Implementing optimal design can provide the most accurate statistical inference with minimal cost. However, optimal designs for high-dimensional models or complicated nonlinear models can be hard to find. I propose a novel swarm algorithm, called competitive swarm optimizer with mutated agents (CSO-MA), to search for optimal designs for high-dimensional and complicated nonlinear models that are useful for biomedical studies. They include logistic models, Poisson-type models with multiple interacting covariates and some factors may have correlated random effects. I first show the proposed algorithm outperforms several state-of-the-art algorithms using benchmark functions commonly used in the engineering literature. I then show it can either perform as efficiently as some current algorithms used in statistics for finding optimal designs or outperform several of its competitors. Additionally, I find some of the claimed optimal designs in the literature are not optimal by showing CSO-MA-generated designs have higher statistical efficiency. Since the bulk of design work in the literature concerns low-dimensional models, my work has the potential to break new ground, especially in the era of big data, where, increasingly, it is more realistic to use more complex models to reflect reality.

The proposed algorithm is a general-purpose optimization algorithm, so it is flexible and can find exact and approximate designs, with and without constraints. In particular, it can efficiently search for different types of optimal designs, including Bayesian optimal designs, which are especially challenging to find when there are multiple factors and there are multi-dimensional integrals involved in the optimization problem. The results from my research will provide new, more realistic and better quality statistical experimental designs for biomedical researchers at a minimal cost.

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 [1972], Atkinson Atkinson et al. [2007], and recent papers by in Yang and Stufken [2009], Yang [2010]. This discovery also disproves the conjecture by Wang et al. [2006] 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.

The design phase of a randomized controlled clinical trial is critical to its success. With many non-adaptive designs and an explosive number of adaptive designs introduced to the research community, the number of designs from which a statistician can select has the potential to be overwhelming. At times, a statistician may be uncertain how a newer adaptive design will perform in a particular setting of interest. While regulatory agencies have originally treated adaptive designs with resistance, recent years have seen more acceptance if there is extensive simulation work that shows good control of Type I error.

There are many adaptive designs, and it is important to understand and compare characteristics of competing designs before implementation. However, the overall lack of understanding of the performance of adaptive designs with regard to several design characteristics and the lack of an effective tool to measure overall design quality may have led to clinical trial statisticians implementing traditional designs rather than adopting more innovative methods. Yet adaptive designs have many appealing features that can benefit both the clinical trial sponsor, who funds the trial, and the clinical trial subjects. These strengths include early completion of a trial due to overwhelming efficacy and minimizing the number of subjects assigned to an inferior treatment arm.

The aim of this dissertation is to introduce methodology that provides statisticians and other clinical trial stakeholders with a tool that can measure the overall quality of a design and thereby facilitate comparison across competing designs. The methodology utilizes desirability functions to measure various statistical and non-statistical features that contribute to the quality of a design. Specifically, individual desirability functions evaluate a library of components including statistical considerations, such as treatment group size imbalance, probability of covariate imbalance, accidental bias, control for chronological bias, Type I error and power, and ethical considerations, such as minimizing the expected number of failures and total sample size needed in the whole trial. The proposed strategy is to compute an overall desirability score for each design, use it to rank the clinical trial designs of interest, and select the most relevant and efficient design for the trial's various objectives. To facilitate use of the proposed methodology, the project includes the development of an online interactive tool for the user to incorporate input before desirability functions are generated to help the user select the most appropriate design for the trial.