Understanding Linkages Among Mixture Models

The methodological literature on mixture modeling has rapidly expanded in the past 15 years, and mixture models are increasingly applied in practice. Nonetheless, this literature has historically been diffuse, with different notations, motivations, and parameterizations making mixture models appear disconnected. This pedagogical review facilitates an integrative understanding of mixture models. First, 5 proto-typic mixture models are presented in a unified format with incremental complexity while highlighting their mutual reliance on familiar probability laws, common assumptions, and shared aspects of interpretation. Second, 2 recent extensions— hybrid mixtures and parallel-process mixtures—are discussed. Both relax a key assumption of classic mixture models but do so in different ways. Similarities in construction and interpretation among hybrid mixtures and among parallel-process mixtures are emphasized. Third, the combination of both extensions is motivated and illustrated by means of an example on oppositional defiant and depressive symptoms. By clarifying how existing mixture models relate and can be combined, this article bridges past and current developments and provides a foundation for understanding new developments. Over the past 15 years, the number of finite mixture modeling applications has dramatically increased in the social sciences (Collins & Lanza, 2010; Muthén & Muthén, 2000; Nagin & Odgers, 2010). Mixture models are now more commonly used for accommodating discrete population heterogeneity than are non-model-based classification methods such as cluster analysis and taxometric