= Group sequential design
A <group sequential design> evaluates accumulating data at several planned <interim analysis>[interim analyses] instead of committing to one final test. At each analysis, a prespecified rule can stop for efficacy, stop for <futility boundary>[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 random variables>[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 probability distribution>[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 <probability>[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 statistic>[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 interval>[confidence intervals]; fixed-sample formulas do not automatically retain their usual interpretation after selection.
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