Much of the literature on optimal design of experiments has focussed on experiments where the behaviour of the system is approximated by a linear model, such as a low-order polynomial. In many areas such as pharmacology and chemistry, such approximations are not appropriate, as the underlying mechanisms produce highly nonlinear or categorical responses. This thesis addresses some issues with the optimal design of experiments in these situations.
Commonly used criteria for the 'optimal' design of experiments relate to optimality in terms of efficient estimation of model parameters. However, quite often another important objective of an experiment is to select the model structure which best describes the underlying behaviour of the system. We examine existing criteria for model discrimination for both nonlinear and generalised linear models, and combine them with criteria for parameter estimation in order to create designs which address both objectives. We show that these designs can be quite efficient in terms of each of the criteria.
Further complications in the design process arise with the use of mixed effects models, that is when some model parameters are allowed to vary randomly between blocks or clusters of units. The Fisher information matrix is involved in the calculation of many optimality criteria, and this matrix cannot be written down in closed form for many nonlinear and generalised linear mixed effects models. We instead rely on approximations to the information matrix to generate optimal designs. This thesis gives details of the use of an existing approximation to the matrix in the optimal design of a complex pharmacokinetic experiment involving nonlinear mixed effects models. We also investigate several alternative approximations to the matrix for logistic regression with random coefficients, with an application in pharmacodynamics: the design of a cross-over trial with a binary response
Regardless of the criterion used to select a particular design, we require a method to search the design space for points which maximise or minimise the criterion. The models considered in this thesis are assumed to have predictor variables taken over a continuous range, so combinatorial optimisation techniques such as the tabu algorithm are not appropriate. Instead we make use of the simulated annealing algorithm (modified for continuous variables) and a relatively new algorithm known as the cross-entropy method. Both algorithms are implemented in programs written for the MATLAB package.