Sample size optimization for clinical trials using graphical approaches for multiplicity adjustment

Graphical approach provides a useful framework for multiplicity adjustment in clinical trials with multiple endpoints. When designing a graphical approach, initial weight and transition probability for the endpoints are often assigned based on clinical importance. For example, practitioners may prefer putting more weights on some primary endpoints. The clinical preference can be formulated as a constrain in the sample size optimization problem. However, there has been a lack of theoretical guidance on how to specify initial weight and transition probability in a graphical approach to meet the clinical preference but at the same time to minimize the sample size needed for a power requirement. To fill this gap, we propose statistical methods to optimize sample size over initial weight and transition probability in a graphical approach under a common setting, which is to use marginal power for each endpoint in a trial design. Importantly, we prove that some of the commonly used graphical approaches such as putting all initial weights on one endpoint are suboptimal. Our methods are flexible, which can be used for both single-arm trials and randomized controlled trials with either continuous or binary or mixed types of endpoints. Additionally, we prove the existence of optimal solution where all marginal powers are placed exactly at the prespecified values, assuming continuity. Two hypothetical clinical trial designs are presented to illustrate the application of our methods under different scenarios. Results are first presented for a design with two endpoints and are further generalized to three or more endpoints. Our findings are helpful to guide the design of a graphical approach and the sample size calculation in clinical trials.