Often there are two or more positively associated response variables which measure different aspects of a condition which is to be evaluated across groups with respect to co-variates. For example, in studying diabetes, one might record the number of first degree relatives diagnosed with diabetes, and a fasting glucose value. The literature shows these variables to have a positive association. Suppose it is desired to distinguish ethnic groups. Separate univariate analyses could be performed, but it would be a more powerful technique to utilize a multivariate model, fitting it separate to the groups, and comparing the groups by comparing the model parameters. In this case, the multi- variate model would have to apply to joint measurement of a discrete and a continuous variable. Multi-variate models which involve discrete as well as continuous components, is an area with little development.
The aim of this proposal is to develop such multivariate models along very specific lines. The author has researched a new family of multivariate discrete models, called cluster-sum distributions, which may be thought of as multivariate type of Poisson-stopped sum distribution. Marginal distributions can be any collection of univariate Poisson-stopped sum distributions, including the negative binomial, the Neyman Type A, the Polya-Aeppli, and the Lagrangian Poisson. In general, these distributions are under-utilized in applications. The technique for generating these multivariate discrete distributions suggests a very natural way to generate multivariate distributions with discrete components, continuous components, and mixed type components (those having a discrete as well as a continuous part). Theoretical development of such distributions will be pursued, as will investigations to determine the modeling potential of such distributions in real data.
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