Using modern methods for missing data analysis with the social relations model: A bridge to social network analysis

Social network analysis identifies social ties, and perceptual measures identify peer norms. The social relations model (SRM) can decompose interval-level perceptual measures among all dyads in a network into multiple person- and dyad-level components. This study demonstrates how to accommodate missing round-robin data using Bayesian data augmentation, including how to incorporate partially observed covariates as auxiliary correlates or as substantive predictors. We discuss how data augmentation opens the possibility to fit SRM to network ties (potentially without boundaries) rather than round-robin data. An illustrative application explores the relationship between sorority members’ self-reported body comparisons and perceptions of friends’ body talk.     

An introduction to exponential random graph (p) models for social networks

This article provides an introductory summary to the formulation and application of exponential random graph models for social networks. The possible ties among nodes of a network are regarded as random variables, and assumptions about dependencies among these random tie variables determine the general form of the exponential random graph model for the network. Examples of different dependence assumptions and their associated models are given, including Bernoulli, dyad-independent and Markov random graph models. The incorporation of actor attributes in social selection models is also reviewed. Newer, more complex dependence assumptions are briefly outlined. Estimation procedures are discussed, including new methods for Monte Carlo maximum likelihood estimation. We foreshadow the discussion taken up in other papers in this special edition: that the homogeneous Markov random graph models of Frank and Strauss [Frank, O., Strauss, D., 1986. Markov graphs. Journal of the American Statistical Association 81, 832–842] are not appropriate for many observed networks, whereas the new model specifications of Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handock, M. New specifications for exponential random graph models. Sociological Methodology, in press] offer substantial improvement.     

Three Likelihood-Based Methods For Mean and Covariance Structure Analysis With Nonnormal Missing Data

Survey and longitudinal studies in the social and behavioral sciences generally contain missing data. Mean and covariance structure models play an important role in analyzing such data. Two promising methods for dealing with missing data are a direct maximum-likelihood and a two-stage approach based on the unstructured mean and covariance estimates obtained by the EM-algorithm. Typical assumptions under these two methods are ignorable nonresponse and normality of data. However, data sets in social and behavioral sciences are seldom normal, and experience with these procedures indicates that normal theory based methods for nonnormal data very often lead to incorrect model evaluations. By dropping the normal distribution assumption, we develop more accurate procedures for model inference. Based on the theory of generalized estimating equations, a way to obtain consistent standard errors of the two-stage estimates is given. The asymptotic efficiencies of different estimators are compared under various assumptions. We also propose a minimum chi-square approach and show that the estimator obtained by this approach is asymptotically at least as efficient as the two likelihood-based estimators for either normal or nonnormal data. The major contribution of this paper is that for each estimator, we give a test statistic whose asymptotic distribution is chi-square as long as the underlying sampling distribution enjoys finite fourth-order moments. We also give a characterization for each of the two likelihood ratio test statistics when the underlying distribution is nonnormal. Modifications to the likelihood ratio statistics are also given. Our working assumption is that the missing data mechanism is missing completely at random. Examples and Monte Carlo studies indicate that, for commonly encountered nonnormal distributions, the procedures developed in this paper are quite reliable even for samples with missing data that are missing at random.     

Settings in Social Networks: A Measurement Model

A class of statistical models is proposed that aims to recover latent settings structures in social networks. Settings may be regarded as clusters of vertices. The measurement model is based on two assumptions. (1) The observed network is generated by hierarchically nested latent transitive structures, expressed by ultrametrics, and (2) the expected tie strength decreases with ultrametric distance. The approach could be described as model–based clustering with an ultrametric space as the underlying metric to capture the dependence in the observations. Bayesian methods as well as maximum–likelihood methods are applied for statistical inference. Both approaches are implemented using Markov chain Monte Carlo methods.     

Considering clustering measures: Third ties,means, and triplets

Measures that estimate the clustering coefficients of ego and overall social networks are important to social network studies. Existing measures differ in how they define and estimate triplet clustering with implications for how network theoretic properties are reflected. In this paper, we propose a novel definition of triplet clustering for weighted and undirected social networks that explicitly considers the relative strength of the tie connecting the two alters of the ego in the triplet. We argue that our proposed definition better reflects theorized effects of the important third tie in the social network literature. We also develop new methods for estimating triplet, local and global clustering. Three different types of mathematical means, i.e. arithmetic, geometric, and quadratic, are used to reflect alternative theoretical assumptions concerning the marginal effect of tie substitution.     

A Multivariate, Multilevel Rasch Model with Application to Self–Reported Criminal Behavior

In studying correlates of social behavior, attitudes, and beliefs, a measurement model is required to combine information across a large number of item responses. Multiple constructs are often of interest, and covariates are often multilevel (e.g., measured at the person and neighborhood level). Some item–level missing data can be expected. This paper proposes a multivariate, multilevel Rasch model with random effects for these purposes and illustrates its application to self–reports of criminal behavior. Under assumptions of conditional independence and additivity, the approach enables the investigator to calibrate the items and persons on an interval scale, assess reliability at the person and neighborhood levels, study the correlations among crime types at each level, assess the proportion of variation in each crime type that lies at each level, incorporate covariates at each level, and accommodate data missing at random. Using data on 20 item responses from 2842 adolescents ages 9 to 18 nested within 196 census tracts in Chicago, we illustrate how to test key assumptions, how to adjust the model in light of diagnostic analyses, and how to interpret parameter estimates.     

The Statistical Evaluation of Social Network Dynamics

A class of statistical models is proposed for longitudinal network data. The dependent variable is the changing (or evolving) relation network, represented by two or more observations of a directed graph with a fixed set of actors. The network evolution is modeled as the consequence of the actors making new choices, or withdrawing existing choices, on the basis of functions, with fixed and random components, that the actors try to maximize. Individual and dyadic exogenous variables can be used as covariates. The change in the network is modeled as the stochastic result of network effects (reciprocity, transitivity, etc.) and these covariates. The existing network structure is a dynamic constraint for the evolution of the structure itself. The models are continuous-time Markov chain models that can be implemented as simulation models. The model parameters are estimated from observed data. For estimating and testing these models, statistical procedures are proposed that are based on the method of moments. The statistical procedures are implemented using a stochastic approximation algorithm based on computer simulations of the network evolution process.     

Introduction to stochastic actor-based models for network dynamics

Stochastic actor-based models are models for network dynamics that can represent a wide variety of influences on network change, and allow to estimate parameters expressing such influences, and test corresponding hypotheses. The nodes in the network represent social actors, and the collection of ties represents a social relation. The assumptions posit that the network evolves as a stochastic process ‘driven by the actors’, i.e., the model lends itself especially for representing theories about how actors change their outgoing ties. The probabilities of tie changes are in part endogenously determined, i.e., as a function of the current network structure itself, and in part exogenously, as a function of characteristics of the nodes (‘actor covariates’) and of characteristics of pairs of nodes (‘dyadic covariates’). In an extended form, stochastic actor-based models can be used to analyze longitudinal data on social networks jointly with changing attributes of the actors: dynamics of networks and behavior.     

Network sampling coverage II: The effect of non-random missing data on network measurement

Missing data is an important, but often ignored, aspect of a network study. Measurement validity is affected by missing data, but the level of bias can be difficult to gauge. Here, we describe the effect of missing data on network measurement across widely different circumstances. In Part I of this study (Smith and Moody, 2013), we explored the effect of measurement bias due to randomly missing nodes. Here, we drop the assumption that data are missing at random: what happens to estimates of key network statistics when central nodes are more/less likely to be missing? We answer this question using a wide range of empirical networks and network measures. We find that bias is worse when more central nodes are missing. With respect to network measures, Bonacich centrality is highly sensitive to the loss of central nodes, while closeness centrality is not; distance and bicomponent size are more affected than triad summary measures and behavioral homophily is more robust than degree-homophily. With respect to types of networks, larger, directed networks tend to be more robust, but the relation is weak. We end the paper with a practical application, showing how researchers can use our results (translated into a publically available java application) to gauge the bias in their own data.     

Quantifying the magnitude and potential influence of missing data in campus sexual assault surveys: A systematic review of surveys, 2010–2016

Objective: To understand how missing data may influence conclusions drawn from campus sexual assault surveys. Methods: We systematically reviewed 40 surveys from 2010–2016. We constructed a pseudo-population of the total population targeted across schools, creating records proportional to the respective response rate and reported sexual assault prevalence. We simulated the effects of 9 scenarios where the sexual assault prevalence among nonresponders differed from that of responders. Results: The surveys represented a total female undergraduate population of 317,387 with only 77,966 (24.6%) survey responses. Among responders, 20.4% reported experiences of sexual assault. However, prevalence of sexual assault could theoretically range from 5.0 to 80.4% under extreme assumptions about prevalence in nonresponders. Smaller, but still significant differences were observed with less extreme assumptions. Conclusions: Missing data are widespread in campus sexual assault surveys. Conclusions drawn from these incomplete data are highly sensitive to assumptions about the sexual assault prevalence among nonresponders.     

Estimating Self‐Selection of Immigrants: Comparing Earnings Differentials Between Natives and Immigrants in the US and Israel

Immigrants’ economic assimilation in host countries is determined by patterns of self‐selection on both – observed attributes (mainly human capital) and unobserved attributes of the immigrants from their source countries. In the present study immigrants’ economic assimilation in the United States and Israel are compared. More specifically, the study compares the impact of immigrants’ unobserved characteristics on their earnings in both countries by applying a model for decomposing difference in differentials. It makes use of United States and Israeli decennial census data for comparing self‐selection patterns on unobserved attributes of Jewish immigrants from the former Soviet Union (FSU) who arrived in the United States and Israel during the 1970s. The results indicate that FSU immigrants who chose the United States have significantly higher levels of unobserved earnings determinants than those who chose Israel. These results are discussed in light of migration theories.     

Analytic strategies for longitudinal networks with missing data

Missing data are often problematic when analyzing complete longitudinal social network data. We review approaches for accommodating missing data when analyzing longitudinal network data with stochastic actor-based models. One common practice is to restrict analyses to participants observed at most or all time points, to achieve model convergence. We propose and evaluate an alternative, more inclusive approach to sub-setting and analyzing longitudinal network data, using data from a school friendship network observed at four waves (N = 694). Compared to standard practices, our approach retained more information from partially observed participants, generated a more representative analytic sample, and led to less biased model estimates for this case study. The implications and potential applications for longitudinal network analysis are discussed.     

A framework for the comparison of maximum pseudo-likelihood and maximum likelihood estimation of exponential family random graph models

The statistical modeling of social network data is difficult due to the complex dependence structure of the tie variables. Statistical exponential families of distributions provide a flexible way to model such dependence. They enable the statistical characteristics of the network to be encapsulated within an exponential family random graph (ERG) model. For a long time, however, likelihood-based estimation was only feasible for ERG models assuming dyad independence. For more realistic and complex models inference has been based on the pseudo-likelihood. Recent advances in computational methods have made likelihood-based inference practical, and comparison of the different estimators possible.     

Representing degree distributions, clustering, and homophily in social networks with latent cluster random effects models

Social network data often involve transitivity, homophily on observed attributes, community structure, and heterogeneity of actor degrees. We propose a latent cluster random effects model to represent all of these features, and we develop Bayesian inference for it. The model is applicable to both binary and non-binary network data. We illustrate the model using two real datasets: liking between monks and coreaderships between Slovenian publications. We also apply it to two simulated network datasets with very different network structure but the same highly skewed degree sequence generated from a preferential attachment process. One has transitivity and community structure while the other does not. Models based solely on degree distributions, such as scale-free, preferential attachment and power-law models, cannot distinguish between these very different situations, but the latent cluster random effects model does.     

Because they were there: Access,deliberation, and the mobilization of networks for support

When people need help, what is the process through which they decide whom in their network to turn to? Research on social support has described a process that is deliberative in nature: people determine their needs, assess who in their network has the needed attributes—such as skill, trustworthiness, intimacy, and accessibility—and then activate that tie. Nevertheless, research in behavioral economics and other fields has shown that people make many decisions not deliberatively but intuitively. We examine this possibility in the context of social support by focusing on one factor: accessibility. Although researchers have argued that people weigh the accessibility of potential helpers as they do any other attribute, accessibility may be not only an attribute of the helper but also a condition of the situation. We develop a framework to make this question tractable for survey research and evaluate competing hypotheses using original data on an analytically strategic sample of ∼2000 college students, probing concrete instances of social support. We identify and document not one but three decision processes, reflective, incidental, and spontaneous activation, which differ in the extent to which actors had deliberated on whether to seek help and on whom to approach before activating the tie. We find that while the process was reflective (consistent with existing theory) when skill or trustworthiness played a role, it was significantly less so (consistent with the alternative) when accessibility did. Findings suggest that actors decide whom in their network to mobilize through at least three systematically different processes, two of which are consistent less with either active “mobilization” or explicit “help seeking” than with responsiveness to opportunity and context.     

Alternative estimation methods for identifying contagion effects in dynamic social networks: A latent-space adjusted approach

Contagion effects, also known as peer effects or social influence process, have become more and more central to social science, especially with the availability of longitudinal social network data. However, contagion effects are usually difficult to identify, as they are often entangled with other factors, such as homophily in the selection process, the individual’s preference for the same social settings, etc. Methods currently available either do not solve these problems or require strong assumptions. Following Shalizi and Thomas (2011), I frame this difficulty as an omitted variable bias problem, and I propose several alternative estimation methods that have potentials to correctly identify contagion effects when there is an unobserved trait that co-determines the influence and the selection. The Monte-Carlo simulation results suggest that a latent-space adjusted estimator is especially promising. It outperforms other estimators that are traditionally used to deal with the unobserved variables, including a structural equation based estimator and an instrumental variable estimator.     

CAPACITY CONSTRAINED EXPORTERS: IDENTIFYING INCREASING MARGINAL COST

This study revisits a central assumption of standard trade models: constant marginal cost technology. The presence of increasing marginal costs for exporters introduces significant market interdependence across borders missing from traditional models of international trade that rely on constant marginal cost technology. Such market interdependence represents an additional channel through which local shocks are transmitted globally. To identify increasing marginal cost at the level of the firm, we build in flexible production assumptions that nest increasing, decreasing, and constant marginal cost technology to an otherwise standard international trade model. We derive an estimating equation that can be taken directly to the data. Our structural equation explicitly guides our inference on the shape of the marginal cost curve from estimated coefficients. The results suggest that increasing marginal cost is predominant at the firm level. Moreover, utilizing plant‐level information on physical and financial capacity constraints, we find that the degree of increasing marginal cost is significantly exacerbated by both types of constraints. The evidence suggests that access to larger markets through greater international integration may not have the expected welfare gains typically predicted in standard models. (JEL F12, F14)     

Imputation of attributes in networked data using Bayesian autocorrelation regression models

Misspecification in network autocorrelation models poses a challenge for parameter estimation, which is amplified by missing data. Model misspecification has been a focus of recent work in the statistics literature and new robust procedures have been developed, in particular cutting feedback. This paper shows how this helps in a misspecified network autocorrelation model. Where model misspecification is mild and the traits are fully observed, Bayesian imputation is routine. In settings with high missingness, Bayesian inference can fail, but a closely related cut model is robust. We illustrate this on a data set of graduate students using a Facebook-like messaging app.     

Regression Models With Parametrically Weighted Explanatory Variables

This paper describes linear regression models with parametrically weighted explanatory variables and related logistic regression models that estimate parameters characterizing (1) the effects of weighted variables on the dependent variable and (2) weights for the components of weighted variables. The models also characterize parsimoniously the interaction effects between weighted variables and covariates on the dependent variable by the use of various constraints on parameters. In particular, the models are concerned with testing the significance of variation with covariates in the weights of weighted variables separately from the significance of variation with those covariates in the effects of weighted variables.
The usefulness of these models in sociological research is demonstrated by an illustrative analysis of the class identifications of married working women using education, occupational prestige, and income as three variables weighted between own and spousal attributes, and using year, age, race, part–time–full–time distinction, and employment status as covariates.