Department of Statistics. Bayesian Modeling for a Generalized Social Relations Model. Tyler McCormick. Introduction.

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1 A University of Connecticut and Columbia University A models for dyadic data are extensions of the (). y i,j = a i + b j + γ i,j (1) Here, y i,j is a measure of the tie from actor i to actor j. The random effects a i and b j are for actors i and j respectively and a random interaction γ i,j describes the adjustment that actor i makes when interacting with j. This model can be easily generalized to include fixed effects, generating the following form of the linear mixed model. tmccormick@stat.uconn.edu y i,j = β x i,j + ɛ i,j ɛ i,j = a i + b j + γ i,j () 1 / 17 / 17 A The can also be equivalently parameterized using a grand-mean parametrization as in Gill and Swartz (006). In this case all random effects are assumed to have zero mean. The distributions of the random effects are assumed to be as follows: ( σ (a i, b j ) MVN(0, Σ a,b ), Σ a,b = a ρσa,b ) ρσa,b σb ( σ (γ i,j, γ j,i ) MVN(0, Σ γ ), Σ γ = γ ρσγ ρσγ σγ ) (3) A Typically, these variance/covariance parameters are of primary interest and define the dependence structure of the observations. Dyadic dependence in the form of mutuality or reciprocity is modeled by ρ. The same general form of the model can be applied to dichotomous our count data via the Linear (see for example McCullagh and Nelder, 1989). 3 / 17 4 / 17

2 Typically we would fit this model using: A Analysis of Variance (ANOVA) Maximum Likelihood (ML) or Restricted Maximum Likelihood (RML) Let s consider another option, fully Analysis A A demonstration of the paradigm. 5 / 17 6 / 17 A A potential limitation of the is its inability to consider third or higher order dependence structure. The is generalized by Hoff 1 (005). In the, we assume that the dependence between i and j is modeled based on the ɛ i,j parameter. We say that ɛ i,j is composed of three dyad-specific components, each of which is an unobserved random effect, and the dependence between these components determines the nature of the dependence of the responses. A We here generalize this notion by adding f, a function of k-dimensional latent variables {z i, z j }. In other words, {z i, z j } are characteristics of the actors that are not observed, but define the dependence between the y i,j s when subjected to some function f. ing these latent variables is less straightforward than the unobserved dyad-specific effects presented in the. If we assume, however, that actors that are more similar in the unobserved variables {z i, z j } are more likely to have similar y i,j s, then there are certain functions that will induce the desired type of dependence. 7 / 17 1 Software to fit the model described here using the statistical software R is available from Peter Hoff s website at 8 / 17

3 A The full parametrization of our model thus becomes: θ i,j = β 0 + β d x i,j + β s x i + β r x j + ɛ i,j ɛ i,j = a i + b j + γ i,j + f (z i, z j ) f (z i, z j ) = z i z j (4) A Transitivity is induced via the triangle inequality. Consider the case where f is simple distance, then z i z j z i z k + z k z j (5) Notice that z i z k and z k z j are small when strong ties are present from i to k and from k to j. From the triangle inequality, z i z j must also be small, an indicator of strong ties between i and j. We choose f (z i, z j ) = z i z j or the inner product. In addition to transitivity, using the inner product also allows a representation of clustering and balance. 9 / / 17 -Roommates A 11 / 17 To determine the potential contribution of higher-order effects to the, we undertake a model-selection experiment using the Curry and Emerson data. If adding higher-order effects via the latent dimension represents a significant improvement to the, we would expect our model selection criterion to prefer models with latent dimension greater than zero. We use a four-fold cross validation procedure that is typically used to assess the predictive ability of the model. We chose this procedure because it is simple to implement and is consistent with the method proposed by Hoff (005). Higher (or less negative) values of our selection criteria (log posterior predictive probability) are preferred. A 1 / 17

4 -Log Posterior Predictive Probability - A Group k0 k1 k Table: Log Posterior Predictive Probability, with roommates A 13 / / 17 - A Group k0 k1 k Table: -Log Posterior Predictive Probability, No Roommates A A few references- Hoff, P.D. (005) Bilinear Mixed-Effects s for Dyadic Data Journal of the American Statistical Association, vol. 100, no. 469, Shortreed, S., Handcock, M.S. and Hoff, P.D. (006) Positional estimation within the latent space model for networks Methodology, vol. no. 1, Hoff, P.D., Raftery, A.E., and Handcock, M.S. (00) Latent Space es to Network Analysis Journal of the American Statistical Association, vol. 97, no. 460, / / 17

5 A A few more- Gill, P. S. and Swartz, T. B. (006) analysis of dyadic data. Gill, P. S. and Swartz, T. B. (004). analysis of directed graphs data with applications to social networks. Applied Statistics: Journal of the Royal Statistical Society Series C, 53, Gill, P. S. and Swartz, T. B. (001) Statistical analyses for round robin interaction data, The Canadian Journal of Statistics,Vol. 9, No, pp / 17