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.     

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.     

EFFECT DISPLAYS FOR MULTINOMIAL AND PROPORTIONAL-ODDS LOGIT MODELS

An "effect display" is a graphical or tabular summary of a statistical model based on high-order terms in the model. Effect displays have previously been defined by Fox (1987, 2003) for generalized linear models (including linear models). Such displays are especially compelling for complicated models—for example, those including interactions or polynomial terms. This paper extends effect displays to models commonly used for polytomous categorical response variables: the multinomial logit model and the proportional-odds logit model. Determining point estimates of effects for these models is a straightforward extension of results for the generalized linear model. Estimating sampling variation for effects on the probability scale in the multinomial and proportional-odds logit models is more challenging, however, and we use the delta method to derive approximate standard errors. Finally, we provide software for effect displays in the R statistical computing environment.     

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.     

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.     

Overcoming the Reference Category Problem in the Presentation of Statistical Models

Effects of categorical variables in statistical models typically are reported in terms of comparison either with a reference category or with a suitably defined "mean effect," for reasons of parameter identification. A conventional presentation of estimates and standard errors, but without the full variance–covariance matrix, does not allow subsequent readers either to make inference on a comparison of interest that is not presented or to compare or combine results from different studies where the same variables but different reference levels are used. It is shown how an alternative presentation, in terms of "quasi standard errors," overcomes this problem in an economical and intuitive way. A primary application is the reporting of effects of categorical predictors, often called factors, in linear and generalized linear models, hazard models, multinomial–response models, generalized additive models, etc. Other applications include the comparison of coefficients between related regression equations—for example, log–odds ratios in a multinomial logit model—and the presentation of multipliers or "scores" in models with multiplicative interaction structure.     

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.     

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.     

A Goodness-Of-Fit Test for the Latent Class Model When Expected Frequencies Are Small

In this paper, a goodness-of-fit test for the latent class model is presented. The test uses only the limited information in the second-order marginal distributions from a set of k dichotomous variables, and it is intended for use when k is large and the sample size, n, is moderate or small. In that situation, a 2^k contingency table formed by the full cross-classification of k variables will be sparse in the sense that a high proportion of cell frequencies will be equal to zero or 1, and the chi-square approximation for traditional goodness-of-fit statistics such as the likelihood ratio will not be valid. The second-order marginal frequencies, which correspond to the bivariate distributions, are rarely sparse even when the joint frequencies have a high proportion of zero cells. Results from Monte Carlo experiments are presented that compare the rates of Type I and Type II errors for the proposed test to the rates for traditional goodness-of-fit tests. Results show that under commonly encountered conditions, a test of fit based on the limited information in the second-order marginals has a Type II error rate that is no higher than the error rate found for full-information test statistics, and that the test statistic given in this paper does not suffer from ill effects of sparseness in the joint frequencies.     

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.     

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.     

COUPLING COUPLES WITH COPULAS: ANALYSIS OF ASSORTATIVE MATCHING ON RISK ATTITUDE

We investigate patterns of assortative matching on risk attitude, using self‐reported (ordinal) data on risk attitudes for males and females within married couples, from the German Socio‐Economic Panel over the period 2004–2012. We apply a novel copula‐based bivariate panel ordinal model. Estimation is in two steps: first, a copula‐based Markov model is used to relate the marginal distribution of the response in different time periods, separately for males and females; second, another copula is used to couple the males' and females' conditional (on the past) distributions. We find positive dependence, both in the middle of the distribution, and in the joint tails, and we interpret this as positive assortative matching (PAM). Hence we reject standard assortative matching theories based on risk‐sharing assumptions, and favor models based on alternative assumptions such as the ability of agents to control income risk. We also find evidence of “assimilation”; that is, PAM appearing to increase with years of marriage. (JEL C33, C51, D81)     

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.     

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.     

The determinants of heroin flows in Europe: A latent space approach

This study utilises recent advances in statistical models for social networks to identify the factors shaping heroin trafficking in relation to European countries. First, it estimates the size of the heroin flows among a network of 61 countries, before subsequently using a latent space approach to model the presence of trafficking and the amount of heroin traded between any two given countries. Many networks, such as trade networks, are intrinsically weighted, and ignoring edge weights results in a loss of relevant information. Traditionally, the gravity model has been used to predict legal trade flows, assuming conditional independence among observations. More recently, latent space position models for social networks have been used to analyze legal trade among countries, and, mutatis mutandis, can be applied to the context of illegal trade to count both edge weights and conditional dependence among observations. These models allow for a better understanding of the generative processes and potential evolution of heroin trafficking routes. This study shows that geographical and social proximity provide fertile ground for the formation of heroin flows. Opportunities are also a driver of drug flows towards countries where regulation of corruption is weak.     

The Role of Family Background in the Heterogeneity of Self-Employment in Some Transition Countries

Aim of this paper is to shed light on how some determinants, especially in the spheres of family background, differently affect the heterogeneous category of self-employment across four transition countries of Central and Eastern Europe (Czech Republic, Hungary, Poland and Slovak Republic), where more or less restrictive policies towards start-ups have been implemented during the pre-1989 years, different liberalization processes have gradually been carried out and distinct policy interventions to support self-employment have been adopted in the post-1989 period. At this end, three-stage structural multinomial logit models as discrete choice models are estimated on 2005 EU-SILC data, which also allows to account for generational changes over time. Country-specific profiles of self-employment are drawn and, even though the self-employment is often devised in a dualist perspective, which stresses its marginal nature as refuge from poverty rather than a way to accumulate capital, significant differentiations within the ranks of self-employed also exist.     

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.     

ON THE ASSIGNMENT OF INDIVIDUALS TO LATENT CLASSES

Consider an m-way cross-classification table (for m = 3, 4, … ) of m dichotomous variables that describes (1) the 2 ^m possible response patterns to a set of m questions (where the response to each question is binary), and (2) the number of individuals whose responses to the m questions can be described by a particular response pattern, for each of the 2 ^m possible response patterns. Consider the situation where the data in the cross-classification table are analyzed using a particular latent class model having T latent classes (for T = 2, 3, …), and where this model fits the data well. With this latent class model, it is possible to estimate, for an individual who has a particular response pattern, what is the conditional probability that this individual is in a particular latent class, for each of the T latent classes. In this article, the following question is considered: For an individual who has a particular response pattern, can we use the corresponding estimated conditional probabilities to assign this individual to one of the T latent classes? Two different assignment procedures are considered here, and for each of these procedures, two different criteria are introduced to help assess when the assignment procedure is satisfactory and when it is not. In addition, we describe here the particular framework and context in which the two assignment procedures, and the two criteria, are considered. For illustrative purposes, the latent class analysis of a classic set of data, a four-way cross-classification of some survey data, obtained in a two-wave panel study, is discussed; and the two different criteria introduced herein are applied in this analysis to each of the two assignment procedures .     

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.     

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)