Control of Directional Errors in Fixed Sequence Multiple Testing

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.