The current volume, Advances in Latent Variable Mixture Models, contains chapters by all of the speakers who participated in the 2006 Cilvr conference, providing not just a snapshot of the event, but more importantly chronicling the state of the art in latent variable mixture model research. The volume starts with an overview chapter by the Cilvr conference keynote speaker, Bengt Muthén, offering a “lay of the land” for latent variable mixture models before the volume moves to more specific constellations of topics. Part I, Multilevel and Longitudinal Systems, deals with mixtures for data that are hierarchical in nature either due to the data's sampling structure or to the repetition of measures (of varied types) over time. Part Ii, Models for Assessment and Diagnosis, addresses scenarios for making judgments about individuals' state of knowledge or development, and about the instruments used for making such judgments. Finally, Part Iii, Challenges in Model Evaluation, focuses on some of the methodological issues associated with the selection of models most accurately representing the processes and populations under investigation. It should be stated that this volume is not intended to be a first exposure to latent variable methods. Readers lacking such foundational knowledge are encouraged to consult primary and/or secondary didactic resources in order to get the most from the chapters in this volume. Once armed with the basic understanding of latent variable methods, we believe readers will find this volume incredibly exciting.

any form reserved. IA395-Handcock.indb 7 vii 10/17/07 1:15:31 PM viii G.R. HANCOCK and K.M. SAMUELSEN ing on a specific aspect of latent variable methods, but where that focus is deemed to have far-reaching methodological and applied implications. The theme for the inaugural conference, held at the University of Maryland on May 18 and 19, 2006, was Mixture Models in Latent Variable Research. With mixture-related training workshops the day before and the day after the conference, the

10,000. It is clear from these plots that for most values of α1 the conditional probability is either 0 or 1. This is especially so when the variance of α1 is 10,000. When this probability is 0 or 1 then Cij cannot vary across individuals in the cluster as it is completely determined by the value of the random effect α1 in the j th cluster. When the variance of α1 is large the influence of the random intercept α1 on Cij is large, which makes the class variables within a cluster so highly

estimates reflect the probability of Time 2 Alcohol use latent class conditional on the pair of Time 1 latent class memberships and Time 2 Tobacco use. “No tobacco” and “No alcohol” classes at Time 1. If one remained in the “No tobacco” class at Time 2, then there was an 87% chance that person would also remain in the “No alcohol” LC. However, if one changed to the IA395-Handcock.indb 95 10/17/07 1:16:25 PM 96 B.P. FLAHERTY “Tried tobacco” LC, then that person was over 20% less likely to

Multiple Event Indicators If there is more than one measure of event status at each time period, we can extend the model previously presented to include M event indicators at each time point j, Umj. Figure 5.4 displays the path diagram for this extension. The measurement model for E is now defined by the set of event indicators, and the relationship between each of the Umj terms and Ej is IA395-Handcock.indb 135 10/17/07 1:16:54 PM 136 K.E. MASYN U11 U21 UM1 U12 E1 U22 UM2 U1J U2J

plate that signifies replication over tasks, which indicates that the psychometric properties of each task depend on the features of the task. Letting Q denote the collection of qj over tasks, the corresponding probability model is now: P (x, Θ, B, η, t, Q) = ∏ ∏ P (x ij | θi , β j ) × P (θi | η) × P (β j | q j , t) × i (6.11) j P (η) × P (t), which makes explicit that the task parameters are conditional on the task features. The dependence of bj on qj reflects the additional information