Optimal design of non-linear multi-factor experiments

Abstract

Optimal design is useful in improving the efficiencies of experiments with respect to a specified optimality criterion, which is often related to one or more statistical models assumed. In particular, sometimes chemical and biological studies could involve multiple experimental factors. Often, it is convenient to fit a nonlinear model in these factors. This nonlinear model can be either mechanistic or empirical, as long as it describes the unknown mechanism behind the response surface. In this thesis, our main interest is in exact optimal design of experiments for nonlinear multifactor models. In order to search for optimal designs, we can use the conventional point or coordinate exchange approach, which however can incorporate a new continuous optimisation method. On the basis of this idea, we further develop and implement a multistage hybrid method to construct local and (pseudo-)Bayesian optimal designs. The recommended hybrid exchange algorithm overcomes the shortcomings of the modified Fedorov exchange algorithm and the coordinate exchange algorithm, contributing to improved properties of the experimental designs obtained. In addition, Bayesian optimal design with respect to an expected criterion function is based on the assumed parameter prior distributions for the nonlinear model. To limit the time for approximating expected criterion values in the algorithm, we use some efficient numerical integration methods (e.g. a Gauss-Hermite quadrature), which are much superior to the traditional pseudo-Monte Carlo method.

We demonstrate the hybrid exchange algorithm by means of several examples relevant to Michaelis-Menten kinetics and other biochemical applications. Under some of these circumstances, we consider hybrid nonlinear models which can be adopted to be fitted to the data of new experiments, the tailor-made optimal designs of which are therefore found and compared with each other. In order to normalise the error structure of such a hybrid model, sometimes the Box-Cox transformation can be applied and the result would be a transform-both-sides (TBS) model. Optimal designs for either untransformed models or TBS models can be used for future experiments, as well as for comprehensive studies of complicated mechanisms.

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