  Sequential likelihood ratio testing is found to be most powerful in
sequential studies with early stopping rules when grouped data come from the
one-parameter exponential family.
  First, to obtain this elusive result, the probability measure of a group
sequential design is constructed with support for all possible outcome events,
as is useful for designing an experiment prior to having data. This
construction identifies impossible events that are not part of the support. The
overall probability distribution is dissected into stage specific components.
These components are sub-densities of interim test statistics first described
by Armitage, McPherson and Rowe (1969) that are commonly used to create
stopping boundaries given an $\alpha$-spending function and a set of interim
analysis times. Likelihood expressions conditional on reaching a stage are
given to connect pieces of the probability anatomy together.
  The reduction of the support caused by the adoption of an early stopping rule
induces sequential truncation (not nesting) in the probability distributions of
possible events. Multiple testing induces mixtures on the adapted support. Even
asymptotic distributions of inferential statistics are mixtures of truncated
distributions. In contrast to the classical result on local asymptotic
normality (Le Cam 1960), statistics that are asymptotically normal without
stopping options have asymptotic distributions that are mixtures of truncated
normal distributions under local alternatives with stopping options; under
fixed alternatives, asymptotic distributions of test statistics are degenerate.
