cm de altura (mayores) Piezas: 10

5.7 Cost considerations for a Bayesian D-optimal

design

In this section cost considerations are applied to nding Bayesian D-optimal designs, and an example of a Bayesian D-optimal design is found for c1.

A Bayesian D-optimal design for c1 was found for the prior distributions Ea ∼ N (4.91 × 104_{, 4000}2_{), k}

r ∼ G(m2k/vk,vk/mk), with a mean of mk= 4.59 × 10−4 and variance of vk = (4 × 10−4)2, τ ∼ G(1, 2), ρ ∼ U(0, 2), log(r) ∼ U(−2.5, 2.5) and log(q) ∼ U(−2.5, 2.5). This prior was chosen as it was found to be robust to a large range of mean and correlation parameters in Chapters 3 and 4. The optimal design was found for m = 1, . . . , 10, and the optimal value of m for α was found, displayed in Figure 5.2.

Figure 5.2: The optimal value of m for a Bayesian D-optimal design for given α and prior distributions given in Section 5.7

the sixth set of assumptions on the parameter values, as m = 2 when α = 1, but it takes until α = 16 for the optimal m = 10.

Locally D-optimal designs are found based on the unrealistic assumption that the parameter values are known before design, which Bayesian optimal design compensates for. However, Bayesian D-optimal designs will take longer to nd due to computational issues, and including a cost constraint will increase the amount of time required to nd the optimal design. For c1, we needed to nd 9 optimal designs (for m = 2, . . . , 10), and for c2 it becomes much harder to obtain the Bayesian D- optimal design. This is an area on which future research could focus.

5.8 Conclusion

In this chapter cost considerations were included while nding locally D-optimal and Bayesian D-optimal designs. The optimal number of observations that should be taken per run of the process was shown to depend on the cost function used

and the strength and type of correlation. When serial correlation is present and strong, designs with fewer observations per run are preferred, while if run-to-run error is present designs with more observations per experimental run are preferred. As observations per run become cheaper with respect to the cost of a run of the process, the optimal number of observations per run increases, as would be expected. Including a cost for length of experimentation was shown to strongly aect designs when run-to-run error was believed to be present, as the optimal designs would normally take observations as late as possible in time. It was also demonstrated that, while approximate designs cannot always be directly converted into exact designs which use all available resource, several methods, such as appending additional random or uniformly spread observation times to the runs of the under-spending exact design obtained from the approximate design had high D-eciencies.

Cost considerations were also considered for a Bayesian D-optimal design, and the results obtained were similar to those for the locally D-optimal designs.

This study into including cost considerations while designing has some short comings. Although the impact on the optimal allocation of cost for a range of dierent assumptions on the mean and correlation parameters was explored, the impact of mis-specifying these assumptions on the mis-allocation of resource was not investigated, and could provide an avenue for future research. Methods for rounding approximate designs were explored, and though no simple method of doing so was provided, it is unlikely that one exists. Two cost functions were investigated, which probably cover most of the cost concerns experimenters might have, although methods of obtaining exact designs from approximate designs for the second cost function, c2, were not investigated, and doing so may prove dicult in practice. The errors were also assumed to be homoscedastic, and error heteroscedasticity could have a large impact on the optimal allocation of resource.

Chapter 6

Conclusions and Future Work

6.1 Conclusions

In this thesis a variety of issues have been explored which have been raised by nding optimal designs for experiments to estimate parameters in the mean function of non-linear models. These models arise in chemical kinetics, and we have studied an example provided by chemists at GlaxoSmithKline. The goals enumerated in Section 1.4 were investigated in full.

Analytical results were obtained for nding locally D-optimal designs for the considered example when independent, normally distributed errors were assumed, and the response and expected response were transformed to reduce heteroscedas- ticity. It was discovered that the optimal temperatures at which to conduct experiments were at the maximum and minimum permitted values, with the optimal times of observation given by Equation (2.5). Additionally, as errors became more heteroscedastic, the optimal design took observations later in time.

These results were applied to assist in obtaining locally D-optimal designs when correlation between observations from the same run was present, where the statistical model included both serial correlation and run-to-run error. It was discovered that the stronger serial correlation was, the further apart in time observations on the same run were made. When run-to-run error was present and the corresponding variance component was large, optimal designs made observations at the maximum

and minimum permitted times as well as observations close to those specied by Equation (2.5).

Mis-specifying the mean parameters, correlation parameters or transformation parameter was found to have a large impact on D-eciency. For the mean parameters, mis-specifying kr had more of an impact than mis-specifying Ea over the range of parameter values considered.

Bayesian D-optimal designs, were found for prior distributions specied across the mean, correlation and transformation parameters. The robustness of the design's performance to mis-specication of the parameter distributions was then assessed via a large simulation study, which demonstrated that the Bayesian D-optimal designs performed well over a range of scenarios, out-performing the locally D-optimal designs and ad hoc designs used by experimenters in the past. The simulation study also demonstrated that failing to include correlation and heteroscedasticity during analysis when either was present would lead to a large drop in the accuracy of the parameter estimates obtained.

Cost considerations were also incorporated into the design criterion. Two cost functions were considered. Both included the cost of taking multiple observations during a run of a process. The second cost function included a cost penalty for the length of time that a process was conducted. For both cost functions, the magnitude of serial correlation between observations from the same run was found to strongly aect the optimal number of observations per run: the stronger the correlation, the fewer observations per run required in the optimal design. The second cost function led to optimal designs with observations taken earlier in time.

An investigation was also conducted into how a practitioner, given an approxi- mate design, could nd exact designs to use in an experiment. It was demonstrated that methods such as appending additional random time points to runs from under spending designs could give a reasonable approximation to an optimal exact design.