1 error are correctly calculated. With the use of Monte-Carlo methods, data generated from stage-specific normal distributions with GSD-determined early stopping inherently generate draws from correct mixtures. Similarly, numeric integration \[see @jennison1999\] also correctly estimates type error and power by the use of the recursive sub-density formula of @Armitage1969.

Currently, the design of many group sequential trials starts with choosing an $\alpha$-spending function. Then, “Information fraction” considerations are used to define times or sample sizes of interim analyses that secure the overall power at a desired level. Statistical power to detect one or another treatment effect is only described. It is important to note that stage-specific sample sizes in such study designs are not driven by desired stage-specific power requirements, but fully determined by a (1) chosen spending functions, (2) “information fraction” for interim looks, and (3) a “maximum information” to reach a desired overall power.

In contrast, this manuscript proposes GSDs driven by *a sequence of ordered alternative hypotheses* attainable with a predetermined statistical power while controlling type 1 error.

New GSDs Determined by Ordered Alternatives
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Consider a study design in which stage-specific sample sizes $\{n_k\}$ and and critical values $\{c_k\}$ are determined by an $\alpha$-spending function and 