The Value of Cognitive Diversity: The Correlation of Local Aggregates with World Standards.1

James Shilts Boster

University of Connecticut

Abstract

As one increases the size of a pool of informants performing a similarity judgment task, the mean correlation of the aggregated responses to a world standard is

equal to

Nr

r

r

xxxx

xy

−+

1, where rxy is the average individual informant’s correlation with

the world standard, rxx is the average correlation among informants on the similarity judgment task, and N is the number of informants in the pool of aggregated responses. In six studies spanning three domains (color, verbs of disintegration, emotions), the r2 of this relationship is above .999. This result can be interpreted as one of the consequences of the fact that cases of cross-cultural universals are rooted in individual cognition. Further, the strength of the relationship suggests that the existence of cross-cultural universals imposes a limit on the magnitude of intra-cultural variation: The square root of the average correlation among informants on the similarity judgment task can be no lower that the average individual correlation with the world standard. At this limit of minimum possible intra-cultural variation, there could be no aspect of the local cultural pattern that was not subsumed by the universal, because the correlation of an infinite sized aggregate with the world standard would be one. Finally, the result also shows that it is precisely the disagreement among informants that allows their aggregation to so closely approximate the world standard.

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Culture is the information pool that emerges when members of a community attempt to make sense of the world and each other as they struggle and collaborate with each other to get what they want (e.g., food, sex, power, acceptance, etc.). Because individuals construct their conceptions of the world from their own experiences and for their own motivations, their understandings vary from one another depending on the characteristics of the individuals, the nature of the domain learned, and the social situations in which learning takes place. (modified from Boster, 1986).

Cognitive anthropologists since Roberts (1964) have seen culture as an information pool. The metaphor is an apt though sloppy one: culture is a fluid with nothing to constrain it, save a container. The metaphor frees us from the expectation that cultural boundaries (the limits of the information pools) will coincide with the boundaries of social groups – some information will spill over from one group to another. Culture is a collective representation, an aggregation of what it is that individuals have learned, but individuals in different social groups may come to similar understandings based on common inferences from experience (Boster, 1987). Indeed, cognitive anthropologists have documented a number of cases of strong cross-cultural universals2 in which particular domains are understood in fundamentally similar ways by different social groups. These semantic universals have been most thoroughly explored in the case of color classification (Berlin and Kay, 1969; Kay and McDaniel, 1978; Kay, Berlin, and Merrifield, 1991, Kay and Maffi, 1999) and folk biological classification (Bulmer, 1970; Berlin, Breedlove, and Raven, 1973; Boster, Berlin, and O’Neill, 1986; Boster and D’Andrade, 1989), although they have been described for other domains as well. 3

The common denominator of cases of cross-cultural universals is that the universals are rooted in the details of individua l cognition: Linguistic communities agree because the individual members of them discern the same underlying pattern in the phenomenon. Usually, this is a result of the fact that the phenomenon has some kind of intrinsic structure independent of the observer (e.g., the biological world) and that humans share a facility for discerning that structure (Boster and D’Andrade, 1989). In other cases, such as color classification, the source of structure in experience that leads to the existence of cross-cultural universals is controversial. Some argue that the phenomenon has no intrinsic structure per se (the photons of light that produce the color spectrum continuously vary in their energies) and that the universal appearance of structure is a product of the way that the neurophysiology of perception interacts with the color spectrum. Others assert that the dimensions of contrast in colo r lexicons directly reflect dimensions of difference in reflected light.4 Regardless of the source of the structure in experience that leads to agreement, the fact that cross-cultural universals generally have their origin in individual cognition has a number of consequences. These consequences have been best documented in the case of color classification. First, differences among languages are mirrored by variation among individual speakers of the same language : Speech communities often contain speakers at adjacent stages in the evolutionary sequence with the younger speakers tending to have more advanced color lexicons than older speakers (Kay, 1975; Dougherty 1977). Second, individuals ’

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conceptions of the internal structure of color categories are often congruent with the evolution of color lexicons: The bifocality of the Tarahumaran GRUE term “siyoname” in green and blue anticipates next stage in color term evolution (Burgess, Kempton, and MacLaury, 1983). Third, individuals respond to the fundamental color foci regardless of the degree of elaboration of their color lexicons: Although the Dani only have two color terms (“warm-light” and “cool-dark”), unnamed foci are more salient and more easily remembered than surrounding nonfocal colors (Heider, 1972). Fourth, aggregates of individuals agree more than individuals do: The level of cross-cultural variation in the selection of foci is small in comparison to the level of intra-cultural variation (Berlin and Kay, 1969; Boster, 1991). Finally, individuals appear to respond to the structure in the domain similarly to the way that aggregates do: Individuals faced with the task of successively dividing color categories recapitulate the stages by which languages evolve (Boster, 1986).

These last two points are both important and general. To rephrase, aggregates agree more than individuals do and aggregates agree because individuals do. Each individual can be seen as fallibly capturing aspects of a universal structure – aggregates agree more than individuals do because the noise in the individual judgments tends to cancel and the common signal reinforces. We can make the simplifying assumption that the informants are more or less equivalent to each other – each contributes their own piece of the overall signal, dusted with their own bit of individual noise or error.

To illustrate this principle, I asked four informants, ‘Andy,’ ‘Beth,’ ‘Carl,’ and

‘Dee,’ to guess the distance from Storrs, CT to New York, Chicago, Miami, and Los Angeles and they answered as follows:

Distance to ‘Andy’ ‘Beth’ ‘Carl’ ‘Dee’ Average Actual New York 100 60 140 105 101 140 Chicago 300 200 1500 700 675 926 Miami 700 1400 1400 800 1075 1434 Los Angeles 1000 3600 3080 3000 2670 2976 Each of them individually does quite respectably, making guesses that are correlated an average of .971 with the actual distances. (They also agree substantially with each other, their guesses have an average correlation of .926 with each other.) But if we aggregate the guesses of pairs of informants, we do even better: The pairs are correlated an average of .987 with the true distances. Aggregating sets of three informants does better still with an average correlation of .993, and aggregating all four informants does best of all with a correlation of .996. In other words, even in a case in which informants individually produce guesses that are highly correlated with the actual distances, the aggregation of their guesses produces an even higher correlation. 5

To state this more formally, if individuals are responding to some universally

perceptible structure in a similarity judgment task, then the correlation of an aggregate of individuals with a world standard should increase as the size of the aggregate increases according to the following formula:

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Nr

r

r

xxxx

xy

−+

1 [1]

where rxy is the average individual informant’s correlation with the world standard, rxx is the average correlation among informants on the similarity judgment task, and N is the number of informants in the pool of aggregated responses (Kelley, 1923:200, cited in Guilford, 1936:422).6 This is the general formula for the correlation between a criterion (in this case, the world standard) and the sum or average of a number of equally weighted scores (in this case, the aggregated similarity matrices). The world standard is a similarity matrix that represents how items in the domain are categorized by speakers of a number of different languages, and therefore captures the universally perceptible structure. As we increase the number of informants in the set of aggregated responses, the correlation of the aggregate to the world standard steadily increases: To start, when we have a single

informant, N is one and the expression becomes xyxxxx

xy

xxxx

xyr

rrr

rr

r=

−+=

−+

11

1

In other words, the expression becomes simply the correlation of that single informant with the world standard (as it should). At its maximum value, when we have an infinite

number of aggregated informant responses, the term N

rxx−1 approaches zero, and the

correlation of the aggregate to the world standard becomes

xx

xy

rr

[2] 7

If we apply formula [1] to the guessed distances example above, we obtain

4926.1926.

971.−+

or an estimate of .999, not far from the observed correlation of the

aggregate to the actual distances of .996. To test this relationship, I compare similarity judgments and world standards for a number of domains. The domains are color, the facial expression of emotion, and disintegration events (e.g., scenes of cutting, breaking, tearing, etcetera). In each domain, a world standard was derived from how speakers of different languages named or identified a collection of items from the domain. After all, language speakers assign items to labeled categories based on the features of the items – items named the same share more features and are judged more similar than items that are distinguished. But the features chosen as the basis of category membership may vary from language to language, so that the naming responses from a number of different languages should better reflect the universally perceptible structure than those from any single language. In comparing the world standard with the similarity judgments, we are comparing two kinds of similarity structures: one derived from the linguistic classification of the items from speakers of many languages, the other derived from a non-linguistic8 categorization of the items by speakers of a single language.

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Methods The steps in the process of creating the world standard are as shown in the cartoon

on the top of Figure 1 and are as follows: First, the items in the domain are named by speakers of a number of languages in a naming or identification task. These identification task responses are then used to produce a mapping matrix. The columns of the mapping matrix correspond to the items used in the task, the rows correspond to the terms used in each language to name the items, and the cell entries are the frequencies by which the item corresponding to the column is given the term corresponding to the row. The columns of the mapping matrix are then inter-correlated to produce an item-by-item world standard similarity matrix of items that captures how similarly the items are named across the languages sampled.9

Figure 1.

Identification Task Mapping Matrix World Standard

Similarity Judgment Task Similarity Matrix

‘red’‘yellow’‘green’

‘blue’

11

11

1.0

1.0

1.0

1.0

.10 -.04

44

44

22

1 111

1 11 1

33

.10

-.04

-.05

-.05

.00

.00 -.02

-.02

.69

.69

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Similarity judgments from individuals were elicited using the successive pile sort (Boster, 1987, 1994). In this task, informants are presented with a number of items and asked to sort them into groups according to which they think are most similar to one another. After the initial sort, they are asked to successively merge their groups until all of the items are merged together. They then are asked to return to their initial groups and to successively split the groups until all of the items have been split apart. The procedure elicits from each informant a complete binary tree expressing the ranked similarity of each pair of items on a scale from 1 to N-1, where N is the number of items. Thus, the similarity matrix is directly derived from the informant’s responses, as shown in the cartoon at the bottom of Figure 1. The similarity of each pair is equal to the lowest node of the binary tree that includes both items or, alternatively, the number of steps in the successive splitting of the completely merged group of items that first separates that pair of items. Items that are split apart when there are only two groups are the least similar (similarity = 1) while the pair of items that are the last to be split apart are the most similar (similarity = N-1). In the domain of color, the world standard was created using the results of the World Color Survey (Kay, Berlin, and Merrifield, 1991, Kay and Maffi, 1999).10 In this research, 2616 speakers of 110 different languages were asked to provide the basic color terms in their own language for each of 330 color chips drawn from the Munsell array. Similarity judgments of the same 330 color chips were elicited from 10 American English speaking students at the University of Connecticut. This study will be referred in Tables and Figures with the symbol WC. The Munsell array is depicted in Figure 2.

Figure 2. World Color Survey Munsell Color Chips.

In the domain of the facial expression of emotion, the world standard was created

by asking 260 informants of 5 languages (English, Spanish, Italian, Polish, and Shuar) to identify the emotion expressed in 22 facial gestures posed by a male and a female actor. The facial expressions were chosen to evenly sample Russell’s circumplex emotion space (Russell, 1980). Similarity judgments of the same 11 male faces and the 11 female faces were elicited from 26 American English speaking students at the University of Connecticut and 26 Polish speaking informants from Warsaw and Krakow. In the balance of this paper, the results of this investigation will be treated as four separate studies: Americans judging the similarity of male faces, Americans judging the similarity of female faces, Poles judging the similarity of male faces, and Poles judging the similarity of female faces. These four studies will be referred in Tables and Figures with the symbols AM, AF, PM, and PF respectively. The photographs used are shown in Figure 3.

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Figure 3. Facial Expression Photographs.

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In the domain of disintegration-events, the world standard was created by asking 91 informants of 28 languages to name the action depicted in 61 video clips of an actor disintegrating (e.g., cutting, breaking, smashing, poking, tearing, etc.) a variety of objects (pottery, yarn, vegetables, sticks, cloth, etc.) (Bohnemeyer, Bowerman and Brown, 2001; Majid, Van Staden, Boster, and Bowerman, 2004). Similarity judgments of the same 61 video clips were elicited from 15 American English speaking students at the University of Connecticut. Informants were first shown each video clip, and then given a still photograph capturing the most salient moment of the disintegration event to be used in the pile sort. This study will be referred in Tables and Figures with the symbol DE. The still photographs used are shown in Figure 4.

Figure 4. Disintegration Events Still Photographs

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Table 1.

Study

Number of informants (languages) contributing

to World standard

Number of stimulus items

Number of informants

participating in the similarity judgment task

(N)

Number of combinations of

informants (2N-1)

Color (WC) 2616 (110) 330 10 1023 American male emotion faces (AM) 260 (5) 11 26 67,108,863

American female emotion faces (AF)

260 (5) 11 26 67,108,863

Polish male emotion faces (PM) 260 (5) 11 26 67,108,863

Polish female emotion faces (PF)

260 (5) 11 26 67,108,863

Disintegration events (DE) 91 (28) 61 15 32,767

Results

The next step in the analysis was to produce every possible combination of informants in the similarity judgment task and to correlate each aggregated similarity matrix with the world standard. The number of possible combinations of N informants is 2N-1. For example, for three informants, there are 3 ways of choosing a single informant, 3 ways of choosing a pair of informants, and one way of choosing all three informants; 3+3+1 = 23-1 = 7. Thus, there are 1023 possible combinations of the 10 informants in the world color chip similarity judgment task and more than 67 million possible combinations of the 26 informants in the facial expression similarity judgment task, as shown in Table 1.11 The average correlation of each size aggregate from 1 to N was recorded and compared with the expected value given by formula [1]. As shown in Table 2, the r2 of the fit between the observed average correlation to the world standard and the expected value given by formula [1] in all cases is above .999, ranging from .99903 to .99999. It appears that the formula very accurately captures the way that the correlation of aggregated responses with a world standard improves with increased size of the aggregate. This is illustrated in Figure 5 which shows the increase in the correlation of the aggregate to the world standard with increasing aggregate size from the actual data fitted to the curve derived from formula [1]. The two constants in formula [1] affect the shape of the curves differently: Increasing rxy increases the height of the intercept while increasing rxx decreases the steepness of the slope of the curve.

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Table 2.

Study

Average individual r with World

standard (rxy)

Average individual r with other informants

(rxx)

Correlation of aggregate

with infinite informants

(xx

xy

rr

)

Fit of Data to Model (r2)

Color (WC) .36 .40 .57 0.99999 American male emotion faces (AM)

.60 .43 .92 0.99986

American female emotion faces (AF) .68 .55 .91 0.99972

Polish male emotion faces (PM) .59 .38 .97 0.99990

Polish female emotion faces (PF) .70 .59 .91 0.99995

Disintegration events (DE) .48 .35 .80 0.99903

Figure 5.12

-1/sqrt(N)

PF

AF

PM

AM

DE

WC

-1.0 -0.8 -0.6 -0.4 -0.2 0.00.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

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Discussion Given the close fit of the data to formula [1], it is useful to reexamine it and unpack its implications for the relationship between intra-cultural variation and cross-cultural universals.

Nr

r

r

xxxx

xy

−+

1 [1]

A dominant theme in recent research on the pattern of intra-cultural variation has

been to emphasize the importance of agreement. The central idea has been that intra-cultural variation is often patterned in such a way that indicates that there is a single cultural system that is known to varying degrees by different individuals. Those informants who agree most with each other can be inferred to best understand the cultural system (Boster, 1985). This insight was formalized in Romney et al.’s (1986) cultural consensus model. They labeled a measure of the average agreement among informants (the first factor score of a minimum-residual factor analysis of the inter-informant agreement matrix), the informant’s “cultural competence.” An informant’s first factor score is roughly equal to the square root of the informant’s average correlation with other informants while the average first factor score is roughly equal to the square root of the average inter- informant correlation, rxx.

There has been a tendency to view this agreement or “cultural competence” as an

unalloyed good. Thus, D’Andrade (1987) presents evidence that those who more often give modal responses on a variety of tasks (including some that have no obvious “correct” answer like a word-association task) tend to be more reliable, consistent, normal, educated, intelligent, and experienced than other informants. However, an examination of the role of rxx in either formula [1] or [2] shows that competence or agreement may be the enemy of validity – the lower the agreement among the judges of similarity, the higher the correlation of the aggregated responses to the world standard.13 This principle can be illustrated by the results of the similarity judgments of the color chips, shown in Figure 6. One of the informants, let’s call him ‘John,’ judged the similarity of the color chips in a way that corresponded to the world standard a little less well than the average of the other informants, but also in a way that had the lowest correlation with other informants. To put this another way, John was just as fallible as other informants in capturing aspects of the universal structure (the world standard), but offered a piece of it that was the most different from his fellow informants. As a consequence, if the other informants had to pick another informant who would best improve the correspondence of their aggregated responses to the world standard, they would be best off in choosing John. In this case, the most valuable informant for improving the fit with the world standard is the one who disagrees the most with other informants, not the one closest to the consensus or the one closest to the world standard. (The appendix presents a formula for predicting the effect on the aggregate correlations with the world standard of these individual differences.)

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Figure 6.14

0.30 0.35 0.40 0.45 0.50Agreement with other informants

0.30

0.35

0.40

0.45

0.50

‘John’

World Color (WC)

However, in practice this effect may be overwhelmed by another phenomenon: Informants who agree the most to their own aggregate also agree the most with the world standard – it may be rare to find informants like John who capture a significant portion of the signal that is maximally independent of that caught by other informants. This point is illustrated by the results of the American similarity judgments of the female emotion faces (AM), shown in Figure 7. Here, neither ‘Ann’ (the informant who agrees the most with the world standard) nor ‘Zoë’ (the informant who agrees the least with other informants) is the most valuable informant for improving the fit with the world standard. Rather it is ‘Jane,’ who among the informants who have high correlations with the world standard has the lowest agreement with other informants. It is the optimal combination of high agreement with the world standard and low agreement with other informants that makes an informant the most valuable contributor to the aggregate. Intriguingly, ‘Ann’ and ‘Zoë’ make essentially equivalent contributions to increasing the magnitude of the correlation of the aggregates they join with the world standard: ‘Ann’ by contributing a strong signal highly correlated with the world standard and ‘Zoë’ by contributing a signal that while weakly correlated with the world standard is highly independent of that offered

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by other informants. Again, aggregates agree more than individuals do because the noise in the individual judgments tends to cancel and the common signal reinforces, but it helps if the signals are as strong and as independent of one another as possible.

Figure 7.15

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Agreement with other informants

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Americanfemale (AF)

‘Jane’‘Ann’

‘Zoë’

This observation prompts us to reassess whether or not agreement or “cultural

competence” is an unalloyed good, and to argue that the value of agreement depends on one’s goals as an ethnographer. If one seeks to estimate the judgments of a group with a minimum number of informants, then agreement is good because one needs fewer informants to get a highly reliable picture of the group aggregate. However, if one seeks to discover the maximum possible concordance between an aggregated structure and a world standard, then high agreement within the group limits what one learns from each additional informant, and the correspondence between the aggregated similarity judgments and the world standard will be lower. Again, the high agreement of the informants diminishes the correlation with the world standard: Their reliability is an enemy of validity. For this purpose, there is a great value in cognitive diversity.

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However, there is a limit on cognitive diversity, else the aggregation of an infinite number of informants would produce a correlation greater than one with the world

standard. This follows from formula [2], if 1≤xx

xy

rr

, then xxxy rr ≤ . In other words, the

square root of the average inter-informant correlation on the similarity judgment task can not be lower than the average individual correlation with the world standard: The existence of cross-cultural universals imposes a limit on the possible magnitude of intra-cultural variation. Furthermore, at the limit when decreasing xxr approaches rxy, there would be no pattern in the local similarity judgments that is not part of the universal pattern, because the similarity structure based on an infinite number of informants would have a correlation of 1.0 with the world standard. Only when the agreement among informants, as assessed by xxr , is substantially greater than rxy, could one argue that a substantial portion of the informants’ judgments represent a local culturally specific pattern not subsumed by the universal one. This is clearly not the case with the four studies of the facial expression of emotion. In each study, the expected correlation of an infinite sized aggregate of informants to the world standard is above .9. This leaves very little room for the existence of a culturally unique pattern in the similarity judgments expressing a peculiarly American or Polish point of view.

Conclusion

In sum, as one increases the size of a pool of informants performing a similarity

judgment task, the mean correlation of the aggregated responses to a world standard is

equal to

Nr

r

r

xxxx

xy

−+

1, where rxy is the average individual informant’s correlation with

the world standard, rxx is the average correlation among informants on the similarity judgment task, and N is the number of informants in the pool of aggregated responses. The r2 of this relationship was found to be above .999 in six studies spanning three domains (color, verbs of disintegration, facial expression of emotions). This result can be interpreted as one of the consequences of the fact that cases of cross-cultural universals are rooted in individual cognition: Cross-cultural universals occur when some structure in experience is available to individuals irrespective of their cultural or linguistic background. In these cases, the individual informants can be seen as fallibly capturing aspects of a universal structure. Unless there is perfect agreement among informants, groups of individuals are always closer to the world standards than are individuals because the error or noise in the individual judgments tends to cancel and the common signal reinforces. Because they more faithfully converge on the common signal, we can say that the aggregates are “smarter” or more accurate than the individuals are. But the formula also shows the diminishing returns of each additional informant: we learn less and less from each successive informant. Cognitive diversity is finite. Indeed, given the fact that no correlation can be greater than one, the existence of cross-cultural universals can be seen as imposing a limit on the magnitude of intra-cultural variation: xxxy rr ≤ . In other words, the square root of the average correlation among informants can be no lower that the average individual correlation with the world standard. In the case of

5/6/2004 15

minimum possible intra-cultural variation, there could be no aspect of the local cultural pattern that was not subsumed by the universal, because the correlation of an infinite sized aggregate with the world standard would be one. Further, the result also shows that it is precisely the disagreement among informants that allows their aggregation to so closely approximate the world standard. In demonstrating the value of cognitive diversity, this result harkens back to Anthony Wallace’s early and prescient observation (1961) that some degree of cognitive diversity is necessary to the functioning of society. Here it is shown to be also necessary for the common phenomenon that groups are more accurate than individuals are.

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Literature Cited.

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Berlin, Brent and Paul Kay. 1969. Basic Color Terms. Berkeley, CA: University of California Press.

Bohnemeyer, Jurgen, Melissa Bowerman and Penelope Brown 2001. Cut and break clips, version 3. In Stephen C. Levinson and Nicholas Enfield (Eds.), Field Manual 2001. Language & Cognition Group, Max Planck Institute for Psycholinguistics. Boster, James S. 1985. 'Requiem for the Omniscient Informant': There's Life in the Old Girl Yet. In Directions in Cognitive Anthropology. Janet Dougherty (ed.). University of Illlinois Press. pp. 177-197. Boster, James S. 1986. Can Individuals Recapitulate the Evolutionary Development of Color Lexicons? Ethnology 25(1):61-74. Boster, James S. 1987. Agreement between Biological Classification Systems is Not Dependent on Cultural Transmission. American Anthropologist. 89(4):914-919. Boster, James S. 1991. The Information Economy Model Applied to Biological Similarity Judgment. In Socially Shared Cognition. Lauren Resnick, John Levine, and Stephanie Teasley (eds.). Washington, DC: American Psychological Association. pp. 203-225. Boster, James S. 1994. The Successive Pile Sort. Cultural Anthropology Methods 6(2):7-8. Boster, James S., Brent Berlin, & John P. O'Neill. 1986. The Correspondence of Jivaroan to Scientific Ornithology. American Anthropologist 88(3):569-583. Boster, James S. and Roy G. D'Andrade. 1989. Natural and Human Sources of Cross-Cultural Agreement in Ornithological Classification. American Anthropologist 91(1):132-142. Brown, Roger W. and Eric H. Lenneberg. 1954. A study of language and cognition. Journal of Abnormal and Social Psychology 49: 454-462. Bulmer, Ralph. 1970. Which came first, the chicken or the egghead? In Pouillon, J., Maranda, P., ed., Échanges et communications. Vol. II. Mouton and Co., The Hague, Netherlands. pp. 1069-1091.

Burgess, Donald, Willett Kempton, and Robert MacLaury. 1983. Tarahumara Color Modifiers: Category Structure Presaging Evolutionary Change. American Ethnologist 10(1): 133-149.

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D’Andrade, Roy. 1987. Modal Responses and Cultural Expertise. American Behavioral Scientist: 31(2):194-202.

DeValois, Russell L., Israel Abramov and Gerald H. Jacobs. 1966. Analysis of response patterns of LGN cells. Journal of the Optical Society of America 56(7):966-977.

Dougherty, Janet W.D. 1977. Color categorization in West Futunese: Variability and change. In Blount, Ben and Mary Sanches eds. Sociocultural Dimensions of Language Change, pp.103-118.

Guilford, J. P. 1936. Psychometric Methods. New York: McGraw-Hill.

Heider, Eleanor Rosch (1972 Universals in color naming and memory. Journal of Experimental Psychology 93: 1-20.

Jameson, Kimberly and Roy G. D’Andrade. 1997. It’s not really red, green, yellow, blue: an inquiry into perpetual color space. In C.L Hardin and L. Maffi (eds.), Color Categories in Thought and Language . Cambridge: Cambridge University Press. Kay, Paul. 1975. Synchronic variability and diachronic change in basic color terms. Language in Society 4: 257-70. Kay, Paul, Brent Berlin, and William R. Merrifield. 1991. Biocultural implications of systems of color naming. Journal of Linguistic Anthropology 1(1): 12-25. Kay, Paul, Brent Berlin, Louisa Maffi, and William Merrifield. 1997. Color naming across languages. In C.L Hardin and L. Maffi (eds.), Color Categories in Thought and Language . Cambridge: Cambridge University Press. Kay, Paul and Louisa Maffi. 1999. Color Appearance and the Emergence and Evolution of Basic Color Lexicons. American Anthropologist 101:743-760. Kay, Paul and Chad McDaniel. 1978. The Linguistic Significance of the Meanings of Basic Color Terms. Language 54:610-64. Kay, Paul and Willett M. Kempton. 1988. What is the Sapir-Whorf hypothesis? American Anthropologist 86: 65-79. Kelley, Truman Lee. 1923. Statistical Method. New York: MacMillan. Majid, Asifa, Miriam van Staden, James S. Boster, and Melissa Bowerman. 2004. Categorization of events: A cross- linguistic perspective Nunnally, Jum C. and Ira H. Bernstein. 1994. Psychometric Theory (3rd ed.) New York: McGraw-Hill.

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Roberts, John. 1964. The self-management of culture. In Explorations in Cultural Anthropology: Essays in Honor of George Peter Murdoch, W. Goodenough, Ed. McGraw-Hill, London, UK. Romney, A. Kimball and Tarow Indow. 2002. A model for the simultaneous analysis of reflectance spectra and basis factors of Munsell color samples under D65 illumination in three-dimensional Euclidean space. Proceedings of the National Academy of Sciences 99(17): 11543-11546 Romney, A. Kimball, Susan C. Weller, and William H. Batchelder. 1986. Culture as consensus : A theory of culture and informant accuracy. American Anthropologist 88(2), 313-338.

Russell, James A. (1980). A circumplex model of affect. Journal of Personality and Social Psychology, 39, 1161-1178. Shepard, Roger N. 1992. The Perceptual Organization of Colors: An Adaptation to Regularities of the Terrestrial World? In Barkow, Jerome H., Leda Cosmides, and John Tooby (eds) The Adapted Mind: Evolutionary Psychology and the Generation of Culture Oxford: Oxford University Press. pp. 495-532 Wallace, Anthony F. C. 1961. Culture and Personality. New York: Random House.

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Appendix The derivation of the formula [1] in the context of psychometric theory is as follows: The correlation of two variables with an increase of N is equal to the ratio of the square roots of the changed reliability to the original reliability. (Nunnally and Bernstein, 1994: p. 258).

xx

xxxyNxy

rr

rr'

)( =

If one uses the Spearman-Brown prophecy formula to estimate the changed reliability, the formula becomes

xx

xx

xx

xyr

rNNr

r)1(1 −+

Dividing above and below by xxr , it becomes

xxxy

rNN

r)1(1 −+

Dividing within the radical above and below by N, it becomes

NrN

rxx

xy

)1(1 −+

Using the distributive law and rearranging terms, it becomes

Nr

NNr

rxxxx

xy

−+

1

Finally, simplifying the expression on the left, it becomes

Nr

r

rxx

xx

xy

−+

1 [1]

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The expected correlation with the world standard of an aggregate including a specific informant is given by the following formula:

II

rN

rNrI

I

rN

rNrI

I

rN

rNrI

xxxxxx

xxxxxx

xyxyxy

')1(

)'()1(

1')1(

)'()1(

')1(

)'()1(

+−−

−−

++

−−

−

+−−

−

[3]

where N is the total number of informants, rxy is the average correlation of informants with the world standard, rxx is the average correlation among informants, rxx' is the average inter- informant correlation of the specific informant, rxy' is the correlation of the specific informant with the world standard, and I is the size of the aggregate that includes the specific informant. It can be seen that when I = 1, the formula resolves to rxy', while

when I = N, the formula resolves to

Nr

r

rxx

xx

xy

−+

1, which is formula [1]. Formula [3]

simply adjusts formula [1] to take into consideration the changed magnitudes of rxy and rxx when we specify one of the informants who is a contributor to the aggregate.

To compute the average correlation of an aggregate including a specific informant, one sums across all size aggregates from 1 to N, multiplying by the number of aggregates of that size that include the specific informant, and dividing the summation by the total number of combinations including that informant, 2N-1.

12

')1(

)'()1(

1')1(

)'()1(

')1(

)'()1(

1

)!1())!1()1(()!1(

−

+−−

−−

++

−−

−

+−−

−

=

−−−−−∑

N

xxxxxx

xxxxxx

xyxyxy

II

rN

rNrI

I

rN

rNrI

I

rN

rNrIN

I

IINN

In practice, this formula does not provide as accurate estimates as [3] above, because the small errors in estimation given by [3] are magnified by the very large number of

aggregates of intermediate size. The term )!1())!1()1((

)!1(−−−−

−IIN

Nreaches a maximum

when the aggregate size is half the total number of informants. With 26 informants and an aggregate size of 13, it equals 5,200,300.

5/6/2004 21

Endnotes 1 I would like to express my sincerest appreciation to Roy D’Andrade and to David A. Kenny who were invaluable in the development of these ideas. I am also grateful for comments by Cornelia Dayton, Douglas Hume, Kristin Kostick, Kateryna Maltseva, John Shaver, and Asha Srinivasan. A portion of this research was supported by grants from the National Science Foundation. 2 It could be argued that the term “cross-cultural universal” is a misnomer and should be replaced with the term “cross-social universal,” because these are cases in which some cultural content is shared between social groups. It is not the culture that is “cross” but the societies. However, I will respect common usage and continue use of the term “cross-cultural universal.” 3 The existence of the universals does not mean that there are no cultural differences; indeed in each of the domains there is evidence of strong commonalities based in attributes of our common humanity, but the phenomena are always understood in locally culturally-specific terms. The task for the ethnographer is not to maintain either of the absurd positions of radical universalism or radical relativism – that differences or similarities are non-existent – but to tease out the pattern and content of those similarities and differences.

4 The controversy regarding the source of the universals in color classification can be summarized as follows: Kay and McDaniel (1978), relying on DeValois et. al, (1966) argued that the unique hue points of red, yellow, green, and blue were given by the pattern of firing of opponent process cells in the Lateral Geniculate Nuclei. However, Jameson and D’Andrade (1997) pointed out that the opponent processes gave unique hue points in the wrong places: The axes of the system are closer to cherry – teal versus chartreuse – violet, rather than red – green versus yellow – blue. Thus, they argue that the universals in color classification cannot be traced solely to a pattern imposed by the human neurophysiology of color vision, though they remained agnostic about the precise source of the structure. At the other extreme, Shepard (1992) argues that the perceptual organization of color is an adaptation to regularities of the illumination by the sun of the earth which has three degrees of freedom: 1) light-dark variation (midday vs. shade vs. moonlight); 2) yellow-blue variation (blue – blocked sunlight vs. yellow – direct sunlight); and 3) red-green variation (the shortest wavelengths in red are the least scattered by sunlight). The three types of cones (sensitive to long, medium, and short wavelengths respectively), according to Shepard, are an adaptive response to these three degrees of freedom. Romney and Indow (2002) studying the reflectance spectra of Munsell chips found three axes: one that roughly corresponds to the mean power of the spectral reflectance (approximate Munsell value), a second that goes from Munsell red to blue-green, and a third tha t goes from Munsell green-yellow to purple. It is intriguing that these are almost precisely the dimensions that Jamieson and D’Andrade (1997) had found for the opponent-process system. Although these are not the dimensions that Shepard (1992) asserts are the dimensions of regularities in sunlight, it does provide some clear evidence that the human neurophysiology of color vision has evolved to extract the

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maximum amount of information from reflected light – the perceptual organization of color is virtually precisely matched to the dimensional structure of reflected light. This suggests that the structure in experience that leads to the universals in color classification is in the phenomenon itself after all, even if they do not match the Hering model oppositions of red -green and yellow-blue.

5 Note that the aggregate of all the guesses is highly correlated with the true distances even though it has a systematic bias, underestimating the distances by about 20%. 6 This is the formula that Kelley (1923) gives for the “correlation between a criterion and the sum or average of a number of equally weighted scores. One can derive the same formula using the Spearman-Brown prophecy model and assuming that adding informants is comparable to adding items to a test, as shown in the appendix. Guilford (1936) appreciated the analogy of informants to test items, but as far as I have been able to determine, it has dropped out of more recent treatments. I am grateful to David A. Kenny and Roy D’Andrade for alerting me to this formula. 7 This is related to the correction for attenuation familiar to psychometricians (Nunnally and Bernstein, 1994: 257). 8 The successive pile sort is “non- linguistic” in the sense that informants are not explicitly required to give a linguistic label for the items in the pile sort. This is not to say that informants cannot be influenced by the categories labeled by the words in their language as they are sorting the items, only that they are not requested to give the category labels as part of the task. This follows the distinction introduced by Brown and Lenneberg (1954) and elaborated by Kay and Kempton (1988) in describing the way to appropriately test the Sapir-Whorf hypothesis. 9 The results are not dependent on this particular way of producing a world standard – alternative methods such as counting the number of informants who called each pair of items by the same term gave similar results. 10 The data are available on- line at http://www.icsi.berkeley.edu/wcs/data.html. 11 The value 2N-1 is related to the sums of the rows of Pascal’s triangle, a common method of calculating the number of combinations of a given number of items. 1 1 1 2 1 1 3 3 1 1 4 6 4 1 The first value in each row is the number of ways of taking zero items, the second value is the number of ways of taking two items, etcetera. One can see that each row sums to 2N, subtracting 1 for the ways of taking no items gives us 2N-1.

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12 The horizontal axis is

N1−

so as to produce a scale that increases with increasing N but

compresses high values of N. The vertical axis is the magnitude of the correlation of the aggregate with the world standard. The points are the observed aggregate correlation while the line is derived from the relationship predicted by formula [1]. The meanings of the labels (PM, PF, etc.) are given in the first column of Table 1, and in the description of the methods of data collection. 13 Roy D’Andrade has pointed out to me (D’Andrade, personal communication) that Guilford (1936) makes a very similar observation about the reliability and validity of mental tests. Guilford states (1936:423)

“A close study of the preceding formulas, particularly of formula (204), will show that, for a given validity rxy, the smaller the reliability rxx of the test, the higher will be the upper limit of validity when the test is lengthened. In other words, an unreliable test gains proportionally more in validity by lengthening than does a test that is already very reliable.”

14 The horizontal axis is the average correlation of each informant with other informants, while the vertical axis is the correlation of each informant with the world standard in the World Color similarity judgment task. The slanting striations indicate the average correlations of aggregates that include the informants, ranging from a high of .51 at the upper left to a low of .49 at the lower right. The informant ‘John’ discussed in the text is indicated. 15 As in Figure 6, the horizontal axis is the average correlation of each informant with other informants, while the vertical axis is the correlation of each informant with the world standard in the American Female (AM) emotion face similarity judgment task. The slanting striations indicate the average correlations of aggregates that include the informants, ranging from a high of .89 at the upper left to a low of .875 at the lower right. The informants ‘Jane,’ ‘Ann,’ and ‘Zoë’ discussed in the text are indicated.