Optimal experimental designs for models with random effects have received increasing attention in recent years. Binary data models, especially logistic, form the main part of the presented research. The main goal of this thesis is to develop optimal experimental designs for the Poisson regression models with random intercept and random slope. An introduction will be presented about fundamental concepts including linear models, generalized linear models, linear mixed models and generalized linear mixed models. Complications in the design process arise with the use of random effects, i.e. when some model parameters are allowed to vary randomly between subjects. In fact the Fisher information matrix can not be written down in closed form for generalized linear mixed models due to the random effects. Therefore we apply a different estimating method to derive an approximating information matrix. This method is called the quasi-likelihood method and the information matrix based on this method is the quasi-information matrix. Some properties of the quasi-score function are studied as a special case of the estimating function. A simulated example shows that the quasi-likelihood estimations are close to the MLE of the unknown model parameters, especially when the variance of random effects is small. Using the quasi-likelihood method, the quasi-information matrices are obtained for different Poisson models. Convex design theory for ordinary linear models could not be extended to the proposed models due to the fact that the quasi-information matrices are not additive because of the existence of random effects in the models. We obtain some new theorems that allow us to apply convex design theory to our models. Besides this, equivalence theorems, similar to the ones known for ordinary linear models, are derived for our situations. The best experimental settings to do an experiment are usually selected via a real-valued function of the respective information matrix. In this work, we derive different representation of these functions based on the quasi-information matrices. Some examples from the models are presented to illustrate proposes. This thesis is closed with a discussion of future work.
Quasi-likelihood, Poisson regression, Random intercept, random slope, Optimal design, Convex design theory