Random Effects Models for Network Data

One impediment to the statistical analysis of network data has been the difficulty in modeling the dependence among the observations. In the very simple case of binary (0-1) network data, some researchers have parameterized network dependence in terms of exponential family representations. Accurate parameter estimation for such models is quite difficult, and the most commonly used models often display a significant lack of fit. Additionally, such models are generally limited to binary data. In contrast, random effects models have been a widely successful tool in capturing statistical dependence for a variety of data types, and allow for prediction, imputation, and hypothesis testing within a general regression context. We propose novel random effects structures to capture network dependence, which can also provide graphical representations of network structure and variability. 1 Network Dependence Network data typically consist of a set of n nodes and a relational tie yi,j, measured on each ordered pair of nodes i, j = 1, . . . , n. This framework has many applications, including the study of war, trade, the behavior of epidemics, the interconnectedness of the World Wide Web, and telephone calling patterns. It is often of interest to relate each network response yi,j to a possibly pair-specific vector valued predictor variable xi,j . A flexible framework for doing so is the generalized linear model (see, for example McCullagh and Nelder 1983), in which the expected value of the response is modeled as a function of a linear predictor β xi,j , where β is an unknown vector of regression coefficients to be estimated from the data. The ordinary regression model E(yi,j) = β xi,j is perhaps the most commonly used model of this type. A generalized linear model for binary (0-1) data is logistic regression, which relates the expectation of the response to the regression variable via the relation g(E[yi,j]) = β xi,j, where g(p) = log p 1−p . As an example of the use of such statistical models, consider the analysis of strong friendship ties among 13 boys and 14 girls in a sixth-grade classroom, as collected by Hansell (1984). Each student was asked if they liked each other student “a lot”, “some”, or “not much”. A strong friendship tie is considered present if a student likes another student “a lot.” Also recorded is the sex of each student. The data, presented in Figure 1, suggest a general preference for same-sex friendship ties. Of potential interest is statistical estimation of this preference, as well as a confidence interval for its value. One approach for such statistical analysis would be to formulate the logistic regression model g(E[yi,j|xi,j , β]) = β0 + β1xi,j , where xi,j is one if children i and j are of the same sex, and zero otherwise, and β = (β0, β1) are parameters to be estimated. Estimation of regression coefficients β typically proceeds under the assumption that the observations are conditionally independent given β and the xi,j ’s. However, this assumption is often violated by many network datasets. For example, the data on friendship ties display several types of dependence: Within-node dependence: The number of ties sent by each student varies considerably, ranging from 0 to 19 with a mean of 5.8 and a standard deviation of 4.7 (the standard deviation of the number of ties received was 3.2). This node level variability suggests that responses from the same individual are positively dependent, in that the probability that yi,j = 1 (i sends a tie to j), is high if we know yi,k = 1 for lots of other nodes k, and lower if yi,k is mostly zero. More formally, we may wish to have a model in which Pr(yi,j = 1|yi,1, . . . , yi,j−1, yi,j+1, . . . , yi,n) is an increasing function of yi,k, k 6= j.