 Group sequential design

A group sequential design evaluates accumulating data at several planned interim analyses instead of committing to one final test. At each analysis, a prespecified rule can stop for efficacy, stop for futility, or continue collecting observations. The design specifies both the timing of analyses and the sequential stopping boundaries.

Repeated inspection changes the probability of rejecting a Null hypothesis. Even two independent tests, each at significance level $0.05$, reject at least once with probability $1-0.95^2=0.0975$ under their common Null hypothesis. Tests on accumulating observations are dependent, so their rejection boundaries require their actual joint distribution rather than that independence calculation. An Alpha-spending function distributes the permitted type I error over information fractions; it does not make unchanged fixed-sample critical values valid at every analysis.

For a simple valid construction with $K$ planned analyses, choose separate rejection events whose null probabilities are each at most $\alpha /K$. The union bound makes the probability of any rejection at most $\alpha$, whatever their dependence. More efficient boundaries exploit the dependence between successive test statistics. A Futility boundary removes rejection paths, and a Conditional error principle can justify a later design adaptation while preserving the original error guarantee. The possibility of early stopping also affects point estimation and confidence intervals; fixed-sample formulas do not automatically retain their usual interpretation after selection.