Data Analysis Using Finite Mixture Models

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Finite mixture models are increasingly popular in the social and medical sciences for analyzing data thought to arise from a population consisting of categorical types. This research project considers Bayesian methods for analyzing data using finite mixture models. The application of classical methods to such models is difficult because the likelihoods for such models are nonstandard: they are inherently multimodal due to symmetries in the labeling of the mixture components, may be multimodal even for a single fixed labeling of the components, and fail to satisfy the regularity conditions required for classical likelihood ratio tests. The principal scientific questions of interest concern drawing inferences about model parameters, determining the number of mixture components, and stochastically classifying sampling units into the mixture components. Modern advances in statistical computing, such as the EM and ECM algorithms, data augmentation, and Gibbs sampling, are used to obtain inferences about the posterior distribution of model parameters; however special care is needed in applying these methods because of the difficulties mentioned above. This research describes methods for distinguishing between the two types of modes described above, and for carrying out the data analysis given the existence of multiple modes. Draws from the posterior distribution of the model parameters can be used to obtain draws from the posterior predictive distribution of replicate experiments similar to the current experiment. The posterior predictive distribution of test statistics, or of other discrepancy measures, can be used to evaluate the fit of a model, e.g., comparing two component and three component mixture models even though the problem is irregular. Additionally, averaging over a prior distribution on plausible alternatives to the existing model can be used to estimate the sample size required to assess the appropriateness of the existing model against such alternatives and therefore can be used to inform design decisions. It is not uncommon to consider a particular class of statistical models, called mixture models, that assume the population of interest consists of a number of relatively homogeneous subpopulations. Mixture models are useful when a relatively complex model would be required to describe the pattern of data that is observed within the entire population, whereas a relatively simple model applies within each subpopulation. Classical approaches are of limited use in such cases, e.g., classical methods do not apply to the crucial question of determining whether and how many subpopulations are in evidence. This research proposal aims to develop new methods for analyzing data using mixture models and for assessing the adequacy of such models. These new methods take advantage of recent theoretical and computational advances to draw accurate inferences about the important features of the mixture models. The basic approach is to average over all descriptions of the population that are supported by the data and thereby provide an accurate assessment of the variation and patterns to be expected in the population.