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 groupWoman 9 in both groups

And they specified positions in each:

1-4 & 13-15 Core5-7 & 11-12 Primary8-9 & 9,10, 16, 17, 18 Secondary

DGG described the groups they saw:

Women 1-9 in one group 9-18 in the other groupWoman 9 in both groups

And they specified positions in each:

1-4 & 13-15 Core5-7 & 11-12 Primary8-9 & 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

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.”

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

Code Analysis Closeness to the Matching Criterion 3 P&C72 Phillips & Conviser, Information Theory (corrected) .968 8 BCH91 Bonacich, Correspondence Analysis .968 11 FR193 Freeman, Genetic Algorithm 1 .968 16 BE297 Borgatti & Everett, Taboo Search .968 17 BE397 Borgatti & Everett, Genetic Algorithm .968 19 ROB00 Roberts, SVE with Dual Normalization .968 18 S&F99 Skvoretz & Faust, p* Model .957 14 FW293 Freeman & White, Galois Sub-Lattice .954 4 BGR74 Breiger, Algebra .933 21 NEW01 Newman, Weighted Co-Attendance .932 5 BBA75 Breiger, Boorman & Arabie, CONCOR .927 9 FRE92 Freeman, G-Transitivity .926 7 DOR79 Doreian, Algebraic Topology .923 1 DGG41 Davis, Gardner and Gardner, Intuition .920 13 FW193 Freeman & White, Galois Lattice .917 15 BE197 Borgatti & Everett, Bi-Cliques .916 2 HOM50 Homans, Intuition .854 12 FR293 Freeman, Genetic Algorithm 2 .842 6 BCH78 Bonacich, Boolean Algebra .841 20 OSB00 Osbourn, VERI Algorithm .543

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:

Procedure N Average Score

Statistical model 1 .957

Eigen structure 3 .954

Optimal partition 5 .941

Transitivity 1 .926

Cliques 1 .916

Algebraic duality 6 .914

Intuition 2 .887