Conditionally Independent Dyads (CID) network models: A latent variable approach to statistical social network analysis

Latent variable network models that accommodate edge correlations implicitly, by assuming an underlying latent factor, are increasing in popularity. Although, these models are examples of what is a growing body of research, much of the research is focused on proposing new models or extending others. There has been very little work on unifying the models in a single framework. In this paper, we present a complete framework that organizes existing latent variable network models within an integrative generalized additive model. Our framework is called Conditionally Independent Dyad (CID) models, and includes existing network models that assume dyad (or edge) independence conditional on latent variables and other components in the model. We further discuss practical aspects of model fitting such as posterior parameter estimation via MCMC, identifiability of parameters, approaches to handle missing data and model selection via cross-validation, for the proposed additive CID models. Finally, by presenting several data examples, we illustrate the utility of the proposed framework and provide advice on selecting components for building new CID models.     

Latent Class Factor and Cluster Models, Bi-Plots, and Related Graphical Displays

We propose an alternative method of conducting exploratory latent class analysis that utilizes latent class factor models, and compare it to the more traditional approach based on latent class cluster models. We show that when formulated in terms of R mutually independent, dichotomous latent factors, the LC factor model has the same number of distinct parameters as an LC cluster model with R+1 clusters. Analyses over several data sets suggest that LC factor models typically fit data better and provide results that are easier to interpret than the corresponding LC cluster models. We also introduce a new graphical "bi-plot" display for LC factor models and compare it to similar plots used in correspondence analysis and to a barycentric coordinate display for LC cluster models. New results on identification of LC models are also presented. We conclude by describing various model extensions and an approach for eliminating boundary solutions in identified and unidentified LC models, which we have implemented in a new computer program.     

FOUR USEFUL FINITE MIXTURE MODELS FOR REGRESSION ANALYSES OF PANEL DATA WITH A CATEGORICAL DEPENDENT VARIABLE

This paper describes and contrasts two useful ways to employ a latent class variable as a mixture variable in regression analyses of panel data with a categorical dependent variable. One way is to model unobserved heterogeneity in the trajectory, or change in the distribution, of the dependent variable. Two models that accomplish this are the latent trajectory model and latent growth curve model for a categorical dependent variable having ordered categories. Each latent class here represents a distinct trajectory of the dependent variable. The latent trajectory model introduces covariate effects on the composition of latent classes, while the latent growth curve model introduces covariate effects on both the "intercept" and the "slope" of growth in logit, which may vary among latent classes.
The other useful way is to model unobserved heterogeneity in the state dependence of the dependent variable. Two models that accomplish this are introduced for a simultaneous analysis of response probability and response stability, and the latent class variable is employed to distinguish two latent populations that differ in the stability of responses over time. One of them is the switching multinomial logit model with a time-lagged dependent variable as its separation indicator, and the other is the mover-stayer regression model.
By applying these four models to empirical data, this paper demonstrates the usefulness of these models for panel-data analyses. Example programs for specifying these models based on the LEM program are also provided.     

MULTIPLE IMPUTATION OF INCOMPLETE CATEGORICAL DATA USING LATENT CLASS ANALYSIS

We propose using latent class analysis as an alternative to log-linear analysis for the multiple imputation of incomplete categorical data. Similar to log-linear models, latent class models can be used to describe complex association structures between the variables used in the imputation model. However, unlike log-linear models, latent class models can be used to build large imputation models containing more than a few categorical variables. To obtain imputations reflecting uncertainty about the unknown model parameters, we use a nonparametric bootstrap procedure as an alternative to the more common full Bayesian approach. The proposed multiple imputation method, which is implemented in Latent GOLD software for latent class analysis, is illustrated with two examples. In a simulated data example, we compare the new method to well-established methods such as maximum likelihood estimation with incomplete data and multiple imputation using a saturated log-linear model. This example shows that the proposed method yields unbiased parameter estimates and standard errors. The second example concerns an application using a typical social sciences data set. It contains 79 variables that are all included in the imputation model. The proposed method is especially useful for such large data sets because standard methods for dealing with missing data in categorical variables break down when the number of variables is so large.     

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.     

USING INSTRUMENTAL VARIABLE TESTS TO EVALUATE MODEL SPECIFICATION IN LATENT VARIABLE STRUCTURAL EQUATION MODELS

Structural equation modeling (SEM) with latent variables is a powerful tool for social and behavioral scientists, combining many of the strengths of psychometrics and econometrics into a single framework. The most common estimator for SEM is the full-information maximum likelihood (ML) estimator, but there is continuing interest in limited information estimators because of their distributional robustness and their greater resistance to structural specification errors. However, the literature discussing model fit for limited information estimators for latent variable models is sparse compared with that for full-information estimators. We address this shortcoming by providing several specification tests based on the 2SLS estimator for latent variable structural equation models developed by Bollen (1996) . We explain how these tests can be used not only to identify a misspecified model but to help diagnose the source of misspecification within a model. We present and discuss results from a Monte Carlo experiment designed to evaluate the finite sample properties of these tests. Our findings suggest that the 2SLS tests successfully identify most misspecified models, even those with modest misspecification, and that they provide researchers with information that can help diagnose the source of misspecification.     

NEW SPECIFICATIONS FOR EXPONENTIAL RANDOM GRAPH MODELS

The most promising class of statistical models for expressing structural properties of social networks observed at one moment in time is the class of exponential random graph models (ERGMs), also known as p * models. The strong point of these models is that they can represent a variety of structural tendencies, such as transitivity, that define complicated dependence patterns not easily modeled by more basic probability models. Recently, Markov chain Monte Carlo (MCMC) algorithms have been developed that produce approximate maximum likelihood estimators. Applying these models in their traditional specification to observed network data often has led to problems, however, which can be traced back to the fact that important parts of the parameter space correspond to nearly degenerate distributions, which may lead to convergence problems of estimation algorithms, and a poor fit to empirical data.
This paper proposes new specifications of exponential random graph models. These specifications represent structural properties such as transitivity and heterogeneity of degrees by more complicated graph statistics than the traditional star and triangle counts. Three kinds of statistics are proposed: geometrically weighted degree distributions, alternating k -triangles, and alternating independent two-paths. Examples are presented both of modeling graphs and digraphs, in which the new specifications lead to much better results than the earlier existing specifications of the ERGM. It is concluded that the new specifications increase the range and applicability of the ERGM as a tool for the statistical analysis of social networks.     

Latent Class Marginal Models for Cross-Classifications of Counts

The standard latent class model is a finite mixture of indirectly observed multinomial distributions, each of which is assumed to exhibit statistical independence. Latent class analysis has been applied in a wide variety of research contexts, including studies of mobility, educational attainment, agreement, and diagnostic accuracy, and as measurement error models in social research. One of the attractive features of the latent class model in these settings is that the parameters defining the individual multinomials are readily interpretable marginal probabilities, conditional on the unobserved latent variable(s), that are often of substantive interest. There are, however, settings where the local-independence axiom is not supported, and hence it is useful to consider some form of local dependence. In this paper we consider a family of models defined in terms of finite mixtures of multinomial models where the multinomials are parameterized in terms of a set of models for the univariate marginal distributions and for marginal associations. Local dependence is introduced through the models for marginal associations, and the standard latent class model obtains as a special case. Three examples are analyzed with the models to illustrate their utility in analyzing complex cross-classifications.     

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.     

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.     

Latent Variable Modeling of Longitudinal and Multilevel Data

An overview is given of modeling of longitudinal and multilevel data using a latent variable framework. Particular emphasis is placed on growth modeling. A latent variable model is presented for three-level data, where the modeling of the longitudinal part of the data imposes both a covariance and a mean structure. Examples are discussed where repeated observations are made on students sampled within classrooms and schools.     

The factor structure of the pregnancy anxiety scale.

This paper presents a measurement model of a ten-item scale of maternal anxiety during pregnancy. Using confirmatory factor analysis, its reliability and validity are examined in a hospital sample of mothers (N = 266) surveyed postpartum in Galveston, Texas. According to several indices of overall fit as well as individual parameter estimates, a latent internal structure of three interrelated dimensions is confirmed for the scale items. These first-order constructs are anxiety about being pregnant, childbirth, and hospitalization. This model exhibits a considerably better fit than both a no-factors model and a model in which the dimensions are uncorrelated. Finally, several exogenous constructs expected to be associated with pregnancy anxiety--age, marital status, and worry over health--exert significant effects on dimensions of the model or on a second-order factor.     

Positive Adult Support and Depression Symptoms in Adolescent Females: The Partially Mediating Role of Eating Disturbances

This study examined linkages between depression symptoms (DEP) and positive adult support (PAS) in female adolescents and the partially mediating influence of eating disturbances (ED). Structural equation modeling was used to establish measurement models for each of the latent constructs, determine the relationships among the latent constructs, and examine the overall model fit of the data. The relationships among the latent constructs of ED, PAS, and DEP were tested using a mediation model. Results indicate that there is a significant, positive relationship between DEP and PAS and that ED are a partial mediator of this relationship. This study provides evidence for the importance of evaluating how ED can influence the trajectory of depression in the lives of adolescent females.     

Quality of prediction: A critique of Bronner

In this comment on Bronner's article ‘Decision style in transport mode choice’ two problems are discussed. The first and minor one deals with the choice of criterion in the model tests for the quality of prediction. The second and major problem concerns the invalidity of the model tests, which is due to the psychometrically incorrect classification procedure adopted by Bronner. This issue is extensively discussed because some of Bronner's conclusions can be shown to evolve from his incorrect assignment procedure.As an alternative method of attitude assignment the use of a matching constant is suggested. In order to evaluate predictive quality of models, it is recommended to consider mode choice as the criterion variable instead of attitudes within travel mode categories.     

Algebraic Representations of Beliefs and Attitudes: Partial Order Models for Item Responses

A partial order of discrete beliefs based on a generalization of item order in Guttman scaling generates a nonunidimensional collection of latent belief states that can be represented by a distributive lattice. By incorporating misclassification errors under local independence assumptions, the lattice structure is transformed into a latent class model for observed response states. We apply this model to survey responses dealing with government welfare programs and suggest that our approach can retrieve information where unidimensional and multidimensional models do not fit. The concluding section discusses directions for future work.     

Poverty dynamics corrected for measurement error

We use latent class models to correct measurement error in estimates of the dynamics of relative income poverty in ten EU countries measured over four waves of the European Community Household Panel. A latent mover-stayer Markov model gives an acceptable fit to all ten transition tables. We focus in more detail on four countries – Denmark, the Netherlands, Italy and the UK – and show that mobility in poverty transition tables is over-estimated by between 25 and 50 percent if measurement error is ignored. In addition, once error is corrected, poverty rates show less cross-national variation.     

In search of the wage-labour/service contract: new evidence on the validity of the Goldthorpe class schema

In this paper we examine new empirical evidence on the coherence and magnitude of the main classes in the Goldthorpe class schema. Particular attention is paid to issues that have recently been a source of academic dispute: the coherence and size of the service class and the distinction between the service class and intermediate classes. Using recently available British data collected by the Office for National Statistics we examine: (i) the extent to which measures of class-relevant job characteristics are empirically discriminated by the categories of the schema; (ii) the structure of a 'contract type' dimension of employment relations conceived of as a categorical latent variable; and (iii) the association between this latent variable and both the Goldthorpe class schema and a related measure socio-economic group (SEG). We find that the data are consistent with the existence of a three category latent 'contract type' variable largely corresponding to the notions of service, intermediate and wage-labour contracts explicit in discussions of the theoretical rationale for the Goldthorpe schema. We further find a substantial degree of fit between the latent 'contract types' and the schema. However, the service class fault line appears to lie within class I and II of the schema rather than between them and the intermediate classes which suggests a revised, smaller service class would better capture the reality of the contemporary British occupational structure.     

Unionization and productivity in commercial banking

This paper examines the net effect of unions on productivity in the commercial banking industry. The focus of the study is on three methodological issues. First, an attempt is made to determine whether individual unions have a differential impact on banking productivity. The influence of unions on output per man-hour was initially estimated by including a union dummy variable in a Cobb-Douglas formulation of bank production. Separate binary variables were then entered into alternative specifications of the model to test the heterogeneity hypothesis. This hypothesis postulates differential productivity effects among the individual unions operating in the commercial banking sector. Second, the sample banks were paired on a case-by-case basis to assure the homogeneity of the two groupings: i.e., union and nonunion. Sample homogeneity is necessary because of the assumptions of identical production functions and output prices between the groups. Third, a complete covariance model was specified in order to estimate the impact of unionization on each parameter of the production function. In general, the unionized banks were less productive than their nonunion peers. It should be noted, however, that the standard errors were large in all the specifications. Moreover, the labor relations problems associated with one union had a large impact on the sector results.     

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.