Abstract
The use of finite mixture modeling (FMM) to identify unobservable or latent groupings of individuals within a population has increased rapidly in applied prevention research. However, many prevention scientists are still unaware of the statistical assumptions underlying FMM. In particular, finite mixture models (FMMs) typically assume that the observed indicator variables are normally distributed within each latent subgroup (i.e., within-class normality). These assumptions are rarely met in applied psychological and prevention research, and violating these assumptions when fitting a FMM can lead to the identification of spurious subgroups and/or biased parameter estimates. Although new methods have been developed that relax the within-class normality assumption when fitting a FMM, prevention scientists continue to rely on FMM methods that assume within-class normality. The purpose of the current article is to introduce prevention researchers to a FMM method for heavy-tailed data: FMM with Student t distributions. We begin by reviewing the distributional assumptions that underlie FMM and the limitations of FMM with normal distributions. Next, we introduce FMM with Student t distributions, and show, step by step, the analytic and substantive results of fitting a FMM with normal and Student t distributions to data from a smoking-cessation trial. Finally, we extend the results of the applied example to draw conclusions about the use of FMM with Student t distributions in applied settings and to provide guidelines for researchers who wish to use these methods in their own research.
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Notes
These data are part of a larger simulation study conducted to highlight the potential dangers of fitting a FMM-n to non-normally distributed datasets. The results of this study are available in the online supplemental material.
Soft-randomization scheme: uses initialization values for the ECM algorithm between 0 and 1. Hard-randomization scheme: uses initialization values of either 0 or 1 for the ECM algorithm.
The k-means initialization procedure derives initialization values from a k-means clustering procedure and uses the parameter estimates derived from the k-means analysis as starting values in the ECM algorithm.
A model’s entropy is an aggregate measure of a model’s classification uncertainty and is derived from each individual’s posterior probability of membership in a particular subgroup. Entropy scores range from 0.00–1.00 with higher values (> .80) indicating that there is adequate separation between the identified subgroups (Asparouhov and Muthén 2018). R code to derive entropy scores from a fitted FMM is available in the online supplemental material.
To assign each individual to a unique subgroup, we took a “classify-analyze” approach where participants were assigned to the subgroup corresponding to their highest posterior probabilities. This approach was deemed appropriate because the majority of the identified models had an entropy ≥ 0.90 (Clark and Muthén 2009).
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Acknowledgments
The author would like to thank Daniel Bolt, Kristin Litzelman, Robert Nix, and David Epstein for helpful comments and feedback on earlier drafts of this article.
Funding
This research was supported by a University of Wisconsin’s School of Human Ecology Summertime Academic Research (STAR) award and a training grant in connection with grant P50 DA019706 from the National Institute on Drug Abuse awarded to Michael Fiore and Timothy Baker.
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Burgess-Hull, A.J. Finite Mixture Models with Student t Distributions: an Applied Example. Prev Sci 21, 872–883 (2020). https://doi.org/10.1007/s11121-020-01109-3
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DOI: https://doi.org/10.1007/s11121-020-01109-3