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SESSION I Social Network Theo PerspecOves

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- ~ Photograph by Ben Shahn' Batches, MS, October, 1933 Library of Congress, Prints & Photographs Division, FSA-OWI Collection, [reproduction number, LC-USF33-006093-M4] Finding Social Groups: A Meta-Analysis of the Southern Women Natal Linton C. Freeman University of California, InJirze I. Introcluction For more than 100 years, sociologists have been concerned with relatively small, cohesive social groups (Tonnies. ~ ~ 887] 1940; Durkheim ~ ~ 893] ~ 933; Spencer ~ 895-97; Cooley, 19091. The groups that conceIn sociologists are not simply categorieslike redheads or people more than six feet tall. Instead they are social collectivities characterized by interaction and interpersonal ties. Concern with groups of this sort has been and remains at the very core of the field. These early writers made no attempt to specify exactly what they meant when they referred to groups. But in the 1930s, investigators like Roethlisberger and Dickson (1939) and Davis, Gardner and Gardner (1941) began to collect systematic data on ' The author owes a considerable debt to Moms H. Sunshine who read an earlier draft and made extensive suggestions al] of which improved this manuscript. DYNAMIC SOCIAL NETWO=MODEL~G ED ISIS 39

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interaction and interpersonal ties. Their aim was to use the data both to assign individuals to groups and to determine the position of each individualas a core or penpheral group member. But, to assign individuals to groups and positions, they needed to specify the sociological notions of group and position in exact terms. Over the years a great many attempts have been made to specify these notions. In the present paper ~ will review the results of 21 of these attempts. All 21 tried to specify the group structure in a single data set. And 11 of the 21 also went on once ~ttP.mnt~.~1 to specify core and peripheral positions. ~ ^r--~~ My approach to this review is a kind of meta-analYsis. Schmid. Koch and ~ ~ . LaVange (1991) define meta-analysis as ". . . a statistical analysis of the data from some collection of studies in order to synthesize the results." And that is precisely my aim here. A typical meta-analysis draws on several data sets from a number of independent studies and brings them together in order to generalize their collective implications. Here am also trying to discover the collective implications of a number of studies. But instead of looking at the results produced by several data sets, ~ will be looking at the results produced by several different analytic methods. In this meta-analysis I will compare the groups and the positions that have been specified by investigators who examined data collected by Davis, Gardner and Gardner (1941) EDGG] in their study of southern women. My comparison draws on a number of techniques, including consensus analysis (Batchelder and Romney, 1986, 1988, 1989), canonical analysis of asymmetry (Gower, 1977) and dynamic paired-comparison scaling (Batchelder and Bershad, 1979; Batchelder, Bershad and Simpson, 19921. ~ will address two questions: (~) do the several specifications produce results that converge in a way that reveals anything about the structural form of the data? And, (2) can we learn anything about the strengths and weaknesses of the various methods for specifying groups and positions? 2. Southern Women Data Set In the 1930s, five ethnographers, Allison Davis, Elizabeth Stubbs Davis, BurIeigh B. Gardner, Mary R. Gardner and J. G. St. CIair Drake. collected data on stratification in Natchez, Mississippi (Warner, 1988, p. 931. They produced the book cited above EDGG] that reported a comparative study of social class in black and in white society. One element of this work involved examining the correspondence between people's social class levels and their patterns of informal interaction. DOG was concerned with the issue of how much the informal contacts made by individuals were established solely (or primarily) with others at approximately their own class levels. To address this question the authors collected data on social events and examined people's patterns of informal contacts. In particular, they collected systematic data on the social activities of 18 women whom they observed over a nine-month penod. Dunn;, that periods various subsets of 40 D YN~MIC SOCIAL NETWORK MODELING ED CAL YSIS

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these women had met Art a series of 14 informal social events. The participation of women in events was uncovered using "interviews, the records of participant observers, guest lists, and the newspapers" (DOG, p. 1491. Homans (1950, p. 82), who presumably had been in touch with the research team, reported that the data reflect joint activities like, "a day's work behind the counter of a store, a meeting of a women's club, a church supper, a card party, a supper party, a meeting, of the Parent-Teacher Association, etc." This data set has several interesting properties. It is small and manageable. It embodies a relatively simple structural pattern, one in which, according to DOG, the women seemed to organize themselves into two more or less distinct groups. Moreover, they reported that the positions~ore and penpheral~f the members of these groups could also be determined in teens of the ways in which different women had been involved in group activities. At the same time, the DOG data set is complicated enough that some of the details of its patterning are less than obvious. As Homans (1950, p. 84) put it, "The patient is frayed at the edges." And, finally, this data set comes to us in a two-mode- woman by event fonn. Thus, it provides an opportunity to explore methods designed for direct application to two-mode data. But at the same time, it can easily be transformed into two one-mode matrices (woman by woman or event by event) that can be examined using tools for one-mode analysis. Because of these properties, this DGG data set has become something of a touchstone for comparing analytic methods in social network analysis. Davis, Gardner and Gardner presented an intuitive interpretation of the data, based in part on their ethnographic experience in the community. Then the DGG data set was picked up by Homans (1950) who provided an alternative intuitive interpretation. In 1972, Phillips and Conviser used an analytic tool, based on information theory, that provided a systematic way to reexamine the DGG data. Since then, this data set has been analyzed again and again. It reappears whenever any network analyst wants to explore the utility of some new tool for analyzing data. 3. The Data Source Figure I, showing which women attended each event, is reproduced from DGG (p. 148~. DGG examined the participation patterns of these women along with additional infonnation generated by interviews. As ~ discussed in Section ~ above, they had two distinct goals in their analysis: (~) they wanted to divide women up into groups within on the basis of their co-attendance at events, and (2) they wanted to determine a position- in the core or periphery for each woman. As they put it (p. 1501: Where it is evident that a group of people participate together in these informal activities consistently, it is obvious DYNAMIC SOCIAL NETWo=MOD~L~G ED ^4YSIS 41

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42 that a cliques had been isolated. Interviewing can then be used to clarify the relationship. Those individuals who participate together most often and at the most intimate affairs are called core members; those who participate with core members upon some occasions but never as a group by themselves alone are called primary members; while individuals on the fringes, who participate only infrequently, constitute the secondary members of a clique. Not ~ P~aU" OCR for page 37
We are faced with a dilemma then. Are we to believe the data presented in Figure ~ or those presented in Figure 2? Homans (1950) was the first to use these data in a secondary analysis. He reported additional details that suggest that he was probably in touch with the original research team. And in his report he used the data as they are displayed in Figure I. Moreover, there is additional ancillary evidence for the correctness of the data as presented in Figure I. It tutus out that there is a contradiction in the presentation of Figure 2 that makes it difficult to accept the data presented there as correct. Compare, for example, the participation patterns displayed by the two women designated with red arrows in Figure 2: woman ~ ~ and woman ~ 6. According to Figure 2, these two women displayed identical patterns of participation. Yet, in that figure, woman ~ ~ was classified as a primal member of her "clique'. while woman 16 was called secondary. This contradiction implies that the correct data are those shown in Figure i. T~ Or s~s=P cow: Ape.... i.. polyp ~ - ~- ~ry. . . Go- ~ ~ ~ ~ ~ Body. al 13 4 l IS tS L7 8 By 10 11 12 13 t~ IS t. (~7 1 ~~ ~ PI 1 2 3 4 5 ~ 7 8 4 10 11 ~ 13 ~ C C ~ C - ~ C ~ ~ C - ~ ~ C C - - C C ~ ~ ~ C C C C - ~ C C C ~ C - p, p ~ _ p _ _ p _ p p _ p _ ~ P ~ p _ - S - S S S - S S & S S ~ - - ~ P P p _ ~ At. P P P - P P P C ~ C ~ - ~ C C - ~ ~ C ~ C ~ - ~ ~ C lO.-CI S S $ S - _ [S - S! q~ _ _ . _ Figure 2. Participation of the Southern Women in Events Most analysts have apparently reached this conclusion. Most have used the data as shown in Figure I. A few, however, have analyzed the data as presented in Figure 9. and this produces a problem for any attempt to compare the results of one analysis with those of another. When the two analyses are based on the use of different data sets comparisons are, of course, not possible. DYNAMIC SOCIAL NETWORK MODEM AND ISIS 43

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~ have assumed that the "correct" data are those shown in Figure I. For the relatively small number of results that have been produced by analyses of the data of Figure 2, I: have asked the original analysts to redo their analyses using the Figure ~ data or ~ have redone them myself. 4. Finding Groups and Positions in the DOG Data 4.l Davis, Gardner and Gardner's Intuition-Based Groups (DGG41) In their own analysis Davis Gardner and Gardner did not use any systematic analytic procedures. They relied entirely on their general ethnographic knowledge of the community and their intuitive grasp of the patterning in Table ~ to make sense of the data. As Davis and Warner (1939) descnbed it, they drew on ". . . records of overt behavior and verbalizations, which cover more than five thousand pages, statistical data on both rural and urban societies, as well as newspaper records of social gatherings . . ." DOG drew on all this material and used it both to assign the women to groups and to determine individuals' positions within groups. They indicated that the eighteen women were divided into two overlapping groups. They assigned women 1 through 9 to one group and women 9 through IS to another. They assigned three levels in terms of core~penphery participation in these groups. They defined women ~ through 4 and 13 through 15 as core members of their respective groups. Women 5 through 7 and ~ ~ and 12 they called primary. Women ~ and 9 on one hand and 9, 10, ~6, 17 and ~~ on the other were secondary. Note that woman 9 was specified as a secondary member of both groups because, they said, "in interviews" she was "claimed by both" (DOG. p. ~ 5 ~ ). 4.2 Homans' Intuition-Based Analysis (HOM50) Like Davis, Gardner and Gardner before him, Homans ~950) interpreted these data from an intuitive perspective. Unlike those earlier investigators, however, Homans did not have years of ethnographic experience in Natchez to draw upon. His intuitions, therefore, had to be generated solely by inspecting the DOG data and, presumably, by conversations with the ethnographers. Homans implied that he had re-analyzed the data using, a procedure introduced by Forsyth and Katz (1946) whom he cited. Forsyth and Katz had suggested permuting the rows and columns of a data matrix so as to display its group structure as clusters around the principal diagonal of the matrix (upper left to lower right). Their procedure required that both the rows and columns be rearranged until as far as possible more or less solid blocks of non-blank cells are gathered together. Such blocks of cells, they suggested, represent "well-knit~' groups. l:t is doubtful that Homans actually used the Forsyth and Katz procedure. DOG had already arranged the matrix in such a way that it displayed group structure. Seemingly they had anticipated the Forsyth and Katz approach by six years. Homans. 44 DYNAMIC SOCIAL NETWORK MODEM ED ISIS

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then, did not rearrange the data matrix at all; he simply copied the arrangement of Figure l, exactly as it was reported by DOG. In any case, after inspecting the arrangement shown in Figure I, Homans grouped ~ 6 of the women and distinguished two levels of core and penpheral positions. In his report Homans (p. 84) report wrote: we generalize these observations by saying that the women were divided into two groups. The pattern is frayed at the edges, but there is a pattern. The first seven women, Evelyn through Eleanor, were clearly members of one group; numbers ~ ~ through ~ 5, Myra through Helen, were just as clearly members of another. Some women participated about equally with both groups but not very much with either; Pear! twoman S] is an example. And some participated, though not very often, only with the second group. Pearl, Olivia, Flora [women 8, ~ 7 and ~ 8] and their like are marginal group members. This statement is somewhat ambiguous. ~ does assign women ~ through ~ to one group and ~ ~ through 15, along with 8, 17 and IS to the other. Because woman ~ (PearI) is assigned to both, the two groups overlap. In addition Homans characterized women 8 17 and IS to "marginal positions" but it is difficult to know whet he intended by the phrase '`and their like.'? His statement, moreover, makes no mention at all of woman 9 (Ruth) or womanI6 (Dorothy). They were simply not assigned to either group or to any position. 4.3 Phillips and Conviser's Analysis Based on Information Theory (P&C72) Phillips and Conviser (1972) were the first to use a systematic procedure in the attempt to uncover the group structure in the DGG data.3 They reasoned that a collection of individuals is a group to He extent that all of the members of the collection attend the same social events. So, to examine the DGG data. they needed an index of the variability of attendance. They chose the standard information theoretic measure of entropy, H (Shannon, 1964~. H provides an index of the viability of a binary (yes/no) variable. In this case, it was applied to all the women (and all the events) in the DGG data. Thus, H was used to provide an index of the degree to which different collections of women attended different sets of events (and different sets of events attracted different collections of women). Phillips and Conviser set about to find groups by comparing various ways of partitioning the women into subsets. They argued that any given partitioning produced social groups if the entropy H summed for all of the subsets was less than the entropy for 3 It should be noted that Phillips and Conviser attributed the southern women data to Homans. Nowhere in their paper did they acknowledge DGG. DYNAMIC SOCIAL NETWORK MODEL~G ED ISIS 45

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the total set of women. In such an event, the women assigned to each subset would be relatively homogeneous with respect to which events they attended. They evaluated the utility of any proposed partitioning by cadculating the information theoretic measure a. a is an index of the degree to which the overall entropy of the total collectivity H is reduced by calculating the value Hi within each of the i designated subsets and summing the results (Garner and McGill, ~ 956~. a is large when all the women who are classed into each subset are similar with respect to their attendance patterns. It is maximal only when the within-subset patterns are all identical. To employ this approach then, it is necessary to partition the women in all possible ways. calculate a for each partitioning, and see which partitioning produces the largest value. There is, however, a major difficulty with this approach. The number of possible partitionings grows at an exponential rate with an increase in the number of individuals examined. The number grows so rapidly that the partitions cannot all be examined, even with as few as ~ ~ women to be considered. So Phillips and Conviser worked out a way to simplify the problem. Like Homans, they cited again the procedure suggested by Forsyth and Katz (19461. That procedure rearranges the rows and columns in the data matrix in such a way that women who attended the same events and events that were attended by the same women are grouped together. When this is done, only those women who are close together in the matrix are eligible to be in the same group. That being the cases only those relatively few partitionings that include or exclude individuals in successive positions in the data matrix need to be considered. As ~ indicated above, the DOG data had already been arranged in the desired order by the original authors. So, like Homans, Phillips and Conviser did not actually have to rearrange them. They could proceed directly to partitioning. They began by partitioning the women into two classes (1 versus 2 through IS, ~ and 2 versus 3 through ~ 8, ~ through 3 versus 4 through ~ 8, etc.~. They reported that, of all these two-group partitions, the split of ~ through ~ ~ versus ~ 2 through ~ ~ yielded the largest value of a. In checking their results, however, T discovered that their result was based on an error in calculation. When ~ recalculated ~ discovered that the maximum value of a is actually achieved with the ~ through 9 versus JO through ~~ split. This approach simply partitions; it cannot distinguish core or peripheral positions, nor can it permit overlapping. 4.4 Breiger's Matrix Algebraic Analysis (BGR74) Breiger (1974) used matrix algebra to show that the original two-mode, woman by event DGG data matrix could be used to generate a pair of matrices that are, 46 DYNAMIC SOCIAL NETWORK MODELING kD ISIS

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mathematically, dual. First, multiplying the original matrix by its transpose produces a woman by woman matrix in which each cell indicates the number of events co-attended by both the row and the column women. Second, multiplying the transpose by the original matrix yields an event-by-event matrix where each cell is the number of women who attended both the row event and the column event. The woman-by-woman matrix is shown in Figure 3.4 And its dual, the event-by-event matrix. is shown in Figure 4. 1 1 1 1 ~ 1 1 ~ 1 1 2 3 4 ~ ~ 7 8 9 0 ~ 2 3 4 ~ 6 7 ~ E L T ~ ~ F E P R V M K S N H D O F 1 EVELYN 8 ~ 7 ~ 3 4 3 3 3 2 2 2 2 2 1 2 1 1 2 LAURA 6 7 6 6 3 4 4 2 3 2 1 1 2 2 2 ~ O O 3 THERESA' 7 6 8 6 4 4 4 3 4 3 2 2 3 3 2 2 1 1 4 BRENDA 6 6 ~ 7 ~ 4 4 2 3 2 1 1 2 2 2 1 0 C~LOi It 3 3 4 4 4 2 2 0 2 1 0 0 1 1 1 ~ ~ 0 6 FRANCES 4 4 4 4 2 4 3 2 2 1 1 1 1 1 1 1 0 0 7 ELEANOR 3 4 4 4 2 3 4 2 3 2 1 1 2 2 2 1 0 0 8 PEARL 3 2 3 2 0 2 2 3 2 2 2 2 2 2 1 2 1 1 9 RUTH 3 3 4 3 2 2 3 2 4 3 2 2 3 2 2 2 1 1 10 VERNE 2 2 3 2 1 1 2 2 3 4 3 3 4 3 3 2 1 1 11 MYRA 2 1 2 1 0 1 1 2 2 3 4 4 4 3 3 2 1 1 12 KATHERINE 2 1 2 1 0 1 1 2 2 3 ~ 6 6 ~ 3 2 1 1 13 SYLVIA 2 2 3 2 1 1 2 2 3 4 4 6 7 6 4 2 1 1 14 NORA 2 2 3 2 1 1 2 2 2 3 3 ~ 6 ~ 4 1 2 2 15 HELEN 1 2 2 2 1 1 2 1 2 3 3 3 4 4 ~ 1 1 1 16 DOROTHY 2 1 2 1 0 1 ~ 2 2 2 2 2 2 1 1 2 1 1 17 OLIVIA 1 0 1 0 0 ~ ~ 1 1 1 ~ 1 1 2 1 1 2 2 18 FLORA 1 ~ 1 ~ O ~ 0 1 1 1 1 1 1 2 ~ 1 2 2 _ Figure 3. The One-Mode, Woman by Woman, Matrix Produced by Matrix Multiplication ~ 2 3 4 5 S 7 ~ 9 10- 11 12 13 14 E1 E2 E3 E4 E5 ES E7 E8 E9 E10 E11 E12 E13 E14 1 E1 3 2 3 2 3 3 2 3 1 ~ ~ ~ 0 0 2 E2 ~ 3 3 2 3 3 2 3 2 ~ 0 0 0 0 3 E3 3 3 6 4 6 ~ 4 ~ 2 ~ 0 ~ 0 0 4 E4 2 2 ~ 4 4 3 3 3 2 0 0 Q O O 5 ES 3 3 6 4 8 6 ~ 7 3 0 a G ~ 0 E6 3 3 ~ 3 6 ~ ~ 7 4 1 1 1 1 1 7 E7 2 2 4 3 6 ~ 10 8 ~ 3 2 4 2 2 8 E8 3 3 ~ 3 7 7 8 14 9 4 1 ~ 2 2 ~ E9 1 2 2 ~ 3 4 ~ 9 12 4 3 ~ 3 3 10 E10 ~ ~ 0 0 C 1 3 4 4 ~ 2 ~ 3 3 11 E11 ~ 0 0 ~ 0 1 2 1 3 2 4 2 1 1 12 E12 0 0 0 ~ 0 1 4 ~ 5 ~ 2 6 3 3 13 E13 ~ 0 0 0 0 1 2 2 3 3 1 3 3 3 14 E 14 0 ~ 0 ~ 0 1 2 2 3 3 1 3 3 3 Figure 4. The One-Mode, Event by Event, Matrix Produced by Matrix Multiplication Breiger renamed DOG's "Myra." He listed her as "Myrnaq in his tables and diagrams. Breiger s designation has been picked up in a number of later works includin;, the data set released as part of the UCINET progran1 (Borgatti, Everett and Freeman. 1992). DYNAMIC SOCIAL NETWORK HOD~:~ING ED ISIS 47

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woman 6, but the paired-comparison analysis places woman 5 closer to the core than woman 6. To evaluate the effectiveness of each of the ~ ~ orders provided by the analytic procedures, ~ used gamma. Gamma provides an order-based measure of agreement. T compared each of the orders suggested by the procedures with the idealized orders provided by canonical analysis and p~red-compar~son analysis. Because the two idealized orders are so similar. their gammas with the orders ~ ~ ~ . ~ ~ . . . p~uuuc<;u by one analytic procedures were, of course, nearly Identical. The results for both the canonical and the paired-comp~son standards are shown in Figure 14. For both model-based standards. Homans' order produced the highest gamma. One must be careful, however, in looking at these values because different gamma calculations may be built on vastly different numbers of observations. In this case, the value of I.0 associated with Homans' work was based on only 17 comparisons in the order of the women. In contrast the values associated with the two analyses by Newman were based on 58 and 59 comparisons respectively. Because Homans' report contained relatively less information about who was in the core and who was peripheral it generated fewer predictions about positions. The predictions it did make happened to agree with the positional information produced by both cntena. But the Newman analyses both produced large numbers of predictions, and they were still mostly in agreement with those produced by the cntena. Beyond Newman. the orders produced hv n~vi.c GY~rrlner ~nr1 f'~~rAnPr ~ J ~ ~7 _ _A ~ A, ~ ,~ ~ ~ of_ A A_ themselves, by Doreian, by Freeman and White in their first analysis, and by Skvoretz and Faust are consistently in agreement with the cntena. Their gammas are all above .9 and they are all based on at least 43 comparisons. At the opposite extreme, both Bonacich analyses and the Borgatti and Everett bi-cTique analysis do not agree very well with the cntena. ID Code Analysis (;amma with Number of 4:;amma wilt, Number of Cannonical Comps Paired-Companson Comps 1 DGG41 cams, Gardner and Gardner, Eggnog. 0.962 52 0.923 52 2 HOMED Homans, Intuidon 1.000 17 1.000 17 6 BCH78 Bonacich, Boolean Algebra -~.111 9 0.~00 10 7 DOR79 Doreian, PJgebra~c Topology 0.329 28 0.929 28 8 BCH91 Bonacich, Correspondence Analysis 0.313 67 0.324 68 12 FWl;93 Freeman and White, Full Lath ce 0.953 43 0.~53 43 13 FVV293 Freeman and White, Su~Latice 0.867 45 O.870 46 14 BEl97 Borgati and Everett, Bi-Clique O.385 39 0.350 40 17 S&F99 Skvore~and Faust,p~ 0.932 59 0.933 60 18 ROB00 Roberts, Correspondence Analysis 0.844 64 0.846 65 20 N~1 Newman, Weighted C~P4tendance 0.~6 58 O.932 59 Figure 14. Gammas Showing the Degree to which 11 Analvses A~ree`1 with the Twn Standards In Assigning individuals to Core and Peripheral Positions DY7 JAMIC SOCIAL NETWORK MODELING AND ANALYSIS 67

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So again, we have been able to uncover something close to a consensus this time with respect to core and peripheral positions. And we have again been able to find out something about the extent to which each of the analytic procedures approaches that consensus. 6. Summary and Discussion 6.l Assignment to Groups Each of the ~ ~ analyses reported here assigned the DGG women to groups. Consensus analysis determined the agreement among the assignments. It turned out that there was a strong core of agreement among most of the analytic devices. The agreement was substantial enough to allow the model to be used specify a partition of the women into groups~ne that captured the consensus of all the analyses. At the same time, the consensus analysis was also able to provide ratings of the "competence' of each of the analytic procedures. The consensual assignment of women to groups and the "competence" ratings of the analytic methods were reported above. The "competence" scores were reflected in the first axis of an singular value decomposition of the matches generated by the methods in assigning pairs of women to the same or to different groups. In that earlier examination ~ reported only the first axis. But here. it is instructive to examine the second and third axes. They are shown in Figure 15. Fues3 BCH78 ~ ~ BeR74 ~ DOR78 BE197 ~~ r FRE8Q Baled nOM5O S&F Romo BE3er Bat ~ P"" SINEWS FR193 ~1 FWI93 Figure 15. Axes 2 and 3 Produced by the Singular Value Decomposition of the Matches in the Assignments of Women 68 DYNAMIC SOCIAL NETWORK HODE~G ED ISIS

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The arrangement of points representing the analytic methods in Figure 15 tells a good deal about both the partitioning of women to groups and the competence of the methods. The consensus put women ~ through A in the first group and women 10 through ~ ~ in the second. In Figure 15, the basis for determining why this was the "best" partition becomes apparent. That partition was specified exactly by six of the analytic procedures: P&C72 (The corrected version of Phillips and Conviser's inflation theoretic algorithms, BCH91 (Bonacich's correspondence analysis), FRI93 (Freeman~s first genetic a:[gonthm solution), BE297 (Borgatti and Everett's taboo search), BE397 (Borgatti and Everett's genetic aIgorithm) and ROB00 (Roberts' correspondence analysis of normalized data). These analyses are all placed at a single point at the upper right of the figure. Other analyses that produced results that were quite close to that ideal pattern are clustered closely around that point. For example, BBA75 and NEW01 produced the same pattern with only one exception. They both assi;,ned woman ~ to the second group. ~ ~ ~ ~ ~ ~ A: ~ ~ 1~ ~ C) _ _ ~ ~ ~ _ .1 1 =~ r ~ ~~ ~ asslg~lt;u corn women ~ ana a to me second group. DGG41 put woman 9 in both groups. And FW193 put woman 16 in both. Finally, S&F99 deviated only by failing to include woman 16 in either group. Thus, in addition to the six "perfect" partitionings, six additional procedures came very close to the ideal and are clustered in the region surrounding these "perfect" solutions. This clustering is the key. It shows a clear consensus around the I-9, 10-~8 division. This consensus is really remarkable in view of the immense differences among the analytic procedures used. Figure 16 re-labels all the points such that their departure from the "perfect" partitioning is displayed. Note that the I-9, 10-IS partition is labeled "PERFECT." Note also that departures from that ideal are generally placed farther from the PERFECT point as the degree of their departure grows. They are, moreover, segregated in terms of the kinds of departure they embody. All the points that fall on the left of the vertical axis involve methods that failed to assign two or more of the women to groups. Overall, those points are arranged in such a way that those falling further to the left are those that are missing more women. Immediately to the right of the vertical, are the methods that located women in the "wrong" group. And their height indicates the number of women classified in "error." All the way to the right are the methods that assigned women to both groups. And, to the degree that they assigned more women in that way, they are farther to the right. Finally, it should be noted that there are two analyses that assigned women to multiple groups on the left. But it is clear from their placement that this analysis was more responsive to their inability to place women in groups than it was to their dual assignments. DYNAMIC SOCKS NETWO=MODEL~G ED TRYSTS 69

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2 MISSING AND 3 MISSING AND 1 MISSING FRFE T 20VERLAP 1 OV~RW P C 6 MISSING 4 MISSING | \ | / 1 WRONG SIDE / O \ 0 2 MISSING O ~ WRONG SIDE / O O O MISSING ~ l I ~ OWR~P 2 WRONG SIDE 3 MISSING ~ 6 WRONG SIDE Figure 16. Axes 2 and 3 of the Matches Labeled by Structural Form OveratI, then, it is clear that there was a consensus about assigning women to groups. Six methods agreed, and most of the others departed relatively little from that agreed-upon pattern. 6.2 Positions in Groups In assigning positions to individuals, ~ used two, quite different, scaling techniques. One was based on a dominance model. For each pair of women A and B. the mode! placed A closer to the core than B. if and only if more procedures placed A closer to the core than B. The other was probability-based. It placed A closer to the core than B with some probability based on the proportion of procedures that placed A closer to the core than B. Despite their differences, the results of these two methods turned out to be almost identical. They were similar enough that either could be taken as providing something very close to an optimum assignment of individuals to positions. The effectiveness of each of the analytic procedures was evaluated by their monotone correlations with these optima. The results were very similar; the correlation between the gammas produced by the dominance model and those produced by the probability mode! was .983. So, even with without consensus amon, the procedures, 1 was able to find the a3reed-upon order~ore to penphe~y and to evaluate the ability of each method to uncover that order. 70 DYNAMIC SOCIAL NETWO~MODEL~G kD ISIS

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6.3 A Final Word As a whole T believe that this meta-analysis has been productive. As far as assigning women to groups was concerned, the results were dramatic. When it came to assigning women to positions, the results were less dramatic, but still fairly convincing. We end, then, with four strong results: (~) We have a consensual partitioning of women into groups. (2) We have a consensual assignment of women into core and peripheral positions. (3) We have a rating of the methods in teens of their competence in assigning women to groups. And (4) we have a rating of the methods in terms of their ordinal correlations with the standard positional assignments. ~ would like to wind up with two additional comparisons of the analytic procedures examined here. These final comparisons will be restricted to the published analyses of the DGG data; they will not include the unpublished analyses involving my use of Osbourn's VERI or the analysis Newman ran at my request. The first comparison is based on time. Figure ~ 7 shows the average competence ratings of procedures published at various points of time. The data in Figure 17 show an interesting secular trend. There has been a slow but consistent trend toward increasing competence through time. Thus, the overall tendency in published reports using the DOG data is clearly in the direction of greater competence. In addition. it is possible to make some generalizations about the adequacy of the various kinds of analytic procedures that have been used to find groups and positions using the DOG data. Six procedures (BGR74, BCH7S, DOR79, E&B93, FWI93 and FW293) all took essentially algebraic approaches. Five (P&C72, FRI93, FR293, BEl97 and BE297) used venous aigonthms to search for an optimal partition. Three analyses (BBA75, BCH91 and ROB00) employed various versions of singular value decomposition. Two (DGG41 and HOM50) were based simply on the authors' intuititions. And three developed unique approaches. One (FRE92) looked at a kind of transitivity. A second (BEl97) dealt with overlapping bictiques. And the third (S&F99) developed a statistical model. D YIJAMIC SOCIAL NETWORK MODELING AND ANAL YSIS 71

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1 0.8 O.6 0.4 0.2 to . . . I:. ' .: , :: -, .: ,-., .: , ,~ .. . .. . .. . . . I:' '' : I:- . ' ' ( ,,,,:~`,,~"i , I'd ,': 'a ,.,,:'~',^~,'~'.','i,' ~ v~ . ',: ''j,,''''f''"'.,.,,';,;,. ',':. .': ;' , (- '. '.:' ' :' ' ' ' ' ' ~1 ~ i . ' . ~ In,, ~~ .d F) ~ ~ . ~ . ~ q)y i`, ,~ ,~ ~ .' ~ '.~ ~ . ,.,, <, _ : :~ i. ~ ;. : .: . :.'^.,':; ,~:'.:~ . ~ :<~;:~ :., ;:.i-~~;^~ I .'. at ~~:~':< .. " ,"':"':'"''' 'a , ant: :,'~., A ''_"-,','[ ' . '': :~:~:~ ~ W ` ~ ~~ : ; ~,W,,,~ .,,{ - . . .,,,,, Al . . ~ 1 St.,; ,; '~ ~~] ~ ~ ~~ :~ of; a,' ~ ~ ~ ~~: ~ , . ~ f ~ 2~r ' ~ ' '.,''.~'.;~~2 A; I' ' W.>' ~ ,' ,; ~'~v~' ', i~ ~~ 40s and SOs 70s 91-93 97-00 Figure 17. Competences of the Procedures Over Time All in all, then, we have used seven distinct classes of procedures in analyzing the DOG data. Figure ~ ~ shows the relative success of each class in terms of its average competence. Procedure N Average Score Statistical hJlodel 1 0.957 Eigen Structure 3 0.954 C: ptimal Partition ~ 0.941 Transitivity 1 0.926 Cliques 1 0.916 Algebraic Duality 6 0.914 Intuition 2 0.887 Figure IS. Average Competences of the Various Classes of Procedures A number of features of Figure 17 are worth noting. First, the statistical mode! of the DOG data developed by Skvoretz and Faust was the winner. It won despite the fact that. unlike most of the other procedures, it was not explicitly designed to uncover groups. Group structure emerged as a sort of bi-product of a broader structural analysis. 72 DYNAMIC SOCIAL N~:TWORKMODEL~G AD ISIS

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The statistical model is not' however, the undisputed champion. It is followed so closely by the three singular value decomposition analyses, that it has to share the crown with them. And the five partitioning programs are right up there near the top. There seems to be a step between all those procedures and the next three. Clearly transitivity, bicliques and the algebra-based approaches did not do as well. And, finally, the intuitive judgments fall at the bottom. In part that position is due to the vagaries of Homans' report, but DGG themselves did very little better. This result is particularly interesting given the fact that Davis, Gardner and Gardner's interpretation of their own data is often taken as privileged. The assumption has been that because they had a huge amount of ethnographic experience in the community, DGG had an edge- they somehow knew the ' true" group structure. But, particularly in the light of the present results, there is no compelling reason to award DGG any special exalted status vis-a-vis their ability to assign individuals to groups. Indeed, their very intimacy with these IS women might have led to various kinds of biased judgments. References Batchelder, W. H. and N. J. Bershad 1979 The statistical analysis of a Thurstonian mode! for rating chess players. Journal of Mathematical Psychology, 19:39-60. Batchelder, W. H., N. J. Bershad and R. S. Simpson 1992 Dynamic p~red-comparison scaling. Journal of Mathematical Psychology, 36: ~ 85-212. Batchelder, W. H. and A. K. Romney 986 The statistical analysis of a general Condorcet mode! for dichotomous choice situations. In B. Grofman and G. Owen (Eds.) hnformation Pooling and Group Decision Malting. Greenwich, Connecticut: JAl Press, Inc. Batchelder, W. H. and A. K. Romney 1988 Test theory without an answer key. Psychometr~ka 53:71-92. Batchelder, W. H. and A. K. Romney 1989 New results in test theory without an answer key. In E. Roskam (Ed.) Advances in Mathematical Psychology Vol. H. Heidelberg New York: Springer VerIag. Birkhoff, G. 1940 Lattice theory. New York: American Mathematical Society. Bonacich, P. 1978 Using boolean algebra to analyze overlapping memberships. DYNAMIC SON NETWO=MODEf~G ED TRYSTS 73

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