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An Alpha-spending function assigns the cumulative Type I error budget to successive information fractions in a group sequential design. Its total at the final analysis is the overall significance level. The rejection boundaries must be calibrated using the joint distribution of the sequential test statistics; independent fixed-level tests do not automatically spend the assigned budget correctly.

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.

Choose a group sequential design. Randomize patients among the three arms, schedule interim analyses after prespecified amounts of information, and calculate the two Wald test statistics at each look. Prespecify efficacy boundaries and a futility rule; allow early stopping of an ineffective experimental arm or of the trial when sufficiently convincing efficacy evidence is obtained. Calibrate the boundaries for both repeated looks and the two shared-control comparisons so that the familywise error rate remains controlled. An alpha-spending function is one way to allocate the error probability across looks.

Two potential benefits are lower expected recruitment and shorter duration when a treatment is clearly effective or futile, and earlier identification of benefit or lack of benefit, reducing continued exposure to an ineffective experimental treatment. These benefits concern expected performance; the maximum sample size may still be required.

Two drawbacks are the extra operational work needed for rapid, reliable outcomes and confidential interim analyses, and more complicated inference: repeated testing requires adjusted boundaries, and ordinary fixed-sample effect estimates and confidence intervals can be misleading after data-dependent stopping. Continuing two comparisons against a shared control also requires care if an arm is stopped.

A second opportunity to reject raises the trial's overall Type I error above the nominal 10% in general. If $A$ and $B$ denote rejection at the first and second analyses, the overall error isThe cumulative-data test statistics are correlated, so neither 20% nor $1-0.9^2=19\%$ is the general answer. The latter requires independent tests, which is not the situation here.

The rejection thresholds and the stopping/continuation rule must be calibrated jointly. A prespecified group sequential design or alpha-spending function can allocate a total error budget of 10% across the analyses, using their joint null distribution. A conservative alternative planned in advance is two tests each at 5%, controlled by the Bonferroni inequality. An unplanned change to a design with a remaining error budget can be justified by the conditional error principle. Here, however, an original single-look test at 10% has no remaining conditional rejection probability after its final nonrejection. Keeping its entire original rejection region and adding a new positive-probability rejection opportunity cannot in general retain a 10% overall error rate. The two-look rule therefore needs prospective joint calibration; an already observed result cannot retrospectively create an unused error budget.