| Abstract: |
Holland and Leinhardt (1981) proposed a simple exponential model called the p1model for analyzing digraphs that arise in studies of networks. A digraph consists of a set of g nodes and a g × g adjacency matrix (Xij), where Xij= 1 if node i relates to node j and Xij= 0 otherwise. The underlying cell probabilities pij= Pr(Xij= 1) are to be estimated from these dichotomous responses. The p1model imposes an additive structure on a log-odds version of the pij. It provides information about the abilities of an individual node to attract and to produce relational ties, as well as the tendency of a pair of nodes to reciprocate ties. For digraphs of realistic sizes, the maximum likelihood estimates (MLE’s) of the p1exponential parameters are often unsatisfactory, particularly when some of the row and column marginal totals of the adjacency matrix are small. A Bayesian approach, using an exchangeable normal prior on the parameters representing the attractiveness and expansiveness characteristics of the nodes, is proposed. The Bayesian p1model explicitly recognizes the association between these two characteristics of a node, an important feature ignored by its fixed effects counterpart. An algorithm for finding the MLE’s of the covariance components based on a marginal likelihood is presented. An approximate posterior estimation procedure for the exponential parameters is proposed. Using an empirical example, it is shown that the Bayesian p1model can yield answers quite different from those of the fixed effects model. © 1976 Taylor & Francis Group, LLC. |
