Optimal experimental designs for generalized linear models have received increasing attention in recent years. Yet, most of the current research focuses on binary data models especially the one-variable first-order logistic regression model. This research extends this topic to count data models. The primary goal of this research is to develop efficient and robust experimental designs for the Poisson regression model in toxicity studies. D-optimal designs for both the one-toxicant second-order model and the two-toxicant interaction model are developed and their dependence upon the model parameters is investigated. Application of the D-optimal designs is very limited due to the fact that these optimal designs, in terms of ED levels, depend upon the unknown parameters. Thus, some practical designs like equally spaced designs and conditional D-optimal designs, which, in terms of ED levels, are independent of the parameters, are studied. It turns out that these practical designs are quite efficient when the design space is restricted. Designs found in terms of ED levels like D-optimal designs are not robust to parameters misspecification. To deal with this problem, sequential designs are proposed for Poisson regression models. Both fully sequential designs and two-stage designs are studied and they are found to be efficient and robust to parameter misspecification. For experiments that involve two or more toxicants, restrictions on the survival proportion lead to restricted design regions dependent on the unknown parameters. It is found that sequential designs perform very well under such restrictions. In most of this research, the log link is assumed to be the true link function for the model. However, in some applications, more than one link functions fit the data very well. To help identify the link function that generates the data, experimental designs for discrimination between two competing link functions are investigated. T-optimal designs for discrimination between the log link and other link functions such as the square root link and the identity link are developed. To relax the dependence of T-optimal designs on the model truth, sequential designs are studied, which are found to converge to T-optimal designs for large experiments. / Ph. D.

The Design of an experiment involves selection of levels of one or more factor in order to optimize one or more criteria such as prediction variance or parameter variance criteria. Good experimental designs will have several desirable properties. Typically, one can not achieve all the ideal properties in a single design. Therefore, there are frequently several good designs and choosing among them involves tradeoffs. This dissertation contains three different components centered around the area of optimal design: developing a new graphical evaluation technique, discussing designs for non-regular regions for first order models with interaction for the two- and three-factor case, and using the standard designs in the case of generalized linear models (GLM). The Fraction of Design Space (FDS) technique is proposed as a new graphical evaluation technique that addresses good prediction. The new technique is comprised of two tools that give the researcher more detailed information by quantifying the fraction of design space where the scaled predicted variance is less than or equal to any pre-specified value. The FDS technique complements Variance Dispersion Graphs (VDGs) to give the researcher more insight about the design prediction capability. Several standard designs are studied with both methods: VDG and FDS. Many Standard designs are constructed for a factor space that is either a p-dimensional hypercube or hypersphere and any point inside or on the boundary of the shape is a candidate design point. However, some economic, or practical constraints may occur that restrict factor settings and result in an irregular experimental region. For the two- and three-factor case with one corner of the cuboidal design space excluded, three sensible alternative designs are proposed and compared. Properties of these designs and relative tradeoffs are discussed. Optimum experimental designs for GLM depend on the values of the unknown parameters. Several solutions to the dependency of the parameters of the optimality function were suggested in the literature. However, they are often unrealistic in practice. The behavior of the factorial designs, the well-known standard designs of the linear case, is studied for the GLM case. Conditions under which these designs have high G-efficiency are formulated. / Ph. D.

Traditional single stage design optimality procedures can be used to efficiently generate data for an assumed model y = f(x^(m),b) + ε. The model assumptions include the form of f, the set of regressors, x^(m) , and the distribution of ε. The nature of the response, y, often provides information about the model form (f) and the error distribution. It is more difficult to know, apriori, the specific set of regressors which will best explain the relationship between the response and a set of design (control) variables x. Misspecification of x^(m) will result in a design which is efficient, but for the wrong model. A Bayesian two stage design approach makes it possible to efficiently design experiments when initial knowledge of x^(m) is poor. This is accomplished by using a Bayesian optimality criterion in the first stage which is robust to model uncertainty. Bayesian analysis of first stage data reduces uncertainty associated with x^(m), enabling the remaining design points (second stage design) to be chosen with greater efficiency. The second stage design is then generated from an optimality procedure which incorporates the improved model knowledge. Using this approach, numerous two stage design procedures have been developed for the normal linear model. Extending this concept, a Bayesian design augmentation procedure has been developed for the purpose of efficiently obtaining data for variance modeling, when initial knowledge of the variance model is poor. / Ph. D.

Graphical summaries are becoming important tools for evaluating designs. The need to compare designs in term of their prediction variance properties advanced this development. A recent graphical tool, the Fraction of Design Space plot, is useful to calculate the fraction of the design space where the scaled prediction variance (SPV) is less than or equal to a given value. In this dissertation we adapt FDS plots, to study three specific design problems: robustness to model assumptions, robustness to measurement error and design properties for generalized linear models (GLM). This dissertation presents a graphical method for examining design robustness related to the SPV values using FDS plots by comparing designs across a number of potential models in a pre-specified model space. Scaling the FDS curves by the G-optimal bounds of each model helps compare designs on the same model scale. FDS plots are also adapted for comparing designs under the GLM framework. Since parameter estimates need to be specified, robustness to parameter misspecification is incorporated into the plots. Binomial and Poisson examples are used to study several scenarios. The third section involves a special type of response surface designs, mixture experiments, and deals with adapting FDS plots for two types of measurement error which can appear due to inaccurate measurements of the individual mixture component amounts. The last part of the dissertation covers mixture experiments for the GLM case and examines prediction properties of mixture designs using the adapted FDS plots. / Ph. D.