An additive boundary for group sequential designs with connection to conditional error

Group sequential designs allow stopping a clinical trial for meeting its efficacy objectives based on interim evaluation of the accumulating data. Various methods to determine group sequential boundaries that control the probability of crossing the boundary at an interim or the final analysis have been proposed. To monitor trials with uncertainty in group sizes at each analysis, error spending functions are often used to derive stopping boundaries. Although flexible, most spending functions are generic increasing functions with parameters that are difficult to interpret. They are often selected arbitrarily, sometimes using trial and error, so that the corresponding boundaries approximate the desired behavior numerically. Lan and DeMets proposed a spending function that approximates in a natural way the O'Brien-Fleming boundary based on the Brownian motion process. We extend this approach to a general family that has an additive boundary for the Brownian motion process. The spending function and the group sequential boundary share a common parameter that regulates how fast the error is spent. Three subfamilies are considered with different additive terms. In the first subfamily, the parameter has an interpretation as the conditional error rate, which is the conditional probability to reject the null hypothesis at the final analysis. This parameter also provides a connection between group sequential and adaptive design methodology. More choices of designs are allowed in the other two subfamilies. Numerical results are provided to illustrate flexibility and interpretability of the proposed procedures. A clinical trial is described to illustrate the utility of conditional error in boundary determination.