Likelihoods for fixed rank nomination networks

Abstract

Many studies that gather social network data use survey methods that lead to censored, missing, or otherwise incomplete information. For example, the popular fixed rank nomination (FRN) scheme, often used in studies of schools and businesses, asks study participants to nominate and rank at most a small number of contacts or friends, leaving the existence of other relations uncertain. However, most statistical models are formulated in terms of completely observed binary networks. Statistical analyses of FRN data with such models ignore the censored and ranked nature of the data and could potentially result in misleading statistical inference. To investigate this possibility, we compare Bayesian parameter estimates obtained from a likelihood for complete binary networks with those obtained from likelihoods that are derived from the FRN scheme, and therefore accommodate the ranked and censored nature of the data. We show analytically and via simulation that the binary likelihood can provide misleading inference, particularly for certain model parameters that relate network ties to characteristics of individuals and pairs of individuals. We also compare these different likelihoods in a data analysis of several adolescent social networks. For some of these networks, the parameter estimates from the binary and FRN likelihoods lead to different conclusions, indicating the importance of analyzing FRN data with a method that accounts for the FRN survey design.

Multiplex congruity: Friendship networks and perceived popularity as correlates of adolescent alcohol use

“…Analyzing a binary network of friendship nomination but ignoring the rank information can result in misleading statistical inferences (Hoff et al, 2013). To address this concern, this study extends the order of friendship nomination to multiplex relationships and takes into account the degree to which one type of relationship is congruent with another type of relationship.…”

Testing and Modeling Dependencies Between a Network and Nodal Attributes

“…In the case of binary data, a thresholding function may be appropriate (corresponding to the probit model), and in the ordinal case the ordered probit can be considered. Hoff et al (2013) discusses additional ℓ functions which account for censoring of binary and ordinal relations when nodes are restricted on the number of relations they can send (i.e. the number of non-zero relations in a row of Y ).…”

“…We use the fixed rank nomination likelihood introduced in Hoff et al (2013) to model the observed network ranks and restriction that at most five friends could be listed on the survey. This likelihood assumes each observed network relation r i,j is the function of an underlying (latent) continuous measure z i,j such that the following relation consistencies are satisfied: $$r_{i,j}>0\Rightarrow z_{i,j}>0,r_{i,j}>r_{i,k}\Rightarrow z_{i,j}>z_{i,k},r_{i,j}=0\text{and student}i\text{listed}<5\text{friends}\Rightarrow z_{i,j}\le 0.$$ The first association is the function used in probit regression which assumes that if a friendship is reported, the latent friendship value must exceed a given threshold (in this case 0).…”

“…We present the simple case here where the observed network Y is continuous, there are no regression terms in the network model, and there is no missing data. For details on accommodating non-continuous network data see Hoff et al (2013) and for including regression terms see Hoff (2005). Model - $$\begin{array}{cc}right{y}_{i,j}& left=\mu +a_i+b_j+{\mathrm{bold-italicu}}_i^T{\mathrm{bold-italic\upsilon }}_j+e_{i,j},\\ right{(e_{i,j},e_{j,i})}^T& left\stackrel{\mathrm{i}\mathrm{i}\mathrm{d}}{~}{\text{normal}}_2(\mathbf{0},{\sigma }_e^2(\begin{array}{cc}\hfill 1\hfill & \hfill \rho \hfill \\ \hfill \rho \hfill & \hfill 1\hfill \end{array}\left)\right)\\ right{({\mathrm{bold-italicx}}_i^T,a_i,b_i,{\mathrm{bold-italicu}}_i^T,{\mathrm{bold-italic\upsilon }}_i^T)}^T& left\stackrel{\mathrm{i}\mathrm{i}\mathrm{d}}{~}{\text{normal}}_{p+2+2k}(\mathbf{0},{\mathrm{normal\Sigma }}_{XN})\end{array}$$ Prior distributions - $$\begin{array}{cc}right{\sigma }_e^{-2}& left~\text{gamma}(1/2,1/2)\\ right\rho & left~\text{uniform}(-1,1)\\ right{\mathrm{normal\Sigma }}_{XN}^{-1}& left~\text{Wishart}(p+2+2k+1,(\begin{array}{cc}\hfill {\mathrm{\Sigma }}_{X0}^{-1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathrm{I}}_{2+2k}\hfill \end{array}\left)\right)\end{array}$$ Markov chain Monte Carlo algorithm - Given initial values of all latent variables { a , b , U, V } and parameters {Σ XN , ρ, $$…”

Networks, Dyads, and the Social Relations Model

“…These kinds of models can be estimated using MLwiN. 11 Hoff et al (2012) have also recently developed a wider range of techniques for nomination networks that also estimates the covariance structures within the SRM framework. 12 Another way this can be done is to estimate a latent space model, which integrates components of the basic SRM by incorporating the decomposition of variance into the error term of a bilinear mixedeffects model.…”