This chapter gives an overview of recent advances in latent variable analysis. Emphasis is placed on the strength of modeling obtained by using a flexible combination of continuous and categorical latent variables.

This article gives an overview of statistical analysis with latent variables. Us-ing traditional structural equation modeling as a starting point, it shows how the idea of latent variables captures a wide variety of statistical concepts, including random e&ects, missing data, sources of variation in hierarchical data, hnite mix-tures, latent classes, and clusters. These latent variable applications go beyond the traditional latent variable useage in psychometrics with its focus on measure-ment error and hypothetical constructs measured by multiple indicators. The article argues for the value of integrating statistical and psychometric modeling ideas. Di&erent applications are discussed in a unifying framework that brings together in one general model such di&erent analysis types as factor models, growth curve models, multilevel models, latent class models and discrete-time survival models. Several possible combinations and extensions of these models are made clear due to the unifying framework. 1.

This paper proposes growth mixture modeling to assess intervention effects in longitudinal randomized trials. Growth mixture modeling represents unobserved heterogeneity among the subjects using a finite-mixture random effects model. The methodology allows one to examine the impact of an intervention on subgroups characterized by different types of growth trajectories. Such modeling is informative when examining effects on populations that contain individuals who have normative growth as well as non-normative growth. The analysis identifies subgroup membership and allows theory-based modeling of intervention effects in the different subgroups. An example is presented concerning a randomized

Growth mixture models are often used to determine if subgroups exist within the population that follow qualitatively distinct developmental trajectories. However, statistical theory developed for finite normal mixture models suggests that latent trajectory classes can be estimated even in the absence of population heterogeneity if the distribution of the repeated measures is nonnormal. By drawing on this theory, this article demonstrates that multiple trajectory classes can be estimated and appear optimal for nonnormal data even when only 1 group exists in the population. Further, the within-class parameter estimates obtained from these mod-els are largely uninterpretable. Significant predictive relationships may be obscured or spurious relationships identified. The implications of these results for applied research are highlighted, and future directions for quantitative developments are suggested. Over the last decade, random coefficient growth modeling has become a centerpiece of longitudinal data analysis. These models have been adopted en-thusiastically by applied psychological researchers in part because they provide a more dynamic analysis of repeated measures data than do many traditional tech-niques. However, these methods are not ideally suited for testing theories that posit the existence of qualita-tively different developmental pathways, that is, theo-ries in which distinct developmental pathways are thought to hold within subpopulations. One widely cited theory of this type is Moffitt’s (1993) distinction between “life-course persistent ” and “adolescent-limited ” antisocial behavior trajectories. Moffitt’s theory is prototypical of other developmental taxono-mies that have been proposed in such diverse areas as developmental psychopathology (Schulenberg,

Sources of population heterogeneity may or may not be observed. If the sources of heterogeneity are observed (e.g., gender), the sample can be split into groups and the data analyzed with methods for multiple groups. If the sources of population heterogeneity are unobserved, the data can be analyzed with latent class models. Factor mixture models are a combination of latent class and common factor models and can be used to explore unobserved population heterogeneity. Observed sources of heterogeneity can be included as covariates. The different ways to incorporate covariates correspond to different conceptual interpretations. These are discussed in detail. Characteristics of factor mixture modeling are described in comparison to other methods designed for data stemming from heterogeneous populations. A step-by-step analysis of a subset of data from the Longitudinal Survey of American Youth illustrates how factor mixture models can be applied in an exploratory fashion to data collected at a single time point. The populations investigated in the behavioral sciences and related fields of research are often heterogeneous. A sample may consist of explicitly defined groups such as experimental and control groups, and the aim is to compare these groups. On the other hand, the sources of population heterogeneity may not be known beforehand. Test scores on a cognitive test may reflect two types of children in the sample: those who master the knowledge required to solve the items (masters) and those who lack this critical knowledge (nonmasters). The interest may be to decide to which of the subpopulations a given child most likely belongs. In addition, it may be of interest to characterize masters and nonmasters using background variables to develop specific

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Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes in SEMM: misspecifica-tion of the structural model, nonnormal continuous measures, and nonlinear rela-tionships among observed and/or latent variables. When the objective of a SEMM analysis is the identification of latent classes, these conditions should be considered as alternative hypotheses and results should be interpreted cautiously. However, armed with greater knowledge about the estimation of SEMMs in practice, re-searchers can exploit the flexibility of the model to gain a fuller understanding of the phenomenon under study. In recent years, many exciting developments have taken place in structural equation modeling, but per-haps none more so than the development of structural equation models that account for unobserved popula-

This study describes trajectories of substance use and dependence from adolescence to adulthood. Identified consumption groups include heavy drinking/heavy drug use, moderate drinking/experimental drug use, and light drinking/rare drug use. Dependence groups include alcohol only, drug only, and comorbid groups. The heavy drinking/heavy drug use group was at risk for alcohol and drug dependence and persistent dependence and showed more familial alcoholism, negative emotionality, and low constraint. The moderate drinking/experimental drug use group was at risk for alcohol dependence but not comorbid or persistent dependence and showed less negative emotionality and higher constraint. Familial alcoholism raised risk for alcohol and drug use and dependence in part because children from alcoholic families were more impulsive and lower in agreeableness. Substance use and substance use disorders show systematic age-related patterns, with adolescent onset, peaks in use and diag-nosed disorders in “emerging adulthood ” (ages 18–25; Arnett, 2000), and declines in use after the mid-twenties (Bachman, Wads-worth, O’Malley, Johnston, & Schulenberg, 1997; Chen & Kandel, 1995). However, despite these overall trends, there is also consid-

Factor mixture models are designed for the analysis of multivariate data obtained from a population consisting of distinct latent classes. A common factor model is assumed to hold within each of the latent classes. Factor mixture modeling involves obtaining estimates of the model parameters, and may also be used to assign subjects to their most likely latent class. This simulation study investigates aspects of model performance such as parameter coverage and correct class membership assignment and focuses on covariate effects, model size, and class-specific versus class-invariant parameters. When fitting true models, parameter coverage is good for most parameters even for the smallest class separation investigated in this study (0.5 SD between 2 classes). The same holds for convergence rates. Correct class assignment is unsatisfactory for the small class separation without covariates, but improves dramatically with increasing separation, covariate effects, or both. Model performance is not influenced by the differences in model size investigated here. Class-specific parameters may improve some aspects of model performance but negatively affect other aspects. Factor mixture models combine latent class analysis and common factor analysis. Factor mixture models are designed for data from possibly heterogenous populations consisting of several latent classes, and are an adequate choice if it is reasonable to assume that observed variables within each class can be modeled using a common factor model. There are two types of latent variables in factor mixture

1st theme is that model-checking procedures may be capable of distinguishing between mixtures of normal and homogeneous nonnormal distributions. Although useful for assessing model quality, it is argued here that currently available pro-cedures may not always help discern between these 2 possibilities. The 2nd theme is that even if these 2 possibilities cannot be distinguished, a growth mixture model may still provide useful insights into the data. It is argued here that whereas this may be true for the scientific goals of description and prediction, the acceptance of a model that fundamentally misrepresents the underlying data structure may be less useful in pursuit of the goal of explanation. We begin by thanking Robert Cudeck, Susan Henly, Bengt Muthén, and David Rindskopf for pro-viding comments on our work (Bauer & Curran, 2003). We could not have asked for a more talented and esteemed group of quantitative methodologists to comment on our article, and we greatly appreciate the