Latent Class Analysis I
Paper-Session
Using Data Augmentation in Latent Class Models for
Longitudinal Data: Estimation, Standard Errors and Hypothesis
Tests Involving Combinations of Parameters
Linda M. Collins, Joseph L. Schafer, Stephanie L. Hyatt & Brian P. Flaherty
This presentation will discuss the use of data augmentation in latent class models. Data augmentation is a special
case of the family of Bayesian procedures known as Markov Chain Monte Carlo methods (Tanner & Wong,
1987; Schafer, 1997). Data augmentation is a stochastic analogue of the EM algorithm. In both the EM algorithm
and data augmentation, the latent classes are considered missing data. With each iteration, the EM algorithm
alternately predicts the missing data (in the E or Expectation step) and based on these predictions re-estimates
the parameters (in the M or Maximization step). In contrast, data augmentation takes a multiple imputation
approach to handling the missing latent data, alternating an imputation step with a posterior step. In the
imputation step, the complete data table, i.e. the crossclassification of the latent classes and observed variables, is
randomly imputed. In the posterior step, Bayes' theorem is used to produce a posterior distribution (based on the
Dirichlet) for the parameters of the latent class model. The objective of the data augmentation procedure is to
produce a set of independent imputed data sets, each of which can be used to compute estimates of the latent
class model parameters. These independent estimates can be combined to produce an estimate of the posterior
mean for each parameter, and the uncertainty associated with each parameter. Data augmentation provides a
very flexible approach to estimation and to the construction of confidence intervals. This flexibility affords the
user the ability to perform hypothesis testing on combinations of parameters. The difference between two
parameters, or the difference between two combinations of parameters, can be tested for significance using data
augmentation. In this presentation, we will give several examples of this in the context of latent class models of
longitudinal data.
A Two-Stage Estimation Strategy for Concomitant-Variable
Latent Class Models
C. Mitchell Dayton
Latent class models incorporating continuous concomitant variables first appeared in the research literature over
ten years ago (Dayton & Macready, 1988a, b). However, little progress has been made on practical
implementation of these models other than for cases where there is a limited number of discrete levels for the
concomitant variable. This paper describes a two-stage estimation strategy in which the parameters for the
structural part of the model related to the form of the relationship between latent class membership and the
concomitant variable(s) are estimated using conditional MLE techniques. Some comparative results are
presented for cases in which full MLE is possible. Also, exemplary applications of the two-stage strategy are
presented.
References:
[1] Dayton, C. M. & Macready, G. B. (1988a). Concomitant-variable latent class models. Journal of the
American Statistical Association, 83, 173-178.
[2] Dayton, C. M. & Macready, G. B. (1988b). "A latent class covariate model with applications to criterionreferenced testing." in R. Langeheine & J. Rost, Latent Class and Latent Trait Models, New York: Plenum
Separating Traited and Untraited Individuals by a Mixture
Distribution Latent State-Trait Model
Michael Eid & Rolf Langeheine
In the debate on the consistency of behavior in personality psychology, the metatrait approach has gained
increasing interest. According to this approach, individuals can differ in their consistency across situations and
their stability across time. Traited individuals show high consistency across situations whereas less traited
behave more situation-specific. In order to separate individuals differing in their traitedness, a mixture
distribution latent state-trait model is presented. In this model, it is assumed that the total population consists of
two subpopulations differing in their consistency across situations. It is shown how this model can be analyzed
with log-linear models with latent variables. Finally, the model is illustrated by an empirical application.
Approximate Estimation in Complex Models by Mean Field
Theory
Keith Humphreys & D. M. Titterington
We will consider estimation in latent structure models with discrete latent state space. For some such models,
those which have a latent state space of high dimension, the implementation of the E step of an appropriate EM
algorithm is, computationally, not feasible. The level of computational difficulty can be substantially reduced by
approximating the EM algorithm using the "mean field approximation" which was initially proposed in the
neural-computing and engineering literatures (Zhang, 1992, for example). In the E step the approximation
essentially replaces expectation based on a joint distribution by expectation based on a simpler, independence
distribution. Parameter estimation using this approximation has been shown to be remarkably effective for a
:range of complex models. We will supplement the work of Dunmur and Titterington (1998) who have already
described the application of the approximation in latent class and latent profile models. We will do this by
considering additional models and by considering the general applicability of extensions to mean field theory.