  This paper addresses the following general scenario: A scientist wishes to
perform a battery of experiments, each generating a sequential stream of data,
to investigate some phenomenon. The scientist would like to control the overall
error rate in order to draw statistically-valid conclusions from each
experiment, while being as efficient as possible. The between-stream data may
differ in distribution and dimension but also may be highly correlated, even
duplicated exactly in some cases. Treating each experiment as a hypothesis test
and adopting the familywise error rate (FWER) metric, we give a procedure that
sequentially tests each hypothesis while controlling both the type I and II
FWERs regardless of the between-stream correlation, and only requires arbitrary
sequential test statistics that control the error rates for a given stream in
isolation. The proposed procedure, which we call the sequential Holm procedure
because of its inspiration from Holm's (1979) seminal fixed-sample procedure,
shows simultaneous savings in expected sample size and less conservative error
control relative to fixed sample, sequential Bonferroni, and other recently
proposed sequential procedures in a simulation study.
