  In this paper, we consider the problem of simultaneously testing many
two-sided hypotheses when rejections of null hypotheses are accompanied by
claims of the direction of the alternative. The fundamental goal is to
construct methods that control the mixed directional familywise error rate,
which is the probability of making any type 1 or type 3 (directional) error. In
particular, attention is focused on cases where the hypotheses are ordered as
$H_1 , \ldots, H_n$, so that $H_{i+1}$ is tested only if $H_1 , \ldots, H_i$
have all been previously rejected. In this situation, one can control the usual
familywise error rate under arbitrary dependence by the basic procedure which
tests each hypothesis at level $\alpha$, and no other multiplicity adjustment
is needed. However, we show that this is far too liberal if one also accounts
for directional errors. But, by imposing certain dependence assumptions on the
test statistics, one can retain the basic procedure. Through a simulation study
and a clinical trial example, we numerically illustrate good performance of the
proposed procedures compared to the existing mdFWER controlling procedures. The
proposed procedures are also implemented in the R-package FixSeqMTP.
