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.     

Log-Multiplicative Association Models As Latent Variable Models For Nominal and/or Ordinal Data

Associations between multiple discrete measures are often due to collapsing over other variables. When the variables collapsed over are unobserved and continuous, log-multiplicative association models, including log-linear models with linear-by-linear interactions for ordinal categorical data and extensions of Goodman's (1979, 1985) RC(M) association model for multiple nominal and/or ordinal categorical variables, can be used to study the relationship between the observed discrete variables and the unobserved continuous ones, and to study the unobserved variables. The derivation and use of log-multiplicative association models as latent variable models for discrete variables are presented in this paper. The models are based on graphical models for discrete and continuous variables where the variables follow a conditional Gaussian distribution. The models have many desirable properties, including having schematic or graphical representations of the system of observed and unobserved variables, the log-multiplicative models can be read from the graphs, and estimates of the means, variances, and covariances of the latent variables given values on the observed variables are a function of the log-multiplicative model parameters. To illustrate some of the advantageous aspects of these models, two examples are presented. In one example, responses to items from the General Social Survey (Davis and Smith 1996) are modeled, and in the other example, panel data from two groups (Coleman 1964) are analyzed.     

Stochastic a posteriori blockmodels: Construction and assessment

In 1983, Holland, Laskey, and Leinhardt, using the ideas of Holland and Leinhardt, and Fienberg and Wasserman, introduced the notion of a stochastic blockmodel. The mathematics for stochastic a priori blockmodels, in which exogenous actor attribute data are used to partition actors independently of any statistical analysis of the available relational data, have been refined by several researchers and the resulting models used by many. Attempts to simultaneously partition actors and to perform relational data analyses using statistical methods that yield stochastic a posteriori blockmodels are still quite rare. In this paper, we discuss some old suggestions for producing such posterior blockmodels, and comment on other new suggestions based on multiple comparisons of model parameters, log-linear models for ordinal categorical data, and correspondence analysis. We also review measures for goodness-of-fit of a blockmodel, and we describe a natural approach to this problem using likelihood-ratio statistics generated from a popular model for relational data.     

A General Class of Nonparametric Models for Ordinal Categorical Data

This paper presents a general class of models for ordinal categorical data that can be specified by means of linear and/or log-linear equality and/or inequality restrictions on the (conditional) probabilities of a multiway contingency table. Some special cases are models with ordered local odds ratios, models with ordered cumulative response probabilities, order-restricted row association and column association models, and models for stochastically ordered marginal distributions. A simple unidimensional Newton algorithm is proposed for obtaining the restricted maximum-likelihood estimates. In situations in which there is some kind of missing data, this algorithm can be implemented in the M step of an EM algorithm. Computation of p-values of testing statistics is performed by means of parametric bootstrapping.     

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.     

Some Log-Linear and Log-Nonlinear Models for Ordinal Scales With Midpoints, With An Application to Public Opinion Data

Ordinal response scales with a middle category are widely used in public opinion studies, psychology, medicine, computed tomography and other fields. The usual models in the statistical literature for ordinal response variables treat the case where the scale has a natural middle category no differently from the case where the scale does not have a middle category. This paper proposes new models for the analysis of ordinal response scales with middle categories, applying these to data collected in 1993-1994 on American opinion toward the balance between environmental quality and economic prosperity. Some of the models should also be useful when the scale does not have a natural middle category. The models are easily used to address issues of concern in empirical work—for example, stochastic ordering among covariate classes and asymmetry about the middle category. Log-linear models are considered in Section 2. The relationship between the normal distribution and a quadratic log-linear model with known scores, discussed in this section, is the basis for Section 3, which considers a log-nonlinear model with unknown scores estimated from the data. Section 4 shows how generalized log-linear and generalized log-nonlinear models can be used to simultaneously study whether the response is below, at, or above the midpoint, and the conditional distribution of responses above (below) the midpoint. These models are also useful when the response scale is viewed as nested and/or the response process is sequential.     

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.     

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.     

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.     

An Analysis of AAMFT Supervisors: Their Identities,Roles, and Resources*

This paper reports data from a comprehensive study of AAMFT approved supervisors. It reviews findings on a variety of demographic variables and identifies significant relationships between these data and patterns of supervisory attitudes and procedures. The discussion presents a demographic profile of the approved supervisors with an analysis of their supervisory models and resources, the extent of their professional activities, and the issue of adjunctive personal psychotherapy for students in clinical training.     

Developing habitat models for waterbirds in urban wetlands: a log-linear approach

Up to 95% of wetlands in southwestern Ontario, Canada, have been drained and converted to other land uses since the mid-1800s. Remaining wetlands now exist within a mosaic of urban and agriculture land cover and may resemble “islands” from the point of view of species that need this habitat. We investigated the relationships between waterbirds and wetland size, isolation, and adjacent forest cover and roads in 19 wetlands within and adjacent to London, Ontario. Waterbirds were grouped into resource-based guilds: Dabblers, Divers, Waders, and Gulls and Terns (because of small samples, the Passerine and Raptor guilds were dropped from the statistical analyses). Because of the high degree of multicollinearity between variables, we used log-linear analysis to develop models that explained variations in species richness in the waterbird guilds. The log-linear technique provided quantification of environmental effects in a richer way than previous research as the interplay between biophysical factors found in reality are partially captured by the inclusion of interaction in the habitat models. All of the models incorporated the biophysical variables of wetland size and adjacent wetland area, forest cover, and roads. Waterbirds in these urban and near-urban landscapes appear to be using all wetlands available to them, no matter how small or how disjunct the wetland. This has implications for wetland evaluations that may not rate wetlands in human-dominated landscapes highly enough to be protected from development.     

Statistics In Sociology, 1950–2000: A Selective Review

Statistical methods have had a successful half-century in sociology, contributing to a greatly improved standard of scientific rigor in the discipline. I identify three overlapping postwar generations of statistical methods in sociology, based on the kinds of data they address. The first generation, which started in the late 1940s, deals with cross-tabulations and focuses on measures of association and log-linear models, perhaps the area of statistics to which sociology has contributed the most. The second generation, which began in the 1960s, deals with unit-level survey data and focuses on LISREL-type causal models and event-history analysis. The third generation, starting to emerge in the late 1980s, deals with data that do not fall easily into either of these categories, either because they have a different form, such as texts or narratives, or because dependence is a crucial aspect, as with spatial or social network data. There are many new challenges, and the area is ripe for statistical research; several major institutions have recently launched new initiatives in statistics and the social sciences.     

Stable exponential random graph models with non-parametric components for large dense networks

Exponential random graph models (ERGM) behave peculiar in large networks with thousand(s) of actors (nodes). Standard models containing 2-star or triangle counts as statistics are often unstable leading to completely full or empty networks. Moreover, numerical methods break down which makes it complicated to apply ERGMs to large networks. In this paper we propose two strategies to circumvent these obstacles. First, we use a subsampling scheme to obtain (conditionally) independent observations for model fitting and secondly, we show how linear statistics (like 2-stars etc.) can be replaced by smooth functional components. These two steps in combination allow to fit stable models to large network data, which is illustrated by a data example including a residual analysis.     

Factors Affecting Juvenile Court Decisions in Israel: A Multivariate Approach*

This paper investigates the factors influencing the disposition of cases in juvenile courts. The factors considered are the prior record of the offender, the nature of the offense, the recommendation of the probation officer, and the age of the offender. These four factors are essential components of the juvenile justice system and as such are perceived to be crucial for the understanding of the court's decision. A multivariate analysis of the data is undertaken using the method of log-linear analysis for hierarchical models. In contrast to other studies that have utilized the same method but with fewer variables, the model arrived at in the paper seems to be more accurate and suggests that while the offense and the prior criminal record are quite important factors, their influence on the disposition of cases is greatly bound by the age of the offender and the recommendation of the probation officer.     

A Review and Reformulation of Adoptive Relationships from an Attachment Perspective

ABSTRACT

From an attachment perspective, the formation and maintenance of adoptive relationships involve many atypical events and circumstances for both adoptive parents and adopted persons. The present analysis reviews the existing theoretical and empirical literature on several of these issues and examines the utility of conceptualizing them from an attachment perspective, focusing in particular on Bartholomew's (1990) categorical model of attachment styles. Inferences regarding the impact of attachment status (both for those who adopt and for those who have been adopted) on interpersonal and family functioning are formulated and critical areas in need of further research are outlined.     

The contribution of E.C. consumer surveys in forecasting consumer expenditures: An econometric analysis for four major countries

In this article we investigate the forecasting performance of alternative models of private consumption using the EEC consumer surveys. Two basic conclusions emerge from the study:(1) in absolute as well as in comparison with a standard economic model, consumption functions incorporating opinion variables perform surprisingly well given the important measurement problems (missing data, qualitative character of the responses, strong collinearity among responses), and(2) consumers' opinions are helpful guides only in very short-term forecasting (between 0 and 3 quarters).The article extends previous investigations both on the basic characteristics of opinion variables and on their relation with actual economic variables.