Model Fit in Structural Equation Models with Censored, Ordinal, and Dichotomous Variables: Testing Vanishing Tetrads

Although the methodology for handling ordinal and dichotomous observed variables in structural equation models (SEMs) is developing rapidly, several important issues are unresolved. One of these is the optimal test statistic to apply as a test of overall model fit. We propose a new "vanishing tetrad" test statistic for such models. We build on Bollen's (1990) simultaneous test statistic for testing multiple vanishing tetrads and on Bollen and Ting's (1993) confirmatory tetrad analysis (CTA) for hypothesis testing of model structures. These and other works on vanishing tetrads assume continuous observed variables and do not consider observed categorical variables. In this paper we present a method to test models when some or all of the observed variables are collapsed or categorical versions of underlying continuous variables. The test statistic that we provide is an alternative "overall fit" statistic for SEMs with censored, ordinal, or dichotomous observed variables. Furthermore, the vanishing tetrad test sometimes permits us to compare the fit of some models that are not nested in the traditional likelihood ratio test. We illustrate the new test statistic with examples and a small simulation experiment comparing it with two other tests of model fit for SEMs with ordinal or dichotomous endogenous variables.     

Missing data in networks: exponential random graph (p1) models for networks with non-respondents

Survey studies of complete social networks often involve non-respondents, whereby certain people within the “boundary” of a network do not complete a sociometric questionnaire—either by their own choice or by the design of the study—yet are still nominated by other respondents as network partners. We develop exponential random graph (p¹) models for network data with non-respondents. We model respondents and non-respondents as two different types of nodes, distinguishing ties between respondents from ties that link respondents to non-respondents. Moreover, if we assume that the non-respondents are missing at random, we invoke homogeneity across certain network configurations to infer effects as applicable to the entire set of network actors. Using an example from a well-known network dataset, we show that treating a sizeable proportion of nodes as non-respondents may still result in estimates, and inferences about structural effects, consistent with those for the entire network.If, on the other hand, the principal research focus is on the respondent-only structure, with non-respondents clearly not missing at random, we incorporate the information about ties to non-respondents as exogenous. We illustrate this model with an example of a network within and between organizational departments. Because in this second class of models the number of non-respondents may be large, values of parameter estimates may not be directly comparable to those for models that exclude non-respondents. In the context of discussing recent technical developments in exponential random graph models, we present a heuristic method based on pseudo-likelihood estimation to infer whether certain structural effects may contribute substantially to the predictive capacity of a model, thereby enabling comparisons of important effects between models with differently sized node sets.     

The VL control measure for symmetric networks

In this paper we measure “control” of nodes in a network by solving an associated optimisation problem. We motivate this so-called VL control measure by giving an interpretation in terms of allocating resources optimally to the nodes in order to maximise some search probability. We determine the VL control measure for various classes of networks. Furthermore, we provide two game theoretic interpretations of this measure. First it turns out that the VL control measure is a particular proper Shapley value of the associated cooperative network game. Secondly, we relate the measure to optimal strategies in an associated matrix search game.     

Recent developments in exponential random graph (p*) models for social networks

This article reviews new specifications for exponential random graph models proposed by Snijders et al. [Snijders, T.A.B., Pattison, P., Robins, G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology] and demonstrates their improvement over homogeneous Markov random graph models in fitting empirical network data. Not only do the new specifications show improvements in goodness of fit for various data sets, but they also help to avoid the problem of near-degeneracy that often afflicts the fitting of Markov random graph models in practice, particularly to network data exhibiting high levels of transitivity. The inclusion of a new higher order transitivity statistic allows estimation of parameters of exponential graph models for many (but not all) cases where it is impossible to estimate parameters of homogeneous Markov graph models. The new specifications were used to model a large number of classical small-scale network data sets and showed a dramatically better performance than Markov graph models. We also review three current programs for obtaining maximum likelihood estimates of model parameters and we compare these Monte Carlo maximum likelihood estimates with less accurate pseudo-likelihood estimates. Finally, we discuss whether homogeneous Markov random graph models may be superseded by the new specifications, and how additional elaborations may further improve model performance.     

A comparative study of social network models: Network evolution models and nodal attribute models

This paper reviews, classifies and compares recent models for social networks that have mainly been published within the physics-oriented complex networks literature. The models fall into two categories: those in which the addition of new links is dependent on the (typically local) network structure (network evolution models, NEMs), and those in which links are generated based only on nodal attributes (nodal attribute models, NAMs). An exponential random graph model (ERGM) with structural dependencies is included for comparison. We fit models from each of these categories to two empirical acquaintance networks with respect to basic network properties. We compare higher order structures in the resulting networks with those in the data, with the aim of determining which models produce the most realistic network structure with respect to degree distributions, assortativity, clustering spectra, geodesic path distributions, and community structure (subgroups with dense internal connections). We find that the nodal attribute models successfully produce assortative networks and very clear community structure. However, they generate unrealistic clustering spectra and peaked degree distributions that do not match empirical data on large social networks. On the other hand, many of the network evolution models produce degree distributions and clustering spectra that agree more closely with data. They also generate assortative networks and community structure, although often not to the same extent as in the data. The ERGM model, which turned out to be near-degenerate in the parameter region best fitting our data, produces the weakest community structure.     

Networks and collective action

This paper proposes a new measure for a group's ability to lead society to adopt their standard of behavior, which in particular takes account of the time the group takes to convince the whole society to adopt their position. This notion of a group's power to initiate action is computed as the reciprocal of the resistance against it, which is in turn given by the expected absorption time of a related finite state partial Markov chain that captures the social dynamics. The measure is applicable and meaningful in a variety of models where interaction between agents is formalized through (weighted) binary relations. Using Percolation Theory, it is shown that the group power is monotonic as a function of groups of agents. We also explain the differences between our measure and those discussed in the literature on Graph Theory, and illustrate all these concerns by a thorough analysis of two particular cases: the Wolfe Primate Data and the 11S hijackers’ network.     

The evaluation and selection of adequate causal models: a compensatory education example

Procedures for ascertaining relative model adequacy in latent variable structural relations models are discussed. Under diverse methods of estimation, this determination may be assessed using the chi square goodness of fit statistic, incremental fit indices for covariance structure models, and latent variable coefficients of determination. An example from evaluation research is taken (cf. Magidson, 1977; Bentler & Woodward, 1978). Numerical sensitivity of parameter estimates under alternative model specifications is demonstrated. Interpretive implications based on these procedures are discussed in terms of parameter sensitivity to alternative model specifications.     

Correlates of chronic disease and patient–provider discussions among middle-aged and older adult males: Implications for successful aging and sexuality

We use data from the Massachusetts Male Aging Study to approximate the total current knowledge about prostate cancer and to evaluate the relative contributions of various risk factors. The sum total of current knowledge is assessed using the area under the receiver operating characteristic curve (AUROC) and by the Hosmer-Lemeshow statistic in a logistic regression model that includes 30 risk factors identified in the literature that are available in the data set. Relative contributions are measured using the adjusted generalized R² (AR²). To measure relative contributions, we group risk factors with similar etiology and then remove groups and compare the AR² attained without each group to the AR² with all variables included. The overall model fits adequately. The A UROC is 0.788, relatively far from its default value of 0.5, and the Hosmer-Lemeshow statistic has a p value of 0.926. We conclude that, while some knowledge about prostate cancer has been accumulated, there are still more risk factors yet unsuspected. The relative importance analysis shows that immutable factors (i.e. age, genotype) contribute 42%, dietary factors 30%, other lifestyle factors 15%, and endocrinological factors 11%. We recommend that age and other unchanging factors should be the primary focus of screening and risk evaluation and that future interventions should center on nutritional behavior.     

Covariate-defined latent space random effects model

Latent factor models are a useful and intuitive class of models; one limitation is their inability to predict links in a dynamic network. We propose a latent space random effects model with a covariate-defined social space, where the social space is a linear combination of the covariates as estimated by an MCMC algorithm. The model allows for the prediction of links in a network; it also provides an interpretable framework to explain why people connect. We fit the model using the Adolescent Health Network dataset and three simulated networks to illustrate its effectiveness in recognizing patterns in the data.     

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.     

Measurement error in network data: A re-classification

Research on measurement error in network data has typically focused on missing data. We embed missing data, which we term false negative nodes and edges, in a broader classification of error scenarios. This includes false positive nodes and edges and falsely aggregated and disaggregated nodes. We simulate these six measurement errors using an online social network and a publication citation network, reporting their effects on four node-level measures – degree centrality, clustering coefficient, network constraint, and eigenvector centrality. Our results suggest that in networks with more positively-skewed degree distributions and higher average clustering, these measures tend to be less resistant to most forms of measurement error. In addition, we argue that the sensitivity of a given measure to an error scenario depends on the idiosyncracies of the measure's calculation, thus revising the general claim from past research that the more ‘global’ a measure, the less resistant it is to measurement error. Finally, we anchor our discussion to commonly-used networks in past research that suffer from these different forms of measurement error and make recommendations for correction strategies.     

Modeling social influence through network autocorrelation: constructing the weight matrix

Many physical and social phenomena are embedded within networks of interdependencies, the so-called ‘context’ of these phenomena. In network analysis, this type of process is typically modeled as a network autocorrelation model. Parameter estimates and inferences based on autocorrelation models, hinge upon the chosen specification of weight matrix W, the elements of which represent the influence pattern present in the network. In this paper I discuss how social influence processes can be incorporated in the specification of W. Theories of social influence center around ‘communication’ and ‘comparison’; it is discussed how these can be operationalized in a network analysis context. Starting from that, a series of operationalizations of W is discussed. Finally, statistical tests are presented that allow an analyst to test various specifications against one another or pick the best fitting model from a set of models.     

Detecting large cohesive subgroups with high clustering coefficients in social networks

Clique relaxations are used in classical models of cohesive subgroups in social network analysis. Clustering coefficient was introduced more recently as a structural feature characterizing small-world networks. Noting that cohesive subgroups tend to have high clustering coefficients, this paper introduces a new clique relaxation, α-cluster, defined by enforcing a lower bound α on the clustering coefficient in the corresponding induced subgraph. Two variations of the clustering coefficient are considered, namely, the local and global clustering coefficient. Certain structural properties of α-clusters are analyzed and mathematical optimization models for determining α-clusters of the largest size in a network are developed and validated using several real-life social networks. In addition, a network clustering algorithm based on local α-clusters is proposed and successfully tested.     

Investigating the factor structure and validity of the family empowerment scale for parents of children with emotional disturbance in middle school

Parent empowerment involves the ability of caregivers to meet the needs of their family while maintaining feelings of control and is particularly important for families of children at-risk. It is necessary to establish reliable and valid tools to measure parent empowerment. The purpose of this study was to examine the internal factor structure, score reliability, and convergent validity of the FES scores with caregivers of middle school youth who had an Individualized Education Plan for Emotional Disturbance (ED) or Other Health Impairment (OHI) due to emotional or behavioral needs. A confirmatory factor analysis (CFA) was used to examine the internal structure the FES. Score reliability was examined by computing coefficient alpha (Cronbach, 1951) for each subscale score and computing Coefficient omega and coefficient omega hierarchical for good-fitting factor models and bifactor models. Convergent validity was examined by generating composite scores for each subscale, followed by computing Pearson correlations between FES subscale scores and scores from the PAM, CGSQ and the SDQ. Results indicated that the hypothesized three-factor model fit the data adequately. FES scores were reliable based on coefficient alpha and omega, and evidence of convergent validity with measures of parent activation, caregiver strain, and child behavior were moderate to strong. The results support the use of the FES with parents of middle school youth who have ED and further validate the three factor structure identified in the initial measure development. Practical and clinical implications of these findings include support for the use of the FES with this particular group of parents.     

Effects of temporal resolution adjustments on dynamic sexual contact models

Simulation and inferential modeling of sexual contact dynamics can be used to help predict the propagation of sexually transmitted infections (STI) such as HIV, by providing information on both cross-sectional network structure and how that structure changes over time. Researchers’ choices regarding the temporal resolution of such network models is often driven by the resolution of the data collected to support model fitting and evaluation. These inherited temporal resolutions can become problematic if they differ from the resolution of other processes to which the network must be related. In such cases, a model “correction” is necessary: specifically, we would like to have a systematic method for adjusting a fitted model to allow it to make predictions at a different timescale than the one on which it was initially based. Here, we introduce a basic set of desiderata for such adjustments, and assess three very simple approaches to timescale adjustments for models of sexual contact networks (SCNs), with focus on models parameterized within the separable temporal exponential family random graph model (STERGM) framework. We also examine the impact of time-scale changes on SCN characteristics themselves, and outline a set of desiderata for what an adjustment strategy may be expected to achieve. Our findings have implications both for timescale correction of dynamic network models, and for dynamic network data collection.     

A new method for finding hierarchical subgroups from networks

We present a new method for decomposing a social network into an optimal number of hierarchical subgroups. With a perfect hierarchical subgroup defined as one in which every member is automorphically equivalent to each other, the method uses the REGGE algorithm to measure the similarities among nodes and applies the k-means method to group the nodes that have congruent profiles of dissimilarities with other nodes into various numbers of hierarchical subgroups. The best number of subgroups is determined by minimizing the intra-cluster variance of dissimilarity subject to the constraint that the improvement in going to more subgroups is better than a network whose n nodes are maximally dispersed in the n-dimensional space would achieve. We also describe a decomposability metric that assesses the deviation of a real network from the ideal one that contains only perfect hierarchical subgroups. Four well known network data sets are used to demonstrate the method and metric. These demonstrations indicate the utility of our approach and suggest how it can be used in a complementary way to Generalized Blockmodeling for hierarchical decomposition.     

Harm reduction for cannabis: Factor analysis of a protective behavioral strategies survey

Objective: To examine the psychometric properties of a measure of cannabis-specific Protective Behavioral Strategies (PBS), which assesses ways in which students may reduce cannabis-related risk. Participants: Respondents to the American College Health Association – National College Health Assessment II (N = 580) during Spring 2015. Methods: Exploratory Factor Analysis (EFA) and Exploratory Structural Equation Modeling (ESEM) were used to identify and replicate the factor structure of the measure. Results: Results support a four-factor model (Respiratory Health, Frequency/Quantity, Socializing, and General Health) with close approximate fit (Χ² (310) = 565.96, p < .001, RMSEA = .038 (.033, .043; 90% CI), CFI = .961, TLI = .929, SRMR = .033). Support for the convergent validity and construct validity of the measure was also found. Conclusions: This is the initial step in the development of a standard, psychometrically validated measure of cannabis PBS that has the potential to inform future research and interventions.     

Closure,connectivity and degree distributions: Exponential random graph (p*) models for directed social networks

The new higher order specifications for exponential random graph models introduced by Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins G.L., Handcock, M., 2006. New specifications for exponential random graph models. Sociological Methodology 36, 99–153] exhibit substantial improvements in model fit compared with the commonly used Markov random graph models. Snijders et al., however, concentrated on non-directed graphs, with only limited extensions to directed graphs. In particular, they presented a transitive closure parameter based on path shortening. In this paper, we explain the theoretical and empirical advantages in generalizing to additional closure effects. We propose three new triadic-based parameters to represent different versions of triadic closure: cyclic effects; transitivity based on shared choices of partners; and transitivity based on shared popularity. We interpret the last two effects as forms of structural homophily, where ties emerge because nodes share a form of localized structural equivalence. We show that, for some datasets, the path shortening parameter is insufficient for practical modeling, whereas the structural homophily parameters can produce useful models with distinctive interpretations. We also introduce corresponding lower order effects for multiple two-path connectivity. We show by example that the in- and out-degree distributions may be better modeled when star-based parameters are supplemented with parameters for the number of isolated nodes, sources (nodes with zero in-degrees) and sinks (nodes with zero out-degrees). Inclusion of a Markov mixed star parameter may also help model the correlation between in- and out-degrees. We select some 50 graph features to be investigated in goodness of fit diagnostics, covering a variety of important network properties including density, reciprocity, geodesic distributions, degree distributions, and various forms of closure. As empirical illustrations, we develop models for two sets of organizational network data: a trust network within a training group, and a work difficulty network within a government instrumentality.     

A note on determinants of household saving behavior

By using cross-sectional interview data form 429 randomly selected households, four different saving estimates, namely reported bank savings, repayments of debts, total savings and a liquidity estimate, are computed and analyzed separately. Both socioeconomic characteristics and “softer” variables, such as attitudes and expectations, are used as explanatory variables.In a regression analysis the predictors used, fail to explain bank savings, while the traditional socioeconomic variables are found to influence the debt measure. The behavioral predictors are more successful in explaining the liquidity measure. When the regression result with the liquidity measure as the dependant variable (with the highest R² value) is decomposed using a hypothetical causal model and path analysis, it appears that the effects of household income and educational level of the family head are substantial. However, the intervening behavioral variables are still significant.     

Categorical attribute based centrality: E–I and G–F centrality

In a paper examining informal networks and organizational crisis, Krackhardt and Stern (1988) proposed a measure assessing the extent to which relations in a network were internal to a group as opposed to external. They called their measure the E–I index. The measure is now in wide use and is implemented in standard network packages such as UCINET ( Borgatti et al., 2002). The measure is based on a partition-based degree centrality measure and as such can be extended to other centrality measures and group level data. We explore extensions to closeness, betweenness and eigenvector centrality, and show how to apply the technique to sets of subgroups that do not form a partition. In addition, the extension to betweenness suggests a linkage to the Gould and Fernandez brokerage measures, which we explore.