The theory of optimal experimental design provides insightful guidance on resource allocation for many dose-response studies and clinical trials. However, as more and more complicated models are developed, finding optimal designs has become an increasingly difficult task; therefore, the availability of an efficient and easy-to-use algorithm to find optimal designs is important for both researchers and practitioners. In recent years, nature-inspired algorithms like Particle Swarm Optimization(PSO) have been successfully applied to many non-statistical disciplines, such as computer science and engineering, even though there is no unified theory to explain why PSO works so well. To date, there is virtually no work in the mainstream statistical literature that applies PSO to solve statistical problems.
In my dissertation, I review PSO methodology and show it is an easy and effective algorithm to generate locally D- and c-optimal designs for a variety of nonlinear statistical models commonly used in biomedical studies. I develop a new version of PSO called Ultra-dimensional PSO (UPSO) to find D-optimal designs for multi-variable exponential and Poisson regression models with up to five variables and all pairwise interactions. I use the proposed novel search strategy to find minimally supported D-optimal designs and ascertain conditions under which such optimal designs exist for such models. A remarkable discovery in my work is that locally D-optimal designs for such models can have many more support points than the number of parameters in the model. This result is both new and interesting because almost all D-optimal designs have equal or just one or two more number of points than the the number of parameters in the mean response function, see the examples in monographs by Fedorov , Atkinson Atkinson et al. , and recent papers by in Yang and Stufken , Yang . This discovery also disproves the conjecture by Wang et al.  that for M-variable interaction model (M > 2), D-optimal designs are also minimally and equally supported and have a similar structure as D-optimal designs for 2-variable model.
In addition to single objective optimal designs, I apply PSO to find optimal designs for estimating parameters and interesting characteristics continuation-ratio (CR) model with non-constant slopes. Such a model has a great potential in dose finding studies because it takes both efficacy and toxicity into consideration. The optimal design I am interested in constructing is a three-objective optimal design, which provides efficient estimates for efficacy, adverse effect and all parameters in the CR model. This work is quite new because there are virtually no three-objective designs for a trinomial model reported in the literature. Through multiple objective efficiency plots, practitioners can construct the desired compound optimal design by selecting appropriate weighted average of three optimal criteria in a more flexible and informative way.
I also conduct simulation studies for parameters selection in PSO, and compare the performance of PSO with other popular deterministic and metaheuristic algorithms in terms of the CPU time and the precision of the generated designs. I show that PSO outperforms its competitors for finding D- and c-optimal designs for different models I considered in my dissertation.