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.     

Multilevel Latent Class Models

The latent class (LC) models that have been developed so far assume that observations are independent. Parametric and nonparametric random–coefficient LC models are proposed here, which will make it possible to modify this assumption. For example, the models can be used for the analysis of data collected with complex sampling designs, data with a multilevel structure, and multiple–group data for more than a few groups. An adapted EM algorithm is presented that makes maximum–likelihood estimation feasible. The new model is illustrated with examples from organizational, educational, and cross–national comparative research.     

Covariance Models For Latent Structure In Longitudinal Data

We present several approaches to modeling latent structure in longitudinal studies when the covariance itself is the primary focus of the analysis. This is a departure from much of the work on longitudinal data analysis, in which attention is focused solely on the cross-sectional mean and the influence of covariates on the mean. Such analyses are particularly important in policy-related studies, in which the heterogeneity of the population is of interest. We describe several traditional approaches to this modeling and introduce a flexible, parsimonious class of covariance models appropriate to such analyses. This class, while rooted in the tradition of mixed effects and random coefficient models, merges several disparate modeling philosophies into what we view as a hybrid approach to longitudinal data modeling. We discuss the implications of this approach and its alternatives especially on model interpretation. We compare several implementations of this class to more commonly employed mixed effects models to describe the strengths and limitations of each. These alternatives are compared in an application to long-term trends in wage inequality for young workers. The findings provide additional guidance for the model formulation process in both statistical and substantive senses.     

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.     

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.     

Random-Effects Modeling of Categorical Response Data

In many applications observations have some type of clustering, with observations within clusters tending to be correlated. A common instance of this occurs when each subject in the sample undergoes repeated measurement, in which case a cluster consists of the set of observations for the subject. One approach to modeling clustered data introduces cluster-level random effects into the model. The use of random effects in linear models for normal responses is well established. By contrast, random effects have only recently seen much use in models for categorical data. This chapter surveys a variety of potential social science applications of random effects modeling of categorical data. Applications discussed include repeated measurement for binary or ordinal responses, shrinkage to improve multiparameter estimation of a set of proportions or rates, multivariate latent variable modeling, hierarchically structured modeling, and cluster sampling. The models discussed belong to the class of generalized linear mixed models (GLMMs), an extension of ordinary linear models that permits nonnormal response variables and both fixed and random effects in the predictor term. The models are GLMMs for either binomial or Poisson response variables, although we also present extensions to multicategory (nominal or ordinal) responses. We also summarize some of the technical issues of model-fitting that complicate the fitting of GLMMs even with existing software.     

A PARTIAL INDEPENDENCE ITEM RESPONSE MODEL FOR SURVEYS WITH FILTER QUESTIONS

In many surveys, responses to earlier questions determine whether later questions are asked. The probability of an affirmative response to a given item is therefore nonzero only if the participant responded affirmatively to some set of logically prior items, known as "filter items." In such surveys, the usual conditional independence assumption of standard item response models fails. A weaker "partial independence" assumption may hold, however, if an individual's responses to different items are independent conditional on the item parameters, the individual's latent trait, and the participant's affirmative responses to each of a set of filter items. In this paper, we propose an item response model for such "partially independent" item response data. We model such item response patterns as a function of a person-specific latent trait and a set of item parameters. Our model can be seen as a generalized hybrid of a discrete-time hazard model and a Rasch model. The proposed procedure yields estimates of (1) person-specific, interval-scale measures of a latent trait (or traits), along with person-specific standard errors of measurement; (2) conditional and marginal item severities for each item in a protocol; (3) person-specific conditional and marginal probabilities of an affirmative response to each item in a protocol; and (4) item information and total survey information. In addition, we show here how to investigate and test alternative conceptions of the dimensionality of the latent trait(s) being measured. Finally, we compare our procedure with a simpler alternative approach to summarizing data of this type.     

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.     

MEASURING AND ANALYZING CLASS INEQUALITY WITH THE GINI INDEX INFORMED BY MODEL-BASED CLUSTERING

The most widely used measure for studying social, economic, and health inequality is the Gini index/ratio. Whereas other measures of inequality possess certain useful characteristics, such as the straightforward decomposability of the generalized entropy measures, the Gini index has remained the most popular, at least in part due to its ease of interpretation. However, the Gini index has a limitation in measuring inequality. It is less sensitive to how the population is stratified than how individual values differ. The twin purposes of this paper are to explain the limitation and to propose a model-based method—latent class/clustering analysis for understanding and measuring inequality. The latent cluster approach has the major advantage of being able to identify potential "classes" of individuals who share similar levels of income or one or more other attributes and to assess the fit of the model-based classes to the empirical data, based on different cluster distributional assumptions and the number of latent classes. This paper distinguishes class inequality from individual inequality, the type that is better captured by the Gini. Once the classes are estimated, the membership of estimated classes obtained from the best fitting model facilitates the decomposition of the Gini index into individual and class inequality. Class inequality is then measured by two relative stratification indices based on either the relative size of the Gini between-class components or the relative number of stratified individuals. Therefore, the Gini index is extended and assisted by model-based clustering to measure class inequality, thereby realizing its great potential for studying inequality. Income data from France and Hungary are used to illustrate the application of the method.     

Regression Analysis of Multivariate Binary Response Variables Using Rasch-Type Models and Finite-Mixture Methods

A model is considered for the regression analysis of multivariate binary data such as repeated-measures data (for example, panel data) or multiple-indicators with measures of some underlying characteristic such as attitude or ability (for example, surveys or tests). The model is related to the usual Rasch model, the usual latent-class model, and other familiar models such as logistic regression. In addition to a regression specification, the model includes parameters that describe heterogeneity not accounted for by the predictors. In contrast to most other approaches, a nonparametric specification of the latent mixing distribution is used, leading to a formulation based on scaled latent classes. We examine the relationship between this model and several other models, give a tractable formulation of the likelihood function and likelihood equations, present an algorithm for maximum-likelihood estimation, and analyze marginal and conditional latent structures. The approach is illustrated with longitudinal data from the German Socioeconomic Panel.     

MEANINGFUL REGRESSION AND ASSOCIATION MODELS FOR CLUSTERED ORDINAL DATA

Many proposed methods for analyzing clustered ordinal data focus on the regression model and consider the association structure within a cluster as a nuisance. However, the association structure is often of equal interest—for example, temporal association in longitudinal studies and association between responses to similar questions in a survey. We discuss the use, appropriateness, and interpretability of various latent variable and Markov models for the association structure and propose a new structure that exploits the ordinality of the response. The models are illustrated with a study concerning opinions regarding government spending and an analysis of stability and change in teenage marijuana use over time, where we reveal different behavioral patterns for boys and girls through a comprehensive investigation of individual response profiles.     

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.     

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.     

Modelling financial satisfaction across life stages: A latent class approach

We explore the determinants of financial satisfaction using a modelling framework which allows the drivers of financial satisfaction to vary across life stages. Given that financial satisfaction is measured as an ordered variable, our modelling approach is based on a latent class ordered probit model with an ordered probit class assignment function. Our analysis of household survey data indicates that four life stages are supported by the data. Our results suggest that such flexibility is important in understanding the drivers of financial satisfaction over the life cycle since there is a substantial amount of parameter heterogeneity across the four classes.     

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.     

Human Capital,Social Support,and Economic Well-being among Rural,Low-income Mothers: A Latent Growth Curve Analysis

The purpose of this study was to better understand human capital and social support in the long-term economic well-being of rural, low-income mothers in the US. Three waves of data from a multi-state, longitudinal investigation tracking the well-being of rural families, known as “Rural Families Speak,” were used to test two latent growth curve models of economic well-being. Results indicated that human capital alone is not a good predictor of economic well-being over time for this sample. A model of economic well-being that includes both social support and human capital provides a better fit for these data. Findings suggest that social support is a key contributor to long-term economic success for this sample. Implications for public policy are presented.

ELECTRIC AND PLUG‐IN HYBRID VEHICLE DEMAND: LESSONS FOR AN EMERGING MARKET

Understanding demand in the new plug‐in hybrid electric vehicle (PHEV) market is critical to designing more effective adoption policies. We use stated preference data from an innovative choice experiment to estimate demand for PHEVs relative to battery electric vehicles (BEVs) and to explore heterogeneity in demand for these vehicles. We find the gap between willingness to pay for PHEVs and their price premium over conventional vehicles is on the order of current subsidies, while that of BEVs is an order of magnitude larger. We use a latent class model to show PHEVs draw a different consumer segment into the market. (JEL Q5, R41)     

Brain Emotion Systems,Personality, Hopelessness,Self/Other Perception,and Gambling Cognition: A Structural Equation Model

The aim of this study was to explore the relations between gambling, brain emotion systems, personality, self/other perception, and hopelessness in an Italian community. Dimensions of gambling, positive and negative emotions, self/other perception, personality and hopelessness were assessed in a community sample of 235 adults aged 19–59 years. Two structural models were tested. We found a significant correlation between problem gambling and impulsivity, which in association with aggressivity and negative personality dimensions may help explain the psychopathology factor, i.e. a latent variable involving neurotic personality, hopelessness, high sensation seeking, low metacognitive responsiveness, and disorganized patterns of interpersonal relationships. These results contribute to develop a theoretical framework of gambling in relation with personality factors and provide a new approach for clinical intervention of problem gambling that relies on a solid multidimensional perspective.     

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.     

Measurement quality issues in dyadic models of relationships

Interpersonal relationships are an important and integral part of numerous social science research agendas. Analytical tools have been created in the last 10 years that model dyadic interactions. In particular, this article focuses on the dyadic models of Fienberg and Wasserman [Fienberg, S.E., Wasserman, S., 1981. Categorical data analysis of single sociometric relations. In: Leinhardt, S. (Ed.), Sociological Methodology. Jossey-Bass, San Francisco.], Holland and Leinhardt [Holland, P.W., Leinhardt, S., 1981. An exponential family of probability densities for directed graphs. Journal of the American Statistical Association 76 (1981) 33–51.], Iacobucci and Wasserman [Iacobucci, D., Wasserman, S., 1988. A general framework for the statistical analysis of sequential dyadic interaction data. Psychological Bulletin 103 (1988) 379–390.] and Wasserman and Iacobucci [Wasserman, S., Iacobucci, D., 1986. Statistical analysis of discrete relational data. British Journal of Mathematical and Statistical Psychology 39 (1986) 41–64.]. However, measurement issues like reliability and validity, as discussed by Allen and Yen [Allen, M.J., Yen, W.M., 1979. Introduction to Measurement Theory. Brooks/Cole, Monterey, CA, 1979.], Nunnally [Nunnally, J., 1978. Psychometric Theory, 2nd edn. McGraw-Hill, New York, NY, 1978.] and Uebersax [Uebersax, J.S., 1988. Validity inferences from interobserver agreement. Psychological Bulletin 104 (1988) 405–416.], have not been considered in conjunction with these models, and little is known about the empirical performance of the dyadic models under sub-optimal measurement quality conditions. We offer two essential approaches to ascertaining the level of measurement error in the observed indicators of social ties and relationships. The first approach combines latent class and social network models in one integrated framework and allows for the simultaneous study of measurement and dyadic structural issues. The second approach is an alternative that may be more useful to social science researchers, both because the method is more accessible and because researchers could apply the techniques to data they have already partially analyzed. This approach is a two-staged procedure whereby in the first stage, a probability model based on latent class analysis is estimated which provides an indication of the measurement quality in the data. In the second stage, traditional social network models are estimated. To investigate the implications of different levels of measurement error for interpreting the nature of the network ties and the dyadic parametric performance, we also designed a Monte Carlo experiment. Measurement error is simulated as the likelihood of a binary relational choice (for simplicity) being inaccurately classified, where incorrect diagnoses can result from poor interitem agreement (i.e., unreliability) or poor interrater agreement. The simulation can be used by researchers in combination with the two-stage approach. The results of the simulation provide guidelines for situations when social network models can withstand a reasonable degree of sub-optimal measurement quality and highlight adverse conditions which can significantly affect the performance of the modeling approach. Further, the simulation shows that sample size assists in reducing the chances of making Type II errors, but it does not compensate for biases in parameter estimates in the presence of increasing error. Finally, the measurement and dyadic analytical methods are applied to a real dataset describing interorganizational relational activity using multiple raters. Recommendations are offered to guide the researcher in making decisions about research design when using dyadic models.