Nonparametric Mixture Models

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While many individual human differences are obvious to the casual observer - differences in height, weight, hair color or complexion, for example - arguably the more important individual differences, from a behavioral science perspective, are latent and not observed. This project will develop statistical methods for separating a given population into its distinct component distributions, including estimation of the number of components and nonparametric estimation of the component distributions. These different components reflect the latent individual differences in the population. A principal way in which this research differs from other research on mixture models is that the focus is on weakening the often strong and possibly unrealistic parametric assumptions that have, in the past, possibly limited the wider application of mixture theory. Specifically, this research will compare existing Bayesian methods to determine the number of components in a nonparametric mixture and develop new Bayesian methods. When covariate information is available, the project will study mixtures of regressions, in which a goal is the estimation of the mixing proportions as a function of the covariate(s). Finally, the project will use analytical and computer simulation methods to expand previously funded research in nonparametric mixture models to the one-way layout and multiple comparisons, an area in which traditional methods such as analysis of variance and standard nonparametric alternatives can fail to detect certain kinds of group differences. The mixture models developed in this project make it possible to both conceptualize and identify individual latent differences such as differences in thinking styles. These mixtures provide not only a useful conceptual perspective but a practical one as well, in that it is possible to identify, at the level of the individual, which cognitive strategy (for example) a person most likely used in the solution of a problem. Having a framework for conceptualizing and identifying how growing children are different in those latent (unobserved) processes vital for effective learning and school achievement would have important implications for how learning environments are constructed. Yet the techniques developed, largely motivated by psychological concerns, are applicable to any field of scientific empirical inquiry employing statistical data analysis in which latent variables have a measurable impact. Thus, this research has broad implications for a variety of intellectual activities.