Supersaturated Multi-Stratum Designs: Construction and Statistical Modelling

Design of experiments (DOE) is one of the most important tools in scientific research, and it is an important aid in innovation in industries and sciences (applications can, for example, be found in the food industry, materials science, robotics, medicine and biology). Designing the experiment appropriately allows investigators to improve the efficiency of their methods and maximize the information obtained from their experiments. Failure to pay attention to the experimental design can lead to a waste in resources, needless repetition and poor inference. Better designed experiments will lead to reductions in the number of participants involved in clinical trials, savings in the amount of expensive components needed, decreases in the number of prototypes required in engineering experiments, and shorter product and process development times.

However, there is an important gap in the state of the art knowledge in DoE. Consider, for example, the following scenario: in a tribocorrosion experiment, the experimenter wants to optimise DLC (diamond like carbon) coatings for use in orthopedic implants. The experiment involves 6 variables, included the coating structure, the interlayer and the substrate with 3 levels each. Different level combinations of these variables will give different coatings. However, in order to get a fully randomized experiment you need to reset these levels each time, something which is very costly and time consuming. So ideally these will be kept constant for a sequence of runs. In addition, there is 1 easy- to-change variable (immersion time). The experimenter wants to study the main effects of the 7 variables and their interactions. Three samples for every condition are needed to perform all the tests and at most 30 samples can be immersed inside the used bath, whereas the number of effects that require estimation is greater than 30. In this situation, there is not only no efficient design available, but the notion of an efficient design in this context has not even been defined in a meaningful manner. In addition, novel methodology will be needed to analyse the data in order to draw the correct conclusions. There is no guidance for practitioners on how to plan and to analyse such an experiment, potentially slowing down the scientific progress in application areas. The proposed research will fill this important gap. Our approach will provide a new general methodology for setting up and analysing informative experiments, with both restricted randomization (multi-stratum designs) and a large number of factors, larger than the number of observations (supersaturated designs). The class of multi-stratum supersaturated designs is a very recently explored research area and there is a lack of a general methodology to tackle the problem of the construction and analysis of these experiments. Multi-stratum designs are very effective in reducing the cost of an experiment in the presence of hard-to-change factors and/or of multi-stage processes. In addition supersaturated designs (SSDs) compose a large class of factorial designs, which can be used for screening out the important factors from a large set of potentially active ones.

We will develop new Bayesian optimality criteria for designing good experiments, and Bayesian modelling tools to analyse data from supersaturated multi-stratum experiments. This will be interesting and challenging from a methodological point of view, and will also increase scientific understanding in numerous application areas, such as materials and surface engineering, tribological and chemical experiments where similar problems arise. Applying our methodology in real world scenarios will lead to important synergies between the different applied disciplines and Statistics. In particular, it will deliver insights into these important areas, demonstration of the effectiveness of the methodology, and exemplars to aid in dissemination.

The design of multi-stratum supersaturated experiments is an important unexplored research area. Mixed effects models are increasingly being used in many scientific disciplines (e.g. biology and medicine), and so the project will not only have a large impact on statistics, but also on scientific research in many disciplines. Thus, the results of this project will have an immediate and sustainable benefit for national and international science, business and industry.

Design of experiments (DOE) is a well-established important area of research in the UK with applications to many important scientific fields such as materials science, pharmacology, and chemistry. This project will contribute to advancing the field in this area in a new and important direction. DOE has for a long time been a strong area of research and applications in UK, compared with the rest of the world. This project will contribute to the maintenance of this traditionally successful national area. DOE will certainly have academic beneficiaries in other disciplines as well as the economic impact of these cost efficient designs studied here.

The PI has already completed successfully an FP7-PEOPLE-2009-IEF project on "Design of experiments for variance component estimation". The techniques developed by the PI and her collaborators can be applied successfully to split-plot designs, which are a special case of multi-stratum designs. Based on the work undertaken in the previous project we have evidence to believe there is a definite benefit to investigating the proposed methods.

To maximise the impact of research, the results will be made available on open-access repositories including the arXiv, prior to their publication in leading international journals. The programmes for the construction of the optimal designs and the statistical analysis methods will be available on the PI' s web page. The PI and PDRA will also publicise the results by giving seminars and colloquia at universities and conferences, both nationally and internationally. The PI will discuss with Dr. Bradley Jones, Principal Research Fellow in the Research and Development team of the JMP division of the SAS institute, the possible application of the developed methodology to the statistical software JMP. This would be an excellent opportunity for establishing a Knowledge Transfer Partnership (more information online at: http://www.ktponline.org.uk/academics/) with SAS/JMP at the end of the project, possibly including the PDRA. The PI is also in contact with the team of SETsquared (Southampton), a partner organisation between five Universities and several companies, in order to discuss possible collaborations with the member companies that are interested in performing experiments.

An important aspect of this grant proposal is the funding of a post-doctoral researcher to collaborate with the PI. This will provide the PDRA with an opportunity to further his/her academic or industrial career. The statistics Group within the School of Mathematical Sciences provides an excellent environment for developing of young researchers, as evidenced by the number of PDRAs here currently, and the past successes of junior researchers within the group. Supporting an early career researcher in this exciting and growing area of statistics is very important to the continuing position of the UK at the forefront of mathematical and statistical research.