  A classical approach for dealing with the multiple testing problem is to
restrict attention to procedures that control the familywise error rate (FWER),
the probability of at least one false rejection. In many applications, one
might be willing to tolerate more than one false rejection provided the number
of such cases is controlled, thereby increasing the ability of the procedure to
detect false null hypotheses. This suggests replacing control of the FWER by
controlling the probability of $k$ or more false rejections, which is called
the $k$-FWER. In this article, a unified approach is presented for deriving the
$k$-FWER controlling procedures. We first generalize the well-known closure
principle in the context of the FWER to the case of controlling the $k$-FWER.
Then, we discuss how to derive the $k$-FWER controlling stepwise (stepdown or
stepup) procedures based on marginal $p$-values using this principle. We show
that, under certain conditions, generalized closed testing procedures can be
reduced to stepwise procedures, and any stepwise procedure is equivalent to a
generalized closed testing procedure. Finally, we generalize the well-known
Hommel procedure in two directions, and show that any generalized Hommel
procedure is equivalent to a generalized closed testing procedure with the same
critical values.
