in group sequential designs**]{}

[*Keywords:*]{} Adaptive designs, maximum likelihood estimation, asymptotic distribution theory, interim analyses, local alternative hypotheses.

Introduction {#Introduction}
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Group sequential designs (GSDs) optimize resource allocation and benefit from the possibility of efficacy- and/or futility-driven early stopping. Along with these attractive properties, a few challenges require attention from statisticians. Specifically, many group sequential methods rely on the [*[the joint canonical distribution assumption]{}*]{} (Section 3.1, @jennison1999) and/or [*[Brownian motion theory]{}*]{} \[Section 2.1, @proschan2006\] to justify the choice of a GSD. These assumptions state that *stage-specific standardized test statistics* follow a multivariate normal distribution, or converge to multivariate normality asymptotically. Multivariate normality, however, does not hold for *cumulative standardized tests statistics* used in common GSD methods including Pocock, O’Brien & Fleming, and Haybittle-Peto GSDs, not for finite sample sizes, and not asymptotically against local alternatives. The popular SAS’s SEQDESIGN procedure, R’s gsDesign package, Cytel’s EAST and others overcome this difficulty by using a recursive sub-density formula \[@Armitage1969\] for assessing type I and power properties. Nonetheless, the impact of non-normality on other statistical quantities such as Fisher information, the information fraction, and repeated confidence intervals, is not widely recognized.

Multivariate normality of many commonly used stage-specific test statistics $\left(Z_1,\ldots,Z_K\right)$, and consequently 