Optimization problems in optimal regression design, response-adaptive design and categorical random variables

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Optimization has been a basic tool in most of the areas of theoretical and applied sciences. The optimization problems are of great importance, and they have yet to be adequately studied. In this proposal, an important theme is the need for objective methodologies for optimization problems in three areas, namely, optimal design, adaptive design and categorical random variables with potential applications. We start with some estimation problems that can be solved using optimal design theory. One such problem is to determine the maximum likelihood estimates of the cell probabilities under the hypothesis of marginal homogeneity for data in contingency tables. Another problem is constructing designs subject to achieving a given efficiency of a parameter in a regression model. The goal is to implement a design that is efficient for two or more models, to discriminate and select the best one. Then we propose to work on optimal response-adaptive designs for circular data with reference to a practical data set on a comparative prospective randomized interventional trial. Our plan is also to work on response-adaptive designs with applications in surrogate endpoints. Surrogate endpoints are used with growing interest to evaluate the effects of treatments or exposures when availability of true endpoints is less due to cost or time constraint. The proposed work also extends to other applications such as in the case of ordinal categorical random variables. The focus will be given on theoretical formulation of the joint distribution of multivariate ordinal categorical random variables. The proposed research will have significant contribution to statistics and applied sciences. The work on model selection has potential application in problems related to industrial and chemical engineering for carrying out tests for models. The work on adaptive designs with circular data will provide efficient and practical tools to applied researchers in meteorology and astronomy. The statistical devices proposed for surrogate response are needed for efficient use of surrogate endpoints. The proposed research on ordinal categorical random variables will provide important contribution in longitudinal studies and in analyzing clustered data.