Locally optimal designs for generalized linear models with a single-variable quadratic polynomial predictor
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Finding optimal designs for generalized linear models is a challenging problem, especially when the predictor contains higher-order terms. This dissertation includes a review of literature on locally optimal designs for generalized linear models and three manuscripts for models with a single-variable quadratic polynomial predictor. The first manuscript focuses on the classical family of $phi_p$-optimality criteria. Our results establish that $phi_p$-optimal designs can be found within a subclass of designs with either three or four support points, depending on the given value of the parameters. These support points are structurally symmetric in terms of their locations and corresponding weights. Due to the small number of support points with symmetric structure, searching for $phi_p$-optimal designs is simply a constraint optimization problem with at most three variables. The second manuscript considers a broader family of $phi$-optimality criteria, which covers the $phi_p$ family in the first manuscript. Under certain conditions, our results establish that the maximum number of support points needed for forming the complete class can be limited to either three or four, with some support points identified. The symmetric structure as in the first manuscript does not apply. The third manuscript develops a simple-to-use computational tool based on the theoretical results for models with single-variable first-order and second-order predictors. The tool is a graphical user interface created using MATLAB. Users can specify design scenarios to find optimal and efficient designs simply by mouse clicks and keyboard input. No software or programming knowledge is required. Moreover, the interface will be made widely available.