratings of dot patterns and ratings of X-O patterns seemed to be due to the occasional figure-ground confusions with the X-O patterns. Generally speaking, if the pattern of X's was a poor pattern, so also was the pattern of O's. But in a few cases, the pattern of X's was a poor pattern, but the pattern of O's was a good pattern, a fact which led to inconsis- tencies in the ratings and is probably responsible for the lower reliability of ratings of these patterns. Pattern 47 is an example where the pattern of the five X's is poorer than the pattern of the four O's. Groupings The score obtained from the groupings was, for each pattern, the size of the group in which it was placed by S. Since we did not require Ss to group the same patterns twice, we have no direct estimate of the intrasubject reliability of these scores. However, each S did make groupings of both types of patterns. The correlation between these two grouping scores with pooled data was 0.18. The correlation between the two types of patterns for average scores per pattern was 0.65. Both comparable correlations for ratings were considerably higher than these, showing that the ratings are more reliable. We have used the mean size of group for each stimulus pattern, and this measure seems as satisfactory as we could find. The groupings were highly variable from S to S, and the task was not an easy one. For the groupings of dot patterns, to illustrate, the number of groups used was about equally divided between 7, 8, and 9, with one S using 10. The actual number of patterns in a group varied from 1 to 36. The group sizes, however, were not at all evenly distributed. A group size of 2 was fairly common, but then group sizes of 4, 8, 12, 16, and 20 were the most common, due to the fact that most of the Ss kept the reflection and rotation equivalence groups intact. Relations between Ratings and Groupings The major hypothesis for this experiment concerns the relation between these two scores --the mean rating per pattern, and the mean size group per pattern. The over-all linear correlation between these scores for the dot patterns was 0.84; for the X-O patterns it was 0.54. If we obtain a mean score for each
PATTERN GOODNESS AND UNCERTAINTY 449 MEAN MEAN CODE PATTERN RATING GROUP SIZE II -% 1.00 9.35 12 " 1.03 8.25 41 ": 1.55 9.80 42 '' 1.7 I I 2.69 43 " "= 1.74 I 4.04 44 " 1.78 II.26 el 45 :: 1.77 I 0.40 46 "~" 2.24 I 2, 16 47 '" 3.05 I 5.36 48 " : 3.50 16.69 el 8l,: 3.40 14.52 82,: 4.59 15.43 o 85 :' 4.77 16.81 84 ": 4.80 16.37 85 ": 5.19 16.:59 86. 5. I I 16.69 ee 87 ": 5.49 I 5.74 Fic. 1. Mean ratings of pattern goodness and mean size of groupings for the different dot patterns. Each pattern shown is just one from that equivalence group, and the number of patterns in each equivalence group is indicated by the first number of the code. equivalence group, as shown in Fig. 1 for the dot patterns, these correlations become 0.88 for the dot patterns and 0.76 for the X-O patterns. Thus our main hypothesis is confirmed, i.e., that there is a correlation between pattern goodness and the size of the psychologically inferred set of patterns. There is some curvilinearity in the relationship between these two measures, but the exact nature of the relationship between these two measures is not pertinent to the hypothesis. Furthermore, the curvilinear correlations are not substantially higher than these linear correlations. Stimulus Factors Influencing Ratings o/ Goodness Our hypothesis as stated concerned a relation between two behavioral variables: ratings of goodness and size of the similarity group. Since the hypothesis is confirmed, it is worth trying to specify the stimulus characteristics which influence the ratings of goodness, particularly in terms of factors which can be used to specify a size of group. And since the ratings of dot patterns were the most reliable of our measures, an analysis of variance of the mean ratings for the 90 dot patterns was carried outas summarized in Table 1. The variances are shown as per cent of the total, so that they can be easily interpreted as correlation ratios. Furthermore, the nature of this analysis is that the factors for which the analysis is carried out are not orthogonal factors, but rather exist in a hierarchy. Thus the terms in Table 1 are successively indented TABLE I ANALYSIS OF VARIANCE OF RATINGS OF GOODNESS Or DOT PATTERNS Variance Source (%) d/ Equivalence sets 98.1 16 Size of eq. set 73.4 2 Eq. sets of same size 24.7 14 No. of straight lines 19.7 Patterns within eq. sets 1.9 73 Total 100 89
450 GARNER AND CLEMENT to indicate when a further analysis is being done for a variance factor already extracted. First, we can divide the total variance into two major components, that due to differences in equivalence sets as shown in Fig. 1, and that due to differences between patterns within a given equivalence set. Approximately 98 ~o of the variance is attributable to the 17 different equivalence sets, and only 2 ~ to differences between patterns within a given equivalence set. This fact alone makes clear that the criterion of reflection and rotation is a critical one for determining psychological set size and also, of course, the perceived goodness of pattern. In turn, all of the equivalence sets can be divided into three groups according to the size of the equivalence set, since by the criterion of reflection and rotation these patterns had either 1, 4, or 8 equivalent stimuli. This factor, with its 2 degrees of freedom, accounts for 73 per cent of the total variance. Size of the equivalence set is the factor directly relevant to our hypothesis, and this analysis makes it clear that this size can be specified in terms of directly measurable stimulus properties and need not be determined solely from the psychological judgment. This per cent of variance is equivalent to a correlation ratio (eta) of 0.86, which is slightly larger than the linear correlation between ratings of goodness and size of groupings. Thus this stimulus factor accounts for approximately as much of the variability in ratings as do the experimentally obtained groupings. Still, approximately 25% of the variance of the ratings is due to differences between equivalence sets of the same size. Within the equivalence sets of size four and eight, there is one obvious factor which contributes most of this variance (20~ of the total) : the number of straight lines in the pattern. For both sizes of equivalence sets there are patterns with 0, 1, or 2 straight lines, although not in equal numbers, and the ratings are lower (better goodness) when there are straight lines. For example, the two poorest patterns for the fours are 47 and 48, neither of which has a straight line. Also, two of the three poorest patterns of the eights have no straight lines. On the other hand, the best pattern of the eights has two straight lines, and it is the only one of these patterns which does have two straight lines. The nature of this relation is summarized in Table 2, where the mean rating is shown for TABLE 2 MEAN RATINGS OF GOODNESS OF DOT PATTERNS AS A FUNCTION OF T]FIE NUMBER OF STRAIGHT LINES~ FOR Two SIZES OF EQUIVALENCE SET Number of straight lines Size of equivalence set 4 8 0 2.75 5.30 1 2.01 4.84 2 1.69 3.40 each number of straight lines, separately for the two sizes of equivalence set. The relevance of this factor is apparent, but what is also apparent is how much less important is this factor than the number of equivalent stimuli. There is one other factor which affects the ratings to a significant extent, but is nevertheless somewhat trivial, and that is the orientation of the straight line. For example, on the average, a verticle straight line is rated slightly better than a horizontal straight line, and either of them is better than a diagonal straight line. Other factors which we considered but were not related to goodness were density of pattern (i.e., how compact the set of dots is) and continuity. There is, of course, very little variance to be accounted for, since the size of the equivalence set and the number of straight lines together account for 93~ of the total variance of the ratings of goodness. DISCUSSION We feel that the size of the equivalence set is the most fundamental factor involved in the perception of pattern goodness. We have shown that pattern goodness is correlated with experimentally obtained sizes of group-
PATTERN GOODNESS AND UNCERTAINTY 451 ings; but even more importantly, we have shown that pattern goodness is related to the size of the equivalence set as specified in objective terms, i. e., in terms of properties of the stimulus rather than subjective properties of the percept. An argument that the size of the equivalence set is the fundamental factor cannot be made unequivocally, because the reflection and rotation criterion which we have used to determine the size of equivalence set is related to other factors which have been assumed to be related to pattern goodness. In particular, we must consider the factor of symmetry. A dot pattern of the type we have used which is unique by the criterion of rotation and reflection is a pattern which is symmetrical around the horizontal, vertical, and both diagonal axes, that is to say, reflection of the pattern on any of these four axes produces the same pattern. On the other hand, a pattern which is one of an equivalence set of eight is not symmetrical in any axis. Thus from these two cases alone we could.not argue that the critical factor is size of set rather than symmetry, since symmetry and size of set are perfectly correlated factors, and one implies the other. But there is an interesting exception in the equivalence sets of four. Pattern 43 is not symmetrical around any of the four possible axes, so it is like the patterns of eight equivalents in this respect. And yet it is judged one of the best patterns in terms of goodness. All of the other patterns of four equivalents are symmetrical on a single axis. We have been discussing objective symmetry, not judged symmetry, and the objectiv e symmetry is not always perceptually obvious. Patterns 44 and 46, to illustrate, are symmetrical about a diagonal axis which is perpendicular to the main axis of the pattern itself. While we have obtained no direct judgments of symmetry, we suspect that these two patterns would not be judged as symmetric as patterns 47 and 48, which are judged considerably poorer patterns. In other words, insofar as the major factor involved is sym- metry rather than uncertaintyl then it must clearly be objective symmetry rather than judged subjective symmetry that is involved. We have also shown that the ratings of goodness are to some extent influenced by the number of straight lines, but the amount of this influence is small compared to the effect of the size of the equivalence set. As another specific example, compare patterns 43 and 81. Both of these patterns have two straight lines, one diagonal and the other either horizontal or vertical. And yet pattern 81 has a goodness rating twice as great as (is poorer than) pattern 43. Furthermore, pattern 42 has no straight lines, but is rated a very good pattern. In other words, if one attempts to argue that such factors as symmetry or number of straight lines are really the fundamental factors, it becomes necessary to make many ad hoc explanations about particular patterns, the type of explanation which has plagued classical Gestalt psychology for so many years. The single factor of size of the objective equivalence set accounts for most of the variation in ratings of goodness, as the analysis of variance showed, and we feel therefore that it is the most fundamental and general factor. Recently Glanzer and Clark (1963) have argued for a verbal loop hypothesis, basing their argument on the experimental finding that the most accurately reproduced patterns were those which required the shortest verbal description. We would not disagree with this hypothesis as long as it is used to explain accuracy of pattern reproduction, but we would argue that it does not constitute an explanation of the nature of perceived pattern goodness. Rather, we feel that the size of the equivalence set, or more generally, pattern uncertainty, is the fundamental factor, and that the length of verbal description required is simply a necessary concomitant of this factor. To illustrate, let us assume that the 17
452 GARNER AND CLEMENT equivalence sets of Fig. 1 form the basis of the perception of the pattern, and that S is required to describe, for future reproduction, each pattern. First he has to determine into which equivalence group the pattern belongs, and then has to determine which particular pattern in the group is involved. The first step requires 4.1 bits of information for any pattern. But the next step will require no further information for the two unique patterns, 2 more bits for the equivalence sets of four, and 3 more bits for the largest equivalence sets. Therefore there will be a correlation between the pattern goodness and the length of the verbal description, but only because pattern goodness is related to the uncertainty of the particular pattern. In other words, the verbal loop hypothesis is a necessary consequence of the critical role of pattern uncertainty--or more specifically, the size of the equivalence set. It is, however, not an explanation of pattern goodness. We have used a measure of size of equivalence group in this experiment, but the more general factor involved is really perceived pattern uncertainty, and this factor has many connotations of the nature of pattern goodness. A poor pattern is one which is perceived as unstable, as easily changed, and as having many alternatives. A good pattern, on the other hand, is one which is perceived as stable, as not easily changed, and as having few alternatives. The nature of this relationship is easily summarized by noting that the best possible pattern is that perceived as unique. SUMMARY It is argued that when a pattern is perceived, it is perceived not only as itself but also as one of a subset of equivalent patterns. The hypothesis tested in this experiment is that pattern goodness is related inversely to the size of this inferred set of equivalent patterns. Two sets of 90 patterns, either five clots or five X's and four O's, were rated for pattern goodness. The same patterns were also arranged into groups according to a similarity criterion. The high correlations between the rating of goodness and the size of the similarity groupings substantiated the hypothesis. An objective measure of the size of equivalence groups was obtained by using reflection and rotation to determine the number of equivalent patterns. This objective measure accounted for 73% of the total variance of the ratings of figural goodness. It is argued that pattern uncertainty is the fundamental factor in pattern goodness, and that factors such as symmetry are simply concomitants of this factor. REFERENCES ATT~CEAW, F. Some informational aspects of visual perception. Psychol. Rev., 1954, 61,183-193. ArTrCEAW, F. Physical determinants of the judged complexity of shapes. J. exp. Psychol., 1957, 63, 221-227. ATTNEAVE, F., AND ARNOULT, M. D. The quantitative study of shape and pattern perception. PsychoL Bull., 1956, 63, 452-471. GARNER, W. R. Uncertainty and structure as psychological concepts. New York: John Wiley, 1962. GLMqZER, M., AND CLARK, W. ~'~. Accuracy of perceptual recall: An analysis of organization. J. verb. Learn. verb. Behav,, 1963, 1, 289-299. HOCm~RC, J., ANO BROOKS, V. The psychophysies of form: Reversible-perspective drawings of spatial objects. Amer. J. Psychol., 1960, 73, 337-354. HOCttBERO, J., ±'~ND McALISTER, E. A quantitative approach to figural "goodness." J. exp. Psychol., 1953, 46, 361-364. PROKI-IOVlTIK, S. J. Pattern variants on a square field. Psychometrlka, 1959, 24, 329-341. (Received April 22, 1963)