Determining the Number of Components in Mixture Models for Hierarchical Data

Abstract

Recently, various types of mixture models have been developed for data sets having a hierarchical or multilevel structure (see, e.g., Vermunt, Sociological Methodology 33:213–239, 2003; Computational Statistics and Data Analysis 51:5368–5376, 2007). Most of these models include finite mixture distributions at multiple levels of a hierarchical structure. In these multilevel mixture models, selection of the number of mixture component is more complex than in standard mixture models because one has to determine the number of mixture components at multiple levels.

In this study the performance of various model selection methods was investigated in the context of multilevel mixture models. We focus on determining the number of mixture components at the higher-level. We consider the information criteria BIC, AIC, and AIC3, and CAIC, as well as ICOMP and the validation log-likelihood. A specific difficulty that occurs in the application of BIC and CAIC in the context of multilevel models is that they contain the sample size as one of their terms and it is not clear which sample size should be used in their formula. This could be the number of groups, the number of individuals, or either the number of groups or number of individuals depending on whether one wishes to determine the number of components at the higher or at the lower level.

Our simulation study showed that when one wishes to determine the number of mixture components at the higher level, the most appropriate sample size for BIC and CAIC is the number of groups (higher-level units). Moreover, we found that BIC, CAIC and ICOMP detect very well the true number of mixture components when both the components’ separation and the group-level sample size are large enough. AIC performs best with low separation levels and small sizes at the group-level.