Optimal split plot designs for nonlinear models

In industrial experiments, often some factors are more expensive or time-consuming to change from experimental run to experimental run than others. For example, in a chemical reaction the factor solution concentration may be hard to vary whereas it is easy to change the factor solution temperature. In this case, the experimenter seeks to randomise the combinations of levels of solution concentration and temperature so that the factor solution concentration has a minimal number of changes. This is a simple example of a split plot experiment. In the analysis of data from such experiments, it has to be taken into account that the data have not been obtained from a completely randomised design. Otherwise conclusions from the data may be flawed.

In recent years, there has been considerable progress in the literature on modelling and analysing data from split plot experiments. This has been followed by rapid advances on how to design such experiments ``optimally'', i.e. such that the most accurate conclusions can be drawn from the data, often even from fewer observations. Here a design means the combinations of factor levels at which to run the experiment and an appropriate randomisation scheme. However, there is still a substantial gap: All published articles in the literature are on linear models, often linear or quadratic polynomials, although in many situations the responses are known to depend nonlinearly on the combinations of factor levels. This is particularly the case in chemical reactions as they appear for example in the pharmaceutical production and development, where the response surface can often be described by the solution of a set of differential equations. In current practice in the pharmaceutical industry, the response surface is often locally approximated by low order polynomials, which may not always be suitable and even prevent a deeper understanding of the underlying chemical process.

This project will fill the existing gap in the literature, thus providing essential guidance to practitioners in industry and science. In particular, it is intended to result in a better appreciation of appropriate design and analysis of split plot experiments by practitioners. We will start with a few special cases to lay the theoretical foundations and then work towards identifying classes of models for which specific types of designs will be optimal. The main focus will be on the theoretical advancement of the area, which will be complemented by the development of novel algorithms to find the optimal designs and by extensive simulations to assess the designs found and to compare them with designs currently used in practice.