Joint optimisation of interim timing and stopping boundaries in group sequential trials

Group sequential designs (GSDs) are common in confirmatory trials, and their efficiency depends jointly on the timing of interim analyses and the shape of stopping boundaries. Most existing approaches optimise one component while holding the other fixed. Joint optimisation over free critical values has been studied for two- and three-stage efficacy-only designs and for tests with binding futility boundaries. Here we jointly optimise the full interim schedule and the boundary shape within the Hwang-Shih-DeCani and Kim-DeMets spending families, minimising the expected sample size under a specified alternative while controlling overall type I and type II error rates. A convex error-spending constraint prevents excessive early type I error spending. On the canonical information scale, the joint optimum depends only on the error rates, number of analyses and spending family, and is invariant to endpoint type and effect size. Without the convexity constraint, the parametric optima reproduced the unrestricted optima of Brittain & Bailey (1993) to within 0.001 in expected inflation factor. For three or more looks the jointly optimal design Pareto-dominated the scheduling-only optimal Pocock design, achieving the same or smaller expected sample size with uniformly smaller maximum sample size. Under a prespecified 1% tolerance, six analyses, five interim plus a final, captured essentially all efficiency attainable with up to seven analyses, reducing expected sample size by 23% and 33% at 80% and 90% power relative to the fixed-sample design. We illustrate the approach by redesigning the CORRECT trial.