JOURNAL
OF
EXPERIMENTAL
CHILD
A Multidimensional Visual
Memory
19, 327-345 (1975)
PSYCHOLOGY
Scaling of !&Year
PHIPPS
and
of Adults
ARABIE
University
of Minnesota
M. KOSSLYN
STEPHEN Johns
Olds
Study
Hopkins
University
AND KEITH New
School
E. NELSON for
Social
Research
Nonmetric multidimensional scaling is expounded as a methodology for investigating memory development. Judgments of similarities between complex objects were obtained from 5-year-olds and adults under two conditions: (1) when objects were simultaneously present at the time of comparison, and (2) when the objects were not simultaneously present and had to be compared on the basis of shortterm memory. Scaling solutions for similarity judgments by each age group in the two conditions were obtained. Comparison within each age group of the configurations from the memory condition with the simultaneous condition indicated that the internal representations of both children and adults contained much of the information the subjects extracted from the stimuli during “direct” perception. The scaling techniques, which could be widely applied in developmental studies of memory and perception, are explained and critically discussed. One of the most difficult
tasks for experimental psychology is that of obtaining tangible representations of mental structures and processes. This problem is especially acute when one is studying young children, who are less facile with language than are adults. The present paper concerns the use of nonmetric multidimensional scaling for obtaining spatial We are indebted to S. A. Boorman, J. Metzler, D. Osherson, E. J. Shoben, and E. E. Smith for their comments and suggestions on an earlier draft of this paper. For very helpful remarks during discussions of this work, we are grateful to J. B. Arnold, R. C. Atkinson, J. M. Carlsmith, D. C. Olivier, and R. N. Shepard. We would like to thank Tom Schumacher for running some of the subjects in the experiments reported here. This research was supported by NIMH Grant MH21747 and NSF Grant GJ-443X3 to Richard C. Atkinson, NSF Grant GS-2689 to Harrison C. White, NIH Grant HD-06524 to Keith E. Nelson, and funds from the Graduate School of the University of Minnesota. Order of authorship is alphabetical. Send requests for reprints to: Phipps Arabie, Dept. of Psychology, Elliott Hall, University of Minnesota, Minneapolis, MN 55455. 327 Copyright @ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
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representations of perceived similarities among a specific set of stimuli. The particular topic of memory development, as investigated here, is only one of many developmental research areas which are amenable to multidimensional scaling (cf., Kosslyn, Pick, & Fariello, 1974). In the present study, we wished to compare the perception and memory of 5-year-olds and adults in one particular situation, involving the retention of complex visual information which is nonverbally encoded. We attempted to minimize the effects of verbal skills by using unfamiliar and difficult-to-describe stimuli and presenting these stimuli for a relatively brief period of time, while never providing, or encouraging use of, labels, and by allowing subjects to indicate their judgments nonverbally through pointing. The experimental technique which we used to study children’s and adults’ presumably nonverbal memory of visually presented objects was first introduced by Shepard and Chipman (1970). Their paradigm is not commonly used by developmental psychologists, but seems a useful technique for externalizing mental events at different ages. Shepard and Chipman had subjects rank order pairs of states of the U. S. on the basis of similarity when only the names of the states or when only line drawings of them were presented. Spatial representations of these two sets of data then were obtained through the use of nonmetric multidimensional scaling. The most important finding was the high correspondence between the spatial configurations for the two sets of stimuli. Both configurations exhibited clustering of states that were characterized by the same cartographic features (e.g., square states, like Colorado and Oregon, were close together in space). Thus, it appeared that presentation of the names alone evoked internal representations of the states from long-term memory and that those representations bore some functional similarity to line drawings and could be ranked according to the same criteria applied to the actual drawings. In the present experiment, we used the Shepard-Chipman technique, obtaining similarity rankings under stimulus present and memory conditions, but in a short-term memory situation with novel stimuli. Thus, our experiment consists of four cells: children and adults in ‘*memory” and “simultaneous” stimulus presentation conditions. To the extent that the subjects at either age make judgments from memory which parallel their judgments when the stimuli are physically present, then the subjects must have retained much of the same visual information used to make similarity judgments when the stimuli are present. Furthermore, to the extent that the 5-year-olds and adults display similar judgments, then we can infer that their respective internal representations, or perceptions, include much of the same information, and that the judgments themselves are based on similar criteria.
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METHOD Design for Scaling
We wish to represent the subjects’ perceived similarities of our stimulus set in terms of a geometric model in which short distances between stimuli represent high similarity and large distances indicate low similarity between stimuli. Given such a representation in two dimensions, for example, we can compare the “map” of the stimulus space obtained from adults with that from children, as well as comparing the memory and simultaneous conditions. The method we have used to obtain such a representation of perceived similarity is that of nonmetric multidimensional scaling. Originally called “analysis of proximities,” this method was devised and implemented by Shepard (1962a,b). A stronger logical basis for the procedure was provided by Kruskal (1964a,b), who also wrote MDSCAL (Kruskal & Carmone, 1970), the most widely used computer program for nonmetric multidimensional scaling. This procedure is especially appropriate for developmental studies because it uses only the ordinal information available from the interstimulus similarities data. That is, the program uses only the rank order of the n * (n - 1)/2 pairwise similarities (or a subset thereof), as opposed to doing arithmetic with the actual numerical values. Furthermore, the fact that the procedure is “monotone invariant” (i.e., uses only the ordinal information from the data) means that substantially weaker assumptions about the data are entailed than is the case for metric multidimensional scaling (e.g., factor analysis), in which it is assumed that the data are on an interval scale (e.g., Seitz, 1971). Kruskal, in his iterative procedure, also seeks to minimize a badness-offit function called “stress,” which tells how far the data depart from the best-fitting monotone function relating psychological dissimilarity to metric distance. Thus, stress provides an indication as to how well the program is able to construct a metric representation of the relationships implicit in the data, beginning with only a rank ordering of the interstimulus similarities as input. The question naturally arises: How are we to obtain such a similarities matrix from 5-year-old subjects? Clearly, we wanted a very simple and nonverbal task, with the consequence that Shepard and Chipman’s procedure requiring subjects to rank order many pairs of stimuli is inappropriate. There are many other traditional tasks suitable for adult subjects: obtaining category judgments of similarity on (typically) an 1 l-point scale, paired comparisons of differences (Ramsay, 1968), “direct” ratio scaling (Indow & Uchizono, 1960; Ekman, 1963), triads (Torgerson, 1958), and various other techniques. We found the method
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of triads to be attractive insofar as it requires no familarity with the number scale, unlike many other methods. Rather, at the presentation of each triad, the subject has only to decide which two of the three stimuli are most similar and which two are least similar. The main disadvantage is that the standard method entails n * (n - 1). (n - 2)/6 different trials, which is 220 for n = 12 stimuli. An imaginative variation on the traditional method is described by Levelt, van de Geer, and Plomp (1966), who used balanced subsets of 35 triads for 15 stimuli, instead of the full set of 455. (The interesting details of this design are given in Levelt et nl. (1966, pp. 165-166).) In the present experiment, a balanced subset containing only 66 triads of the 12 stimuli was used and the difficulty of the judgment required from the subject was also lessened. Specifically, for a given triad (a,x,b), the subject was asked whether a or b was more similar to the standard x, thus affording a relative comparison of the similarity of the two pairs (a,x) and (b,x). (The traditional judgmental task, as employed by Levelt et al., implicitly compares the similarity of all three pairs in the triad.) We ran a computerized search to find a set of 66 triads (a,x,b) which was balanced in the sense that each of the 66 pairs [(a,x) or (b,x)] occurred exactly twice. Obviously, it would have been desirable to have more than two such presentations per subject of each stimulus pair, but we felt that many more than 66 presentations would place an excessive demand on a 5-year-old subject’s span of attention. Given this balanced set, five different orders were used across subjects, and for each of these orders, no stimulus occurred in two successive triads. In the present study, our intention was to have stimuli which were relatively difficult to encode verbally. For our stimuli we constructed a set of 12 toy structures (made of LEG0 blocks, manufactured by Samsonite and specially painted) looking somewhat like 3-dimensional miniatures of asymmetric office buildings. Each stimulus was 3 in. wide and 34 in. in height. It was our intention that these stimuli would be equally novel to both 5-year-olds and adults and thus would circumvent problems arising from age differences in familiarity with stimulus materials. Holding the number of blocks (20) in each stimulus constant, the two physical dimensions systematically varied were: (A) the number of different colors (2-13, selected from a set of 13 colors); and (B) the number of “crossings” (3-14), that is, how many blocks were crossed completely (overlapped on two sides by a block immediately above). The values of each stimulus along these dimensions certainly can be verbalized, although not very easily within a short period of time. Furthermore, this design for stimulus definition resulted in stimuli quite rich in visual detail. From the 144 possible pairs of values along the two discrete dimensions, 12 stimuli were selected by means of a random
SCALING
VISUAL
TABLE LATIN
SQUARE
TYPE
OF DESIGN
9
Number of different colors (2-13)
6 11 l3
3
FOR THE
8
331
MEMORY
1 12 STIMULI
USED
IN BOTH
CONDITIONS
Number of orthogonal crossings (3-14) 10 7 12 11 5 13 14
4
6
1 2
5 2 ;; L
8 9 12 7
3 4 5
Stimulus Number
6 7 8 9 10 11 12
pairing of the 12 levels of the two dimensions. Since in purpose this selection scheme resembles a Latin square (which reduces the number of stimuli by a factor of n), for convenience it will henceforth be referred to as a Latin square design, although the term is used only in analogy. The specifications of the stimulus set are given in Table 1. Subjects
The children were 28 boys attending Bing Nursery School at Stanford. The median age was 4 years, 9 months, and the range in age was from 4 years, 6 months to 5 years, 2 months. The adult subjects were 20 male undergraduates satisfying a requirement for the introductory psychology course at Stanford. Experimental Procedure
The sessions associated with the two conditions of the experiment were scheduled 1 week apart for each subject. For both conditions, the 5-year-old subjects participated in two sessions on consecutive days, with 33 trials per session. The adults, however, were run in a single session of 66 triads for each condition. Memory condition. The memory condition preceded the simultaneous condition for all the subjects. In the memory condition 66 triads (a,x,b) were presented one at a time, with either stimulus a or b presented for 4 set, then b or a for 4 set, and, finally, the standard for the same duration. After the standard had been removed, the subject’s task was to indicate whether a or b was more similar to the standard x. The apparatus for presenting the stimuli was a small chest, placed at the level of the
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subject’s head, with three open compartments, into each of which the experimenter could place one of the stimuli constituting a triad. The subject’s head was 15-18 in. from each stimulus position. On each trial, the test stimuli, a and b, were presented separately, in either the left or right compartment (randomly determined). The standard x appeared last and was always in the center compartment. The experimenter manually placed the stimuli in the compartments for presentation, and also recorded the responses. All subjects were instructed to watch the first two stimuli carefully and to “try to remember what they looked like.” After the last stimulus (the standard) had been presented for 4 set in the center compartment and taken away, the subject was to point to the position of the stimulus more like the one in the middle. Prior to the first actual test trial of this condition, each of the 12 stimuli was presented, one at a time for 3 set in the middle compartment, and subjects were encouraged to study the stimuli. To ensure that the general procedure was clear to the 5-year-olds, a set of 12 stimuli somewhat similar to those actually presented in the experiment was first used in four practice triads. As noted above, for the 5-year-old subjects, both the memory and simultaneous conditions were split into two sessions each having 33 trials. All of the 66 trials within either condition occurred on consecutive days. Any subject emitting a run of seven responses corresponding exclusively to either side of the chest was discontinued on the presumption of a strong hand-bias. Four potential subjects were eliminated by this rule. At the beginning of the second half of the memory trials, the 5year-old subjects were reminded of the “game played last time.” If a subject seemed uncertain, then the practice triads were once again presented. Simultaneous condition. All subjects were given new instructions at the beginning of this condition. Subjects were informed that all three stimuli would now be presented simultaneously for 12 sec. As before, the subject was instructed to indicate whether the stimulus on the left or right was more similar to the standard. In the simultaneous condition, the same 66 triads in the same order (for a given subject) were again presented, using the same apparatus. The only procedural difference was that all three stimuli of the given triad were presented simultaneously for 12 sec. Once again, the 5-year-olds were reminded of the “game” at the beginning of the second session of trials. RESULTS
Before turning to the spatial configurations from ‘MDSCAL, Table 2 gives the values of r6 (Kendall, 1962) for comparing the raw group
SCALING
VISUAL TABLE
RANK-ORDER MATRICES
5-year-olds Memory 5-year-olds Simultaneous Adults Memory Adults Simultaneous ‘I These correlations scaling. * p i .05. *** p < .OOl.
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MEMORY 2
CORRELATIONS (tnu b) BETWEEN SIMILARITY FOR THE 4 CELLS OF THE EXPERIMENT" 5-year
5-year
Memory
Simultaneous
Adults Memory
Adults Simultaneous
X .416*** ,115 .191*
.416*** X .098 ,023
.I15 ,098 X .518***
.191* .023 .518*** X
are for the “raw”
data, prior to analyses from multidimensional
data (i.e., the similarity matrices, obtained from the 66 triads) for the 4 cells of the experiment.’ Using Kendall’s (1962) test of significance, the rank-order correlations across the conditions are highly significant for both adults (z = 6.15) and for Syear-olds (z = 4.93), and the correlation between Syear-olds in the memory condition and adults in the simultaneous condition is significant at the .025 level (z = 2.26). These findings, then, indicate that both the adults’ and the 5-year-olds’ internal representations seemed to depict an underlying psychological structure highly similar to that evinced by judgments made by each respective age group when stimuli were physically present (simultaneously). This result suggests that subjects’ internal representations of the stimuli were in some way related to subjects’ perception in the simultaneous condition. However, the nonsignificant correlation between age groups in the simultaneous condition suggests that perceptually based judgments for the two groups were quite different. The Spatial Representations
Although the rank-order correlations in Table 2 gave us a quantitative indication as to which cells of the experiment were more closely related, the scaling solutions give a better (qualitative) overall picture. Since for r The similarity matrix S is obtained as follows. Beginning with a lower triangular matrix (i > j) S = 0, we consider each of the 66 triads of the form (i,j,k). Whenever the standard j is judged to be more similar to i than to k, we increment s(ij) by 1. Hence, every entry s(ij) of the individual’s matrix will have a value of 0, 1, or 2, since each stimulus pair is compared twice in the 66 triads. For the group data for a given cell of the experiment, we obtain each entry of the matrix by summing over that entry for individuals’ data.
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AND
in,
.‘.__.
NELSON
h,
ii------..~..?P_.~.-.--
(12, 61
1. Euclidean configuration for Syear-olds in memory condition (stress = ,201). For each stimulus, the number of colors and the number of crossings are given in brackets. Those stimuli having a large number of crossings form a convex cluster. FIG.
n = 12, we have too few stimuli to support (numerically and statistically) a three or higher dimensional solution, we consider only 2dimensional configurations, as shown in Figs. l-4. In using the scaling procedures, the “primary approach” to ties, and stress formula one as the badness-of-fit function (Kruskal & Carroll, 1969) were specified as options for MDSCAL-SM (Kruskal & Carmone, 1970). Each of the (Euclidean) configurations in Figs. l-3 corresponds to the lowest stress value over at least 40 different random initial configurations. According to Klahr (1969), the .05 level of the distribution of stress for scaling 12 stimuli in a 2-dimensional Euclidean space is 0.211, so that our stress values (indicated under each figure) for these three solutions should be considerably better than if the data were random. However, technical
FIG. 2. Euclidean configuration for S-year-olds in the simultaneous condition (stress = ,208). For each stimulus, the number of colors and the number of crossings are given in brackets. Although both colors and crossings offer suitable bases for clusters, the cluster defined by the stimuli having a large number of colors is preferred.
SCALING
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335
FIG. 3. Euclidean configuration for adults in the memory condition (stress = ,195). For each stimulus, the number of colors and the number of crossings are given in brackets. Although both colors and crossings offer suitable bases for clusters, the cluster defined by the stimuli having a small number of crossings is preferred.
arguments have been made elsewhere (Arabie & Boorman, 1973; Arabie, 1973) suggesting that Klahr’s stress criteria are really quite liberal (Fig. 4 will be discussed in detail below). The fact that we chose a Latin square type of design for defining the stimuli entails many problems for interpreting the spatial configurations. Ideally, we would have used the full set from a complete factorial design. However, the combinatorial aspects inherent in the method of scaling employed here, as well as the special difficulties entailed in running Syear-old subjects dictated using a much more restricted group of
FIG. 4. City-block configuration for adults in the simultaneous condition (stress = .217). For each stimulus, the number of colors and the number of crossings are given in brackets. Although both colors and crossings offer suitable bases for clusters, the cluster defined by the stimuli having a large number of crossings is preferred. Axes are included here because the city-block metric, unlike the Euclidean, gives a (mathematically) preferred orientation to the configuration. up to a reflection of the axes.
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stimuli. A Latin square stimulus set renders inapplicable many of the more common methods and approaches (e.g., hierarchical clustering) for interpreting our data. However, we can still ask which of our two physical dimensions (i.e., number of colors or number of orthogonal crossings) seems to be more important for the perceived similarity of the stimuli. This empirical question is related to the probiem of dimensional salience or attribute preference (cf. Odom & Guzman, 1972; Odom, 1972; Suchman & Trabasso, 1966; Trabasso, Stave, & Eichberg. 1969). Dimensional dominance test. The most clear-cut demonstration of one physical dimension’s dominance obviously would be a l-dimensional projection2 in the configuration along which the stimuli were ordered according to one of the dimensions. However, such a strategy does not succeed for any of the configurations in Figs. 14. It was our intention to devise an extremely simple basis for clustering along physical dimensions, such that when compared to the multitude of algorithms for clustering (cf. Lance & Williams, 1967). ours would appear as relatively simple as the sign test is among the many methods for assessing statistical significance. Like the sign test, our method is concerned with answering a simply stated question: Can either (or both) of the dimensions offer a basis for interpreting the spatial solution? The scheme that we have used here for investigating dimensional dominance is based on the simple observation that if the number of crossings, for instance, is important, then stimuli 6 and 7 should be close together, since they have 12 and 11 crossings, respectively. Alternatively, if the number of colors is more important, we would expect stimuli 5 and 6 to be near each other, as they have 3 and 4 different colors, respectively. For a complete set of factorially designed stimuli. we could hope to find a rather obvious pattern of grouping according to one of the physical dimensions. Again, however, with a Latin square set of stimuli, we cannot expect the pattern to be as readily apparent. Nonetheless, one would expect stimuli having a lower value (e.g., the number of different colors) on the more relevant dimension to be closer to each other (more similar) than to the remaining stimuli having higher values. We have somewhat arbitrarily selected “medians” for the range of values on the physical dimensions, and then looked for convex clusters. More specifically, since the number of colors ranges from 2 to 13, we have chosen the “cut” on that dimension to be between 7 and 8 colors. and between 8 and 9 for the number of crossings, which ranges from 3 to 14. Thus. we are looking at four overlapping sets of stimuli (high or low p A unidimensional projection of a 2-dimensional solution sional MDSCAL solution, since nonmetric multidimensional tain difficulties in handling the l-dimensional case.
is to be preferred to a l-dimenscaling algorithms have cer-
SCALING
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337
number of colors and few or many crossings) in each of the configurations to see if any form a convex cluster encompassing no other stimuli. In Figs. 1-4, these clusters have been enclosed by solid lines for number of colors and by broken lines for number of crossings. At least one convex cluster can be constructed on the basis of crossings for each of the four solutions. In Fig. 1, for 5-year-olds in the memory condition, no similar arrangement is possible on the basis of number of colors. In contrast, Fig. 2, for children in the simultaneous condition, shows that both the sets of stimuli having a large number of colors (8-13) and crossings (9-14) allow for convex clustering. Thus, although we can characterize the judgments of 5-year-olds in the memory condition as being dominated by form, we cannot argue that either form or color dominates for that age group in the simultaneous condition. Similarly, our dimensional clustering scheme fails to provide any decisive characterization of either of the solutions for adults, since both physical dimensions serve as suitable bases for convex clusters. A further elaboration of the simple clustering test gives a (somewhat weak) criterion for deciding which is the more important of the two physical dimensions in Figs. 24. Specifically, given that both physical dimensions provide suitable sets of stimuli as candidates for convex clusters, we assign priority to the dimension which defines the set whose complement more nearly forms a convex cluster. In Fig. 2, then, the set having a small number of colors is closer to forming a convex cluster than is the set having a small number of crossings. Thus, for .5-year-olds in the simultaneous condition, we would decide that the color dimension is closer to being dominant. The same type of argument would lead us to conclude that the number of crossings rather than colors is the more important dimension in both adult conditions (Figs. 3. 4). The fact that both color and form seemed to be important physical dimensions in the simultaneous condition would be expected from the results of Odom and Guzman (1972) and Odom (1972). However, the finding that color dimension was more useful here for interpreting our spatial model of the 5year-olds perceptions (simultaneous condition) is somewhat in contrast to other experimenters’ results (e.g., Suchman & Trabasso, 1966; Trabasso, Stave, & Eichberg, 1969). It should be noted though, that our variable is the number of colors rather than salience or identity of colors, as employed by most previous investigators. Minkowski
Metrics
and Psychological
Spaces
With Fig. 4 we introduce another aspect of multidimensional scaling representations that has not yet been mentioned in this paper, and that concerns the underlying metric of the stimulus space. Although it is
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often assumed by default that most psychological spaces are Euclidean. there is no a priori reason why this assumption should be valid. A more general distance formula is given by the Minkowski I’ metric, for which the distance between two points (stimuli y and Z) in a 2-dimensional space is: d(y, 7,) = (i
lyj - q, ,=I
where Y > 1, and i is the index over the psychological dimensions (and we are only considering the 2-dimensional case, given just 12 stimuli). When I’ = 2, we have the familiar Euclidean distance formula. It is traditionally argued (Attneave. 1950; Garner. 1970; Lockhead, 1966; Shepard, 1964; Torgerson, 1958) that Euclidean distance is more appropriate when the stimuli are processed as “integral” or “unanalyzable wholes,” that is, when the dimensions of the stimuli are not perceived independently or separately. For I’ = 1, the distance formula corresponds to the city-block metric, first investigated by Landahl (1945) and Attneave (1950) as being relevant to psychological processes. A very nice psychological interpretation of the Minkowski metrics for the special cases I’ = 1 and 2, as well as infinity has been given by Cross (1965a,b). Omitting the elegant details of Cross’s work while briefly reviewing some of his arguments, we note that for the city-block metric, distances are additive along the axes of the psychological dimensions, with the projections along each dimension being given equal weight in the function determining perceived similarity. Thus, it can be argued that the city-block metric is appropriate for an experimental context where the dimensions of the stimuli are perceived separately and thus are processed independently. As Y increases, the axis along which the two stimuli are most discrepant contributes more heavily to their psychological distance (i.e., increasing the exponent causes the dimension having the greatest projection to be differentially given greater weight). In the limiting case where I’ is infinite (sometimes called the “max metric”), the distance between each pair of points may be regarded as unidimensional since it is determined solely by the axis of greatest separation, although the space in which the configuration is embedded may be of much higher dimensionality. An underlying max metric (e.g., Arnold, 1971) implies an experimental situation where one maximally salient psychological dimension is the basis for each pair-wise comparison; moreover, in a multidimensional space, the dimension perceived as most salient will vary across comparisons of different stimulus pairs. Occasionally, the city-block metric has been found superior to the Euclidean metric for representing psychological processes (e.g., Cross, 1965a.b; Shepard & Cermak, 1973). Comparisons of solutions for the
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MEMORY
TABLE 3 MINIMUM STRESS (FORMULA 1) VALUES OBTAINED FOR EACH CELL OF THE EXPERIMENT, USING BOTH CITY-BLOCK AND EUCLIDEAN METRICS" Metric:
r= 1
r=2
.205 ,223 ,221
,201 ,208 ,195 ,203
Subject kTOUP
Condition
S-year-olds S-year-olds Adults Adults
Memory Simultaneous Memory Simultaneous
,217
(1 Each r = 1 solution is the best stress over at least 120 different random initial configurations, and each r = 2 value is taken over at least 40 different random initial configurations.
two metrics strictly on the basis of stress values have rarely favored the city-block metric, and possible doubt has recently been cast on the face value of such comparisons (Shepard, 1974). The purpose in the present study was to find interpretable configurations (cf., remarks by Shepard, (1969, p. 41)), and we have found that the city-block metric yielded interpretable clusters for adults in the simultaneous condition, and that the Euclidean solution was much less interpretable. Hence we believe it is justifiable to argue that the city-block metric represents the results better for adults in that cell of the experiment. Therefore, all subsequent discussion of the results for that cell refers to the city-block configuration shown in Fig. 4. In passing, it might be noted that of the four cells in the present experiment, “adults in the simultaneous condition” is clearly the cell most similar to the conditions under which other investigators have sought and occasionally found evidence for the city-block metric. The three remaining city-block solutions were uniformly uninterpretable, and none yielded even a single convex cluster, as would be required by the scheme outlined above. Attempts at devising other schemes for interpreting the three city-block solutions also were unsuccessful. Table 3 includes stress values of configurations from each cell for both values of the Minkowski r. The Euclidean case gives lower stresses, although the city-block stresses are all within about 10% of the corresponding Euclidean values. Each value of stress for the city-block case is the minimum value found using at least 120 different random initial configurations. We have dismissed three of the city-block configurations as uninterpretable. However, since the city-block stress values were all within about 10% of the corresponding Euclidean values, a skeptical reader might well ask if the city-block solutions3 were really different from 3 There are no Monte Carlo studies of stress available for the city-block metric.
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TABLE SUMMARY
AND
4
STATISTICS FROM COMPARISONS CORRESPONDING TO THE 4 CELLS
Cell: 5-year-olds in Memory Condition 5-year-olds jn Simultaneous Condition Adults in Memory Condition Adults in Simultaneous Condition
NELSON
OF THE SCALING SOLUTIONS OF THE EXPERIMENT
Pearson R between coordinates of Euclidean solution rotated to corresponding city-block solution ,748 .626 .b27
.729
those of the Euclidean metric. Table 4 gives the values of Pearson R between coordinates of rotated Euclidean solutions and their city-block counterparts. The fact that all values are considerably less than unity indicates that the pairs of configurations are appreciably different. The third special case of the Minkowski metrics considered by Cross, the max metric (v = infinity), has been briefly mentioned. Although somewhat counterintuitive, it is the case that 2-dimensional spatial configurations for r = 1 (city-block) and r = infinity (max metric) differ by only a 45’ rotation. (Euclidean distance is the only r-metric for which distances remain invariant under a rotation of the axes.) A very straightforward derivation of this relationship between the two metrics in the 2dimensional case is given by Arnold (1971, p. 357). The fact that the city-block/max metric solutions are uninterpretable for the Syear-olds is rather interesting when one recalls Vygotsky’s (1962, p. 64) discussion of the “chain complex” and its role in children’s similarity groupings. After the collection state of thinking in complexes, we must place the chain complex-a dynamic, consecutive joining of individual links into a single chain, with meaning carried over from one link to the next . . The decisive attribute keeps changing during the entire process. There is no consistency in the type of bonds or in the manner in which a link in the chain is joined with the one that precedes and the one that follows it.
If one replaces Vygotsky’s “decisive attribute” with “most important dimension,” then all successive pairs of the chain are linked on a unidimensional basis (viz,, along the axis having the longest projection) and Y = infinity is suggested for the underlying psychological space (cf., White, 1963, Chap. 1). Unfortunately, the present results lend no support to this implication of Vygotsky’s theory. Comparing Conjigurations
As has been emphasized, we chose multidimensional principal method of data analysis because it facilitates
scaling as our comparison of
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VISUAL
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MEMORY
psychological spaces across conditions. We have already given the values of rb on our raw data, but those correlations told little more for the adult subjects’ data than one would have anticipated after reading the Shepard-Chipman study. The rank-order correlations did, however, suggest that the 5-year-olds’ internal representations were in some sense related to veridical representations of the stimuli. This inference would have a stronger logical basis if the psychological spaces obtained from scaling techniques were shown to be similar across conditions. In order to render the scaling solutions comparable, we have used the computer program CONGRU (Olivier, 1970) to rotate various pairs of our configurations to maximal congruence (in a least-squares sense) CONGRU allows the user to rotate a Euclidean configuration to match another (target) configuration, and the program also gives the value of Pearson R between coordinates of the two solutions as a symmetric measure of agreement. Table 5 is a quantitative summary of the results from running CONGRU on the four configurations shown in Figs. l-4. The pattern of correlation is very similar to that for rb on the raw data, as shown in Table 2. In both tables, there is a strong correlation for both adult and 5year-old subjects between memory and simultaneous conditions, with the value for adults being larger. Also, the third largest correlation is again that between adults in the simultaneous condition and 5-year-olds in the memory condition. Similarly, as in Table 2, the lowest correlation is that between adult and 5-year-old subjects in the simultaneous condition. Thus, it appears that the two age groups have different perceptual bases for their judgments in the simultaneous condition, as will be discussed below. The scaling configuration shown in Fig. 1 has been rotated to maximal congruence with Fig. 2, while Fig. 3 has been rotated to match Fig. 4 as TABLE
5
PEARSON R PRODUCT-MOMENT CORRELATIONS OF CONFIGURATIONS FOR THE 4 CELLS AFTER ROTATION TO MAXIMAL
5-year-olds Memory 5-year-olds Simultaneous Adults Memory Adults Simultaneous (r = 1)
BETWEEN COORDINATES OF THE EXPERIMENT CONGRUENCE
S-year
5-year
Memory
Simultaneous
Adults Memory
Adults Simultaneous (r = 1)
x .627 ,411 ,555
.627 X ,406 ,405
,411 ,406 X .705
,555 .405 ,705 x
Note: The coordinates for adults in the simultaneous condition are from the city-block solution. (If the best Euclidean solution had been used for adults in the simultaneous condition, the first 3 entries of the last row and column would have been ,352, ,304, ,755.)
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a target configuration. When Figs. 1 and 2 are compared visually, it is clear that stimuli seem to be similarly located quadrant-by-quadrant across the two figures. The same kind of correspondence is in evidence between Figs. 3 and 4. Such a correspondence, of course, contributes to the two highest correlation values in Table 5 (those cutting across conditions within the same age group). However, our correlation for adults across conditions (R = ,705) is not as high as the correlation in the Shepard-Chipman study (0.917). There are various possible reasons for this discrepancy, such as the different methods for obtaining similarities data, or the fact that our adult subjects came from a psychology class pool and were not screened with respect to the task. Still, our correlation is sufficiently high as to suggest that we have successfully replicated the Shepard-Chipman results for adults in a short-term memory task. Similarly, we feel that there is also a demonstrated correspondence between the configurations for children across the two conditions. Considering the relationships between the four remaining pairs of solutions, the rotated configurations (not presented here) on the whole tell very little. The three pairs having the really low correlations (Figs. 1 and 3,2 and 3.2 and 4) show minimal correspondence between configurations. However, the comparison of 5-year-olds in the memory condition with adults in the simultaneous condition (R = .555) is quite interesting, as there is a fairly close quadrant-by-quadrant correspondence in the location of stimuli. Our correlation, then, suggests that adults’ “direct” perception tends, for some reason, to be more similar to children’s memory for stimuli than to children’s direct perception in our experimental context. This finding could be a consequence of the color dimension’s being dominant for 5-year-olds only in the simultaneous condition. DISCUSSION
We found a clear correspondence between memory-based and perceptually based judgments of object similarity by S-year-olds and by adults, although there was a slightly higher degree of correspondence for the adults. These correlations were obtained both for the similarities data (Table 2) prior to multidimensional scaling and for the scaling solutions (Table 5). For each age group these findings indicate that internal representations included much of the same information the subjects used in judging similarity between stimuli in view. As the stimulus objects were novel and complex, and the procedures of this study were designed to limit verbal coding, we further conclude that adults and 5-year-olds 4 Shepard and Chipman (1970, p. 11) used a nonsymmetric coefficient of agreement for correlating their configurations. We have computed their coefficient for our pairs ofconfigurations and found a maximum absolute discrepancy between the computed coefficients and our R’s of 0.05.
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showed quite comparable ability in a particular aspect of memory, the retention of complex visual information which is nonverbally coded. This conclusion, in conjunction with the findings of other recent work, is consistent with the view that performance in numerous memory tasks improves with development not because of augmented capacity for basic information processing, but rather because verbal skills are used increasingly to enhance encoding, retention, and retrieval processes (cf., Flavell, 1970; Kosslyn & Bower, 1974; Nelson, 1971). The fact that children’s perception seemed quite different from adults’ does not complicate our conclusions about internal representation, since we were able to compare each age group’s performance in the stimuluspresent condition. However, in many developmental studies, perceptually based differences might pose difficulties for conclusions about information retention and retrieval, and other aspects of cognitive processes. In the present study, multidimensional scaling techniques afforded the capability of comparing performance across conditions when the stimuli were present versus unavailable. Not only did the analysis yield a 2dimensional representation of the stimulus space for each cell of the experimental design, but in addition the analyses allowed us to compare, for example, the suitability of different models for attentional processes (i.e., those of Vygotsky and of Cross, cited earlier). The same scaling methods could be used in other developmental investigations to provide an increasingly precise understanding of cognitive processes and representations. REFERENCES Arabie, P. Concerning Monte Carlo evaluations of nonmetric multidimensional scaling algorithms. Psychometrika, 1973, 38, 607-608. Arabie, P., & Boorman, S. A. Multidimensional scaling of measures of distance between partitions. Journal of Mathematical Psychology, 1973, 10, 148-203. Arnold, J. B. A multidimensional scaling study of semantic distance. Journal of Experimental Psychology Monograph, 1971, 90, 349-372. Attneave, F. Dimensions of similarity. American Journal of Psychology, 1950, 63, 516-556. Cross, D. V. Multidimensional stimulus control of the discriminative response in experimental conditioning and psychophysics. Technical Report No. 05613-4-F(78[d]), University of Michigan, 1965a. Cross, D. V. Metric properties of multidimensional stimulus generalization. In D. I. Mostofsky (Ed.), Stimulus generalization. Stanford: Stanford University Press, 1965b. Ekman, G. A direct method for multidimensional ratio scaling. Psychometrika, 1963, 28, 33-tl. Flavell, J. H. Developmental studies of mediated memory. In H. W. Reese & I,. P. Lipsitt (Eds.), Advances in child development and behavior (Vol. 5). New York: Academic Press, 1970. Garner, W. R. The stimulus in information processing. American Psychologist, 1970, 25, 350-3.58.
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Indow, T., & Uchizono, T. Multidimensional mapping of Munsell colors varying in hue and chroma. Journal of Experimental Psychology, 1960, 59, 321-329. Kendall, M. G. Rank correlation methods (3rd ed.) London: Griffin, 1962. Klahr, D. A Monte Carlo investigation of the statistical significance of Kruskal’s nonmetric scaling procedure. Psychometrika. 1969, 34, 319-330. Kosslyn, S. M., & Bower, G. H. The role of imagery in sentence memory: A developmental study. Child Development, 1974, 45, 30-38. Kosslyn, S. M., Pick, H. L.. Jr., & Fariello, G. R. Cognitive maps in children and men. Child
Development,
1974, 45, 707-716.
Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964a, 29, 1-27. Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method. Psychometrika, I964b, 29, 115-129. Kruskal, J. B., & Carmone, F. How to use M-D-SCAL (Version 5M) and other useful information. Cambridge: Marketing Science Institute, 1970. Kruskal, J. B., & Carroll, J. D. Geometrical models and badness-of-fit functions. In P. R. Krishnaiah (Ed.), Mulfivariute analysis II. New York: Academic Press. 1969. Lance. G. N., & Williams, W. T. A general theory of classificatory sorting strategies II. Clustering systems. Computer Journal, 1967, 10, 271-277. Landahl, H. D. Neural mechanisms for the concepts of difference and similarity. Bulletin of Mathematical Biophysics, 1945, 7, 83-88. Levelt, W. J. M., Van de Geer, J. P., & Plomp, R. Triadic comparisons of musical intervals. British Journal of Mathematical and Statistical Psychology, 1966, 19, 163-179. Lockhead. G. R. Effects of dimensional redundancy on visual discrimination. Journal of Experimental Psychology, 1966,72, 95-104. Nelson, K. E. Memory development in children: Evidence from nonverbal task. Psychonomic Science, 1971, 25, 346-348. Odom, R. D. Effects of perceptual salience on the recall of relevant and incidental dimensional values: A developmental study. Journal of Experimental Psychology, 1972, 92, 285-291. Odom, R. D.. & Guzman, R. D. Development of hierarachies of dimensional salience. Developmental
Psychology,
1972,
6, 271-287.
Olivier, D. C. Metrics for comparison of multidimensional scalings. Unpublished, Harvard University, 1970. Ramsay, J. 0. Economical method of analyzing perceived color differences. Journal of the Optical Society of America, 1968, 58, 19-22. Seitz, V. R. Multidimensional scaling of dimensional preferences: A methodological study. Child Development, 1971, 42, 1701-1720. Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika, 1%2a, 27, 125-140: II. Psychometrika, 1962b, 27, 219-246. Shepard, R. N. Attention and the metric structure of the stimulus space. Journal of Mathematical Psychology, 1964, 1, 54-87. Shepard, R. N. Some principles and prospects for the spatial representation of behavioral science data. Paper presented at the Mathematical Social Science Board Advanced Research Seminar, Irvine, California, June 13-18, 1969. Shepard, R. N. Representation of structure in similarity data: Problems and prospects. Psychometrika, 1974, 39, 373-421. Shepard, R. N., & Cermak, G. W. Perceptual-occognitive explorations of a toroidal set of free-form stimuli. Cognitive Psychology, 1973, 4, 351-377.
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Shepard, R. N., & Chipman, S. Second-order isomorphism of internal representations: Shapes of states. Cognitive Psychology, 1970, 1, 1-17. Suchman, R. G., & Trabasso, T. Color and form preference in young children. Journal of Experimental Child Psychology, 1966, 3, 177-187. Torgerson, W. S. Theory and methods of scaling. New York: Wiley, 1958. Trabasso, T., Stave, M., & Eichberg, R. Attribute preference and discrimination shifts in young children. Journal of Experimental Child Psychology. 1969, 8, 195-209. Vygotsky, L. S. Thought and language. (Ed. & trans. by E. Hanfmann & G. Vakar). Cambridge: MIT Press, 1962. White, H. C. Anatomy of kinship. Englewood Cliffs, NJ: Prentice-Hall, 1963. RECEIVED: May 24, 1974; REVISED: August 13, 1974
OF
EXPERIMENTAL
CHILD
A Multidimensional Visual
Memory
19, 327-345 (1975)
PSYCHOLOGY
Scaling of !&Year
PHIPPS
and
of Adults
ARABIE
University
of Minnesota
M. KOSSLYN
STEPHEN Johns
Olds
Study
Hopkins
University
AND KEITH New
School
E. NELSON for
Social
Research
Nonmetric multidimensional scaling is expounded as a methodology for investigating memory development. Judgments of similarities between complex objects were obtained from 5-year-olds and adults under two conditions: (1) when objects were simultaneously present at the time of comparison, and (2) when the objects were not simultaneously present and had to be compared on the basis of shortterm memory. Scaling solutions for similarity judgments by each age group in the two conditions were obtained. Comparison within each age group of the configurations from the memory condition with the simultaneous condition indicated that the internal representations of both children and adults contained much of the information the subjects extracted from the stimuli during “direct” perception. The scaling techniques, which could be widely applied in developmental studies of memory and perception, are explained and critically discussed. One of the most difficult
tasks for experimental psychology is that of obtaining tangible representations of mental structures and processes. This problem is especially acute when one is studying young children, who are less facile with language than are adults. The present paper concerns the use of nonmetric multidimensional scaling for obtaining spatial We are indebted to S. A. Boorman, J. Metzler, D. Osherson, E. J. Shoben, and E. E. Smith for their comments and suggestions on an earlier draft of this paper. For very helpful remarks during discussions of this work, we are grateful to J. B. Arnold, R. C. Atkinson, J. M. Carlsmith, D. C. Olivier, and R. N. Shepard. We would like to thank Tom Schumacher for running some of the subjects in the experiments reported here. This research was supported by NIMH Grant MH21747 and NSF Grant GJ-443X3 to Richard C. Atkinson, NSF Grant GS-2689 to Harrison C. White, NIH Grant HD-06524 to Keith E. Nelson, and funds from the Graduate School of the University of Minnesota. Order of authorship is alphabetical. Send requests for reprints to: Phipps Arabie, Dept. of Psychology, Elliott Hall, University of Minnesota, Minneapolis, MN 55455. 327 Copyright @ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
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representations of perceived similarities among a specific set of stimuli. The particular topic of memory development, as investigated here, is only one of many developmental research areas which are amenable to multidimensional scaling (cf., Kosslyn, Pick, & Fariello, 1974). In the present study, we wished to compare the perception and memory of 5-year-olds and adults in one particular situation, involving the retention of complex visual information which is nonverbally encoded. We attempted to minimize the effects of verbal skills by using unfamiliar and difficult-to-describe stimuli and presenting these stimuli for a relatively brief period of time, while never providing, or encouraging use of, labels, and by allowing subjects to indicate their judgments nonverbally through pointing. The experimental technique which we used to study children’s and adults’ presumably nonverbal memory of visually presented objects was first introduced by Shepard and Chipman (1970). Their paradigm is not commonly used by developmental psychologists, but seems a useful technique for externalizing mental events at different ages. Shepard and Chipman had subjects rank order pairs of states of the U. S. on the basis of similarity when only the names of the states or when only line drawings of them were presented. Spatial representations of these two sets of data then were obtained through the use of nonmetric multidimensional scaling. The most important finding was the high correspondence between the spatial configurations for the two sets of stimuli. Both configurations exhibited clustering of states that were characterized by the same cartographic features (e.g., square states, like Colorado and Oregon, were close together in space). Thus, it appeared that presentation of the names alone evoked internal representations of the states from long-term memory and that those representations bore some functional similarity to line drawings and could be ranked according to the same criteria applied to the actual drawings. In the present experiment, we used the Shepard-Chipman technique, obtaining similarity rankings under stimulus present and memory conditions, but in a short-term memory situation with novel stimuli. Thus, our experiment consists of four cells: children and adults in ‘*memory” and “simultaneous” stimulus presentation conditions. To the extent that the subjects at either age make judgments from memory which parallel their judgments when the stimuli are physically present, then the subjects must have retained much of the same visual information used to make similarity judgments when the stimuli are present. Furthermore, to the extent that the 5-year-olds and adults display similar judgments, then we can infer that their respective internal representations, or perceptions, include much of the same information, and that the judgments themselves are based on similar criteria.
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METHOD Design for Scaling
We wish to represent the subjects’ perceived similarities of our stimulus set in terms of a geometric model in which short distances between stimuli represent high similarity and large distances indicate low similarity between stimuli. Given such a representation in two dimensions, for example, we can compare the “map” of the stimulus space obtained from adults with that from children, as well as comparing the memory and simultaneous conditions. The method we have used to obtain such a representation of perceived similarity is that of nonmetric multidimensional scaling. Originally called “analysis of proximities,” this method was devised and implemented by Shepard (1962a,b). A stronger logical basis for the procedure was provided by Kruskal (1964a,b), who also wrote MDSCAL (Kruskal & Carmone, 1970), the most widely used computer program for nonmetric multidimensional scaling. This procedure is especially appropriate for developmental studies because it uses only the ordinal information available from the interstimulus similarities data. That is, the program uses only the rank order of the n * (n - 1)/2 pairwise similarities (or a subset thereof), as opposed to doing arithmetic with the actual numerical values. Furthermore, the fact that the procedure is “monotone invariant” (i.e., uses only the ordinal information from the data) means that substantially weaker assumptions about the data are entailed than is the case for metric multidimensional scaling (e.g., factor analysis), in which it is assumed that the data are on an interval scale (e.g., Seitz, 1971). Kruskal, in his iterative procedure, also seeks to minimize a badness-offit function called “stress,” which tells how far the data depart from the best-fitting monotone function relating psychological dissimilarity to metric distance. Thus, stress provides an indication as to how well the program is able to construct a metric representation of the relationships implicit in the data, beginning with only a rank ordering of the interstimulus similarities as input. The question naturally arises: How are we to obtain such a similarities matrix from 5-year-old subjects? Clearly, we wanted a very simple and nonverbal task, with the consequence that Shepard and Chipman’s procedure requiring subjects to rank order many pairs of stimuli is inappropriate. There are many other traditional tasks suitable for adult subjects: obtaining category judgments of similarity on (typically) an 1 l-point scale, paired comparisons of differences (Ramsay, 1968), “direct” ratio scaling (Indow & Uchizono, 1960; Ekman, 1963), triads (Torgerson, 1958), and various other techniques. We found the method
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of triads to be attractive insofar as it requires no familarity with the number scale, unlike many other methods. Rather, at the presentation of each triad, the subject has only to decide which two of the three stimuli are most similar and which two are least similar. The main disadvantage is that the standard method entails n * (n - 1). (n - 2)/6 different trials, which is 220 for n = 12 stimuli. An imaginative variation on the traditional method is described by Levelt, van de Geer, and Plomp (1966), who used balanced subsets of 35 triads for 15 stimuli, instead of the full set of 455. (The interesting details of this design are given in Levelt et nl. (1966, pp. 165-166).) In the present experiment, a balanced subset containing only 66 triads of the 12 stimuli was used and the difficulty of the judgment required from the subject was also lessened. Specifically, for a given triad (a,x,b), the subject was asked whether a or b was more similar to the standard x, thus affording a relative comparison of the similarity of the two pairs (a,x) and (b,x). (The traditional judgmental task, as employed by Levelt et al., implicitly compares the similarity of all three pairs in the triad.) We ran a computerized search to find a set of 66 triads (a,x,b) which was balanced in the sense that each of the 66 pairs [(a,x) or (b,x)] occurred exactly twice. Obviously, it would have been desirable to have more than two such presentations per subject of each stimulus pair, but we felt that many more than 66 presentations would place an excessive demand on a 5-year-old subject’s span of attention. Given this balanced set, five different orders were used across subjects, and for each of these orders, no stimulus occurred in two successive triads. In the present study, our intention was to have stimuli which were relatively difficult to encode verbally. For our stimuli we constructed a set of 12 toy structures (made of LEG0 blocks, manufactured by Samsonite and specially painted) looking somewhat like 3-dimensional miniatures of asymmetric office buildings. Each stimulus was 3 in. wide and 34 in. in height. It was our intention that these stimuli would be equally novel to both 5-year-olds and adults and thus would circumvent problems arising from age differences in familiarity with stimulus materials. Holding the number of blocks (20) in each stimulus constant, the two physical dimensions systematically varied were: (A) the number of different colors (2-13, selected from a set of 13 colors); and (B) the number of “crossings” (3-14), that is, how many blocks were crossed completely (overlapped on two sides by a block immediately above). The values of each stimulus along these dimensions certainly can be verbalized, although not very easily within a short period of time. Furthermore, this design for stimulus definition resulted in stimuli quite rich in visual detail. From the 144 possible pairs of values along the two discrete dimensions, 12 stimuli were selected by means of a random
SCALING
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TABLE LATIN
SQUARE
TYPE
OF DESIGN
9
Number of different colors (2-13)
6 11 l3
3
FOR THE
8
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1 12 STIMULI
USED
IN BOTH
CONDITIONS
Number of orthogonal crossings (3-14) 10 7 12 11 5 13 14
4
6
1 2
5 2 ;; L
8 9 12 7
3 4 5
Stimulus Number
6 7 8 9 10 11 12
pairing of the 12 levels of the two dimensions. Since in purpose this selection scheme resembles a Latin square (which reduces the number of stimuli by a factor of n), for convenience it will henceforth be referred to as a Latin square design, although the term is used only in analogy. The specifications of the stimulus set are given in Table 1. Subjects
The children were 28 boys attending Bing Nursery School at Stanford. The median age was 4 years, 9 months, and the range in age was from 4 years, 6 months to 5 years, 2 months. The adult subjects were 20 male undergraduates satisfying a requirement for the introductory psychology course at Stanford. Experimental Procedure
The sessions associated with the two conditions of the experiment were scheduled 1 week apart for each subject. For both conditions, the 5-year-old subjects participated in two sessions on consecutive days, with 33 trials per session. The adults, however, were run in a single session of 66 triads for each condition. Memory condition. The memory condition preceded the simultaneous condition for all the subjects. In the memory condition 66 triads (a,x,b) were presented one at a time, with either stimulus a or b presented for 4 set, then b or a for 4 set, and, finally, the standard for the same duration. After the standard had been removed, the subject’s task was to indicate whether a or b was more similar to the standard x. The apparatus for presenting the stimuli was a small chest, placed at the level of the
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subject’s head, with three open compartments, into each of which the experimenter could place one of the stimuli constituting a triad. The subject’s head was 15-18 in. from each stimulus position. On each trial, the test stimuli, a and b, were presented separately, in either the left or right compartment (randomly determined). The standard x appeared last and was always in the center compartment. The experimenter manually placed the stimuli in the compartments for presentation, and also recorded the responses. All subjects were instructed to watch the first two stimuli carefully and to “try to remember what they looked like.” After the last stimulus (the standard) had been presented for 4 set in the center compartment and taken away, the subject was to point to the position of the stimulus more like the one in the middle. Prior to the first actual test trial of this condition, each of the 12 stimuli was presented, one at a time for 3 set in the middle compartment, and subjects were encouraged to study the stimuli. To ensure that the general procedure was clear to the 5-year-olds, a set of 12 stimuli somewhat similar to those actually presented in the experiment was first used in four practice triads. As noted above, for the 5-year-old subjects, both the memory and simultaneous conditions were split into two sessions each having 33 trials. All of the 66 trials within either condition occurred on consecutive days. Any subject emitting a run of seven responses corresponding exclusively to either side of the chest was discontinued on the presumption of a strong hand-bias. Four potential subjects were eliminated by this rule. At the beginning of the second half of the memory trials, the 5year-old subjects were reminded of the “game played last time.” If a subject seemed uncertain, then the practice triads were once again presented. Simultaneous condition. All subjects were given new instructions at the beginning of this condition. Subjects were informed that all three stimuli would now be presented simultaneously for 12 sec. As before, the subject was instructed to indicate whether the stimulus on the left or right was more similar to the standard. In the simultaneous condition, the same 66 triads in the same order (for a given subject) were again presented, using the same apparatus. The only procedural difference was that all three stimuli of the given triad were presented simultaneously for 12 sec. Once again, the 5-year-olds were reminded of the “game” at the beginning of the second session of trials. RESULTS
Before turning to the spatial configurations from ‘MDSCAL, Table 2 gives the values of r6 (Kendall, 1962) for comparing the raw group
SCALING
VISUAL TABLE
RANK-ORDER MATRICES
5-year-olds Memory 5-year-olds Simultaneous Adults Memory Adults Simultaneous ‘I These correlations scaling. * p i .05. *** p < .OOl.
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MEMORY 2
CORRELATIONS (tnu b) BETWEEN SIMILARITY FOR THE 4 CELLS OF THE EXPERIMENT" 5-year
5-year
Memory
Simultaneous
Adults Memory
Adults Simultaneous
X .416*** ,115 .191*
.416*** X .098 ,023
.I15 ,098 X .518***
.191* .023 .518*** X
are for the “raw”
data, prior to analyses from multidimensional
data (i.e., the similarity matrices, obtained from the 66 triads) for the 4 cells of the experiment.’ Using Kendall’s (1962) test of significance, the rank-order correlations across the conditions are highly significant for both adults (z = 6.15) and for Syear-olds (z = 4.93), and the correlation between Syear-olds in the memory condition and adults in the simultaneous condition is significant at the .025 level (z = 2.26). These findings, then, indicate that both the adults’ and the 5-year-olds’ internal representations seemed to depict an underlying psychological structure highly similar to that evinced by judgments made by each respective age group when stimuli were physically present (simultaneously). This result suggests that subjects’ internal representations of the stimuli were in some way related to subjects’ perception in the simultaneous condition. However, the nonsignificant correlation between age groups in the simultaneous condition suggests that perceptually based judgments for the two groups were quite different. The Spatial Representations
Although the rank-order correlations in Table 2 gave us a quantitative indication as to which cells of the experiment were more closely related, the scaling solutions give a better (qualitative) overall picture. Since for r The similarity matrix S is obtained as follows. Beginning with a lower triangular matrix (i > j) S = 0, we consider each of the 66 triads of the form (i,j,k). Whenever the standard j is judged to be more similar to i than to k, we increment s(ij) by 1. Hence, every entry s(ij) of the individual’s matrix will have a value of 0, 1, or 2, since each stimulus pair is compared twice in the 66 triads. For the group data for a given cell of the experiment, we obtain each entry of the matrix by summing over that entry for individuals’ data.
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in,
.‘.__.
NELSON
h,
ii------..~..?P_.~.-.--
(12, 61
1. Euclidean configuration for Syear-olds in memory condition (stress = ,201). For each stimulus, the number of colors and the number of crossings are given in brackets. Those stimuli having a large number of crossings form a convex cluster. FIG.
n = 12, we have too few stimuli to support (numerically and statistically) a three or higher dimensional solution, we consider only 2dimensional configurations, as shown in Figs. l-4. In using the scaling procedures, the “primary approach” to ties, and stress formula one as the badness-of-fit function (Kruskal & Carroll, 1969) were specified as options for MDSCAL-SM (Kruskal & Carmone, 1970). Each of the (Euclidean) configurations in Figs. l-3 corresponds to the lowest stress value over at least 40 different random initial configurations. According to Klahr (1969), the .05 level of the distribution of stress for scaling 12 stimuli in a 2-dimensional Euclidean space is 0.211, so that our stress values (indicated under each figure) for these three solutions should be considerably better than if the data were random. However, technical
FIG. 2. Euclidean configuration for S-year-olds in the simultaneous condition (stress = ,208). For each stimulus, the number of colors and the number of crossings are given in brackets. Although both colors and crossings offer suitable bases for clusters, the cluster defined by the stimuli having a large number of colors is preferred.
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FIG. 3. Euclidean configuration for adults in the memory condition (stress = ,195). For each stimulus, the number of colors and the number of crossings are given in brackets. Although both colors and crossings offer suitable bases for clusters, the cluster defined by the stimuli having a small number of crossings is preferred.
arguments have been made elsewhere (Arabie & Boorman, 1973; Arabie, 1973) suggesting that Klahr’s stress criteria are really quite liberal (Fig. 4 will be discussed in detail below). The fact that we chose a Latin square type of design for defining the stimuli entails many problems for interpreting the spatial configurations. Ideally, we would have used the full set from a complete factorial design. However, the combinatorial aspects inherent in the method of scaling employed here, as well as the special difficulties entailed in running Syear-old subjects dictated using a much more restricted group of
FIG. 4. City-block configuration for adults in the simultaneous condition (stress = .217). For each stimulus, the number of colors and the number of crossings are given in brackets. Although both colors and crossings offer suitable bases for clusters, the cluster defined by the stimuli having a large number of crossings is preferred. Axes are included here because the city-block metric, unlike the Euclidean, gives a (mathematically) preferred orientation to the configuration. up to a reflection of the axes.
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stimuli. A Latin square stimulus set renders inapplicable many of the more common methods and approaches (e.g., hierarchical clustering) for interpreting our data. However, we can still ask which of our two physical dimensions (i.e., number of colors or number of orthogonal crossings) seems to be more important for the perceived similarity of the stimuli. This empirical question is related to the probiem of dimensional salience or attribute preference (cf. Odom & Guzman, 1972; Odom, 1972; Suchman & Trabasso, 1966; Trabasso, Stave, & Eichberg. 1969). Dimensional dominance test. The most clear-cut demonstration of one physical dimension’s dominance obviously would be a l-dimensional projection2 in the configuration along which the stimuli were ordered according to one of the dimensions. However, such a strategy does not succeed for any of the configurations in Figs. 14. It was our intention to devise an extremely simple basis for clustering along physical dimensions, such that when compared to the multitude of algorithms for clustering (cf. Lance & Williams, 1967). ours would appear as relatively simple as the sign test is among the many methods for assessing statistical significance. Like the sign test, our method is concerned with answering a simply stated question: Can either (or both) of the dimensions offer a basis for interpreting the spatial solution? The scheme that we have used here for investigating dimensional dominance is based on the simple observation that if the number of crossings, for instance, is important, then stimuli 6 and 7 should be close together, since they have 12 and 11 crossings, respectively. Alternatively, if the number of colors is more important, we would expect stimuli 5 and 6 to be near each other, as they have 3 and 4 different colors, respectively. For a complete set of factorially designed stimuli. we could hope to find a rather obvious pattern of grouping according to one of the physical dimensions. Again, however, with a Latin square set of stimuli, we cannot expect the pattern to be as readily apparent. Nonetheless, one would expect stimuli having a lower value (e.g., the number of different colors) on the more relevant dimension to be closer to each other (more similar) than to the remaining stimuli having higher values. We have somewhat arbitrarily selected “medians” for the range of values on the physical dimensions, and then looked for convex clusters. More specifically, since the number of colors ranges from 2 to 13, we have chosen the “cut” on that dimension to be between 7 and 8 colors. and between 8 and 9 for the number of crossings, which ranges from 3 to 14. Thus. we are looking at four overlapping sets of stimuli (high or low p A unidimensional projection of a 2-dimensional solution sional MDSCAL solution, since nonmetric multidimensional tain difficulties in handling the l-dimensional case.
is to be preferred to a l-dimenscaling algorithms have cer-
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number of colors and few or many crossings) in each of the configurations to see if any form a convex cluster encompassing no other stimuli. In Figs. 1-4, these clusters have been enclosed by solid lines for number of colors and by broken lines for number of crossings. At least one convex cluster can be constructed on the basis of crossings for each of the four solutions. In Fig. 1, for 5-year-olds in the memory condition, no similar arrangement is possible on the basis of number of colors. In contrast, Fig. 2, for children in the simultaneous condition, shows that both the sets of stimuli having a large number of colors (8-13) and crossings (9-14) allow for convex clustering. Thus, although we can characterize the judgments of 5-year-olds in the memory condition as being dominated by form, we cannot argue that either form or color dominates for that age group in the simultaneous condition. Similarly, our dimensional clustering scheme fails to provide any decisive characterization of either of the solutions for adults, since both physical dimensions serve as suitable bases for convex clusters. A further elaboration of the simple clustering test gives a (somewhat weak) criterion for deciding which is the more important of the two physical dimensions in Figs. 24. Specifically, given that both physical dimensions provide suitable sets of stimuli as candidates for convex clusters, we assign priority to the dimension which defines the set whose complement more nearly forms a convex cluster. In Fig. 2, then, the set having a small number of colors is closer to forming a convex cluster than is the set having a small number of crossings. Thus, for .5-year-olds in the simultaneous condition, we would decide that the color dimension is closer to being dominant. The same type of argument would lead us to conclude that the number of crossings rather than colors is the more important dimension in both adult conditions (Figs. 3. 4). The fact that both color and form seemed to be important physical dimensions in the simultaneous condition would be expected from the results of Odom and Guzman (1972) and Odom (1972). However, the finding that color dimension was more useful here for interpreting our spatial model of the 5year-olds perceptions (simultaneous condition) is somewhat in contrast to other experimenters’ results (e.g., Suchman & Trabasso, 1966; Trabasso, Stave, & Eichberg, 1969). It should be noted though, that our variable is the number of colors rather than salience or identity of colors, as employed by most previous investigators. Minkowski
Metrics
and Psychological
Spaces
With Fig. 4 we introduce another aspect of multidimensional scaling representations that has not yet been mentioned in this paper, and that concerns the underlying metric of the stimulus space. Although it is
338
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often assumed by default that most psychological spaces are Euclidean. there is no a priori reason why this assumption should be valid. A more general distance formula is given by the Minkowski I’ metric, for which the distance between two points (stimuli y and Z) in a 2-dimensional space is: d(y, 7,) = (i
lyj - q, ,=I
where Y > 1, and i is the index over the psychological dimensions (and we are only considering the 2-dimensional case, given just 12 stimuli). When I’ = 2, we have the familiar Euclidean distance formula. It is traditionally argued (Attneave. 1950; Garner. 1970; Lockhead, 1966; Shepard, 1964; Torgerson, 1958) that Euclidean distance is more appropriate when the stimuli are processed as “integral” or “unanalyzable wholes,” that is, when the dimensions of the stimuli are not perceived independently or separately. For I’ = 1, the distance formula corresponds to the city-block metric, first investigated by Landahl (1945) and Attneave (1950) as being relevant to psychological processes. A very nice psychological interpretation of the Minkowski metrics for the special cases I’ = 1 and 2, as well as infinity has been given by Cross (1965a,b). Omitting the elegant details of Cross’s work while briefly reviewing some of his arguments, we note that for the city-block metric, distances are additive along the axes of the psychological dimensions, with the projections along each dimension being given equal weight in the function determining perceived similarity. Thus, it can be argued that the city-block metric is appropriate for an experimental context where the dimensions of the stimuli are perceived separately and thus are processed independently. As Y increases, the axis along which the two stimuli are most discrepant contributes more heavily to their psychological distance (i.e., increasing the exponent causes the dimension having the greatest projection to be differentially given greater weight). In the limiting case where I’ is infinite (sometimes called the “max metric”), the distance between each pair of points may be regarded as unidimensional since it is determined solely by the axis of greatest separation, although the space in which the configuration is embedded may be of much higher dimensionality. An underlying max metric (e.g., Arnold, 1971) implies an experimental situation where one maximally salient psychological dimension is the basis for each pair-wise comparison; moreover, in a multidimensional space, the dimension perceived as most salient will vary across comparisons of different stimulus pairs. Occasionally, the city-block metric has been found superior to the Euclidean metric for representing psychological processes (e.g., Cross, 1965a.b; Shepard & Cermak, 1973). Comparisons of solutions for the
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TABLE 3 MINIMUM STRESS (FORMULA 1) VALUES OBTAINED FOR EACH CELL OF THE EXPERIMENT, USING BOTH CITY-BLOCK AND EUCLIDEAN METRICS" Metric:
r= 1
r=2
.205 ,223 ,221
,201 ,208 ,195 ,203
Subject kTOUP
Condition
S-year-olds S-year-olds Adults Adults
Memory Simultaneous Memory Simultaneous
,217
(1 Each r = 1 solution is the best stress over at least 120 different random initial configurations, and each r = 2 value is taken over at least 40 different random initial configurations.
two metrics strictly on the basis of stress values have rarely favored the city-block metric, and possible doubt has recently been cast on the face value of such comparisons (Shepard, 1974). The purpose in the present study was to find interpretable configurations (cf., remarks by Shepard, (1969, p. 41)), and we have found that the city-block metric yielded interpretable clusters for adults in the simultaneous condition, and that the Euclidean solution was much less interpretable. Hence we believe it is justifiable to argue that the city-block metric represents the results better for adults in that cell of the experiment. Therefore, all subsequent discussion of the results for that cell refers to the city-block configuration shown in Fig. 4. In passing, it might be noted that of the four cells in the present experiment, “adults in the simultaneous condition” is clearly the cell most similar to the conditions under which other investigators have sought and occasionally found evidence for the city-block metric. The three remaining city-block solutions were uniformly uninterpretable, and none yielded even a single convex cluster, as would be required by the scheme outlined above. Attempts at devising other schemes for interpreting the three city-block solutions also were unsuccessful. Table 3 includes stress values of configurations from each cell for both values of the Minkowski r. The Euclidean case gives lower stresses, although the city-block stresses are all within about 10% of the corresponding Euclidean values. Each value of stress for the city-block case is the minimum value found using at least 120 different random initial configurations. We have dismissed three of the city-block configurations as uninterpretable. However, since the city-block stress values were all within about 10% of the corresponding Euclidean values, a skeptical reader might well ask if the city-block solutions3 were really different from 3 There are no Monte Carlo studies of stress available for the city-block metric.
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TABLE SUMMARY
AND
4
STATISTICS FROM COMPARISONS CORRESPONDING TO THE 4 CELLS
Cell: 5-year-olds in Memory Condition 5-year-olds jn Simultaneous Condition Adults in Memory Condition Adults in Simultaneous Condition
NELSON
OF THE SCALING SOLUTIONS OF THE EXPERIMENT
Pearson R between coordinates of Euclidean solution rotated to corresponding city-block solution ,748 .626 .b27
.729
those of the Euclidean metric. Table 4 gives the values of Pearson R between coordinates of rotated Euclidean solutions and their city-block counterparts. The fact that all values are considerably less than unity indicates that the pairs of configurations are appreciably different. The third special case of the Minkowski metrics considered by Cross, the max metric (v = infinity), has been briefly mentioned. Although somewhat counterintuitive, it is the case that 2-dimensional spatial configurations for r = 1 (city-block) and r = infinity (max metric) differ by only a 45’ rotation. (Euclidean distance is the only r-metric for which distances remain invariant under a rotation of the axes.) A very straightforward derivation of this relationship between the two metrics in the 2dimensional case is given by Arnold (1971, p. 357). The fact that the city-block/max metric solutions are uninterpretable for the Syear-olds is rather interesting when one recalls Vygotsky’s (1962, p. 64) discussion of the “chain complex” and its role in children’s similarity groupings. After the collection state of thinking in complexes, we must place the chain complex-a dynamic, consecutive joining of individual links into a single chain, with meaning carried over from one link to the next . . The decisive attribute keeps changing during the entire process. There is no consistency in the type of bonds or in the manner in which a link in the chain is joined with the one that precedes and the one that follows it.
If one replaces Vygotsky’s “decisive attribute” with “most important dimension,” then all successive pairs of the chain are linked on a unidimensional basis (viz,, along the axis having the longest projection) and Y = infinity is suggested for the underlying psychological space (cf., White, 1963, Chap. 1). Unfortunately, the present results lend no support to this implication of Vygotsky’s theory. Comparing Conjigurations
As has been emphasized, we chose multidimensional principal method of data analysis because it facilitates
scaling as our comparison of
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psychological spaces across conditions. We have already given the values of rb on our raw data, but those correlations told little more for the adult subjects’ data than one would have anticipated after reading the Shepard-Chipman study. The rank-order correlations did, however, suggest that the 5-year-olds’ internal representations were in some sense related to veridical representations of the stimuli. This inference would have a stronger logical basis if the psychological spaces obtained from scaling techniques were shown to be similar across conditions. In order to render the scaling solutions comparable, we have used the computer program CONGRU (Olivier, 1970) to rotate various pairs of our configurations to maximal congruence (in a least-squares sense) CONGRU allows the user to rotate a Euclidean configuration to match another (target) configuration, and the program also gives the value of Pearson R between coordinates of the two solutions as a symmetric measure of agreement. Table 5 is a quantitative summary of the results from running CONGRU on the four configurations shown in Figs. l-4. The pattern of correlation is very similar to that for rb on the raw data, as shown in Table 2. In both tables, there is a strong correlation for both adult and 5year-old subjects between memory and simultaneous conditions, with the value for adults being larger. Also, the third largest correlation is again that between adults in the simultaneous condition and 5-year-olds in the memory condition. Similarly, as in Table 2, the lowest correlation is that between adult and 5-year-old subjects in the simultaneous condition. Thus, it appears that the two age groups have different perceptual bases for their judgments in the simultaneous condition, as will be discussed below. The scaling configuration shown in Fig. 1 has been rotated to maximal congruence with Fig. 2, while Fig. 3 has been rotated to match Fig. 4 as TABLE
5
PEARSON R PRODUCT-MOMENT CORRELATIONS OF CONFIGURATIONS FOR THE 4 CELLS AFTER ROTATION TO MAXIMAL
5-year-olds Memory 5-year-olds Simultaneous Adults Memory Adults Simultaneous (r = 1)
BETWEEN COORDINATES OF THE EXPERIMENT CONGRUENCE
S-year
5-year
Memory
Simultaneous
Adults Memory
Adults Simultaneous (r = 1)
x .627 ,411 ,555
.627 X ,406 ,405
,411 ,406 X .705
,555 .405 ,705 x
Note: The coordinates for adults in the simultaneous condition are from the city-block solution. (If the best Euclidean solution had been used for adults in the simultaneous condition, the first 3 entries of the last row and column would have been ,352, ,304, ,755.)
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a target configuration. When Figs. 1 and 2 are compared visually, it is clear that stimuli seem to be similarly located quadrant-by-quadrant across the two figures. The same kind of correspondence is in evidence between Figs. 3 and 4. Such a correspondence, of course, contributes to the two highest correlation values in Table 5 (those cutting across conditions within the same age group). However, our correlation for adults across conditions (R = ,705) is not as high as the correlation in the Shepard-Chipman study (0.917). There are various possible reasons for this discrepancy, such as the different methods for obtaining similarities data, or the fact that our adult subjects came from a psychology class pool and were not screened with respect to the task. Still, our correlation is sufficiently high as to suggest that we have successfully replicated the Shepard-Chipman results for adults in a short-term memory task. Similarly, we feel that there is also a demonstrated correspondence between the configurations for children across the two conditions. Considering the relationships between the four remaining pairs of solutions, the rotated configurations (not presented here) on the whole tell very little. The three pairs having the really low correlations (Figs. 1 and 3,2 and 3.2 and 4) show minimal correspondence between configurations. However, the comparison of 5-year-olds in the memory condition with adults in the simultaneous condition (R = .555) is quite interesting, as there is a fairly close quadrant-by-quadrant correspondence in the location of stimuli. Our correlation, then, suggests that adults’ “direct” perception tends, for some reason, to be more similar to children’s memory for stimuli than to children’s direct perception in our experimental context. This finding could be a consequence of the color dimension’s being dominant for 5-year-olds only in the simultaneous condition. DISCUSSION
We found a clear correspondence between memory-based and perceptually based judgments of object similarity by S-year-olds and by adults, although there was a slightly higher degree of correspondence for the adults. These correlations were obtained both for the similarities data (Table 2) prior to multidimensional scaling and for the scaling solutions (Table 5). For each age group these findings indicate that internal representations included much of the same information the subjects used in judging similarity between stimuli in view. As the stimulus objects were novel and complex, and the procedures of this study were designed to limit verbal coding, we further conclude that adults and 5-year-olds 4 Shepard and Chipman (1970, p. 11) used a nonsymmetric coefficient of agreement for correlating their configurations. We have computed their coefficient for our pairs ofconfigurations and found a maximum absolute discrepancy between the computed coefficients and our R’s of 0.05.
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showed quite comparable ability in a particular aspect of memory, the retention of complex visual information which is nonverbally coded. This conclusion, in conjunction with the findings of other recent work, is consistent with the view that performance in numerous memory tasks improves with development not because of augmented capacity for basic information processing, but rather because verbal skills are used increasingly to enhance encoding, retention, and retrieval processes (cf., Flavell, 1970; Kosslyn & Bower, 1974; Nelson, 1971). The fact that children’s perception seemed quite different from adults’ does not complicate our conclusions about internal representation, since we were able to compare each age group’s performance in the stimuluspresent condition. However, in many developmental studies, perceptually based differences might pose difficulties for conclusions about information retention and retrieval, and other aspects of cognitive processes. In the present study, multidimensional scaling techniques afforded the capability of comparing performance across conditions when the stimuli were present versus unavailable. Not only did the analysis yield a 2dimensional representation of the stimulus space for each cell of the experimental design, but in addition the analyses allowed us to compare, for example, the suitability of different models for attentional processes (i.e., those of Vygotsky and of Cross, cited earlier). The same scaling methods could be used in other developmental investigations to provide an increasingly precise understanding of cognitive processes and representations. REFERENCES Arabie, P. Concerning Monte Carlo evaluations of nonmetric multidimensional scaling algorithms. Psychometrika, 1973, 38, 607-608. Arabie, P., & Boorman, S. A. Multidimensional scaling of measures of distance between partitions. Journal of Mathematical Psychology, 1973, 10, 148-203. Arnold, J. B. A multidimensional scaling study of semantic distance. Journal of Experimental Psychology Monograph, 1971, 90, 349-372. Attneave, F. Dimensions of similarity. American Journal of Psychology, 1950, 63, 516-556. Cross, D. V. Multidimensional stimulus control of the discriminative response in experimental conditioning and psychophysics. Technical Report No. 05613-4-F(78[d]), University of Michigan, 1965a. Cross, D. V. Metric properties of multidimensional stimulus generalization. In D. I. Mostofsky (Ed.), Stimulus generalization. Stanford: Stanford University Press, 1965b. Ekman, G. A direct method for multidimensional ratio scaling. Psychometrika, 1963, 28, 33-tl. Flavell, J. H. Developmental studies of mediated memory. In H. W. Reese & I,. P. Lipsitt (Eds.), Advances in child development and behavior (Vol. 5). New York: Academic Press, 1970. Garner, W. R. The stimulus in information processing. American Psychologist, 1970, 25, 350-3.58.
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Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 1964a, 29, 1-27. Kruskal, J. B. Nonmetric multidimensional scaling: A numerical method. Psychometrika, I964b, 29, 115-129. Kruskal, J. B., & Carmone, F. How to use M-D-SCAL (Version 5M) and other useful information. Cambridge: Marketing Science Institute, 1970. Kruskal, J. B., & Carroll, J. D. Geometrical models and badness-of-fit functions. In P. R. Krishnaiah (Ed.), Mulfivariute analysis II. New York: Academic Press. 1969. Lance. G. N., & Williams, W. T. A general theory of classificatory sorting strategies II. Clustering systems. Computer Journal, 1967, 10, 271-277. Landahl, H. D. Neural mechanisms for the concepts of difference and similarity. Bulletin of Mathematical Biophysics, 1945, 7, 83-88. Levelt, W. J. M., Van de Geer, J. P., & Plomp, R. Triadic comparisons of musical intervals. British Journal of Mathematical and Statistical Psychology, 1966, 19, 163-179. Lockhead. G. R. Effects of dimensional redundancy on visual discrimination. Journal of Experimental Psychology, 1966,72, 95-104. Nelson, K. E. Memory development in children: Evidence from nonverbal task. Psychonomic Science, 1971, 25, 346-348. Odom, R. D. Effects of perceptual salience on the recall of relevant and incidental dimensional values: A developmental study. Journal of Experimental Psychology, 1972, 92, 285-291. Odom, R. D.. & Guzman, R. D. Development of hierarachies of dimensional salience. Developmental
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Olivier, D. C. Metrics for comparison of multidimensional scalings. Unpublished, Harvard University, 1970. Ramsay, J. 0. Economical method of analyzing perceived color differences. Journal of the Optical Society of America, 1968, 58, 19-22. Seitz, V. R. Multidimensional scaling of dimensional preferences: A methodological study. Child Development, 1971, 42, 1701-1720. Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function. I. Psychometrika, 1%2a, 27, 125-140: II. Psychometrika, 1962b, 27, 219-246. Shepard, R. N. Attention and the metric structure of the stimulus space. Journal of Mathematical Psychology, 1964, 1, 54-87. Shepard, R. N. Some principles and prospects for the spatial representation of behavioral science data. Paper presented at the Mathematical Social Science Board Advanced Research Seminar, Irvine, California, June 13-18, 1969. Shepard, R. N. Representation of structure in similarity data: Problems and prospects. Psychometrika, 1974, 39, 373-421. Shepard, R. N., & Cermak, G. W. Perceptual-occognitive explorations of a toroidal set of free-form stimuli. Cognitive Psychology, 1973, 4, 351-377.
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Shepard, R. N., & Chipman, S. Second-order isomorphism of internal representations: Shapes of states. Cognitive Psychology, 1970, 1, 1-17. Suchman, R. G., & Trabasso, T. Color and form preference in young children. Journal of Experimental Child Psychology, 1966, 3, 177-187. Torgerson, W. S. Theory and methods of scaling. New York: Wiley, 1958. Trabasso, T., Stave, M., & Eichberg, R. Attribute preference and discrimination shifts in young children. Journal of Experimental Child Psychology. 1969, 8, 195-209. Vygotsky, L. S. Thought and language. (Ed. & trans. by E. Hanfmann & G. Vakar). Cambridge: MIT Press, 1962. White, H. C. Anatomy of kinship. Englewood Cliffs, NJ: Prentice-Hall, 1963. RECEIVED: May 24, 1974; REVISED: August 13, 1974
