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A futility boundary specifies an interim result below which a clinical trial stops without rejecting the null hypothesis. Removing only potential rejection paths cannot increase the Type I error. With a finite boundary and a nonsingular joint normal test-statistic distribution, it strictly reduces that error relative to the corresponding fixed-final-boundary test.

A least-favourable configuration maximizes a test procedure’s Type I error probability over allowed null hypothesis parameters. For one-sided normal comparisons with a fixed covariance matrix, coupling by common centered variables shows monotonicity in the means. For the familywise error rate, mixed true and false null configurations must also be checked.

Generate independent replicated datasets under a fully specified null hypothesis and calculate the same predictive discrepancy statistic for them and the observation. Their null exchangeability gives a finite-simulation rank test. The displayed add-one tail probability is valid with conservative treatment of ties, and avoids reporting zero solely because no replicate exceeds the observation.