Sociological Methodology

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.

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.

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.

It may often be the case that the beliefs about which survey researchers query respondents are composed of discrete components, such that holding all of the components is necessary to give a “yes” response. Simple logical relations, which some researchers have proposed may structure belief data, may obtain between these components, and not between the beliefs that are actually measured. This paper demonstrates that an algebraic inversion of a data matrix, first used in test theory by Haertel and Wiley (1993), can be seen as a unique and interpretable decomposition that can recover information regarding the compositional formulas of the measured beliefs as well as the logical relations between the unobserved components. The inversion is illustrated with a set of data from the GSS. Finally, the conditions under which related techniques are then helpful or not helpful for analyzing survey data are discussed.

Survey and longitudinal studies in the social and behavioral sciences generally contain missing data. Mean and covariance structure models play an important role in analyzing such data. Two promising methods for dealing with missing data are a direct maximum-likelihood and a two-stage approach based on the unstructured mean and covariance estimates obtained by the EM-algorithm. Typical assumptions under these two methods are ignorable nonresponse and normality of data. However, data sets in social and behavioral sciences are seldom normal, and experience with these procedures indicates that normal theory based methods for nonnormal data very often lead to incorrect model evaluations. By dropping the normal distribution assumption, we develop more accurate procedures for model inference. Based on the theory of generalized estimating equations, a way to obtain consistent standard errors of the two-stage estimates is given. The asymptotic efficiencies of different estimators are compared under various assumptions. We also propose a minimum chi-square approach and show that the estimator obtained by this approach is asymptotically at least as efficient as the two likelihood-based estimators for either normal or nonnormal data. The major contribution of this paper is that for each estimator, we give a test statistic whose asymptotic distribution is chisquare as long as the underlying sampling distribution enjoys finite fourth-order moments. We also give a characterization for each of the two likelihood ratio test statistics when the underlying distribution is nonnormal. Modifications to the likelihood ratio statistics are also given. Our working assumption is that the missing data mechanism is missing completely at random. Examples and Monte Carlo studies indicate that, for commonly encountered nonnormal distributions, the procedures developed in this paper are quite reliable even for samples with missing data that are missing at random.

Combining innovations in hazard modeling with those in multilevel modeling, we develop a method to estimate discrete-time multilevel hazard models. We derive the likelihood of and formulate assumptions for a discrete-time multilevel hazard model with time-varying covariates at two levels. We pay special attention to assumptions justifying the estimation method. Next, we demonstrate file construction and estimation of the models using two common software packages, HLM and MLN. We also illustrate the use of both packages by estimating a model of the hazard of contraceptive use in rural Nepal using time-varying covariates at both individual and neighborhood levels.

Users of intensity-regression techniques in event-history analysis based on occurrence and exposure matrices frequently encounter zero-valued exposures where there is no observation for selected combinations of factor levels, either because they are logically impossible or because the observational design leaves them out. For example, higher educational levels cannot be attained at very young ages, and observations collected over a given calendar period cannot contain data for older birth cohorts at younger ages or data for younger cohorts at older ages. Unless the exposure matrix is too badly full of zeros, this does not necessarily cause problems for the estimation of regression parameters, but interpretation of results may have to be carried out with extra care. In this paper, we discuss why this is so and show how deft handling of systematic patterns of zero exposures can facilitate analysis.

This paper shows how the socialization process (viewed by role theorists as a “role-person merger”) can be represented and analyzed as a fuzzy (dynamical) system. Given a self-concept (a fuzzy set of roles) and social norms (logical implications from roles to actions), an individual infers actions (through approximate reasoning). Given these actions, alters make biased attributions (about the roles in the individual's self-concept) that are gradually internalized by the individual. This feedback loop creates a fuzzy system (a role-space vectorfield) that generates a set of stable long-run selves (role-space attractors). I illustrate this general “endogenous-self” framework with a model based loosely on Tally's Corner; the analysis examines how the individual's long-run self-concept is influenced by a constraint on employment opportunity.