Conditional frequentist tests of a precise hypothesis versus a composite al
ternative have recently been developed, and have been shown to be equivalen
t to conventional Bayes tests in the very strong sense that the reported fr
equentist error probabilities equal the posterior probabilities of the hypo
theses. These results are herein extended to sequential testing, and yield
fully frequentist sequential tests that are considerably-easier to use than
are conventional sequential tests. Among the interesting properties of the
se new tests is the lack of dependence of the reported error probabilities
on the stopping rule, seeming to lend frequentist support to the stopping r
ule principle.