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FOREWORD
Latent variables of one form or another have proven invaluable in many
scientific disciplines. Sociology is no exception, with its long tradition
of latent structure modeling. In this volume of Sociological Methodology the chapters by Anderson and Vermunt; Martin and Wiley; and Yuan
and Bentler advance this tradition, albeit in different ways. A related
but not identical line of work that is becoming increasingly important
involves the incorporation of so-called random effects into regressiontype models, be they linear regressions, event-history regressions, or
categorical response regressions. The contributions by Agresti, Booth,
Hobert, and Caffo; Arminger, Clogg, and Cheng; and Barber, Murphy,
Axinn, and Maples advance this line of work. Thus six of the eight chapters in this volume are concerned with latent variables in one guise or
another.
The publication of this volume also marks the conclusion of the late
Clifford C. Clogg’s research career. The opening chapter is the result of a
long and fruitful collaboration between Cliff and his dear friend and colleague, Gerhard Arminger. We are delighted to have the opportunity to
publish the results of the work that they were deeply engaged in at the time
of Cliff ’s sudden and unexpected death in May 1995. (Please see the opening chapter of Sociological Methodology 1996 for a tribute to Cliff ’s life
and work.)
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Contents
Chapters 1 and 2 are concerned with models and estimation procedures
for categorical responses when one or more of the predictors is a latent
variable or random effect. Rasch models and more general latent trait
models, which arise out of psychometrics and educational testing, are
well-known examples. Recently, advances in statistical computing and
power have led researchers to put forth and consider broad generalizations of these models, applicable in a variety of substantively interesting
and useful contexts.
Arminger, Clogg, and Cheng, building on Lindsay, Clogg, and Grego
(1991), consider semiparametric estimation of Rasch models with covariates, applying their model in the context of panel data. It is important to
note that minimal assumptions are made about the distribution of the random effects in the model. Thus the models should be especially attractive
to empirical researchers in the usual case where it is difficult to justify on
substantive grounds important assumptions about the distribution of the
random effects that are invoked in order to use such methods. The authors
also provide an algorithm for fitting the models that can be viewed as a
compromise between unidimensional and full Newton-Raphson.
In Chapter 2, Agresti, Booth, Hobert, and Caffo give a state of the
art review of the generalized linear mixed models (GLMMs) approach to
random-effects modeling of categorical responses. The models are especially useful when the observed responses do not arise from independent
observations, as in longitudinal studies, family studies, and other settings
where observations are either clustered and0or serially correlated. Applications of this type of modeling are starting to become more common in
sociology and the authors cite a number of such articles. Further, they
present nine different examples to illustrate the potential utility of the models to a broad range of sociological topics. Readers will find this paper
useful not only for the breadth of applications considered but also because
it includes detailed discussion of practical computational tools for fitting
GLMMs. In particular, relatively detailed discussion and example code for
SAS Institute’s PROC NLMIXED is provided in the paper, and the SAS
macro %GLIMMIX is discussed as an appropriate maximum-likelihood
method. Other computing programs, including EGRET, MIXOR, BUGS,
HLM, and LogExact, are also discussed. Finally, the reader will benefit
from relating example 5 in this chapter to the previous chapter by Arminger, Clogg, and Cheng.
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In Chapter 3, Anderson and Vermunt take up a latent variable model
that is directly related to association models for multivariate categorical
responses. The authors start with a general model for multiple correlated
latent variables, and they develop both graphical representations for the
models and corresponding log-multiplicative association models. Interestingly, both this chapter and its predecessor use the now classic Coleman
(1964) panel data on membership and attitude with respect to the “leading
crowd,” and a comparison of the two analyses will give the reader deeper
insights into both sets of methodological contributions and their respective
utility in substantive work.
In Chapter 4, Martin and Wiley continue the line of research they
reported on in Sociological Methodology 1999. There they developed
latent class models for sets of dichotomous items where each item measured a particular dichotomous belief. Beliefs were related through relations of precedence (which impose restrictions on the response patterns)
and the statistical model was completed by imposing restrictions on the
permissible patterns of measurement error. In this paper, however, the
observed dichotomous items (macrobeliefs) may measure more than one
latent microbelief. To illustrate the proposed procedures, the authors analyze items from the General Social Survey on beliefs regarding economic equality.
In Chapter 5, Yuan and Bentler propose a new minimum x 2 estimator for parameters in covariance structure models when some of the
data are missing. The estimator has desirable asymptotic properties when
the missing data are missing completely at random; it is not assumed that
the data are drawn from a multivariate normal distribution. They also compare the performance of this estimator with several other estimators appropriate for this situation, and they develop a suitable test statistic for
testing the fit of the model using this estimator.
Random-effects models are also the subject of Chapter 6, in which
Barber, Murphy, Axinn, and Maples discuss a discrete-time survival model
where the random effects are assumed to follow a normal distribution. The
authors show that standard software for “multilevel” models can be used to
fit the model. To illustrate, they model the hazard for adopting contraception using the logistic model with the woman’s education, number of previous children and two terms for time as covariates; the intercept and the
parameters for education and number of previous children are then allowed to depend upon characteristics of the neighborhood in which the
woman lives.
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In Chapter 7, Hoem considers the analysis of event-history data
when certain combinations of the regressors are impossible or unobserved,
implying there is no exposure information for such combinations. Consider, for example, an event-history analysis of divorce including mother’s
parity and the age of youngest child as predictors. Should childless women
be excluded from such an analysis, or are there special considerations that
must be taken into account to include them in the model, and ultimately in
the interpretation of the model coefficients? This is one of Hoem’s examples, and it provides an excellent context for illustrating the issues and
Hoem’s proposed solution.
Sociologists have long noted that persons do not simply and automatically enact the norms that are bundled together into a role, implying
that traditional versions of role theory are too strong. But sociologists continue to find the basic ideas underlying role theory attractive, as evidenced
by attempts to relax such theories to permit more nuanced patterns of behavior. A typical way to weaken a deterministic theory that does not successfully account for the evidence is to make the theory probabilistic; thus,
for example, Stryker (1980) has put forth a version of role theory that is
essentially probabilistic. In Chapter 8, Montgomery approaches this problem using fuzzy sets. Whereas in classical set theory, the function that
describes whether or not an element belongs to a given set is binary (say 0
for not belonging, 1 for belonging), in fuzzy set theory, this function is
allowed to take on all values in the (closed) unit interval. Thus, using fuzzy
set theory, a person could, for example, “nearly” hold (not hold) the role of
father. Montgomery then works through some of the implications of this
point of view and contrasts his work with other approaches—for example,
affect control theory and rational choice.
ACKNOWLEDGMENTS
We thank the reviewers and advisory editors who so graciously and
generously gave of their time to help us and the authors. The tradition of
scholarly excellence for which Sociological Methodology is known and
respected is in no small part due to the efforts of these persons and their
counterparts before them. We are especially grateful to Carson Chase Hicks
for her outstanding contributions, as managing editor, to the production of
this volume. Likewise, we sincerely appreciate the efforts of the copy editor, Stephanie Argeros-Magean, and the help of Janet Cronin and Roberta
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Spinosa-Millman of Blackwell Publishers. Sociological Methodology is
sponsored and supported by the American Sociological Association.
REFERENCES
Coleman, James S. (1964). Introduction to Mathematical Sociology. Glencoe, IL: The
Free Press.
Lindsay, Bruce G., Clifford C. Clogg, and John M. Grego. 1991. “Semi-Parametric
Estimation in the Rasch Model and Related Exponential Response Models, Including a Simple Latent Class Model for Item Analysis.” Journal of the American Statistical Association 86:96–107.
Stryker, Sheldon. 1980. Symbolic Interactionism: A Social Structural Version. Menlo
Park, CA: Benjamin0Cummings.