  Ordered testing procedures are multiple testing procedures that exploit a
pre-specified ordering of the null hypotheses, from most to least promising. We
analyze and compare the power of several recent proposals using the asymptotic
framework of Li & Barber (2015). While accumulation tests including ForwardStop
can be quite powerful when the ordering is very informative, they are
asymptotically powerless when the ordering is weaker. By contrast, Selective
SeqStep, proposed by Barber & Cand\`es (2015), is much less sensitive to the
quality of the ordering. We compare the power of these procedures in different
r\'egimes, concluding that Selective SeqStep dominates accumulation tests if
either the ordering is weak or non-null hypotheses are sparse or weak.
Motivated by our asymptotic analysis, we derive an improved version of
Selective SeqStep which we call Adaptive SeqStep, analogous to Storey's
improvement on the Benjamini-Hochberg procedure. We compare these methods using
the GEOQuery data set analyzed by Li & Barber (2015) and find Adaptive SeqStep
has favorable performance for both good and bad prior orderings.
