Finding Social Groups: A Meta-Analysis of the Southern Women Data Linton C. Freeman Photograph by Ben Shahn, Natchez, MS, October, 1935

In 1933 W. Lloyd Warner was teaching at Harvard. He decided to send four graduate students, Allison and Elizabeth Davis and Burleigh and Mary Gardner to study race and social class in Natchez, Mississippi. They collected systematic two mode data on the participation of 18 women in 14 small informal social events.

p.148

Davis, Gardner and Gardner sought: 1. To specify tightly knit groups 2. To assign women to core and peripheral positions in their assigned groups They said:

Where it is evident that a group of people participate together in these informal activities consistently, it is obvious that a clique 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. p. 150

DGG described the groups they came up with: Women 1-9 in one group 9-18 in the other group Woman 9 in both groups And they specified positions in each: 1-4 & Core 5-7 & Primary 8-9 & 9,10, 16, 17, 18 Secondary

DGG described the groups they saw: Women 1-9 in one group 9-18 in the other group Woman 9 in both groups And they specified positions in each: 1-4 & Core 5-7 & Primary 8-9 & 9,10, 16, 17, 18 Secondary

Since then: 21 procedures have been used to assign women to groups, and 11 to assign positions in the groups They are:

DGG 1941 Intuition Homans 1951 Intuition Phillips and Conviser 1972 Information Theory Breiger 1974 Matrix Algebra Breiger, Boorman & Arabie 1975 Computational Bonacich 1978 Boolean Algebra Doreian 1979 Algebraic Topology Bonacich 1991 Correspondence Analysis Freeman 1992 G-Transitivity Everett & Borgatti 1993 Regular Coloring Freeman 1993 Genetic Algorithm I & II Freeman & White 1993 Galois Lattices I & II Borgatti & Everett 1997 Bipartite Analyses I, II & III Skvoretz & Faust 1999 p* Model Roberts 2000 Normalized SVD Osbourn 2000 VERI Procedure Newman 2001 Weighted Proximities

And they assigned women to groups:

Group Assignments

And they assigned positions:

Core/Periphery Assignments

Here I will do a kind of meta-analysis: one data set several analytic procedures. Schmid, Koch, and LaVange (1991): Meta-analysis is “... a statistical analysis of the data from some collection of studies in order to synthesize the results.”

The Question of Group Membership

Batchelder, Romney and Weller—Consensus Analysis Gets: “true” answers (consensus) “competence of judges” (approach to consensus)

(Based on iteration to maximum likelihood)

Then calculate matches and covariance. If they agree, factor analyze and the first factor estimates “competence.”

Here the correlation between matches and covariance is.967

The Question of Core and Periphery

Two Methods for Ordering: Gower’s (1977) canonical analysis of asymmetry (algebraic-deterministic) Batchelder and Bershad’s (1979) dynamic paired-comparison scaling (probabilistic)

Here, I’ll try to interpret dimensions 2 and 3 of the principal components analysis. Here they are:

They show a consistent pattern in terms of the way they depart from the consensual pattern:

Through time there has been a very slow, but steady movement toward the consensual pattern

And, we can evaluate the several families of approaches to uncovering groups:

ProcedureNAverage Score Statistical model1.957 Eigen structure3.954 Optimal partition5.941 Transitivity1.926 Cliques1.916 Algebraic duality6.914 Intuition2.887