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Item Optimal Designs for the Hill Model with Three Parameters(North Dakota State University, 2012) Dockter, Travis JonOptimal designs specify design points to use and how to distribute subjects over these design points in the most efficient manner. The Hill model with three parameters is often used to describe sigmoid dose response functions. In our paper, we study optimal designs under the Hill model. The first is D-optimal design, which works best to study the model to fit the data. Next is c-optimal design, which works best to study a target dose level, such as ED50 - the dose level with 50% maximum treatment effect. The third is a two-stage optimal design, which considers both D-optimality and c-optimality. In order to compare the optimal designs, their design efficiencies are compared.Item Robust D-Optimal Design for Multiple Nominal Parameter Values under the 5PL-1P Model(North Dakota State University, 2018) Liang, CuipingA robust D-optimal design that works well for multiple nominal parameter values is presented in this paper. In general, D-optimal design works very well for estimating the model parameters, but it is very sensitive to multiple nominal model parameter values when the response is modeled by nonlinear models. The 5PL-1P model is considered in this study to describe a dose-response function. The sensitivity of the D-optimal design to the model parameter values under the 5PL-1P model is studied. The robust D-optimal design that can reduce the impact of the multiple nominal model parameter values is proposed using the Bayesian technique. Lastly, we compare performances of the proposed design to other well-known designs for estimating the model parameters under the 5PL-1P model.Item Ds-Optimal Design for Model Discrimination in a Probit Model(North Dakota State University, 2014) Liu, RuifengIn toxicology studies, dose response functions with a downturn at higher doses are often observed. For such response functions, researchers often want to see if the downturn of the response is signifcant. A probit model with a quadratic term is adopted to demonstrate the dose response with a downturn. Under the probit model, we obtain optimal designs to study the signifcance of the downturn and their efficiencies are compared. Our approach identites the upper bound of the number of optimal design points and searches for the optimal design numerically based on the upper bound.Item Robust D-Optimal Design for Response Functions with a Downturn(North Dakota State University, 2013) Carter, Jessica AnneResearchers studying dose-response relationships must allocate limited resources to design points in order to maximize the information gained from the study. D-optimal design is a well-described design that works efficiently to study model parameters. In order to find the D-optimal design, the model that describes the dose-response relationship has to be known. In cases where dose-response relationships show a downturn at high doses, scientists sometimes ignore the downturn to study only the increasing part of the response curve. Here we have two model choices; one describes the overall dose-response relationship, and the other describes only the increasing part of the response curve. The D-optimal designs for these two models will be different and the D-optimal design for one model may not work efficiently for the other model. This research studies robust D-optimal design, a design that works efficiently for both models.