chapter  11
24 Pages

Finite Mixtures of Structured Models

ByMarco Alfó, Sara Viviani

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 11.2 Existing Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.3 Overdispersed Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.4 Clustered Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

11.4.1 Longitudinal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 11.4.2 Multivariate Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 11.4.3 Further Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

11.5 ML Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 11.5.1 The EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.5.2 The Choice of the Number of Components . . . . . . . . . . . . . . . . . . . . . . 233 11.5.3 The Provision of Standard Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.5.4 Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

11.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 11.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

In this chapter, we describe developments in finitemixturemodels for structured data, with a particular focus on clustered, multilevel, longitudinal, and multivariate data. Literature on finite mixtures in generalized linear models is now quite extensive and entails application areas such as marketing (Wedel and DeSarbo 1995), biostatistics (Wang et al. 1996), econometrics (Deb and Trivedi 1997), machine-learning (Jacobs et al. 1991), just to mention a few. After a brief introduction, we will discuss some examples of the use of finite mixtures in heterogeneous generalized linear models, with a particular emphasis on model definition. We will also provide a brief review of available software and some suggestions on potential research areas.