A p* primer: Logit models for social networks

A major criticism of the statistical models for analyzing social networks developed by Holland, Leinhardt, and others [Holland, P.W., Leinhardt, S., 1977. Notes on the statistical analysis of social network data; Holland, P.W., Leinhardt, S., 1981. An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association. 76, pp. 33-65 (with discussion); Fienberg, S.E., Wasserman, S., 1981. Categorical data analysis of single sociometric relations. In: Leinhardt, S. (Ed.), Sociological Methodology 1981, San Francisco: Jossey-Bass, pp. 156-192; Fienberg, S.E., Meyer, M.M., Wasserman, S., 1985. Statistical analysis of multiple sociometric relations. Journal of the American Statistical Association, 80, pp. 51-67; Wasserman, S., Weaver, S., 1985. Statistical analysis of binary relational data: Parameter estimation. Journal of Mathematical Psychology. 29, pp. 406-427; Wasserman, S., 1987. Conformity of two sociometric relations. Psychometrika. 52, pp. 3-18] is the very strong independence assumption made on interacting individuals or units within a network or group. This limiting assumption is no longer necessary given recent developments on models for random graphs made by Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association. 81, pp. 832-842] and Strauss and Ikeda [Strauss, D., Ikeda, M., 1990. Pseudolikelihood estimation for social networks. Journal of the American Statistical Association. 85, pp. 204-212]. The resulting models are extremely flexible and easy to fit to data. Although Wasserman and Pattison [Wasserman, S., Pattison, P., 1996. Logit models and logistic regressions for social networks: I. An introduction to Markov random graphs and p*. Psychometrika. 60, pp. 401-426] present a derivation and extension of these models, this paper is a primer on how to use these important breakthroughs to model the relationships between actors (individuals, units) within a single network and provides an extension of the models to multiple networks. The models for multiple networks permit researchers to study how groups are similar and/or how they are different. The models for single and multiple networks and the modeling process are illustrated using friendship data from elementary school children from a study by Parker and Asher [Parker, J.G., Asher, S.R., 1993. Friendship and friendship quality in middle childhood: Links with peer group acceptance and feelings of loneliness and social dissatisfaction. Developmental Psychology. 29, pp. 611-621].