Dynamic Social Network Modeling and Analysis Workshop Summary and Papers (2003) / Chapter Skim
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Random Effects Models for Network Data
Random Effects Models for Network Data
Pages 303-312
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From page 303... ...
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 corn also provide graphical representations of network structure and variability.
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From page 304... ...
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~.
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From page 305... ...
2 Network Random Effects Models Generalized linear models, or glm's, are ubiquitous tools which extend linear regression models to non-normal data and transformable additive covariate effects (McCullagh and Nelder, 1983)
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From page 306... ...
One approach is to presume reciprocity, transitivity, and balance arise due to the existence of unobserved node characteristics, and that nodes relate preferentially to other nodes with similar values of those characteristics This motivates letting f be a measure of "similarity" between the random effects zi and Zj, which gives rise to a "latent position" interpretation as discussed in Hoff et se.
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From page 307... ...
Additionally, plotting estimates and confidence regions for the zi's gives a graphical, model-based representation of the network data 3 Parameter Estimation Given network data Y = {Yi,j} and possible regressor variables X = {xi,j}, the goal is to make statistical inference on the unknown model parameters, which we generically denote as §. The parameter ~ may include the regression coefficients 3, the variances of the random effects, and possibly the random effects themselves.
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From page 308... ...
data exhibit a tendency of children to form same sex friendship ties, in that 72~o of the ties are sam~sex. We consider a statistical analysis of this preference, in which we estimate the log odds of a same-sex tie, as well as make a confidence interval for its value.
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From page 309... ...
Marginal posterior distributions of IBM, ~27 Cub' ~2 are presented in Figure 2. The results suggest a significant preference for same sex friendship ties, in that the posterior distribution for ,5~ is centered around a median of 1.49, and a 95~o quartile-based confidence interval for 3~ is (0.84, 2.~)
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From page 310... ...
1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 2: Marginal posterior distributions for the classroom data: Dashed lines represent the prior distributions for the variance parameters, solid lines the posterior. Vertical lines give the posterior median.
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From page 311... ...
5 Discussion This article proposes a form of generalized linear mixed-effects mode} for the statistical analysis of network data for which parameter estimation is practical to implement. The approach has some advantages over existing social network models and inferential procedures: the approach allows for prediction and hypothesis testing; lends itself to a model-based method of network visualization; is highly extendable and interpretable in terms of well known statistical pro ceclures such as regression and generalized linear models; and has a feasible means of exact parameter estimation.
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From page 312... ...
Their latent class model, combined with types of random effects models presented here and possibly other random effects structures, could provide a rich class of models for dependent network data. References Besag, J., Green, P., Higdon, D., and Mengersen, K
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