Determining the sample size to meet the inferential objectives of a study is of central importance in experimental design. There is an extensive collection of methods addressing this problem from diverse perspectives. The Bayesian paradigm, in particular, has attracted noticeable attention and includes different perspectives for sample size determination. While traditional Bayesian methods formulate sample size determination as a decision problem that optimizes a given utility functions (Lindley, 1997), practical experimental settings may require a more flexible approach based upon simulating analysis and design objectives (see, e.g., O'Hagan and Stevens, 2001). Building upon the latter approach, we devise a general Bayesian framework for simulation-based sample size determination using Bayesian assurance that can be easily implemented on modest computing architectures. We qualify the need for different priors for the design and analysis stage, working primarily in the context of conjugate Bayesian linear regression models, where we consider known and unknown variances. We also compare the assurance to a utility-based approach that involves the specification of objective functions to determine the rate of correct classification (Inoue, Berry, and Parmigiani, 2005). Throughout, we draw parallels with frequentist solutions, which arise as special cases, and alternate Bayesian approaches with an emphasis on how the numerical results from existing methods arise as special cases in our framework.

We further extend our conjugate linear model's capabilities to encompass the multiple testing framework, where the assurance is now characterized by conditions placed on the Bayesian false discovery rate (FDR). Under this framework, we investigate the effects of multiple comparison adjustments on assurance and sample size determination. Adjustments include enforcing different assigned threshold values for the Bayesian FDR and conditions related to the credible interval condition, and varying the number of pairwise hypothesis tests being conducted. Of particular interest is observing how the number of pairwise tests being conducted affects the assurance under fixed constraints placed on the Bayesian FDR as defined in Muller et al., 2004. We assess how our proposed model performs in commonplace large-scale problems, specifically microarray data. Our methodology is implemented in a study of mammary cancer in the rat, where four distinct patterns of expression are provided. Future tasks involve assessing how our method performs when comparing more than two subgroups and enforcing objective ways of choosing optimal threshold values.

This dissertation captures the vast applicability of the two-stage framework, offering a robust Bayesian approach for sample size determination equipped for addressing a wide selection of problems taking place both within and outside clinical trial settings. There is broad potential for growth and development in the methods introduced, with numerous routes available for future exploration.